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Course 2A/2B Preliminary Work TopicsSHEET SET TOPIC PAGE

A1 Number knowledge (counting, whole, integers, prime, composite, square and number line)

3

A2 Factors and Multiples of numbers 5A3 Fractions and decimals 8A4 Percentages 13A5 Rounding 15A6 Index laws 16A7 BIMDAS 19A8 Ratio and Rates 20A9 Mathematical terminology (sum, product, adjacent,

etc…)23

B1 Understanding and using formulae 24B2 Expanding brackets 27

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B3 Factorising 29B4 Solving equations (manually and by calculator) 31B5 Co-ordinates 33B6 Linear relationships and the rule 34B7 Transformations (rotation, reflection and translation) 36B8 Area 38B9 Pythagoras’ theorem 40C1 Displaying data and Data Analysis (mean, mode,

median)42

C2 Likelihood 50

You should be printing off and completing the work sheets that you need to work on (especially ones that you got wrong in the test). It is also expected that you use these sheets to revise before you sit the post test next week.

Your teacher has a set of the solutions and you are to mark any worksheets you do.

A1 – Number Knowledge

Name of Group Pattern Explanation

Whole numbers 0, 1, 2, 3, 4, … Zero + counting numbers

Counting numbers 1, 2, 3, 4, 5, … Whole numbers greater than zero

Integers …, -2, -1, 0, 1, 2, … A whole number that can be negative, positive or zero

Even numbers 2, 4, 6, 8, 10, … Numbers divisible by 2 with no remainder

Odd numbers 1, 3, 5, 7, 9, 11, … Numbers not divisible by 2 with no remainder

Square numbers 1, 4, 9, 16, … Counting numbers squared (n²)

Triangular numbers

1, 3, 6, 10, 15, … 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, …


Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, … Except for the two terms, each term is the sum of the two terms before it

Palindromic numbers

Examples929, 7337

Numbers that read the same whether they are read forward or backwards

Prime numbers First six numbers 2, 3, 5, 7, 11, 13, …

Prime numbers are numbers greater than 1 that have only 2 factors; 1 and itself

Composite numbers

First six numbers4, 6, 8, 9, 10, 12, …

Are numbers that have more than 2 factors (the number 4 has 3 factors; 1, 2 and 4)

The number line is a line on which each point represents a real number. The scale of a number line can change the one below has a scale of 1 (where each point is 1 away from the other) but it could have a scale of ½ or 0.1 if that is better suited to the situation.

A1 – Practise questions

Put the following numbers into which category or categories they belong to. Remember they may fit into more than one.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 28, 29, 33, 36, 45, 101

Name of Group Numbers in the group

Whole numbers

Counting numbers

Integers

Even numbers

Odd numbers


Square numbers

Triangular numbers

Fibonacci numbers

Palindromic numbers

Prime numbers

Composite numbers

Put the following numbers on the number line below; -4, 2.5, -3/4, 1, 4½

A2 – Factors and Multiples of numbers

A factor is an exact divider of a number.

ExampleList the factors of 1001 divides 100 exactly.1 x 100 = 100Therefore 1 and 100 are both factors of 100

2 divides 100 exactly.2 x 50 = 100Therefore 2 and 50 are also factors of 100

3 doesn’t divide 100 exactly

4 divides 100 exactly.4 x 25 = 100Therefore 4 and 25 are factors of 100

5 divides 100 exactly5 x 20 = 100 Therefore 5 and 20 are factors of 100

6, 7, 8 and 9 don’t divide 100 exactly

10 divides 100 exactly10 x 10 = 100Therefore 10 is a factor of 100

Therefore the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100.

Highest Common Factor (HCF)


This is the largest (highest) factor of two numbers which is common to both. To find the HCF 1. You list the factors of the two numbers

2. Write down the common factors3. The highest number circled is the HCF

Example 1Find the HCF between 28 and 42

Number Factors Common Factors28 1, 2, 4, 7, 14, 28

2, 7, 1442 1, 2, 3, 6, 7, 14, 21, 42

14 is the highest number in the common factor column, therefore the HCF of 28 and 42 is 14.

Example 2Find the HCF between 35 and 50

Number Factors Common Factors35 1, 5, 7, 35

1, 550 1, 2, 5, 10, 25, 50

5 is the highest number in the common factor column, therefore the HCF of 35 and 50 is 5.

A multiple is found when you multiply the original integer by another. An integer ‘b’ is a multiple of ‘a’ if there is an integer ‘d’ where b = d x a

Example 1List the first 6 multiples of 77 x 1 = 7 7 x 4 = 287 x 2 = 14 7 x 5 = 357 x 3 = 21 7 x 6 = 42

The first 6 multiples of 7 are 7, 14, 21, 28, 35 and 42.

Example 2List the first 9 multiples of 55 x 1 = 5 5 x 4 = 20 5 x 7 = 355 x 2 = 10 5 x 5 = 25 5 x 8 = 405 x 3 = 15 5 x 6 = 30 5 x 9 = 45

The first 9 multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40 and 45.

Lowest Common Multiple (LCM)

This is the lowest number that is a multiple of two integers.To find the LCM 1. You list the multiples of the two numbers

2. Write down the common multiples**3. The lowest number circled is the LCM

**You’ll notice you’ll only have to keep going till you have the first common multiple

Example 1Find the LCM of 5 and 7

Number Multiples Common Multiples

55, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70

35, 70

77, 14, 21, 28, 35, 42, 49, 56, 63, 70

35 is the lowest number in the common multiple column, therefore the LCM of 5 and 7 is 35.


Example 2Find the LCM of 4 and 8

Number Multiples Common Multiples

4 4, 8, 12, 16, 20, 24 8, 16, 24

8 8, 16, 24

8 is the lowest number in the common multiple column, therefore the LCM of 4 and 8 is 8.


1. Find the factors of the following numbers

(a) 32

(b) 75

(c) 84

(d) 60

(e) 81

2. Find the highest common factor of the following pairs of numbers

(a) 20 and 30

(b) 24 and 36

(c) 64 and 108

(d) 102 and 153

(e) 120 and 132

3. Find the first 5 multiples of the following numbers

(a) 8

(b) 12

(c) 15

(d) 17

(e) 21

4. Find the lowest common multiple of the following pairs of numbers

(a) 3 and 7/tt/file_convert/5e7934727254574f30332540/document.doc 6

(b) 4 and 6

(c) 8 and 12

(d) 13 and 15 (e) 14 and 17

A3 – Fractions and Decimals

Fractions

Are numbers of the form; where a is the numerator and b is the denominator.

Type of Fraction Explanation Examples

Proper When the denominator is smaller than the numerator. and

Improper When the denominator is larger than the numerator. and

Mixed When there is a whole number part as well as the fraction and

You must be able to add, subtract, multiply and divide fractions.

Steps for adding and subtracting fractions1. If there is a whole number you must make it part of the numerator (see example 3)2. Make it so that the fractions have the same denominator3. Add/Subtract the numerators4. Simplify

Example 1

Add and

+ = + this is done to make the denominators the same

= +

= Now we add the numerators

=


= We need to simplify the improper fraction to a mixed

Example 2

Subtract from

+ = + the denominators are already the same

= Now we add the numerators

= We don’t need to simplify it further

Example 3

Subtract from

- = - the denominators are already the same but we

need to move the whole number so it is in the fraction

= Now we subtract the numerators

=

= Simplify the fraction into a mixed fraction

Steps for multiplying fractions1. If there is a whole number part make it part of the numerator2. Multiply the two denominators to get the new denominator3. Multiply the two numerators to get the new numerator4. Simplify your answer if needed

Example 1

Multiply and

x = multiply the two numerators and denominators

= Now we need to simplify the fraction

=

Example 2

Multiply and

x = x the denominators are already the same but we


= x


= multiply the two numerators and denominators

=


Steps for dividing fractions1. If there is a whole number part make it part of the numerator (see example 2)2. Invert the dividing fraction (this means to flip it)3. Multiply the two denominators to get the new denominator4. Multiply the two numerators to get the new numerator5. Simplify your answer if needed

Example 1

Divide by

= x invert the dividing fraction and now multiply instead of divide


= Now we need to simplify the fraction

= = is the answer in the most simplified form

Example 2

Divide by

= the denominators are already the same but we


=

= x invert the dividing fraction and now multiply instead of divide


=


Finding a fraction of a quantity1. If there is a whole number make it a part of the numerator2. Divide the quantity by the denominator3. Multiply your answer by the numerator4. Put the answer back into ‘real life’ terms (remember to include the units)


Example

Find of $4200

$4200 7 x 3 = $1800 divided by the denominator and multiplied by the numerator

Finding one quantity as a fraction of anotherIf you’re trying to find one quantity ‘a’ as a fraction of another quantity ‘b’

1. Make quantity ‘a’ the numerator and quantity ‘b’ as the denominator.2. Simplify if needed

Example 1Find $12 as a fraction of $30

we made 12 the numerator and 30 the denominator of our fraction

= = is the answer in its simplest form

DecimalsAre numbers which have an integer followed by a decimal point followed by some numbers for example 1.376 and 0.0345. You can tell how many decimal places a number is by how many numbers are after the decimal place (on the left hand side). Remember the place value of each decimal so 0.6 is 6 tenths, you are expected to know this.

Number 10 1 . 1/10 1/100 1/1000 1/1000021.204 2 1 . 2 0 4 0

0.65 0 0 . 6 5 0 03.0067 0 3 . 0 0 6 7

To change a fraction to a decimal 1. Proper or ImproperDivide the numerator by the denominator

2. Mixed fractionDivide the numerator by the denominator and add this to the whole number.

Example 1

Turn into a decimal

3 5 = 0.6

Example 2

Turn into a decimal

(2 8) + 3 = 3.25

To change a decimal to a fraction1. Multiply the decimal by a multiple of 10 (10 for 1dp, 100 for 2dp, 1000 for 3dp, etc…) so that the decimal

is now a whole number2. The number you multiplied by is the denominator of the fraction and the whole number you ended up

with is the numerator3. Simplify if needed

Example 0.35 as a fraction

0.35 x 100 = 35 you multiply by 100 as it had 2 decimal places

= 100 is my denominator and 35 becomes the numerator

= simplify the fraction



1. Calculate the following

(a) + (b) + (c) +

(d) + (e) (f)

(g) (h)

2. Calculate the following by multiplying or dividing

(a) (b) (c)

(d) (e) x (f) x

(g) x (h) x

3. Calculate the following problems

(a) of 35 kilograms (b) of $600 (c) of 28 metres

(d) $20 as a fraction of $72 (e) 500g as a fraction of 2kg (f) 75 cents as a fraction of $5.00

4. Fill in the missing spots of the table

Fraction form Decimal form

0.4

0.02

0.635

1.7

A4 – Percentages

Percentages and Decimals

1. To change from a decimal to a percentage


Multiply the decimal by 100.

Example 0.27 as a percentage0.27 x 100 = 27%

2. To change from a percentage to a decimalDivide the percentage by 100.

Example 125% as a decimal125 100 = 1.25

Percentages and Fractions

1. To change from a fraction to a percentageNumerator denominator x 100

Example

Write as a percentage

34 50 x 100 = 68%

2. To change from a percentage to a fractionPut the percentage given over 100**, then simplify if needed**remember that fractions only use whole numbers

ExampleWrite 12.5% as a fraction

= must multiply both so that the numerator and denominator

are whole numbers. Notice how I multiply by 2 to get 25 and 200

= simplify the fraction

To find a percentage of a quantityQuantity 100 x percentage

Example 40% of $200200 100 x 40 = 80 $80

To find one quantity as a percentage of anotherIf you’re trying to find one quantity ‘a’ as a percentage of another quantity ‘b’ a b x 100 = Percentage

Example15 marks out of a 40 mark test, what percentage did they get15 40 x 100 = 37.5 37.5%

A4 – practise questions

1. Fill in the blanks in the table below

Percentage Decimal Fraction

37%


0.05

75%

1.24

10.5%

250%

60%

2. Find the quantity of the following

(a) 20% of $63

(b) 56% of 2000kg

(c) 89% of 400km

(d) 127% of $4500

3. Write the following percentages

(a) 20 cents of $3.00

(b) 4kg of 28kg

(c) 6.5m of 30m

(d) 450g of 2kg

(e) 17cm of 1.2m

(f) 67 marks out of 75 marks

A5 – Rounding and Truncating

RoundingTo round off (or approximate) a number correct to a given place, we round up if the next figure is 5 or more and round down if the next figure is 4 or lower.

Example

ROUNDING TO… 354.9635 ROUND UP OR DOWN ANSWER3 decimal places (3dp)* 354.963 | 5 Up 354.964

2dp* 354.96 | 35 Down – Stays same 354.961dp* 354.9 | 635 Up 355.0

Whole number 354. | 9635 Up 355Nearest ten 35 | 4.9635 Down 350


Nearest hundred 3 | 54.9635 Up 400

* Note: 3dp = nearest thousandth = accurate to 0.0012dp = nearest hundredth = accurate to 0.011dp = nearest tenth = accurate to 0.1

TruncatingTo truncate a number correct to a given place, we simply cut off the numbers after the given place without any rounding off beforehand.

ExampleThis is the same table from above but notice the difference in the rounding answers compared to the truncating answers.

TRUNCATING TO… 354.9635 ANSWER3 decimal places (3dp)* 354.963 | 5 354.963

2dp* 354.96 | 35 354.961dp* 354.9 | 635 354.9

Whole number 354. | 9635 354Nearest ten 35 | 4.9635 350

Nearest hundred 3 | 54.9635 300

A5 – practise questions

1. Fill in the table

7469.4954 Working space (if needed) Rounded Truncated

3 decimal places (3dp)

2dp

1dp

Whole number

Nearest ten

Nearest hundred

Nearest thousand

A6 – Index Laws and Scientific Notation

where ‘a’ is the base and ‘m’ is the index or power. This means that ‘a’ is multiplied by itself ‘m’ times.

For example in the expression , 2 is the base and 4 is the index. This means 2 is multiplied by itself 4 times.

index form

= 2 x 2 x 2 x 2 expanded form

= 16 evaluated form

What happens when you have a negative power such as , where 4 is the base and -3 is the index.

index form


= expanded form

= evaluated form

Scientific NotationWhen expressing a number in standard form or scientific notation, the number is written as the product of a number between 1 and 10, and a power of 10. Note that when the original number is greater than 1 we have a positive index and when the original is less than 1 we use a negative number.

Example

Common numeral Scientific notation

453

5 670 000

2 350

0.045

0.00003

0.0078

Index LawsYou must be able to calculate and evaluate expressions with indices involved. There are eight laws you follow.

Index Law Example


, ,


1. Fill in the blanks of the following table

Question Common Numeral Scientific Notation

A

B

C

D

E

F

G


H

I

J

2. Expand and evaluate the following problems involving indices.

(a) (b)

(c) (d)

(e) (f)

(g)

A7 – BIMDAS

Order of OperationsMathematicians have agreed on an order of operations so that when there was a problem with more than one operation, they would all get the same answer and confusion wouldn’t occur.

1. Brackets – do the operations in the brackets first so what’s in the brackets is in its most simplified form2. Indices – Follow the index laws (see A6)3. Multiplication and Division – these are done in the same step so if there is more than one you complete

from left to right4. Addition and Subtraction - these are done in the same step so if there is more than one you complete

from left to right

Example 1

= there’s no brackets or indices so we go straight to Multiplication and Division because there is more than one multiplication or division sign so we go left to right, notice we did the 4 x 7 first

= this step we did the 282

= now we are in the addition/subtraction step. Notice how because there is more than one sign we go from left to right doing the 2-14 first.

=

Example 2

= there’s brackets so we must simplify what’s in the brackets first

= now in this step we deal with the indices, notice the 10² becomes 100


= now we do any multiplication or division, so we change 382 to 19

=


Solve the following

(a)

(b)

(c)

(d)

(e)

A8 – Ratio and Rates

RatioExamples of ratio can be found in many real life situations. One situation is when you make a glass of cordial. The label on the back says “mix 1 part cordial with 5 parts water” This example allows us to cover many important aspects of ratio.

1. A ratio is a comparison of two quantities. In the above situation we’re comparing the volume of cordial which needs to be mixed with the volume of water.

2. The order of the numbers in the comparison is important. “ mix 5 parts cordial with 1 part water” is not the same as “mix 1 part cordial with 5 parts water”

3. Notice there is no mention of units. The word “parts” is used so that you know whatever units you’re using with the cordial you must use with the water. The size of the unit is not important.

Ratio is a comparison of numbers in a definite order. The numbers are expressed in the same units and are called

the ‘terms’ of the ratio. The ratio can be written in the form : or .

ExampleLiam says he has four times as much money as Bronwyn.(a) What is the ratio of Liam’s amount of money to Bronwyn’s amount of money?

4 : 1

(b) What is the ratio of Bronwyn’s amount of money to Liam’s amount of money?

1 : 4

(c) If Bronwyn has $14, how much money does Liam have?

1 : 4 this is the ratio of Bronwyn’s money to Liam’s money

= 1 x 14 : 4 x 14 to change Bronwyn’s 1 part to equal $14 we multiply by 14. It is important to note that whatever we do to one term we must do to each term in the ratio.

= 14 : 56

If Bronwyn has $14 that means Liam has $56


(d) If Liam has $36, how much money does Bronwyn have?

4 : 1 this is the ratio of Bronwyn’s money to Liam’s money

36 : Liam has $36 but we don’t know how much Bronwyn has

36 4 = 9 we need to see what we multiplied 4 by to get 36

4 x 9 : 1 x 9 we found out that we multiplied 4 by 9 to get 36 so now we need to multiply both terms of our ratio by 9

36 : 9 multiplying by 9 gives us the ‘equivalent ratio’ on the left

If Liam has $36 that means Bronwyn has $9

Equivalent ratios An equivalent ratio is formed, if the terms of a ratio are multiplied or divided by a number other than zero.

Example1:3, 4:12, 15:45 are all equivalent ratios

RatesWhen two like quantities are compared, a ratio is formed. When two unlike quantities are compared, a rate is formed. Rates are very common place in our everyday living, speed is a rate in which distance is compared with time.

Example 1Express each of the following as a rate in its simplest form(a) A woman travels 210km in 3hours210km in 3h

= 2103km in 33h We divide each by 3 because there was 3 hours (you generally try to get the second quantity to be a 1)

= 70km in 1h

travelling at 70km/h once the second term is a 1 you write the rate in the shorter form (notice instead of 70km per 1 hour we just wrote 70km/h)

(b) 500 sheep are grazed on 200 hectares500 sheep on 200 hectares

= 500200 sheep on 200200 hectares We divide each by 200 because there was 200 hectares (you want to get the second quantity to be a 1)

= 2.5 sheep on 1 hectare

2.5 sheep/hectare remember you write the rate in the shorter form

(c) $160 paid for 20 hours worked$160 for 20h

= $16020 for 2020h divide each by 20 to get the second quantity to be a 1

= $8 for 1h

$8/h

Example 2Convert the following rates to the rate indicated(a) 600L/h = L/min600L/h = 600L/60min we change the 1h into 60min

= 60060L/6060min divide each by 60 to get the second quantity to be a 1

= 10L/1min/tt/file_convert/5e7934727254574f30332540/document.doc 19

10L/min

(b) 600m/min = km/h

600m/min = 0.6km/min we change the 600m into km

0.6 x 60 km / 1 x 60 min we then multiply both terms by 60 (60 minutes in an hour)

36km/60min = 36km/1h we change the 60 minutes into 1 hour

36km/h


1. Complete the following

(a) 2 : 1 = 6 : ___ (b) 3 : 13 = ___ : 52

(c) 24 : 36 = ___ : 3 (d) 20 : 6 = 10 : ___

2. Simplify the following ratios

(a) 15 : 5 (b) 120 : 8

(c) 8 : 20 (d) 15 : 200

3. Simplify the following ratios once you make them the same units

(a) 3cm : 1mm (b) 1.5km : 200m

(c) 3.5L : 500ml (d) 2 days : 6 h

4. Larry buys chickens for $3 and sells them for $5. Find the ratio of:

(a) selling price to cost price (b) selling price to profit

(c) cost price to profit (d) profit to selling price

5. “I can lift twice as much as you” says Ruth to Bronte.

(a) Find the ratio of the amount Ruth can lift compared to the amount that Bronte can lift

(b) If Ruth can lift 30kg what can Bronte lift?

(c) If Bronte can lift 18kg what can Ruth lift?

6. Write the following rates in their simplest form

(a) 120L in 4h (b) 10kg for $5

(c) 5km in 20 min (d) $315 for 7 days

7. Complete the equivalent rates


(a) 2km/min = ___ km/h (b) 3m/s = ___ km/h

(c) $2.45/kg = ___c/kg (d) 700g/min = ___t/day

8. (a) Nails cost $4.60/kg. What is the cost of 20kg?

(b) If $1 Australian = US$0.719, what is the value in American dollars of $500 Australian?

(c) If I can run at 8km/h. How far can I travel in 2.5 hours?

(d) A car uses petrol at a rate of 9.5L/100km. How many litres would the car use if we travelled 350km?

A9 – Mathematical TerminologyThe table below shows the main symbols you should already be quite familiar with. It also shows some of the key words that let you know what operation you should be doing, such as to have 7 lots of is to multiply by 7.

Symbol Terminology Example

Add, sum, increase, exceed, plus Increase 7 by 13

Subtract, minus, decrease, difference, less than, take away

Find the difference between 15 and 3

or Divide, quotient 20 divided by 2

or or Multiply, product, lots of 8 lots of 4

Therefore

Less than 9 56

Less than or equal to f 4 so f could be 4, 3, 2, …

Greater than 54 33

Greater than or equal to g 6 so g could be 6, 8, 12, …

= Equal to, is j = 23

Not equal to m

Infinity

Approximately equal to 3.14


1. Rewrite the following using symbols where possible

(a) 15 is less than 105

(b) Therefore, the square root of 9 is 3


(c) 12 divided by 4 plus 6

(d) 0.7 is equal to 70 percent.

(e) The product of 12 and 4 is greater than the quotient of 12 and 4.

B1 – Understanding and using formulae

Algebraic ExpressionsIf an expression contains one or more variables it is an algebraic expression. Generally in algebraic expressions it will be written in its shortest form.

Example 1. 2.

3.

4.

Like termsIt is important to remember that only like terms can be added or subtracted together. Like terms are terms that have the exact same letter variables.

Example1.

= remember that when a variable doesn’t have a number in front of it there is an assumed 1 so you subtract 1 of f the 3

2.

= even though all the terms have x and y they have to have the exact same powers above each variable to be like terms.

Operations with Negative and Positive numbersIt is assumed that you are familiar with operating with negative numbers, here are the 6 rules

Rule ExampleAdding a negative = subtractingSubtracting a negative = addingNegative x or Negative = PositiveNegative x or Positive = NegativePositive x or Negative = NegativePositive x or Positive = Positive

Understanding and using formulae

is a formula that describes how the speed of something can be found by dividing the distance travelled by

the time it took to travel that distance. When given two out of the three variables you are able to work out the unknown variable.


Example

1. Find the value of the unknown variable given that

(a) s if d=120 and t=3

Substitute the number value for the variables

Solve for the unknown

(b) d if s=120 and t=4

Rearrange the equation to isolate the unknown variable

Substitute the number value for the variables (remember that when variables are next to each other this means to multiply them)


2. Solve for the unknown given that

(a) s if u=1, a=5, t=4 and v=7

Substitute the number value for the variables


(b) u if s=6.5, a=5, t=4, v=4

Rearrange the equation to isolate the unknown variable

Substitute the numbers for the variables



B1 – Practise questions

Use each of the given formulae to find the unknown variable in each case.

1.

(a) Find d if h=7 and t=4

(b) Find d if h=12 and t=3

(c) Rearrange the equation so you can find h

(d) Find h if t=11 and d=16.5

(e) Find h if t=5 and d=18

2.

(a) Find s given that u=3, v=5 and t=10

(b) find s given that u=12, v=-3 and t=8

(c) Rearrange the equation so you can find v

(d) Find v given that s=133, u=22 and t=7

(e) Find v given that s=46, u=19 and t=4

3. The production of 1kg of the metal of % purity costs the company $ where a good approximation of is

given by the formula

(a) The value of when =60

(b) The value of when =73

(c) The cost of producing one kg of the metal of 99% purity

(d) The cost of producing one kg of the metal of 99.9% purity

B2 – Expanding bracketsWhen given an expression like we know that this means or .Expanding brackets means to write an expression without the grouping symbols (brackets). To write it without the grouping symbol you multiply each term inside the brackets by the term outside. /tt/file_convert/5e7934727254574f30332540/document.doc 24

Example1. = multiply each term inside the bracket by the

outside term (remember the index laws)

=

= simplify answer (notice we write -6a² not +(-6a²))

2. = multiply each term inside the bracket by the outside term (remember the index laws)

=

= simplify answer**

3. = multiply each term inside the bracket by the outside term (remember the index laws)

=

= or simplify answer**

4. = remember if there is no number after the minus sign it is the short way of saying multiply by -1

= multiply each term inside the bracket by the outside term (remember the index laws

=

= simplify answer**

** You’ll notice I keep the negative numbers in brackets until the final answer. I do this so that the negative numbers stand out and you’ll be less likely to make the mistake of ignoring the negative numbers.


1. Expand the following

(a) (b)

(c) (d)


(e) (f)

(g) (h)

2. Expand and then simplify each expression

(a)

(b)

(c)

(d)

(e)

3. Expand all brackets, then simplify each expression

(a)

(b)

(c)

(d)

(e)

B3 – FactorisingFactorising an expression is the opposite of expanding an expression. If we had the expression , when we expand it we have . Therefore is the factorised form of the expression and is the expanded form of the expression.

To factorise an expression you follow the following steps1. Find the highest common factor of the terms (this is the outside term of the brackets)2. Find the other factor of each term (other factor is the one that pairs up with the HCF)3. Place them in the correct order inside the brackets.

ExampleFactorise the following1. = HCF( ___ + ___ )

Term Factors Common Factors HCF1, 3, x, 3x, 1, 3 3


1, 3, 9

find the other factor of each term

2. = HCF( ___ + ___ )Term Factors Common Factors HCF

1, 2, 4, m, 2m, 4m 1, 2 2±(1 2, 3, 6)*

* if one factor is positive the other must be negative


simplify the expression (notice +(-3) becomes -3)

3. = HCF( ___ + ___ )Term Factors Common Factors HCF

1, 5, m, 5m, n, 5n, mn, 5mn 1, 5, m, 5m 5m**1, 2, 5, 10, m, 2m, 5m, 10m

**The HCF is the factor with the highest number and variable with the highest power


Divisor Divisibility Test2 The number must be even (end in a 0, 2, 4, 6, or 8)3 The sum of the digits must be divisible by 34 The number formed by the last two digits must be divisible by 45 The last digit must be a 0 or 56 The number must be divisible by 2 and 38 The numbers formed by the last three digits is divisible by 39 The sum of the digits is divisible by 9

10 The last digit must be 011 The sum of the odd digits will equal to the sum of the even digits or will differ by a multiple of 11


Factorise the following1.(a)

(b)

(c)

(d)

(e)

(f)

2.(a)

(b)


(c)

(d)

(e)

(f)

3. (a)

(b)

(c)

(d)

(e)

B4 – Solving equations

Solving equations manuallySolving equations is like balancing scales. With equations we know that one side is equal to the other. The solution of the equation is the value of the variable which “balances” the equation. Often solving an equation requires us to change the equation into a simpler one. We can do this by adding, subtracting, multiplying or dividing both sides of the equation by the same number. It is very important to note that whatever you do to one side you do must to the other side.

When solving equations you perform operations to get the variable by itself on one side of the equation.

Example

1.

We add 7 to both sides

Divide both sides by 3

2.

Add 3 to both sides



3.

Subtract 5 from both sides

Multiply both sides by 2

4.

Subtract 2 from both sides

Subtract 2a from both sides


B4- Practise questions

1. Solve each of these one step equations

(a) (b)

(c) (d)

(e) (f)

2. Solve these one step equations that involve negative integers

(a) (b)

(c) (d)

(e) (f)

3. Solve these equations involving two or more steps

(a) (b)


(c) (d)

(e) (f)

B5 – Co-ordinates

Coordinates are numbers or letters that help us find the position of a point. Coordinates are always written in the form (a, b) where a is the distance across the x axis and b is the distance up or down the y axis.

The coordinates of the points on the left are as follows

A coordinates (3, 2)B coordinates (2, -2)C coordinates (-2, 2)


Plot the points with the following coordinates

A (-6, 3) B (5, -9)C (2, 4) D (-5, -7)E (0, -4) F (-2, 8)G (-1, -10) H (9, 1)I (-2, 0) J (1, 9)


B6 – Linear relationships and the rule

If a group of points lie in a straight line the co-ordinates ( , ) obey the rule of the form and we say the relationship between and is linear. In the above rule you should remember that is the gradient of the line and is the y co-ordinate of the y axis intercept (where the line would cut the y axis) or the vertical intercept.

The gradient is the slope of the line and it is measures the steepness of the line. It is measured by the amount of units a graph will move vertically for each unit it moves horizontally.

The y axis intercept is where the line cuts the y axis. This is the y value when x equals 0. Look at the grid below and see how the 1 and 5 coincide with the y axis intercepts.

Example

This means that the gradient of the line is 3 and the y-axis intercept is -7.

This means that the gradient of the line is -2 and the y-axis intercept is 4.

The following grid shows the graph of and . Note that a positive gradient means the line increases as you move left to right and a negative gradient means the line decreases as you move left to right.


Special Note: This equation has a gradient of 3 and a y intercept of 0

This equation has a gradient of 0 and a y intercept of 4


1. Write the equations of the following lines given the following

(a) gradient of -3, y intercept of 5

(b) gradient of 6, y intercept of -1

(c) gradient of 1, y intercept of 4

(d) gradient of -7, y intercept of -5

(e) gradient of 0.6, y intercept of 3

(f) gradient of -12, y intercept of 15

(g) gradient of 0, y intercept of -20

(h) gradient of -3, y intercept of 0

2. State the gradient and y intercept of the following equations

(a)

(b)

(c)

(d)

(e)

(f)


B7 – Transformations

There are three types of transformations that can occur; rotation, reflection and translation. When an object undergoes any of these transformations, its size and shape remain unchanged. The object is the original shape and the image is the transformed shape.

To rotate means to spin or turn an object. It can be measured clockwise or anticlockwise. A 90° anticlockwise rotation would turn an arrow from pointing north to pointing to the west. When rotating an object we need to know; how big the rotation for example is it 270° or 45°, in which direction we will be rotating; clockwise or anti clockwise and the centre of rotation (this is the point that we’re rotating about).

To translate means to slide an image from one place to another. When translating an object we need to know the distance and direction that we are translating an object. The object X has been translated 4 left and 1 down making the image Y.

To reflect an object means to flip the object about a line. To reflect an object we need to know the line of reflection or mirror line. When we reflect the object the image is exactly the same shape and size as the object. It is important to make sure that the image and object are equal distances from the mirror line.



1. Which type of transformation is shown in each of the following set of pictures?

2. Which type of transformation is associated with each of the following situations?

(a) A golfer swings a club at a golf ball

(b) A woman opens a car door to get into her car

(c) A space shuttle takes off from its launching pad

(d) A boy opens the lid of his toy chest

(e) A card is turned over to show its value

(f) A driver checks his rear view mirror

3. Describe the transformations in the below pictures


B8 – Area

Area of a Rectangle Area of a Triangle

Area of a Parallelogram Area of a Trapezium

Area of a Rhombus or Kite Area of a Circle


B9 – Pythagoras’ theorem


Pythagoras’ theorem says that you if you have two sides of a right angled triangle you can work out the missing third side by the following rule; , where is the hypotenuse (the side opposite the right angle, see the line pointing out from the right angle) and and are the other sides.

To find the hypotenuse

Example

To find the other side

Remember that it doesn’t matter which other side you are finding but it may make it easier for you to substitute into the equation if you label the unknown other side .


1. Find the value of the unknown hypotenuse


2. Find the value of the unknown other side

3. Find the unknown side in the following situations

3. Find the unknown side in these diagrams

C1 – Displaying Data

The following notes are from your 2A text book pages 13-19


C1 – Practise questions


1. Find the mean mode and median of the following data sets

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

2. Draw an example different from the notes of the following type of data display

(a) Pie chart

(b) Bar graph

(c) A pictograph

(d) A dot frequency graph

(e) A stem and leaf diagram

(f) A frequency table with tally column included

(g) A two way classification table

C2 – LikelihoodLikelihood means how likely an event is to occur. We know that some events are more likely to occur than others. For example if Tom gets a prize when he rolls a 6 or 3 on a fair die and Jill gets a prize if she rolls a 1, 2, 4 or 5 it is more likely that Jill will get a prize not Tom. It is important to note that this doesn’t mean that it is impossible for Tom to get the prize just less likely than Jill.

You should be familiar with labelling events as Impossible, unlikely, even chance, very likely and certain.


Example

Likelihood EventImpossible You will swim on the moon tomorrowUnlikely It will rain in JanuaryEven chance When I flip a coin it will be a headVery likely You will eat something todayCertain The sun will rise

C2 – Practise questions

1. State the likelihood that best suits each event

(a) It will rain next Monday

(b) The next teacher I have will be 10 years old

(c) Australia will win the next test cricket match

(d) On my way home I will see a kangaroo

(e) It will be warmer in February than June

(f) If I toss a coin I will get a tail

(g) The next baby born in WA will be a boy

(h) I will see a dolphin when I go to Monkey Mia

2. List the possible outcomes in each experiment below

(a) Rolling a fair die

(b) Tossing a coin

(c) Picking a letter from the alphabet


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