place value and number lines -...

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Number ndash Place value and Number lines

The position of each digit is important as it carries a specific value

654321

Tens

Units

Hundred Thousands

Ten Thousands

Thousands

Hundreds

1234

Tenths

Hundredths

Thousandths

Ten Thousandths

Line numbers up according to place

value when calculating

Each place to the left is 10 times larger

Number lines can be used to count up or down

1 2 3 4 5 6 7

Markings are the same distance apart each time

An increase means moving to the right a decrease means moving to the left

Useful for ordering numbers of different forms

minus4 minus2 0 2 4 6 8

35

10028minus3

1

2620


Number ndash Rounding

Rules of rounding01234 56789 ROUND UPROUND DOWN

Place Value Significant FiguresDecimal Places

14672To the nearest ten

14670To the nearest hundred

14700To the nearest thousand

15000

125298To 1 decimal place

125To 2 decimal places


12530

325484To 1 significant figure

300000

To 3 significant figures

325000Be careful when rounding up a nine (A double rounding will occur)

1 2 3 4

Count Right from the Decimal Point


330000

Count Right from first non-zero Digit

1 2 3 4 5 6


Number ndash Addition and Subtraction

Addition Subtraction

Chunking method

Column method

Addition and subtraction of positive and negative numbers

AddSubtract according to place value436 + 58

430 + 50 = 480

6 + 8 = 14

436 + 58 = 494

563 minus 237

563 minus 200 = 363

363 minus 30 = 333

333 minus 7 = 326

563 minus 237 = 326

Line up numbers according to place value436 + 58 563 minus 237

436058+

4941

563237minus

326

5 1

Carry over if column is greater than 10

Borrow from next column if too small

+ + =

+ minus =

minus minus =

minus + =

Filling the bath analogy

119860119889119889 ℎ119900119905 119908119886119905119890119903

119860119889119889 119888119900119897119889 119908119886119905119890119903

119877119890119898119900119907119890 119888119900119897119889 119908119886119905119890119903

119877119890119898119900119907119890 ℎ119900119905 119908119886119905119890119903


Number ndash Multiplication and Division

Multiplication DivisionColumn method

BoxGrid method

Bus shelter method

Chinese

Tens in top half

Units in bottom half

Starting from the right

Add the diagonals Carry over onto

the next diagonal if you need to

523 times 76

523 times 76

Separate numbers

according to place value

Numbers first then zeros

Double check calculations

Add using column addition

523 times 76

Line up numbers


Multiply by units first then tens etc

Donrsquot forget to add zeros


Know times tablesRemainders carried over

Add zeros when numbers run out

Double decker division

958 divide 8

2016 divide 12 Split into factorsDivide by one factor

Divide answer by second factor

Negative numbers

Signs the same = Positive answer

Signs different = Negative answer


Number ndash BIDMAS

Example 1

BracketsIndicesDivisionMultiplicationAdditionSubtraction

ORDER

Follow the order of BIDMAS to calculate

6 + 3 times 4

6 + 12

6 + 3 times 4Multiplication first

Addition second

18

3 + 8 times 4 + 6 divide 5 minus 2

3 + 8 times 10 divide 5 minus 2

3 + 16 minus 2

3 + 8 times 4 + 6 divide 5 minus 2Brackets first

Multiplication and Division second

Addition and Subtraction last

17

(9 minus 3 times 2) 2divide (10 divide 5)

(3) 2divide (2)

9 divide (2)

45


Follow the order of operations

within brackets

Example 2

Example 3

=

==

=

=

==

=

=

=

=

If you can do nothing for a given step move on to the next


Number ndash Prime Numbers Factors Multiples

Prime numbers MultiplesFactorsA Number is Prime if it has exactly 2 factors

1 and itself

The factors of a number are the numbers which

divide into it exactly

A number that features in the times table of

another number

1 is not a prime

number

2 is the only even prime

number

No other number can divide into it exactly

2 3 5 7 11 13 17 19 2329 31 37 41 43 47

No remainder when divided by a factor

Factor Pairs

Factors of 24

1 times 242 times 123 times 84 times 6

Prime numbers up to 50

1 2 3 4 6 8 12 24

The product of two integers will produce a multiple

Close link to factors

Times table knowledge important

Multiples of 7 between 30 and 60

7 14 21 28 35 42 49 56 63 70

Really useful for Prime Factor Decomposition HCF and LCM

1 is a factor of all whole

numbers

Multiples of 9

9 18 27 36 45 54 63 72 81 90hellip


Number ndash Prime Factor Decomposition HCF LCM

Prime Factor Decomposition

HCFHighest common factor

Tree method and ladder method

LCMLowest common multiple

Break numbers down into prime factors

Divide by a prime

Multiply Primes

Write in index form

The largest number that divides into two or more numbers

Use long format of Prime Factor Decomposition

119867119862119865 119900119891 48 119886119899119889 120

2 times 2 times 3 = 12

119867119862119865 119900119891 84 119886119899119889 980


Prime factor decomposition

Identify shared factors

Multiply values

The smallest number that occurs in the times table of two

or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120

48 = 24 times 3180 = 22 times 32 times 5


Multiply together all prime factors apart

from duplicates

In index form MultiplyHighest Power of each

prime

6 = 2 times 345 3 times 3 times 5

2 times 3 times 3 times 5 = 90

=


Number ndash Powers and Roots

Positive powers

Index notation used to write easily

Numbers multiplied by themselves

Negative powers Roots Fractional powers

4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890

84 = 8 times 8 times 8 times 8

2

3

2

=22

32=

2 times 2

3 times 3=

4

9

Anything to the power of 1 is just itself

51 = 5 281= 28

Know square numbers and cube numbers

Negative powers are fractions

A negative power means ldquoTake the reciprocal and make the

power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal

Apply Positive Power

Apply top and bottom

119909minus119899 ൗ1 119909119899

Roots are the inverse operations to powers

7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905

If you know square numbers and cube

numbers you can find their roots

2-Stage Powers

119909119898119899 = 119899 119909 119898

Numerator and denominator are important

Root by denominator first

Then power of numerator

1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9

Negative Fractional PowersApply reciprocal first


Algebra ndash Notation and Collecting like terms

Algebraic notation Collecting like terms

Algebra is the language we use to communicate mathematical information

Letters used to represent values are known as variables

Notation creates shortcuts

119886 times 119887 becomes 119886119887

119909 + 119909 + 119909 + 119909 becomes 4 119909119910 times 119910 becomes 1199102

coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms

Same rules of BIDMAS applies to Algebra

Collecting like terms enables us to simplify expressions making them easier to use

Terms that contain the exact same variable can be classed as lsquolikersquo terms and be simplified

Watch out for the sign before each term

5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms

Use coefficients to collect like terms

First step in many problems involving Algebra


Algebra ndash Formulae

Introduction Substitution Changing the subject

Explains how to calculate the value of a variable

ldquoThe price of a taxi fare in Manchester depends on the distance driven Each fare is

charged a flat fee of pound2 and then pound3 for each mile drivenrdquo

119862 = 2 + 3119872For any given trip can easily work

out the cost of a taxi

119860 = 1205871199032

Area of circle formula

119878119906119887119895119890119888119905

Replace letters in the formula with numbers you are given

ldquoThe perimeter of a square is 4times the length of its sidesrdquo

119875 = 4119897What is perimeter of a

square with side length 5cm

119897 = 5 119875 = 4 5

119875 = 20119888119898Identify the formula and the values

to substitute in

Substitute values in using brackets

Carry out calculation remembering BIDMAS

Often it is useful to re-arrange a formula to make a different

variable the subject

119875 = 4119897

Make 119897 the subject of the formula

Use inverse operationsdivide 4divide 4

119875

4= 119897

119910 =18119905 minus 3

119901

Make 119905 the subject

119905 =119901119910 + 3

18

times 119901 +3 divide 18

Sometimes a variable will appear more than once in a formula

Make 119909 the subject of the formula119886 = 5119909 + 119909119910 rarr 119886 = 119909(5 + 119910)

119886

5 + 119910= 119909Factorise first


Algebra ndash Laws of indices

Basic Laws of Indices Advanced Laws of Indices

Special indices to consider

Anything to the power 1 = itself

Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1

Anything to the power 0 = 1

1 to the power of anything = 1

These laws can be applied if the bases are the same

119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887

When multiplying powers with the same base ndash Add the powers

When dividing powers with the same base ndash Subtract the powers

When raising the power (brackets) ndash Multiply the powers

1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012

119909minus119899 ൗ1 119909119899Find Reciprocal



119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices

119909119898119899 = 119899 119909 119898 Root by denominator first


Negative Fractional Indices


119909minus119886119887 =

1119887 119909

119886


Algebra ndash Expanding brackets

Single Brackets Double bracketsMultiply terms outside by all terms inside FOIL Method

10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092

Expanding brackets often the first step in simplifying algebra

+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms

L Last terms +441199092

1199092

11119909

4119909

44

Multiply each term in first bracket by each term in second

Grid Method

(119909 + 4)(119909 minus 3) Split brackets up around grid

119909 +4119909

minus3

1199092 4119909

minus3119909 minus12 1199092 +119909 minus12

Multiply each term in the grid2 119909 + 3119910 minus 7(2119909 minus 119910)= 2119909 + 6119910 minus 14119909 + 7119910

Include sign in multiplication

= minus12119909 + 13119910

Then simplify

Then simplify


Algebra ndash Factorising

Highest common factor method Ladder method

The process where an expression has common factors removed and brackets introduced

119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911

Look at whole expression identify HCF and divide out Divide out simple common factors repeatedly

3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)

Look at each term separately divide numbers first then the algebraic terms

181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)


Algebra ndash Linear Equations

Creating equations Advanced equations

A linear equation has the unknown variables as a power of one in the form 119886119909 + 119887 = 0

Solving linear equations

Once the equation has been created it can be solved using a

balance method or inverse method

Expand brackets and simplify (collect like terms)

If 119909 is on both sides eliminate smallest value

Eliminate excess number

Divide and solve for 119909

General 4 step process

ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1

minus2119909minus2119909

minus3 minus3

Equations where fractions are involved

Fractions are divisions and can be eliminated by multiplying119909

2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4

times (3 minus 119910) times (3 minus 119910)

Remove variable from denominator

2119910 = 4 3 minus 119910

Cross-multiplying allows us to move terms in a fraction from one side of

an equation to the other

119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67

The perimeter of the triangle is

67cm Form an equation in 119909


Algebra ndash Quadratics

An equation where the highest power of the variable is 2

1198861199092 + 119887119909 + 119888Factorising 119886 = 1 Quadratics

119909 plusmn 119909 plusmnAim Convert quadratic into double brackets

Sum and product rule

1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)

If c is negative Signs are different

1199092 + 5119909 minus 6 (119909 + 6)(119909 minus 1)Example

1199092 minus 7119909 + 12

Positive c rarr Signs Same

Negative b rarr Both Minus

(119909 minus )(119909 minus )Factors of 12

12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)

Factorising 119886 ne 1 Quadratics

119909 plusmn 119909 plusmnFactors of a to find

possible values= 5119909 plusmn 1119909 plusmn

51199092 minus 14119909 minus 3

3119909 plusmn 2119909 plusmn61199092 + 119909 minus 2 =

6119909 plusmn 1119909 plusmnOR

Then find factors of c and see which satisfy b

Difference of Two Squares (DOTS)

1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)


Algebra ndash Sequences

Introduction Arithmetic progressions Quadratic sequencesA series of numbers patterns

that follow a given ruleFinding the 119899119905ℎ term rule

Find the common difference (this will be your 119899 coefficient)

Write times table underneath sequence (of your 119899 coefficient)

Sequence minus times table (this is your extra bit)

Has the form 1198861198992 + 119887119899 + 119888A second layer difference

Halve 2nd layer difference for 1198992coefficient

Each number in the sequence is known as a lsquotermrsquo

Identify what is happening between each term to generate the rule

General rule is known as 119899119905ℎ term

Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13

Times table 3 6 9 12

Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2

11198992 + 119887119899 + 119888Find linear sequence

Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7

119899119905ℎ term rule of this

= 119899 + 3

11198992 + 1119899 + 3


Statistics ndash Mean Median Mode and Range

Mean Median Mode Range

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904

Collect it all together and share it out evenly

Median = Middle value(Numbers written in order)

Using the mean to find the total amount

Use of formula to find location of median

Mode = Most common valueitem

Occurrence of no mode

Range = Largest - Smallest

Finds the middle valueAverage usually used for

qualitative dataReveals how closefar apart the values are

3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52

8= 65 Median = 5 Mode = 3 119886119899119889 5 Range = 15 minus 3 = 12

Interpreting measures of spread

119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1

2Ezytown FC have scored an

average of 38 goals per game in their last 15 matches How many goals have they scored

38 times 15 = 57119892119900119886119897119904

The median of 45 values would be the 23rd number

when written in order45 + 1

2= 23

If every value appears equally there is no mode

1 1 3 3 7 7

Each value appears twice so there is no mode

The Smaller the range the closer and more lsquoconsistentrsquo

the values are

The Larger the range the more varied and more

lsquoinconsistentrsquo the values are


Statistics ndash Averages from a frequency table

Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904

MedianMedian = Middle value

(Numbers written in order)

ModeMode = Most common

valueitem

Range


Cars Frequency 119891119909

0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80

This column is created by multiplying

the frequency (119891) by the number in

the category (119909)

This enables us to find out

the total amount which is

needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910

The category with the highest frequency

2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205

The median lies between 20th and 21st value

Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27

Add down the frequency column When location

value has been exceeded that is the group where the

median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34

Two extra columns are created the midpoint of each group and the 119891119909

column

Weight Frequency Mid-point 119891119909

0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136

Statistics ndash Averages from a grouped frequency table

Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910

The category with the highest frequency119871119900119888119886119905119894119900119899 =

119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175

The median lies between 17th and 18th value

12

15

22

Add down the frequency column

When location value has been exceeded

that is the group where the median

lies

Median class =

+

+

8 minus 0 = 8119896119892

This enables us to find

out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892


Statistics ndash Representing data

Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels

Bars (equal width) (equal gaps)

Read off barsEllie = 40 hours

Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72

Find degrees per value 360deg divide 119905119900119905119886119897

1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25

Using a symbol to represent certain amount Line charts are useful for displaying time series data

Data points within the lines are important

Lines visualise change

Extract required data carefully


Statistics ndash Cumulative Frequency tables and graphs

Cumulative Frequency tables Cumulative Frequency graphs

Running total (add up as we go down)

Time Frequency Cumulative Frequency

45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52

The difference between cumulative frequency values will tell you the frequency

Points Cumulative Frequency

0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored

Always plot each point at the end of

the group

Draw a smooth line through the points

How many games did the player

score more than 12 points22

30 minus 22 = 8 119892119886119898119890119904

May be asked to calculate percentages

=119860119898119900119906119899119905

119879119900119905119886119897times 100

What percentage of games does the player score less than 12 points

22

30times 100 = 73

Equals

Equals

Equals


Statistics ndash Quartiles and box plots

Quartiles Box plotsBox plots allow us to visualise the spread of

a datasetInterested in specific points along the distribution

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25

The IQR provides a measure of the spread of the middle 50 of

the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1

Also used to compare two or more different datasets

0 10 20 30 40 50 60 70 80 90

Look to compare medians IQR and range

Q3

Q2

It is less affected by outliers than the range


Age Frequency Class Width FD

20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5

Statistics ndash Histograms

A special type of bar chart for grouped data

Create the Frequency

density column

Frequency Density on the vertical axis

Bars can be different widths depending on the group size

119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ

Calculating Frequency from Histograms

dividedividedividedivide

119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25

Class Width Frequency

10 30

15 60

5 25

10 30

10 25

timestimestimestimestimes


Statistics ndash Scatter Graphs

We use scatter graphs when we are interested in the relationship between two variables

Points plotted like coordinates

Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45

Labour Party Electoral Performance

Vote ()

Seat

s w

on

As one value increases so does the other

As one value increases the other decreases

No link between the two variables

One of the main incentives for drawing lines of best fit is to make predictions

Place line of best fit through the middle of the data

(Ignore Outliers)

Predict values by reading off the line

40 = 390 119904119890119886119905119904

Interpolation

Extrapolation

Predictions made within the dataset

Predictions made outsideof the dataset


Number ndash Fractions ndash Simplifying Improper Mixed

Simplifying Improper Fractions Mixed Number

Divide both the numerator and denominator by the same value

Turning into a mixed number

The combination of a WHOLE number and a Fraction

Repeat the process until the fraction is in its simplest form

Even = divide 2

Odd = divide 357hellip

The numerator is larger than the denominator

1199033

Divide numerator by denominatorto get whole number

Remainder forms new numerator

Denominator remains the same

Turning into an improper fraction

Multiply whole number by denominator

Add on the numerator56 + 3

Denominator remains the same59

8Donrsquot forget to simplify your answers where necessary

Useful skills for addingsubtractingmultiplying and dividing fractions


Number ndash Fractions ndash Addition and Subtraction

AddingSubtracting Mixed NumbersAdding or Subtracting fractions requires a

common denominator

2

9+5

9

7

9

When denominators are the same simply add the numerators

+

7

9minus

1

6When denominators are different

multiply the fractions

times 2 times 3

14

18minus

3

18

11

18

minus

Remember to simplify your answers

Method 1 ndash Deal with whole numbers and fractions separately

31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9

Method 2 ndash Convert to improper fractions first then calculate

61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10


Number ndash Fractions ndash Multiplying and Dividing

Multiplying

Multiply across the top and bottom

Dividing

Check to see if you can cross cancel

5

1

2

1

10

3divide

2

3

10

3times

3

2

Multiply by the Reciprocal

Finding the reciprocal of a fraction swaps the numerator and denominator

10

21

21

5divide 3

3

10

1

31

2


50100

Number ndash Fractions ndash Converting Decimal to Fraction

Decimal -gt Fraction Conversions you should know

Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10

Write 032 as a fraction in its simplest form

03225

32 divide 2

divide 2

16divide 2

divide 2

8Know your place values

Place Decimal part over 101001000 etc

Simplify the Fraction

Put back whole numbers if you had them


Number ndash Fractions ndash Converting Fraction to Decimal


Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Equivalent Fraction method

075

025 ൗ1 4

025 + 025 + 025ൗ3 4

Multiply or Divide the fractions so that they can be converted to decimals easily

Division methodDivide the numerator by the denominator

Using Bus shelter division

13

20times 5

100

65= 065

Mixed Numbers and Improper Fractions

Process does not change

Try and work with mixed numbers where possible

344


Number ndash Fractions ndash Converting recurring decimals

More complex recurring decimalsBe in a position to eliminate the Decimal numbers

020 ሶ5= 119909(times 100) (times 100)020 ሶ5

Move recurring decimal up to the decimal point

20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =

Be in a position to eliminate the

Decimal numbers

119909 = 185900

=37

180

Move recurring decimal up to the decimal point by multiplying by 10100 etc

Create equation by labelling the decimal 119909

Be in a position to eliminate the recurring decimal by multiplying again by 10100 etc

Subtract two equations to eliminate recurring decimals and convert into fraction

Simplify your answer


RPR ndash Quantities as fractionspercentages of each other

Quantities as fractions of each other Quantities as percentages of each other

Think of it as a test score

Fractions of amounts

Percentages of amounts

Express 5 as a fraction of 25

5

25

1

5

Nigel earns pound90 and saves pound30 Sanjay earns pound100 and saves pound35 Who has saved a greater proportion of their

earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035

Divide amount by denominator

Then multiply by the numerator

3

5of 60 60 divide 5 = 12 12 times 3 = 36

Express 5 as a percentage of 20

Convert to fraction or decimal then to percentage

5

20

divide 5

divide 5

25

100

times 5

times 525

Equivalent fraction over

100

Method 1

Convert to decimal

Method 2

025 255 divide 20times 100

times 100

Method 1- Unitary method

Find 1 10 5 etc

Method 2- Decimal method

Turn to a decimal divide 100 Then multiply by

amount

Find 35 of 40


RPR ndash Percentages and percentage change

Introduction to percentages Percentage increasedecrease

Calculate percentage of amountAdd on for increase Subtract for decrease

Means out of 100

The common

conversions you should really know

Conversion Flow Chart

Increase 70 by 15

Method 1- Unitary method 10 = 7 5 = 35 15 = 105

Method 2- Decimal method 15 = 015 015 times 70 15 = 105

119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value

Method 3- Calculator method

Increase or Decrease

Add to 100 for increase

Subtract from 100 for decrease

Convert to decimal (divide 100)

This is your multiplier factor

119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058

To find original amount work backwards and divide

by multiplier factor


RPR ndash Simple interest and Compound Growth and Decay

Simple Interest Compound Growth

Follows the I=PRY formula

Interest RatePrincipal Yearstimes times

Calculate the Simple Interest earned on pound350 at a rate of 9 pa for 4 years

pound350times009times4= pound126Interest

Remember to convert the to a decimal

To calculate other parts of the formula you will need to change the subject

How many years would it take for pound45000 to receive pound19800 Simple Interest at a rate

of 55 pa

Interest

RatePrincipal

Years

times

An amount is increased or decreased by a percentage

The process is repeated several times at each interval

The most efficient way to do this is using a Multiplier

Interest 1 plusmn RatePrincipal

Years

times

Growth

Decay

The general formula for compound

growth and decay

pound4000 is invested at a rate of 5 pa for three years Calculate Final value of the investment after three years

pound4000 times 1053= pound463050

A car worth pound15000 depreciates in value at a rate of 15 pa What is the depreciated value of the car after 4 years

pound15000times0854= pound783009



RPR ndash Ratio

Introduction Sharing in a given ratio Map scale factors

Is the relationship between two or more quantities

It is written in the form 119938 ∶ 119939

Compares one part to another part

The ratio of red to blue is 4 ∶ 5

The ratio of blue to red is 5 ∶ 4

Sentence structure is important

Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2

Divide all numbers by the same valueThe ratio of boys to girls in a

Geography class is 15 ∶ 5What fraction of the class is girls

5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4

Share $40 in the ratio 3 ∶ 5Find total

number of partsAdd the ratio

parts together

3 + 5 = 8Find value of

one partDivide amount by number of parts

$40 divide8 = $5Each part of the ratio is worth $5

Multiply by original ratio

3 ∶ 5times $5

$15 ∶ $25

Mark and John have sweets in the ratio 3 ∶ 4 If Mark has 27 sweets How many does John have

27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)

It is the ratio of a distance on the mapmodel to the corresponding

size in real life

Written in the form 120783 ∶ 119951

Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903

Know your conversions

10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898

A map has a scale of 125000 Michael is 6cm from his home

How far from home is he Give your answer in 119896119898

6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000


RPR ndash Proportion

Direct Proportion Inverse ProportionDirect Proportion

As one value increases the other increases at the same rate

Inverse Proportion

As one value increases the other decreases at the same rate

Three Coffees cost pound750 How much would five Coffees cost

Find the value of one coffee then multiply by quantity needed

pound750divide 3 = pound250 119901119890119903 119888119900119891119891119890119890

pound250 times 5 = pound1250

It takes 3 men 4 days to build a wall How long would it take 2 men

Find the time taken by one man then divide by quantity stated

3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904

119910 is directly proportional to 119909

119910 prop 119909

119910 = 119896 times 119909 119896 is the rate of change

Constant of proportionality

Work out the value of 119896

Compare two values

Form equation to solve problems

Solving direct proportion problems119901 is directly proportional to 119905

119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13

119910 is inversely proportional to 119909

119910 prop1

119909119910 =

119896

119909


119896

Solving inverse proportion problems119901 is inversely proportional to 119905

119901 = 16 119905 = 2a) Find 119901 when 119905 = 8

b) Find 119905 when 119901 = 64Compare two values

119901 =119896

11990516 =

119896

2

Work out the value of 119896times 2

32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05


RPR ndash Graphical representations of Proportion

Direct Proportion Inverse Proportion

119910 prop 119909 119910 = 119896 times 119909As one value increases so

does the other

Linear relationships

Non - linear relationships

119910 prop1

119909119910 =

119896

119909


Inverse proportion will always give a Curved graph


Geometry ndash Quadrilaterals

Know the names of these Quadrilaterals and their properties


Geometry ndash Triangles

Know the names of these Triangles and their properties


Geometry ndash Polygons

RegularSide lengths are the

same

Interior and exterior angles the same

IrregularSide lengths are not

ALL the same

Interior and exterior angles the not ALL the

same

Equal sides are marked with a dash through the line

ConvexThe shape is lsquobulgingrsquo

outwards

All angles less than 180deg

ConcaveThe shape has lsquocavedrsquo

inwards

One or more angles is greater than 180deg

They are classified by the number of sides they have

Number of sides Name of shape

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon


Geometry ndash 3D shapes

The same cross sectional area throughoutPrisms

Faces

Faces Edges and Vertices

The flat surface of a 3-D shape

Edges

Vertices

Where two Faces meet

Where two or more edges meet at a point

Shape Faces Edges Vertices

Cube 6 12 8

Cuboid 6 12 8

Triangular Prism 5 9 6

Pentagonal Prism 7 15 10

Square based Pyramid 5 8 5

Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0

Faces + Vertices ndash Edges = 2


Geometry ndash Angle facts

Introduction

A straight line Around a point Vertically opposite

A measure of turn Using a protractor

It has the unit degrees (deg)Reflex

Obtuse

Right

Acute Straight

Full turn

Greater than 0deg less than 90deg

Exactly 90deg


Exactly 180deg


Exactly 360deg

Has a square in the angle to indicate that it is 90deg

Looks like a book closing or crocodile jaws

Looks like a book falling open

A half turn to create a straight line

The larger angle outside the acute or obtuse angle

A movement around a point to create a circle

All angles on a straight line will add up to make 180deg

All angles around a point will add up to make 360deg

Where two straight lines cross opposite angles are

equal

30deg150deg


Geometry ndash Angles in triangles and polygons

Triangles Polygons

All three angles can be orientated to fit on a straight line All angles in a triangle make 180deg

Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg

Exterior angle = Sum of two angles on opposite side

107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle

It has two equal lengths

It has two equal angles

Knowledge of triangles is important

Number of sides

Number of

Sum of interior angles

Regular interior angle

Regular exterior

angle

3 1 180deg 60deg 120deg

4 2 360deg 90deg 90deg

5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg

119899 (119899 minus 2) (119899 minus 2) times 180deg (119899 minus 2) times 180deg

119899360deg divide 119899

The number of triangles in a shape will always be TWO less than the number of sides


Geometry ndash Angles Parallel lines

Alternate Corresponding Co-Interior

Two lines that travel side by side keeping the same distance apart at all times never intersecting

When a line intersects the parallel lines

Eight angles are created

The angles on one parallel

line match up with the angles

on the other parallel line

One Acute angleOne Obtuse angle

Different lines will

create different sets

of angles

Alternate angles are the same

lsquoZrsquo shape lsquoFrsquo shape

Corresponding angles are the same

lsquoCrsquo shape

Co-interior angles make 180deg


Geometry ndash Pythagorasrsquo Theorem

1198882 = 1198862 + 1198872

For any right angled triangle the area of the square drawn on the hypotenuse is equal to the sum of the areas drawn on the other

two sides

Finding the Hypotenuse Finding the Shorter side

If you know the lengths of the two shorter sides you can calculate

the length of the hypotenuse

Square the two sides

Add them

Square root for answer

8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )

If you know the Hypotenuse and a shorter side you can calculate the

length of the other shorter side

Square the two sidesSubtract

themSquare root for answer

119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )


Geometry ndash Trigonometry functions

Sine Function Cosine Function Tangent Function

119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1

you have to press shift first

Work out side lengths and angles when given the opposite and

hypotenuse

Only used to calculate an angle

Trigonometry is used to calculate sides lengths and angles in triangles using three important ratios Sine Cosine and Tangent

The angle is often described as theta which

is the Greek letter (120579)

Work out side lengths and angles when given the adjacent and

hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1




Geometry ndash SohCahToa

Decide which ratio to apply

Label the triangle accurately with OAH

Use the triangles to help

When to use trigonometry

Find a missing side when given a length and an angle

Find an angle when given two lengths

119952119955

Calculating a missing side Calculating a missing angle

42deg

14

119909119874

119867

119860119914 119919

119912

Cover up what youneed to find

119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926

119909 is an angle so use inverse

119909 = sinminus1(14 divide 378)

= 217deg

Use the functions and substitute

Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905


Graphs ndash Coordinates

A set of values that indicate the position of a point

They normally occur in pairs in the form (119909 119910)

(119909 119910)

Direction along the 119909 - axis

Direction updownthe 119910 - axis

Along the corridor

Up Down the stairs

Start from a central point(00) - Origin

Reading the coordinates will lead you to the exact position

(7 minus4) Seven units rightFour units down

(minus26) Two units left Six units up

(minus5minus2) Five units left Two units Down

119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899

Four Quadrants Plotting coordinates


Graphs ndash Equation of a Straight line

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786

Horizontal lines 119910 =

Vertical lines 119909 =

All straight line graphs follow the same rule

119910 = 119898119909 + 119888Gradient 119910 intercept

119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892

Gradient is the lsquosteepnessrsquo of the line

Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5

Equation of line from coordinates

Calculate gradient between points (119898)

Substitute in points and solve (119888)

Find the equation of the line that passes through (02) and (38)

Gradient=1199102minus1199101

1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888


Graphs ndash Midpoints Parallel lines and Perpendicular lines

Midpoints Parallel lines Perpendicular lines

A midpoint is the halfway point between two end points of a line

segment119909119860 + 119909119861

2119910119860 + 119910119861

2

Add up the 119909coordinates and halve it


Find the coordinate of the midpoint joining the points (611) and (15 minus9)

119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1

The distance between two

points will always be the hypotenuse

Parallel lines are lines that run equidistant to each other and never

intersect (cross)

Parallel lines have the same gradientDifferent 119910 minus intercepts

119910 = 119898119909 + 119888Same Different

Find the equation of the line parallel to 119910 = 2119909 + 4 that passes through (42)

Substitute in point and solve (119888)

119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8

Perpendicular lines are lines that intersect (cross) to form 120791120782deg angles

Gradients that are negative reciprocals of each other

Use gradients and coordinate points to calculate the equation of the line

Invert = Reciprocal

1198981 times1198982 = minus1


Graphs ndash Contextual graphs

Distance ndash Time graphs Velocity ndash Time graphs Financial graphsDistance - Time graphs record the journey of an object as it begins to

move away from and return to a point119860

119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890

Not all objects travel at a

constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899

Calculate speed at a specific point

by creating a tangent

Velocity - Time graphs record the velocity of a particle moving along a

straight line

The gradient of the line is the acceleration

Area under graph = Distance travelled

Currency Conversions

Cost Comparisons

Predict future costs


Graphs ndash Quadratic and Cubic graphs

Quadratic graphs Cubic graphs

-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092

Substitute the 119909 values into equation to get 119910

and plot points like coordinates

To plot graphs

The points where the graph crosses

the 119909 axis are known as roots

These are the solutions to the

quadratic when it equals zero

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6

Maximum and minimum points

Read off coordinate point from graph

The line of symmetry runs through these points

Find the midpoint of the roots and substitute into equation to calculate 119910 minuscoordinate

Complete the square in the form (119909 minus 119901)2+119902

(119901 119902)Maxmin point can be found by

Can be defined as an equation where the highest power of the

variable (usually 119909) is 3

119910 = 1199093 + 31199092 + 5119909 minus 20


Graphs ndash Reciprocal and Exponential graphs

Reciprocal graphs Exponential graphs

119910 =1

119909

119910 =4

119909

119910 =6

119909

As 119886 increases the graphs move

further away from the origin

119909-axis and 119910-axis are asymptotes

Two lines of symmetry

119910 = minus1

119909119910 =

1

119909

119910 =119886

119909

Points can be found by substitution

As 119909 continues to increase 119910 continues to rise or fall at a continually faster or slower rate

When 119896 gt 1 and 119909 is positive

the graph will curve upwards

When 0 lt 119896 lt 1 or 119909 is negative the graph will curve

downwards

119910 =1

119896

119909


Graphs ndash Equation of a circle

There is a specific general formula for the equation of a circle

1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin

Find the radius (119903)

Radius = 5

1199092 + 1199102 = 1199032

Finding the radius and equation using Pythagoras

52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41

We may be asked to find the equation of a tangent to a circle at a given point

What is the tangent to the circle 1199092 + 1199102 = 25 at the point (34)

Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3

Circle TheoremA radius always meets a tangent at a right-angle

minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)

= Perpendicular line

119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4


Geometry ndash Perimeter and Area

Perimeter Rectangular areas Triangular areas

The total distance AROUND a 2D shape

Adding all the side lengths together

100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898

The process does not change if we have algebraic terms

= 9119909 + 2119910

The total space taken up by a 2D shape

Multiplying two side lengths together

= 119897 times 119908Area of rectangle

6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982

With compound shapes break it down

119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982

The area of a triangle takes up half the space of the rectangle that is

formed around it

=1

2(119887 times ℎ)Area of triangle

7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982

Be sure to use perpendicular heights

Calculate base times height first

Remember to halve your answer


Geometry ndash Advanced areas

Parallelogram

Trapezium

Imagine a tilted rectangle


= 119887 times ℎℎ

119887

(119886 + 119887)times ℎ=1

2

A more complex formula to know

Add the parallel sides

Halve it

Multiply by height

Calculating area of a triangle using Τ1 2 119886119887 119904119894119899119862

Two sides and the included angle

Label triangle

Substitute values in and calculate

119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)

119860119903119890119886 = ൗ1 2 times 7 times 10 sin(73deg)

119860119903119890119886 = 33471198881198982

We have two sides and the included angle


Geometry ndash Circle Definitions

119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment

The perimeter around the circle

The distance across the centre of the circle

The distance from the centre to the edge of the circle

Part of the area of a circle enclosed by two radii

Part of the circumference of a circle

A straight line that touches the curve of the circle at a point

A straight line segment between two points on the circle edge

The area created by the chord


119860 = 1205871199032

Geometry ndash Area and Circumference

Area

Circumference

Sector

Arc length

Pi times the radius squared

Diameter is double the radius

119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898

The circumference is always about three time the length of the diameter

119860 =119899deg

3601205871199032

Calculate the proportion of the circle required then use area

formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889

Calculate the proportion of the circle required then use circumference formula

85deg

85deg

360deg(120587 times 12)

890119888119898


Geometry ndash Volume

Prisms

The same cross sectional area throughout

119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids

The volume of a pyramid is always Τ120783 120785 of the prism that surrounds it

119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3

Calculate the volume of the square based pyramid if its height measures 15119888119898

121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033

Find the volume of the sphere with diameter 16119888119898 Give your answer to

3 significant figures

119881119900119897119906119898119890 =4

3times 120587 times (8)3

21446605851198881198983

21401198881198983

For a hemisphere donrsquot forget to halve your answer


Number ndash Approximation and Error Intervals (Bounds)

Error IntervalsBy definition a rounded number does not give us

the exact value

Approximation

Estimates tell us the rough value of a calculation

Rounding off makes it easier to calculate

Round values to 1sf

Perform Calculation

841 times 32

000216asymp8 times 3

0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound

The minimum a value might be

The maximum a value might be

Continuous Values (Decimal values)

Halve accuracy level

Add on for Upper bound

Subtract for Lower bound

240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898

Discrete values (Whole values)The number of people on a train is 400 to the nearest 100

400350 449


Geometry ndash Similarity and Congruence

Congruence Similarity in 1D Similarity in more than 1DArea and Volume scale factors will

need to be calculatedCongruent shapes are just exact

replicas of the original

The angles and side lengths remain the same

The shapes may well be orientated differently

The two shapes are congruentThey are reflections of each other

Similar shapes are just enlargements of the original

The angles remain the same but the lengths have been scaled up or down

This scale factor needs to be calculated in order to solve problems involving similar

shapes

Find two comparative lengths

119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4

Area scale factor = 119878119888119886119897119890 1198911198861198881199051199001199032

Volume scale factor = 119878119888119886119897119890 1198911198861198881199051199001199033

Work out all scale factors first

Scale factor = 2

Area scale factor = 2 2 = 4

Volume scale factor = 2 3 = 8


Geometry ndash Transformations

Reflection Rotation Translation EnlargementReflection the

replacement of each point on one side of a line by the point symmetrically placed

on the other side of the line

Need a mirror line

Reflection in the line119909 = minus2

Rotation the action of rotating a shape about an

axis or central point

Need a measure of turn

Need a Centre point

Need a Direction

Use of tracing paper

Rotation90deg

Clockwise

About (minus50)

Translation the action of lsquoslidingrsquo a shape to a new

position

Described using Vector Notation

119929 rarr 119930 =

Translation minus108

Enlargement the action of resizing a shape to the scale factor given from a specific

point

Need a Centre point

Need a scale factor119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

EnlargementScale factor

3

From (3 minus6)

119929 rarr 119930

Use projection

lines


Geometry ndash Congruence criteria for triangles

Side Side Side Side Angle Side Angle Side Angle Right Hypot Side

All three sides of one triangle are equal to the

corresponding sides of the other triangle

Two sides and the included angle are equal

to the corresponding sides and included angle of the

other triangle

Two angles and one side of a triangle are equal to the corresponding angles and side of the other triangle

Each triangle contains a right angle the hypotenuses

are equal in length as well as another equal comparative side


Geometry ndash Constructing bisectors and Loci

Create arcs above and below the line

Set compass to over halfway

Sometimes you will need to

create a separate line segment

Mark off each line segment

Create arcs within the angle

Do not alter the compass at all


The locus of points from a point is a circle

119860 119861

The locus of points between two points is the perpendicular

bisector

The locus of points between two lines is

the angle bisector

Notice how the locus of points around a corner is CURVED

A locus is a series of

points that satisfy a

particular conditionLoci is the

plural and will often involve

several conditions


Algebra ndash Solving by factorising and the Quadratic formula

Solving by factorising The quadratic formula

Substitute values into the formula to generate two answers for 119909

119909 =minus119887 plusmn 1198872 minus 4119886119888

2119886

51199092 + 8119909 minus 4 Identify values of 119886 119887 and c119886 119887 119888

The formula you need to

know

119909 =minus 8 plusmn 8 2 minus 4 5 minus4

2 5

119909 =minus8 plusmn 144

10

119909 = 04 or minus2

Substitute and simplify

Carry out two calculations

1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0

Factorise the quadratic ndash You may need to rearrange first

1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0

Find values for 119909 that will make each bracket = 0

119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1

Expand and rearrange to = 0


Algebra ndash Completing the square and solving quadratics

Changing the form of the quadratic

1198861199092 + 119887119909 + 119888

Completing the square 119886 ne 1

(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify

1199092 + 6119909 minus 2 119909 + 3 2 minus 3 2 minus 2

119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0

coefficient of 119887

Solving equations by completing the square

119909 minus 5 2 minus 10 = 0

119909 minus 5 2 = +10 119909 minus 5 = plusmn 10

119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184

Complete the square Solve equation

31199092 + 6119909 minus 4 = 0Factor out the three

3 1199092 + 2119909 minus4

3= 0

Complete the square on this expression

3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7

3 119909 = minus1 plusmn7

3119909 = 0528 119900119903 minus 2528

Add 7 to both sides

Divide both sides by 3

Square root both sides

Subtract 1from both

sides


Algebra ndash Simultaneous equations

Simultaneous equations

Linear equations (Elimination method)

Linear equations (Substitution method)

Quadratic equations (Substitution method)

Multiply equations to get matching coefficients

4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1

Addsubtract equations

Substitute to find second variable

3119909 + 2(minus1) = 4Substitute 119910 = minus1 into equation 2

3119909 minus 2 = 4 119909 = 2

Matching coefficientsSame signs (Subtract the equations)Opposite signs (Add the equations)

119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7

119910 = 2119909 + 1Rearrange to get a single variable on

its own

3( 2119909 + 1) + 10119909 = 7Substitute 119910 = 2119909 + 1 into equation 1

16119909 + 3 = 7119909 = 025

Substitute equations to find

first variable

Substitute 119909 = 025 into equation 2

119910 = 2(025) + 1 119910 = 15Substitute to find second variable

119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2

119909 + 6 = 1199092 minus 2119909 + 20 = 1199092 minus 3119909 minus 4

119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10

Substitute equation into quadratic and rearrange to = 0

Factorise and find two solutions for variable

Substitute each answer to find other pair of solutions

Equations involving two or more unknowns that are to have the same values in each equation

4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1


Algebra ndash InequalitiesSymbols

Inequalities on a numberline

Solving Quadratic inequalities

Shows the range of values for an inequality

Open circle does not include value

Closed circle includes value

Solving Linear inequalities

Same process as solving linear equations

When we multiply or divide by a negative number we must flip the inequality sign

119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5

Same process as solving equations

Sketch the graph to interpret the range of values required

1199092 le 16

119909 = plusmn4The roots

minus4 le 119909 le 4The range of values

below the line

Two variable inequalitiesInequalities with 2-variables need to be

represented on a graph

Shade the region satisfied by the inequalities 119910 gt minus119909 119910 le 4 119909 lt 3

le - Solid linelt - Dashed line

Draw the line graph for each inequality


Statistics ndash Probability

Introduction

Counting outcomes

Calculating probability Types of events

The likelihood of an event happening 119875(Event)=119899119906119898119887119890119903 119900119891 119904119906119888119888119890119904119904119891119906119897 119900119906119905119888119900119898119890119904

119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904

Simplify answers where

possible

Mutually exclusive

Working out how many combinations there are

1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6

Rolling a die and flipping a coin

This is a sample space diagram

There are 12 possible outcomes from this event

119875(3) =2

8

1

4

The lsquoORrsquo rule (mutually exclusive)

119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability

The lsquoANDrsquo rule (independent)

119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability

Events that cannot happen at the same time

Rolling a die 119875(1 119886119899119889 6)

All probabilities from the event will sum to make 120783

Independent events

Events where the outcome of one doesnrsquot affect the outcomes of the others

Picking a counter out of a bag replacing it and repeating

Dependent events

Events where the outcome of one doesaffect the outcomes of the othersPicking a counter out of a bag not

replacing it and repeating

Calculating expected outcomes

119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904


Statistics ndash Venn Diagrams and Probability trees

Venn diagrams

Intersections cap and Unions cup

A set is a collection of things called elements

119860 = 2 3 5 7 11The set of prime

numbers less than 12

119860 = 2 3 5 7 11 119861 = 1 3 5 7 9

119860 cap 119861 = 3 5 7 minus Intersection of A and B

119860 cup 119861 = 1 2 3 5 7 9 11 minus Union of A and B

119860prime cap 119861 = 1 9

The crossover between sets

All values in the sets

Values in 119861 but not in 119860

Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200

Probability trees are really useful to calculate the probabilities of combined events happening

Multiply along branches

119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15

Add together all combinations

Probability trees where the outcome of one events affects the outcome of the next event eg no replacement weather etc

119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012

When dealing with no replacement remember to reduce the denominator

by one for the second event

119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104

Number ndash Standard Form

AddSubtract Standard formTake numbers out of Standard form

AddSubtract valuesConvert answer back to Standard form

Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000

314 times 10minus4 = 0000314Positive power of 10 = Large number

Negative power of 10 = Small decimal number

(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104

MultiplyDivide Standard formSeparate the numbers and powers of 10

MultiplyDivide numbers Apply laws of indices to power of 10s

Give answer in Standard form

( times ) times ( times )46 3

46 times 3

138

103104 times

107times

138 times 108

156 10minus4 75 10minus7( times ) divide ( times )

156 75divide 10minus4 10minus7divide

0208 103

208 102

times

times

===

times

times


Algebra ndash Algebraic fractions

Multiplication and DivisionAddition and Subtraction

Make sure the denominators are

the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910

Multiply fractions to achieve same denominator

times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5

Same rules apply in Subtraction

times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5

Expand brackets and simplify where possible on the numerators

2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1

minus8 minus 51199092 minus 3119909

119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1

Multiply across the numerators and denominators

Cross cancel terms where possible

Simplify each fraction

Factorise expressions

2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3

To divide multiply by reciprocal of 2nd fraction

4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911


Algebra ndash FunctionsThink of it as a machine that has an input which is

processed by the function to give an output

119891 4 = 6 119891 minus3 = minus1

Substitute input into the function to

generate output value

119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9

Substitute input into the function and calculate

119891 119905 = 31199052 + 2 find 119891 2

119891 2 = 31199052 + 2Substitute input into the

function and calculate

119891 2 = 12 + 2 119891 2 = 14

Inverse functionsA function that performs the

opposite process of the original function

You have been given the output and need to work out

the value of the input

119891 119909 119891minus1 119909Normal function Inverse function

Two ways to solve problems involving inverse functions

Function machine method Subject of Formula method

Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5

The combination of two or more functions to create a new function

119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)

The output of 119892(119909) will form the input of 119891(119909)

119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2


RPR ndash Rates of change

Rate of Change Average rate of changeA rate that describes how one quantity changes in relation to another quantity

It is represented by the Gradient of a line

Gradient =1199102minus1199101

1199092minus1199091

Gradient =119877119894119904119890

119877119906119899

Interpreting Rates of Change

GradientAmount of (119910) per

Amount of (119909)

Rate of change = $50 per month

Instantaneous rate of change

The rate of change over a given interval

The rate of change at a particular moment

Create chord between two intervals

Calculate gradient of chord

Interpret gradient as a rate of change

Create tangent at specific point

Calculate gradient of tangent



Graphs ndash Translations and Reflections

ReflectionTranslation

A translation can be defined as the movement lsquoslidingrsquo of a shape to a new position

119891 119909 + 119886The graph

shifts updown the 119910 minus 119886119909119894119904 by 119886 units

119891 119909 + 119886The graph shifts leftright along the 119909 minus 119886119909119894119904

by 119886 units

+= Left

minus= Right

Adding to the function causes a Translation

Given 119891(119909) Sketch 119891(119909 minus 3) + 5

3 units right 5 units up

119891(119909)

119891(119909 minus 3) + 5

3

5

Reflection The replacement of each point on one side of a line by the point symmetrically placed on

the other side of the line

The graph is reflected in the 119962 minus 119938119961119946119956

119910 = minus119891(119909) 119910 = 119891(minus119909)


The outputs are reversed

The inputs are reversed

119891(119909)119891(minus119909)119891(119909)

minus119891(119909)


Graphs ndash Using graphs to find solutions

When we are given a system of equations we can find the solutions to these equations algebraically (simultaneous equations) or graphically

2119909 + 3119910 = 18

119910 = 119909 + 1

Solve simultaneously or graph each

equation

The point of intersection will be

your solutions The point where 119909 and 119910 have the same value

for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1

Two intersections = two sets of solutions

or

Find the solutions of 119909 when

1199092 minus 2 = 7

119909 = 3

119909 = minus3

Plot graph and read off points

or

Identify and read off the points of intersections for your solutions

Use graph to read off specific values

for 119909 and 119910

Plot each equation

separately


Graphs ndash Estimating gradients Area under a curve

Area under a curveEstimating GradientsCalculate average gradient from

beginning to end These are not very accurate and do not

show the full picture

Break the graph down into smaller pieces to see what is happening

119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904

Find out what is happening at a particular point - Tangents

Estimate because tangents vary

The area under a curve will enable you to estimate the total distance travelled in

velocity time graphs

1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed

Estimate the distance travelled for the first ten seconds

275119898


Geometry ndash Sine and Cosine rules

The Sine rule formula The Cosine rule formula

119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or

Missing length Missing angle

Using Trigonometry to calculate missing angles and side lengths in non right angled triangles

We tend to use the Sine rule if we know an angle and its opposite length

35

sin(125)=

119909

sin 35 times sin(35)

35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28

21119909deg = sinminus1

sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle

If we know two sides AND the included angle

1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90

cos 119909 = minus0303hellip

119909 = cosminus1(minus0303hellip) 119909 = 1077deg


Graphs ndash Trigonometric Graphs

Sine 119909 axis intercept at 0deg 180deg 360deg

Cosine

Tangent

119878119894119899 120579 =119900119901119901

ℎ119910119901Turning points at 90deg 270deg

They are periodicalthey will continue to repeat every 360deg in both directions

There can be multiple answers for values of 119878119894119899 120579 119862119900119904 120579 119879119886119899(120579)

Use symmetry of graphs to solve problems

119862119900119904 120579 =119886119889119895

ℎ119910119901

119909 axis intercept at 90deg 270deg

Turning points at 0deg 180deg 360deg

119879119886119899 120579 =119900119901119901

119886119889119895


Asymptotes at 90deg 270deg


Statistics ndash Types of data and Sampling

Types of dataQuantitative Qualitative

Data that is numeric Data that is descriptive

Discrete

Continuous

Data that can be counted and only has

certain valuesPeople on a bus

Shoe SizeDress size

Data that can be measured to various

levels of accuracyHeight of a treeSpeed of a car

Mass of a person

Categorical data

Data which can be grouped into

categories

Grouped Data

Hair colourFavourite food

Sport

Data which is organised into classes

Primary

Secondary

Data collected by you

Data gathered from another source

SamplingWe collect and analyse data to give us information about a population

Census

Sample

Bias

Data is collected from the WHOLE population

Can take a very long time to collect the information

Data is collected from PART of the population

Quicker to collect the data and the data can be used to describe the whole population

Random Stratified

Each member assigned a numberNumbers randomly generatedThose numbers used in sample

119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890

Your sample is randomly selected Proportionate numbers from each group selected to make sample

Some situations can cause bias and make the sample

unrepresentative

When and where the sample is takenIs the sample large enough

Who is in the sample


Probability and Statistics ndash Frequency and Two-Way Tables

Colour Tally Frequency

Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue

Favourite ColourTally Up Add Tally Marks

Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01

Relative frequency describes

what proportion

selected that colour

If the sample size is large enough we

can interpret relative

frequencies as

probabilities

TwondashWay Frequency Tables

Year 12 Year 13 Total

Male 120 80 200

Female 70 100 170

Total 190 180 370

Provides information about the frequency of two variables

Number of girls in Year 12 is 70


Male 120 80 200

Female 70 100 170

Total 190 180 370

Rows and columns add up

Use to calculate missing values


Geometry ndash Circle Theorems

Angle at the centre is twice the angle at the circumference

119909

2119909

Look for the lsquoArrowrsquo Shape

Angles in the same segment are equal

119909

Look for the lsquoBowrsquo Shape

119909

119909

Angle subtended at circumference by a semicircle is 90deg

Opposite angle to the diameter

90deg

Opposite angles in a cyclic quadrilateral sum to 180deg

A

C

B

D

A + C = 180deg B + D = 180deg

Tangents and radii meet at 90deg

90deg

Tangents from a point have equal length

Alternate Segment Theorem

119909

119909119910

119910

Tangent just touches the

circumference

Tangent


Number ndash Surds and Rationalising the denominator

Surds Rationalising the denominatorSurds are expressions which contain an irrational square

root Rationalising the denominator involves removing all

of the roots from the bottom of a fraction

119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5

Writing in the form 119886 119887

Think square numbers 200

119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100

Choose the largest square factor

100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3

Multiply top and bottom by irrational root

6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2

Multiply top and bottom by Conjugate

(opposite root)

=5 3 minus 2

3 + 2 3 minus 2Expand and simplify

=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7

A more complex denominator


Geometry ndash Vectors

Vectors describe translations

119909119910

Direction along

Direction updown

Represented by arrows

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow

Direction = Where arrow is

pointing

Adding and Subtracting

Start arrow at end of previous vector

119938 119939

119938 + 119939119939 starts at the

end of 119938

Negative vector goes in opposite direction

119939

minus119939 119938 119939

minus119939

Multiplying

Only affects magnitude not direction

3119939 = 119939 + 119939 + 119939

119939

1199393119939

Parallel VectorsSame direction May have different

magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction


Number ndash Units ndash Mass Length Area and Volume

Length

Mass Area and Volume

119898119892 119892 119896119892 119905119900119899119899119890times 1000 times 1000 times 1000

divide 1000 divide 1000 divide 1000

11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions

Area is a 2D measurement formed by multiplying two lengths

100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982

Volume is a 3D measurement formed by multiplying three lengths

10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983

Squared units results in squared conversion factors

Cubed units have cubed conversion factors


Number ndash Units ndash Time and Money

MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds

Conversions between units are not decimal

Convert 350 seconds into minutes and seconds

1 Minute 60 seconds

2 Minutes 120 seconds





5 minutes and 50 seconds

Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898

Adding and Subtracting money requires lining up the decimal point

pound 6 5 2 0pound 1 0 0 0pound 0 6 5+

5857pound

pound1 is worth and can be exchanged for $134

119866119861119875 119880119878119863times 134

divide 134

Money comes in specific denominations


GBPUSD = 1 134

Exchange rate calculations


Geometry ndash Bearings

Angle clockwise from North

119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg

Sentence Structure Important

The bearing ofB from A is 075deg

119873

75deg

B

A

Lines North are Parallel

119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg

120579deg = 180deg minus 75deg = 105deg

Angles around a point

119873 B

A

119873

75deg

105deg

120601

105deg + 120601deg = 360deg120601deg = 360deg minus 105deg = 255deg

255deg

The bearing of A from B is 255deg


Number ndash Rounding

Rules of rounding01234 56789 ROUND UPROUND DOWN

Place Value Significant FiguresDecimal Places

14672To the nearest ten

14670To the nearest hundred

14700To the nearest thousand

15000

125298To 1 decimal place



12530

325484To 1 significant figure

300000


325000Be careful when rounding up a nine (A double rounding will occur)

1 2 3 4

Count Right from the Decimal Point


330000

Count Right from first non-zero Digit

1 2 3 4 5 6




Chunking method

Column method



430 + 50 = 480

6 + 8 = 14

436 + 58 = 494

563 minus 237

563 minus 200 = 363

363 minus 30 = 333

333 minus 7 = 326

563 minus 237 = 326


436058+

4941

563237minus

326

5 1



+ + =

+ minus =

minus minus =

minus + =


119860119889119889 ℎ119900119905 119908119886119905119890119903

119860119889119889 119888119900119897119889 119908119886119905119890119903

119877119890119898119900119907119890 119888119900119897119889 119908119886119905119890119903

119877119890119898119900119907119890 ℎ119900119905 119908119886119905119890119903




BoxGrid method

Bus shelter method

Chinese

Tens in top half





523 times 76

523 times 76

Separate numbers





523 times 76

Line up numbers








958 divide 8



Negative numbers




Number ndash BIDMAS

Example 1


ORDER


6 + 3 times 4

6 + 12


Addition second

18



3 + 16 minus 2




17


(3) 2divide (2)

9 divide (2)

45



within brackets

Example 2

Example 3

=

==

=

=

==

=

=

=

=





1 and itself




another number

1 is not a prime

number


number


2 3 5 7 11 13 17 19 2329 31 37 41 43 47


Factor Pairs

Factors of 24



1 2 3 4 6 8 12 24





7 14 21 28 35 42 49 56 63 70



numbers

Multiples of 9

9 18 27 36 45 54 63 72 81 90hellip








Divide by a prime

Multiply Primes

Write in index form



119867119862119865 119900119891 48 119886119899119889 120


119867119862119865 119900119891 84 119886119899119889 980




Multiply values


or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120




from duplicates


prime



=



Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





Chunking method

Column method



430 + 50 = 480

6 + 8 = 14

436 + 58 = 494

563 minus 237

563 minus 200 = 363

363 minus 30 = 333

333 minus 7 = 326

563 minus 237 = 326


436058+

4941

563237minus

326

5 1



+ + =

+ minus =

minus minus =

minus + =


119860119889119889 ℎ119900119905 119908119886119905119890119903

119860119889119889 119888119900119897119889 119908119886119905119890119903

119877119890119898119900119907119890 119888119900119897119889 119908119886119905119890119903

119877119890119898119900119907119890 ℎ119900119905 119908119886119905119890119903




BoxGrid method

Bus shelter method

Chinese

Tens in top half





523 times 76

523 times 76

Separate numbers





523 times 76

Line up numbers








958 divide 8



Negative numbers




Number ndash BIDMAS

Example 1


ORDER


6 + 3 times 4

6 + 12


Addition second

18



3 + 16 minus 2




17


(3) 2divide (2)

9 divide (2)

45



within brackets

Example 2

Example 3

=

==

=

=

==

=

=

=

=





1 and itself




another number

1 is not a prime

number


number


2 3 5 7 11 13 17 19 2329 31 37 41 43 47


Factor Pairs

Factors of 24



1 2 3 4 6 8 12 24





7 14 21 28 35 42 49 56 63 70



numbers

Multiples of 9

9 18 27 36 45 54 63 72 81 90hellip








Divide by a prime

Multiply Primes

Write in index form



119867119862119865 119900119891 48 119886119899119889 120


119867119862119865 119900119891 84 119886119899119889 980




Multiply values


or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120




from duplicates


prime



=



Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





BoxGrid method

Bus shelter method

Chinese

Tens in top half





523 times 76

523 times 76

Separate numbers





523 times 76

Line up numbers








958 divide 8



Negative numbers




Number ndash BIDMAS

Example 1


ORDER


6 + 3 times 4

6 + 12


Addition second

18



3 + 16 minus 2




17


(3) 2divide (2)

9 divide (2)

45



within brackets

Example 2

Example 3

=

==

=

=

==

=

=

=

=





1 and itself




another number

1 is not a prime

number


number


2 3 5 7 11 13 17 19 2329 31 37 41 43 47


Factor Pairs

Factors of 24



1 2 3 4 6 8 12 24





7 14 21 28 35 42 49 56 63 70



numbers

Multiples of 9

9 18 27 36 45 54 63 72 81 90hellip








Divide by a prime

Multiply Primes

Write in index form



119867119862119865 119900119891 48 119886119899119889 120


119867119862119865 119900119891 84 119886119899119889 980




Multiply values


or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120




from duplicates


prime



=



Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg



Number ndash BIDMAS

Example 1


ORDER


6 + 3 times 4

6 + 12


Addition second

18



3 + 16 minus 2




17


(3) 2divide (2)

9 divide (2)

45



within brackets

Example 2

Example 3

=

==

=

=

==

=

=

=

=





1 and itself




another number

1 is not a prime

number


number


2 3 5 7 11 13 17 19 2329 31 37 41 43 47


Factor Pairs

Factors of 24



1 2 3 4 6 8 12 24





7 14 21 28 35 42 49 56 63 70



numbers

Multiples of 9

9 18 27 36 45 54 63 72 81 90hellip








Divide by a prime

Multiply Primes

Write in index form



119867119862119865 119900119891 48 119886119899119889 120


119867119862119865 119900119891 84 119886119899119889 980




Multiply values


or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120




from duplicates


prime



=



Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





1 and itself




another number

1 is not a prime

number


number


2 3 5 7 11 13 17 19 2329 31 37 41 43 47


Factor Pairs

Factors of 24



1 2 3 4 6 8 12 24





7 14 21 28 35 42 49 56 63 70



numbers

Multiples of 9

9 18 27 36 45 54 63 72 81 90hellip








Divide by a prime

Multiply Primes

Write in index form



119867119862119865 119900119891 48 119886119899119889 120


119867119862119865 119900119891 84 119886119899119889 980




Multiply values


or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120




from duplicates


prime



=



Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg









Divide by a prime

Multiply Primes

Write in index form



119867119862119865 119900119891 48 119886119899119889 120


119867119862119865 119900119891 84 119886119899119889 980




Multiply values


or more numbers

119871119862119872 119900119891 6 119886119899119889 45 119871119862119872 119900119891 48 119886119899119889 120




from duplicates


prime



=



Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Positive powers




4 times 4 times 4 = 43 119875119900119908119890119903

119861119886119904119890


2

3

2

=22

32=

2 times 2

3 times 3=

4

9


51 = 5 281= 28




power positiverdquo

4

5

minus2

=5

4

2

=52

42=25

16Find Reciprocal



119909minus119899 ൗ1 119909119899


7 492

3119862119906119887119890 119903119900119900119905

4119865119900119906119903119905ℎ 119903119900119900119905



2-Stage Powers

119909119898119899 = 119899 119909 119898




1632 =

216

3= 64

27

64

minus23

=64

27

23

=4

3

2

=16

9








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg








119886 times 119887 becomes 119886119887


coefficient

6119909119910 minus 5119886

119887+ 211199094

Expression

6119909119910 minus 5119886

119887+ 211199094

Terms





5119909 + 6119910 minus 2119909 minus 5119910 =3119909 + 119910

5119909119910 + 3119909 minus 2119909119910 + 4119910 = 3119909119910 + 3119909 + 4119910

21199092 + 3119909 + 51199092 minus 5119909 = 71199092 minus 2119909

Identify like terms











119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg










119860 = 1205871199032


119878119906119887119895119890119888119905





119897 = 5 119875 = 4 5


to substitute in





119875 = 4119897



119875

4= 119897

119910 =18119905 minus 3

119901


119905 =119901119910 + 3

18




119886







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







Negative Indices

1199091 = 119909

1199090 = 1

1119909 = 1




119909119886 times 119909119887 = 119909119886+119887

119909119886 divide 119909119887 = 119909119886minus119887

119909119886 119887 = 119909119886times119887




1199113 times 1199117 = 11991110

1199042 divide 1199045 = 119904minus3

1198904 3 = 11989012




119911minus3 =1

119911

3

=1

11991136minus2 =

1

6

2

=1

36

Fractional Indices





119909minus119886119887 =

1119887 119909

119886




10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





10 (119909 + 119910 + 4) = 10119909 +40+10119910

minus61199093119909 (6119909 minus 2) = 181199092


+15119909

F First terms

O Outside terms (119909 + 4)(119909 + 11)

I Inside terms


1199092

11119909

4119909

44


Grid Method


119909 +4119909

minus3

1199092 4119909




= minus12119909 + 13119910

Then simplify

Then simplify





119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






119886119909 + 119886119887119910 + 4119886119911 119886119909 + 119886119887119910 + 4119886119911

12119909 minus 6119910 + 3119911 12119909 minus 6119910 + 3119911

181199092119910 + 6119909119910 minus 241199091199102119911


3(4119909 minus 2119910 + 119911)

HCF = 3 12119909 minus 6119910 + 31199113

4119909 minus 2119910 + 11199113(4119909 minus 2119910 + 119911)

HCF = 119886

119886(119909 + 119887119910 + 4119911) 119886(119909 + 119887119910 + 4119911)

119886

119909 + 119887119910 + 4119911

119886119909 + 119886119887119910 + 4119886119911

HCF = 6119909119910

6119909119910(3119909 + 1 minus 4119910119911)


181199092119910 + 6119909119910 minus 2411990911991021199112

91199092119910 + 3119909119910 minus 1211990911991021199113

31199092119910 + 1119909119910 minus 41199091199102119911119909

3119909119910 + 1119910 minus 41199102119911119910

3119909 + 1 minus 41199101199116119909119910(3119909 + 1 minus 4119910119911)













ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg














ORDER

3 119909 + 1 = 2(119909 + 2)

3119909+3 = 2119909+4

+3= 4119909

119909= 1


minus3 minus3



2= 5

times 2times 2

119909 = 10

2119910

(3 minus 119910)= 4



2119910 = 4 3 minus 119910



119909 + 1

3=119909

22 119909 + 1 = 3119909

3119896 minus 25119896

3119896 + 3 3119896 + 3 + 3k minus 2 + 5119896 = 67









1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg








1199092 + 119887119909 + 119888Add to make b

Multiply to make c

Establish Signs

If c is positive

1199092 + 5119909 + 6

Signs are same

(119909 + 3)(119909 + 2)



1199092 minus 7119909 + 12




12 times 16 times 2

4 times 3

Which pair make 7

(119909 minus 4)(119909 minus 3)




51199092 minus 14119909 minus 3





1198862 minus 1198872 = (119886 + 119887)(119886 minus 119887)

1199092 minus 81 = (119909 + 9)(119909 minus 9)

41199102 minus 25 = (2119910 + 5)(2119910 minus 5)













Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg














Triangular Numbers

Square Numbers

Cube Numbers

1361015

1491625

182764125

Sequence 5 8 11 13


Extra bit +2 +2 +2 +2

+3 +3 +3

3119899 + 2

General formula

5 9 15 23 33 hellip+4 +6 +8 +10

+2 +2 +2


Sequence 5 9 15 23

11198992 1 4 9 16

Subtract 4 5 6 7


= 119899 + 3

11198992 + 1119899 + 3




Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904










3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15 3 3 4 5 5 8 9 15

Mean =52



119872119890119886119899 times 119873119906119898119887119890119903 119900119891 119907119886119897119906119890119904119871119900119888119886119905119894119900119899 =

119899 + 1



38 times 15 = 57119892119900119886119897119904



2= 23


1 1 3 3 7 7



the values are





Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range



0 4 0

1 11 11

2 12 24

3 7 21

4 6 24

Sum 40 80






needed for the mean

times

times

times

times

times

Mean = 80

40= 2

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


2119871119900119888119886119905119894119900119899 =119899 + 1

2

119871119900119888119886119905119894119900119899 =40 + 1

2= 205


Cars Frequency

0 4

1 11

2 12

3 7

4 6

Sum 40

4

15

27



median lies

Median = 2

+

+

4 minus 0 = 4


Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg



Weight Frequency

0119896119892 le 119909 le 2119896119892 12

2119896119892 lt 119909 le 4119896119892 3

4119896119892 lt 119909 le 6119896119892 9

6119896119892 lt 119909 le 8119896119892 10

Sum of Freq 34


column


0119896119892 le 119909 le 2119896119892 12 1 12

2119896119892 lt 119909 le 4119896119892 3 3 9

4119896119892 lt 119909 le 6119896119892 9 5 45

6119896119892 lt 119909 le 8119896119892 10 7 70

Sum of Freq 34 Sum of 119891119909 136


Mean

Mean = 119879119900119905119886119897 119900119891 119886119897119897 119907119886119897119906119890119904

119899119906119898119887119890119903 119900119891 119907119886119897119906119890119904




valueitem

Range


times

times

times

times

Mean = 136

34= 4119896119892

Mean = 119879119900119905119886119897 119900119891 119891119909 119888119900119897119906119898119899

119879119900119905119886119897 119891119903119890119902119906119890119899119888119910


119899 + 1

2

119871119900119888119886119905119894119900119899 =34 + 1

2= 175


12

15

22




lies

Median class =

+

+

8 minus 0 = 8119896119892


out an estimate for

the mean

0119896119892 le 119909 le 2119896119892

4119896119892 lt 119909 le 6119896119892



Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Bar Charts

Pictograms

Pie Charts

Line Charts

Title Axes

Scales Labels



Nationality Guests

Spanish 30

British 24

French 10

German 8

Total 72


1 119866119906119890119904119905 =360deg

72= 5deg

Angle

150deg

120deg

50deg

40deg

Farmer Pumpkins

Harry 20

Sami 50

Doug 40

Rachael 25










45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







45 le 119909 lt 48 3 3

48 le 119909 lt 50 8 11

50 le 119909 lt 52 16 27

52 le 119909 lt 55 12 39

55 le 119909 lt 60 11 50

60 le 119909 lt 70 2 52



0 ndash 4 6

5 ndash 9 18

10 ndash 15 25

16 ndash 20 30

0

5

10

15

20

25

30

0 5 10 15 20C

um

ula

tive

Fre

qu

ency

Points Scored


the group




30 minus 22 = 8 119892119886119898119890119904


=119860119898119900119906119899119905

119879119900119905119886119897times 100


22

30times 100 = 73

Equals

Equals

Equals





0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Cu

mu

lati

ve F

req

uen

cy

Marks

Upper Quartile = 75

Median = 50

Lower Quartile = 25


the data

0 10 20 30 40 50 60 70 80 90

Exam Marks

Minimum Value

Maximum Value

Q1


0 10 20 30 40 50 60 70 80 90


Q3

Q2




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




20 le 119909 lt 25 300 5 60

25 le 119909 lt 30 150 5 30

30 le 119909 lt 40 100 10 10

40 le 119909 lt 60 100 20 5




density column



119865119903119890119902119906119890119899119888119910 119863119890119899119904119894119905119910 =119865119903119890119902119906119890119899119888119910

119862119897119886119904119904 119882119894119889119905ℎ



119865119903119890119902119906119890119899119888119910 = 119865119863 times 119862119897119886119904119904 119882119894119889119905ℎ

Age FD

0 lt 119909 le 10 3

10 lt 119909 le 25 4

25 lt 119909 le 30 5

30 lt 119909 le 40 3

40 lt 119909 le 50 25


10 30

15 60

5 25

10 30

10 25






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






Labour Party

Year Vote () Seats

2015 31 232

2010 29 258

2005 35 356

2001 41 413

1997 43 418

1992 34 271200

250

300

350

400

450

25 30 35 40 45


Vote ()

Seat

s w

on






(Ignore Outliers)


40 = 390 119904119890119886119905119904

Interpolation

Extrapolation










Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg









Even = divide 2



1199033













common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





common denominator

2

9+5

9

7

9


+

7

9minus

1



times 2 times 3

14

18minus

3

18

11

18

minus



31

2+ 4

1

43 + 4 +

1

2+1

4= 7

3

4

52

3minus 2

1

95 minus 2 +

2

3minus1

9= 3

5

9


61

5minus 4

3

4

31

5minus19

4

124

20minus95

20

29

20= 1

9

20

31

5+ 5

9

10

16

5+59

10

32

10+59

10

91

10= 9

1

10



Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Multiplying


Dividing


5

1

2

1

10

3divide

2

3

10

3times

3

2



10

21

21

5divide 3

3

10

1

31

2


50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg



50100



Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10

Using place value

0123403

01 ൗ1 10

ൗ1 10 + ൗ1 10 + ൗ1 10 ൗ3 10


03225

32 divide 2

divide 2

16divide 2

divide 2








Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





Decimal Fraction

05 ൗ1 2

0 ሶ3 ൗ1 3

025 ൗ1 4

02 ൗ1 5

0125 ൗ1 8

01 ൗ1 10


075

025 ൗ1 4

025 + 025 + 025ൗ3 4




13

20times 5

100

65= 065




344




020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





020 ሶ5= 119909(times 100) (times 100)020 ሶ5


20 ሶ5100119909 =(times 10) (times 10)

205 ሶ51000119909 =


Decimal numbers

119909 = 185900

=37

180













5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg









5

25

1

5


earnings

Nigel =30

90=

1

3Sanjay =

35

100=

7

20

0 ሶ3 035



3

5of 60 60 divide 5 = 12 12 times 3 = 36



5

20

divide 5

divide 5

25

100

times 5

times 525


100

Method 1

Convert to decimal

Method 2


times 100


Find 1 10 5 etc



amount

Find 35 of 40





Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






Means out of 100

The common



Increase 70 by 15



119875119890119903119888119890119899119905119886119892119890 119888ℎ119886119899119892119890 =119889119894119891119891119890119903119890119899119888119890

119900119903119894119892119899119886119897 119886119898119900119906119899119905times 100

times119872119906119897119905119894119901119897119894119890119903 119891119886119888119905119900119903

divide 119872119906119897119905119894119901119897119894119890119903 119865119886119888119905119900119903

OriginalValue

New Value







119868119899119888119903119890119886119904119890 119900119891 23 = times 123

119863119890119888119903119890119886119904119890 119900119891 42 = times 058













of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg












of 55 pa

Interest

RatePrincipal

Years

times





Years

times

Growth

Decay


growth and decay







RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg



RPR ndash Ratio








Simplifying Ratios

12 ∶ 8 6 ∶ 4 3 ∶ 2divide 2divide 2



5

20

119892119894119903119897119904119905119900119905119886119897 119901119886119903119905119904

1

4



parts together





3 ∶ 5times $5

$15 ∶ $25


27 divide 3 = 9 119904119908119890119890119905119904 119901119890119903 119901119886119903119905

4 times 9 = 36 (119869119900ℎ119899prime119904 119904119908119890119890119905119904)


size in real life


Map or

Model

times 119904119888119886119897119890 119891119886119888119905119900119903 Real life

divide 119904119888119886119897119890 119891119886119888119905119900119903


10119898119898 = 1119888119898100119888119898 = 11198981000119898 = 1119896119898



6119888119898 times 25000 150000119888119898=

150000119888119898 1500119898 15119896119898divide 100 divide 1000





Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






Inverse Proportion








3119898119890119899 times4 119889119886119910119904 = 12 119889119886119910119904

12 119889119886119910119904 divide 2 119898119890119899 = 6 119889119886119910119904


119910 prop 119909




Compare two values



119901 = 24 119905 = 8a) Find 119901 when 119905 = 7

b) Find 119905 when 119901 = 39

119901 = 119896 times 119905 24 = 119896 times 8

24

8= 119896 3 = 11989624 = 119896 times 8

divide 8

119901 = 3 times 119905 119886) 119901 = 3 times 7 = 21

119887) 39 = 3 times 119905divide 3

119905 = 13


119910 prop1

119909119910 =

119896

119909


119896


119901 = 16 119905 = 2a) Find 119901 when 119905 = 8


119901 =119896

11990516 =

119896

2


32 = 119896


119901 =32

119905119886) 119901 =

32

8= 4

119887) 64 =32

119905119905 =

32

64= 05





does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






does the other



119910 prop1

119909119910 =

119896

119909












same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg











same



ALL the same


same



outwards



inwards




3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon




Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





Faces



Edges

Vertices




Cube 6 12 8

Cuboid 6 12 8




Cone 2 1 1

Cylinder 3 2 0

Sphere 1 0 0




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Introduction




Obtuse

Right

Acute Straight

Full turn


Exactly 90deg


Exactly 180deg


Exactly 360deg










equal

30deg150deg



Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Triangles Polygons


Calculate what you

already know

149deg180deg minus

31deg119909 =

112deg37deg

149deg

+Subtract

from 180deg


107deg55deg 119909deg=+

162deg 119909deg=

Isosceles triangle




Number of sides

Number of



Regular exterior

angle

3 1 180deg 60deg 120deg


5 3 540deg 108deg 72deg

6 4 720deg 120deg 60deg

7 5 900deg 129deg 51deg

8 6 1080deg 135deg 45deg


119899360deg divide 119899














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg














of angles




lsquoCrsquo shape




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




1198882 = 1198862 + 1198872


two sides





Add them


8

6

1198882 = 82 + 62

1198882 = 64 + 36

1198882 = 100119862 = 10

( )





119886

6

1198882 minus 1198872 = 1198862

122 minus 62 = 1198862144 minus 36 = 1198862

108 = 1198862

1039 = 119886( )




119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119904119894119899120579 Value

0deg 0

30deg 05

45deg 2

2= 0707

60deg 3

2=0866

90deg 1

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

To get sinminus1



hypotenuse






hypotenuse

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

To get cosminus1



119888119900119904120579 Value

0deg 1

30deg 3

2= 0866

45deg 2

2= 0707

60deg 05

90deg 0


opposite

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905

119905119886119899120579 Value

0deg 0

30deg 3

3

45deg 1

60deg 3

To get cosminus1











119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg










119952119955


42deg

14

119909119874

119867

119860119914 119919

119912


119909 = 14 divide cos(42deg)= 1884

119931 119912

119926


119909 = tan 64deg times 115= 2358


119930 119919

119926



= 217deg


Alternate process

119904119894119899120579 =119900119901119901119900119904119894119905119890

ℎ119910119901119900119905119890119899119906119904119890

119888119900119904120579 =119886119889119895119886119888119890119899119905

ℎ119910119901119900119905119890119899119906119904119890

119905119886119899120579 =119900119901119901119900119904119894119905119890

119886119889119895119886119888119890119899119905





(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






(119909 119910)



Along the corridor

Up Down the stairs






119910

119909

(minus54)

(77)

(minus3minus4)

(9 minus8)

119874119903119894119892119894119899




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119962 = 120790

119962 = 120785

119962 = minus120787

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

119961 = minus120791 119961 = 120786





119862ℎ119886119899119892119890 119894119899 119910

119862ℎ119886119899119892119890 119894119899 119909

119877119894119904119890 119906119901

119877119906119899 119886119897119900119899119892


Calculated by or

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-3 -2 -1 0 1 2 3 4 5






1199092minus1199091

6

3= 2 (119898)

119910 = 2119909 + 119888 8 = 2(3) + 119888

8 = 6 + 119888 2 = 119888

119910 = 2119909 + 2

119904119906119887119904119905119894119905119906119905119890

119904119900119897119907119890

119910 = 119898119909 + 119888





segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






segment119909119860 + 119909119861

2119910119860 + 119910119861

2




119909 =6 + 15

2

119909 = 105

119910 =11 + minus9

2

119910 = 1105 1




intersect (cross)


119910 = 119898119909 + 119888Same Different



119910 = 2119909 + 119888 2 = 2(4) + 119888

2 = 8 + 119888 minus6 = 119888

119910 = 2119909 minus 6

minus8




Invert = Reciprocal

1198981 times1198982 = minus1





119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






119861

119862

Moving away

Stationary

Returning

Gradient = Speed

119878119901119890119890119889 =119863119894119904119905119886119899119888119890

119879119894119898119890


constant speed

119866119903119886119889119894119890119899119905 =119903119894119904119890

119903119906119899




straight line




Cost Comparisons





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





-25

-20

-15

-10

-5

0

5

10

15

20

25

-5 -4 -3 -2 -1 0 1 2 3 4 5

Positive quadratic

lsquoursquo shape

Negative quadratic

lsquonrsquo shape

1199092

minus1199092



To plot graphs





-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2 -1 0 1 2 3 4 5 6









119910 = 1199093 + 31199092 + 5119909 minus 20




119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119910 =1

119909

119910 =4

119909

119910 =6

119909





119910 = minus1

119909119910 =

1

119909

119910 =119886

119909






downwards

119910 =1

119896

119909




1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





1199092 + 1199102 = 1199032

Algebra Number

1199092 + 1199102 = 25

Centered at Origin


Radius = 5

1199092 + 1199102 = 1199032


52 + 42 = 1199032

41 = 1199032

119903 = 41

1199092 + 1199102 = 41



Gradient of Radius

Gradient of Tangent

Substitute in Point

119866119903119886119889119894119890119899119905(119877119886119889119894119906119904)

=4

3


minus3

4119866119903119886119889119894119890119899119905(119879119886119899119892119890119899119905)


119910 = minus3

4119909 + 119888

Point 34 4 = minus

3

43 + 119888

25

4= 119888

119910 = minus3

4119909 +

25

4






100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







100119898

100119898

35119898 35119898100 + 100 + 35 + 35

270119898


= 9119909 + 2119910




6119888119898

12119888119898 Area = 6119888119898 times 12119888119898

721198881198982


119860 119861 119862

119860 = 601198881198982

119861 = 201198881198982

119862 = 601198881198982

1401198881198982


formed around it

=1


7119898

4119898

119860 =1

2(7119898 times 4119898)=

1

2(281198982)

141198982






Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Parallelogram

Trapezium




119887

(119886 + 119887)times ℎ=1

2



Halve it

Multiply by height



Label triangle


119860119903119890119886 = ൗ1 2119886119887 119904119894119899119862

119860119903119890119886 = ൗ1 2 (7)(10) sin(73deg)


119860119903119890119886 = 33471198881198982




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




119863119894119886119898119890119905119890119903

119878119890119888119905119900119903

Circumference

Diameter

Radius

Sector

Arc

Tangent

Chord

Segment










119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg



119860 = 1205871199032


Area

Circumference

Sector

Arc length



119860 = 120587 times 652

119860 = 120587 times 4225

119860 = 132731198982

119862 = 120587119889

119862 = 2120587119903

119900

119862 = 120587 times 12119888119898

119862 = 377119888119898


119860 =119899deg

3601205871199032


formula

85deg

85deg

360deg(120587 times 62)

2671198881198982

119871 =119899deg

360120587119889


85deg

85deg

360deg(120587 times 12)

890119888119898



Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Prisms


119881119900119897119906119898119890 = 119860119903119890119886 119900119891 119891119886119888119890 times 119889119890119901119905ℎ

119860119903119890119886 119900119891 119891119886119888119890 =1

2(8 times 6)

241198881198982

241198881198982 times 10119888119898 = 2401198881198983

Pyramids


119881119900119897119906119898119890 =119860119903119890119886 119900119891 119887119886119904119890 times ℎ119890119894119892ℎ119905

3


121198881198982 times 15119888119898 = 21601198881198983

21601198881198983 divide 37201198881198983

Spheres

Curved area of cone

119860119903119890119886 = 120587119903119897119897

119903

119881119900119897119906119898119890 =4

31205871199033



119881119900119897119906119898119890 =4


21446605851198881198983

21401198881198983





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





the exact value

Approximation



Round values to 1sf

Perform Calculation

841 times 32


0002

24

0002

24000

2

= 12000

Lower Bound

Upper Bound







240119898 119905119900 119899119890119886119903119890119904119905 10119898

240119898235119898 245119898

235119898 le 119909 lt 245119898


400350 449












shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg













shapes


119878119888119886119897119890 119891119886119888119905119900119903 =119873119890119908 119904ℎ119886119901119890

119874119903119894119892119894119899119886119897 119904ℎ119886119901119890

119878119888119886119897119890 119891119886119888119905119900119903 =12119888119898

3119888119898= 4




Scale factor = 2








Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







Need a mirror line





Need a Centre point

Need a Direction


Rotation90deg

Clockwise

About (minus50)


position


119929 rarr 119930 =



point

Need a Centre point


119874119903119894119892119894119899119886119897 119904ℎ119886119901119890


3

From (3 minus6)

119929 rarr 119930

Use projection

lines








other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg









other triangle















119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg













119860 119861


bisector


the angle bisector






several conditions






2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







2119886



know


2 5


10

119909 = 04 or minus2



1198861199092 + 119887119909 + 119888 = 0 119909 plusmn 119909 plusmn = 0


1199092 + 8119909 + 7 = 0 119909 + 7 119909 + 1 = 0


119909 = minus7 119900119903 minus 1

21199092 minus 2119909 = 3 1 minus 119909

21199092 + 119909 minus 3 = 0

21199092 minus 2119909 = 3 minus 3119909

2119909 + 3 119909 minus 1 = 0

119909 = minus3

2119900119903 + 1





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





1198861199092 + 119887119909 + 119888


(119909 + 119901)2+1199021199092 + 119887119909 + 119888

119909 + ൗ119887 2

2minus ൗ119887 2

2+ 119888

119909 + 119901 2 + 119902

Halve the

Simplify


119909 + 3 2 minus 11

1199092 minus 10119909 + 15 = 0



119909 minus 5 2 minus 10 = 0


119909 = 5 plusmn 10119909 = 5 + 10 = 816

119909 = 5 minus 10 = 184



3 1199092 + 2119909 minus4

3= 0


3 119909 + 1 2 minus 1 2 minus4

3= 0

3 119909 + 1 2 minus7

3= 0

3 119909 + 1 2 minus 7 = 0

Complete the square

Expand

Solve for 119909

3 119909 + 1 2 minus 7 = 0

3 119909 + 1 2 = 7

119909 + 1 2 =7

3

119909 + 1 = plusmn7


3119909 = 0528 119900119903 minus 2528

Add 7 to both sides



Subtract 1from both

sides








4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg









4119909 + 3119910 = 53119909 + 2119910 = 4

times 3

times 4

12119909 + 9119910 = 1512119909 + 8119910 = 16minus

119910 = minus1




3119909 minus 2 = 4 119909 = 2


119910 minus 2119909 = 13119910 + 10119909 = 7 3119910 + 10119909 = 7


its own


16119909 + 3 = 7119909 = 025


first variable



119910 = 119909 + 6119910 = 1199092 minus 2119909 + 2


119909 + 1 119909 minus 4 = 0 119909 = minus1 119900119903 + 4

119910 = (minus1) + 6119910 = 5

119910 = (4) + 6119910 = 10





4119909 + 3119910 = 53119909 + 2119910 = 4

2119909 minus 3119910 = 45119909 + 2119910 = 1

3119910 + 10119909 = 7119910 = 2119909 + 1











119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg












119909 + 5 le 16

119909 le 16 minus 5

119909 le 11

minus2119909 gt 4

119909 lt minus2

4 lt 3119909 + 1 lt 16

1 lt 119909 lt 5



1199092 le 16



below the line








Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Introduction

Counting outcomes



119905119900119905119886119897 119899119906119898119887119890119903 119900119891 119900119906119905119888119900119898119890119904


possible

Mutually exclusive


1 2 3 4 5 6

Heads H 1 H 2 H 3 H 4 H 5 H 6

Tails T 1 T 2 T 3 T 4 T 5 T 6




119875(3) =2

8

1

4


119875 119886 119900119903 119887 = 119875 119886 + 119875 119887

119875 2 119900119903 4 =2

8+1

8

3

8Add each

probability


119875 119886 119886119899119889 119887 = 119875 119886 times 119875 119887

119875 2 119905119886119894119897119904 =1

2times1

2

Flip a coin twice1

4

Multiplyeach

probability


Rolling a die 119875(1 119886119899119889 6)


Independent events



Dependent events




119875 119890119907119890119899119905 times 119899119906119898119887119890119903 119900119891 119905119903119894119886119897119904



Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Venn diagrams



119860 = 2 3 5 7 11The set of prime


119860 = 2 3 5 7 11 119861 = 1 3 5 7 9



119860prime cap 119861 = 1 9




Probability trees

Dependent events

119875 119866 =85

200

119875 119867 cap 119866 =25

200

119875 119867 cap 119866prime =50

200



119875 119877119890119889 119886119899119889 119877119890119889 =4

15

119875 1 119877119890119889 119886119899119889 1 119861119897119906119890 =2

15+

6

15=

8

15



119875 119877119886119894119899 119886119899119889 119897119886119905119890 = 03 times 04 = 012



119875 119874119899 119905119894119898119890 = 018 + 056 = 074


103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg



103104




Basic Structure

119886 times 101198871 le 119886 lt 10 119882ℎ119900119897119890 119899119906119898119887119890119903

283 times 106 = 2830000



(323 times 104) + (82 times 103)

32300 8200+

40500

405 times 104





46 times 3

138

103104 times

107times

138 times 108



0208 103

208 102

times

times

===

times

times





the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






the same

119909

119910+3

119910119909 + 3

119910=

1199092 + 5

1199102+119909

119910


times 119910

times 119910

1199092 + 5

1199102+119909119910

11991021199092 + 5 + 119909119910

1199102

1199092

9 119909 minus 5minus119909 + 4

119909 minus 5


times 9

times 9

1199092

9 119909 minus 5minus9 119909 + 4

9 119909 minus 5

1199092 minus 9119909 minus 36

9 119909 minus 5


2

119909 + 1minus

5119909

119909 minus 4

2 119909 minus 4

119909 minus 4 119909 + 1minus

5119909 119909 + 1

119909 minus 4 119909 + 1


119909 minus 4 119909 + 1

2 119909 minus 4 minus 5119909 119909 + 1

119909 minus 4 119909 + 1





2

119909times1199092

119910

2

119909times1199092

119910

119909

1

2119909

119910

6119886 + 6119887

2times

1

119886 + 1198876 119886 + 119887

2times

1

119886 + 119887

119865119886119888119905119900119903119894119904119890= 3


4119910119911

119909divide1199101199112

10

119870119890119890119901 119888ℎ119886119899119892119890 119891119897119894119901 4119910119911

119909times

10

1199101199112

4119910119911

119909times

10

1199101199112119911

40

119909119911




119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119891 4 = 6 119891 minus3 = minus1



119892 119909 = 4119909 minus 3 find 119892 21

Using functions

119892 21 = 4119909 minus 3

119892 21 = 81 119892 21 = plusmn9


119891 119905 = 31199052 + 2 find 119891 2



119891 2 = 12 + 2 119891 2 = 14








Composite functions

119891 119909 = 5119909 + 2

119910 = 5119909 + 2 119910 minus 2 = 5119909

119910 minus 2

5= 119909119891minus1(119909) =

119909 minus 2

5


119891 119909 = 2119909 + 2 and 119892 119909 = 119909 minus 2Find 119891119892(119909)


119892 119909 = 119909 minus 2

119891 119909 = 2119909 + 2

119891 119909 minus 2 = 2119909 + 2

119891 119909 minus 2 = 2 119909 minus 2 + 2

119891119892 119909 = 2119909 minus 2






1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







1199092minus1199091

Gradient =119877119894119904119890

119877119906119899



Amount of (119909)















119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






119891 119909 + 119886The graph



by 119886 units

+= Left

minus= Right




119891(119909)

119891(119909 minus 3) + 5

3

5




119910 = minus119891(119909) 119910 = 119891(minus119909)




119891(119909)119891(minus119909)119891(119909)

minus119891(119909)




2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





2119909 + 3119910 = 18

119910 = 119909 + 1


equation



for each equation

119909 = 3 119910 = 4

119910 = 1199092 minus 2

119910 = 2119909 + 1


equation

119909 = 3 119910 = 7

119909 = minus1 119910 = minus1


or


1199092 minus 2 = 7

119909 = 3

119909 = minus3


or



for 119909 and 119910

Plot each equation

separately







119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg








119866119903119886119889119894119890119899119905 119860 = ൗ1 3

119866119903119886119889119894119890119899119905 119861 = ൗ5 3

119866119903119886119889119894119890119899119905 119862 = ൗ3 5

03119898119904

17 119898119904

06119898119904





1

2(119886 + 119887) times ℎ

Trapezium Triangle1

2(119887 times ℎ)

Formulae needed


275119898




119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119886

sin119860=

119887

sin119861=

119888

sin119862

sin119860

119886=sin119861

119887=sin119862

119888or




35

sin(125)=

119909


35

sin(125)times sin 35 = 119909 2451119898

sin(119909deg)

28=sin(40)

21times 28

sin 119909 =sin(40) times 28


sin 40 times 28

21

119909 = 59deg

1198862 = 1198872 + 1198882 minus 2119887119888 cos119860

cos119860 =1198872 + 1198882 minus 1198862

2119887119888

Missing length

Missing angle


1199092 = 90 2 + 35 2 minus (2(90)(35) cos(68deg))

1199092 = 6964978hellip

119909 = 8346119896119898

cos 119909 =60 2 + 35 2 minus 90 2

2 60 90






Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





Cosine

Tangent

119878119894119899 120579 =119900119901119901





119862119900119904 120579 =119886119889119895

ℎ119910119901



119879119886119899 120579 =119900119901119901

119886119889119895







Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg






Discrete

Continuous



Shoe SizeDress size



Mass of a person

Categorical data


categories

Grouped Data


Sport


Primary

Secondary




Census

Sample

Bias





Random Stratified


119860119898119900119906119899119905 119894119899 119892119903119900119906119901

119879119900119905119886119897 119899119906119898119887119890119903times 119878119886119898119901119897119890 119904119894119911119890



unrepresentative






Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





Yellow 1

Blue 7

Red 7

Green 3

Pink 2

Sum 20

Yellow Blue Red Red

Red Blue Green Pink

Blue Red Blue Green

Pink Red Blue Red

Red Blue Green Blue


Relative Frequency

120 = 005

720 = 035

720 = 035

320 = 015

220 = 01


what proportion




frequencies as

probabilities



Male 120 80 200

Female 70 100 170

Total 190 180 370




Male 120 80 200

Female 70 100 170

Total 190 180 370






119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119909

2119909



119909


119909

119909



90deg


A

C

B

D

A + C = 180deg B + D = 180deg


90deg



119909

119909119910

119910


circumference

Tangent






119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg







119886 times 119887 = 119886 times 119887

119886

119887=

119886

119887

3 times 7 = 3 times 7 = 21

6

10=

6

10

3

5 =3

5



119878119902119906119886119903119890 119865119886119888119905119900119903119904 = 4 25 100


100 times 2 = 10 2

119886 + 119887 ne 119886 + 119887 5 + 20 = 25

6

3


6

3times

3

3

6 3

3

6 3

9

5

3 + 2times3 minus 2

3 minus 2

5

3 + 2


(opposite root)

=5 3 minus 2


=15 minus 5 2

9 minus 3 2 + 3 2 minus 2=

15 minus 5 2

7





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119909119910

Direction along

Direction updown


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

24

119938 =119938

3minus2

119939 =

119939

2 right

4 u

p

3 right

2 d

ow

n

Notation

119860119861 119938 119938Magnitude =

Length of Arrow


pointing



119938 119939

119938 + 119939119939 starts at the

end of 119938


119939

minus119939 119938 119939

minus119939

Multiplying


3119939 = 119939 + 119939 + 119939

119939

1199393119939


magnitude

119938 minus 3119939 3(119938 minus 3119939)

119939

Same Direction



Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




Length




11198961198921000119892 =

1000119898119892 = 1119892

1000119896119892 = 1 119905119900119899119899119890

Standard Metric

Conversions



1000119898 = 1 119896119898100119888119898 = 1119898

10119898119898 = 1 119888119898Standard Metric

Conversions


100001198881198982 = 111989821001198981198982 = 11198881198982

10000001198982 = 11198961198982


10000001198881198983 = 1119898310001198981198983 = 11198881198983

10000000001198983 = 11198961198983





MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg




MoneyTime

1 Week 7 days

1 Day 24 hours

1 Hour 60 minutes

1 Minute 60 seconds



1 Minute 60 seconds







Add 6 Τ1 4 hours to 4 23 119886119898

4 23 119886119898+6 ℎ119900119906119903119904

10 23 119886119898+15 119898119894119899119904

10 38 119886119898



5857pound


119866119861119875 119880119878119863times 134

divide 134



GBPUSD = 1 134





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg





119873

75deg

075degClockwise

119873

310deg

310deg

Clockwise

Always 3 digits

75deg 075deg

4deg 004deg



119873

75deg

B

A


119873

B

A

119873

75deg

Co-Interior Angles

119873

B

A

119873

75deg

105deg

120579

75deg + 120579deg = 180deg



119873 B

A

119873

75deg

105deg

120601


255deg


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