year 9 - the maths knowledge autumn 1 parallel lines … · 2019. 4. 2. · “same letter, same...

Year 9 - The Maths Knowledge – Autumn 1

(3,1)

(0,0)(-5,-2)

(𝑥, 𝑦)“Along the corridor, up the stairs”

Vertical lines are always 𝑥 = …

Horizontal lines are always y = …

y = 6

𝑥 = 3

y

Gradient

y-intercept

Parallel Lines Investigation

Parallel lines have the SAME gradient

𝑦 = 𝑥 + 2𝑦 = 𝑥

𝑦 = 𝑥 − 2

Perpendicular lines have −1

𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡

𝑦 = 3𝑥 − 3

𝑦 = −1

3𝑥 − 1

A way of writing very small or very big numbers using powers of 1010-3 = 0.00110-2 = 0.0110-1 = 0.1100 = 1101 = 10 102 = 10 x 10 = 100103 = 10 x 10 x 10 = 1000

Examples:4000 = 4 x 103

0.004 = 4 x 10-3

6 300 000 = 6.3 × 106

Always 1 ≤ 𝑛 < 10

The unitary method = find one first

y intercept = 0𝑦 = 2𝑥

e

Year 9 – The Maths Knowledge – Autumn 2

1 Linear/arithmetic sequence

A number pattern which increases (or decreases) by the same amount each time is called a linear sequence. The amount it increases or decreases by is known as the common difference.

2 Geometric sequence

A sequence of numbers where you are multiplying by the same number each time.

3 Term A number in a sequence ORA single number or variable which are separated by + or − signs

4 Like terms "Like terms have what?!““SAME LETTER, SAME INDEX”

7x , x , -2x are like terms because they all have the same variable x6y , 4y, - 3y are like terms because they all have the same variable y

5 Simplify The process to make a very long expressionsimpler. Simpler expressions are easier to solve. You do this by collecting like terms

6 Expand Multiply everything inside the bracket by the term (or number) outside the bracket.

Always use the grid method.Then simplify your terms.

7 Factorise Finding a common factor (numerical or alphabetical or both) in each term, then dividing each term by this common factor. What is ‘left’ is put in brackets.

8 Solve To find the value of the missing number by performing the same operation on each side.

Note* You must always keep your equals sign lined up.

Key Word

Solve the equation to find x 6x + 1 = 13

(-1) (-1)6x = 12

(÷6) (÷6)x = 2

ExampleDefinition

Rearrange equations (to make a specific letter the subject of the equation)To rearrange equations we are reversing the operations so we do BIDMAS in reverse.Use the same method as solving equations

Make t the subject

The common difference between each term is +4 so we compare it to the 4 times tables and write the first part of our nth term rule 4nWe then look at the difference between the four times tables and our sequence. It is one less, so our sequence must be

4n – 1

We can find the value of any term in the sequence by substituting n for the term number. For example, to find the 20th term, I will substitute n as 20.

4 (20) -180 - 179

B I D M A S

Year 9 – Spring 1 - Maths - Geometry

1 EquidistantAt equal distances

13 Similar shapes Identical in shape, angles are the samebut different in size , the ratio between sides is the same

2 Perpendicular At right angles to 14 Congruent shapes Identical in shape and size

3 Bisector Cuts in half 15 Congruency rules SSS SAS ASA RHS

4 Perpendicular bisector 16 SSS3 sides are equal

5 Angle bisector 17 SASSide, Angle ,Side

6 Possible triangles The sum of the two shorter sides must be greater than the longest side

18 ASAAngle, Side, Angle

7 Equilateral triangle Equal angles (60 degrees), equalsides

19 RHS Right angle , Hypotenuse, Side

8 Isosceles triangle 2 equal sides and 2 equal angles 20 PolygonRegular polygon

Any 2D shape formed with straight linesA 2D shape formed with equal straight lines and equal interior angles

9 Scalene triangle No sides and no angles are the same 21 Interior angles The angles inside a shape

10 Right angled triangle A triangle with a right angle 22 Sum of interior angles (Number of sides – 2) x 180

11 Enlargement Changes the size of the shape by a scale factor f rom a centre point

23 Exterior angles

12 Scale factor What all the sides are multiplied by to get the enlargement

24 Exterior angles Sum to 360 degrees

Year 9 – Maths Spring 2 - Equations

1 Solve Find the unknown Solve to find 𝑥:2𝑥 + 1 = 52𝑥 = 4𝑥 = 2

9 > Greater than4 > 3

2 Unknown The letter in an equation 2𝑥 + 1 = 5𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑢𝑛𝑘𝑛𝑜𝑤𝑛

10 < Less than 1 < 2

3 Expand Multiply out the bracket in the expression

2 𝑥 + 5 = 2𝑥 + 10 11 𝑥 > 2 𝑥 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 2

4 Rearrange Make another letter the subject of the equation

Make x the subject2𝑥 + 𝑦 = 𝑧2𝑥 = 𝑧 − 𝑦

𝑥 =𝑧 − 𝑦

2

12 𝑥 ≥ 2 𝑥 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 2

5 Subject The letter of the equation which is on its own on one side

𝑥 =𝑧 − 𝑦

2𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡

13 𝑥 < 2 𝑥 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 2

6 Linear An equation which forms a straight line on a graph

2𝑥 + 5 = 𝑦 14 𝑥 ≤ 2 𝑥 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 2

7 Quadratic An equation containing a power which forms a curved line on a graph

2𝑥2 + 5 = 𝑦 15 2 < 𝑏 < 4 𝑏 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 2 𝑎𝑛𝑑𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 4

8 Simultaneous A pair of equations that have thesame solutions for the unknown

𝑥 + 𝑦 = 102𝑥 + 𝑦 = 14

16 2 ≤ 𝑏 ≤ 4 𝑏 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 2𝑎𝑛𝑑

𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 4

Recap of knowledge

1 Alternateangles

Alternate angles are equal

4 Regular polygon A shape with straight sides Equal sides and equal angles

2 Corresponding angles

Corresponding angles are equal

5 Sum of interior angles (n-2) x 180

3 Co-interior angles

Co-interior angles sum to 180

6 Sum of exterior angles of a polygon add up to 360⁰

Rotation Must include:• Centre of rotation• Direction• Degrees

Example:This shape has been rotated from centre (0,0) anti-clockwise 90⁰

Reflection Must include:• Line of symmetry

Example:This shape has been reflected in the line x = -1

Translation Must include:• Vector

2

52 right, 5 up

−2

−52 left, 5 down

Example:This shape has been translated by vector 7

0

Year 9 - The Maths Knowledge – Summer 1

hypotenuseopposite

adjacent

θ

sin θ = 𝑜𝑝𝑝

ℎ𝑦𝑝

cos θ = 𝑎𝑑𝑗

ℎ𝑦𝑝

tan θ = 𝑜𝑝𝑝

𝑎𝑑𝑗

sin 30 = 1

2

cos 60 = 1

2

SOHCAHTOA


P A

P B

'P A

'P B

'P A B

P A B

Mean Add all the numbers and divide by the frequency

Mode Most common

Median Order the number set and find the middle number

Range The difference between the highest and the lowest number

“Mean is average, mean is averageMode is most, mode is mostMedian’s in the middle, median’s in the middleRange: high low, range: high low”


A way of writing very small or very big numbers using powers of 1010-3 = 0.00110-2 = 0.0110-1 = 0.1100 = 1101 = 10 102 = 10 x 10 = 100103 = 10 x 10 x 10 = 1000

Examples:4000 = 4 x 103

0.004 = 4 x 10-3

6 300 000 = 6.3 × 106

Always 1 ≤ 𝑛 < 10

HIGHER ONLY

1. 𝑎 × 𝑏 = 𝑎𝑏

2.𝑎

𝑏=

𝑎

𝑏

3. 𝑎 + 𝑎 = 2 𝑎

4. 0 = 𝑎 − 𝑎

𝑎−𝑏 =1

𝑎𝑏

𝑎1𝑏 = 𝑏 𝑎

𝑎−1𝑏 =

1𝑏 𝑎

Examples: If you want to increase 40 by 20%:1. Increase = 120% (all of it and 20% more)2. 120% = 1.2 as a decimal3. 40 x 1.2 = 48If you want to decrease 40 by 20%1. Decrease = take away 20% so you have 80% left2. 80% = 0.8 as a decimal3. 40 x 0.8 = 32

Examples: If you want to increase 40 by 20% every year for 3 years:1. Year 1: 40 x 1.2 = 482. Year 2: 48 x 1.2 = 57.63. Year 3: 57.6 x 1.2 = 69.12

QUICK WAY: 40 x 1.23 = 69.12

number of years

Linear sequence nth term: 4n + 2

times table

shift

Quadratic sequence nth term:A sequence beings 4, 13, 26, 43, …The difference between the terms:9, 13, 17, …The second difference (difference between the differences):4, 4, 4, …This tells me that the nth term formula includes:2n2

Subtracting 2n2 from each term, leaves:2, 5, 8, 11…This is a linear sequence with nth term 3n - 1So the nth term for the original sequence is 2n2 + 3n - 1

Rotation Must include:• Centre of rotation• Direction• Degrees

Example:This shape has been rotated from centre (0,0) anti-clockwise 90⁰

Reflection Must include:• Line of symmetry

Example:This shape has been reflected in the line x = -1

Translation Must include:• Vector

2

52 right, 5 up

−2

−52 left, 5 down

Example:This shape has been translated by vector 7

0

Enlargement Must include• Centre of enlargement• Scale Factor

Fractional scale factor makes the image smaller

Negative enlargement inverts the image


hypotenuseopposite

adjacent

θ

sin θ = 𝑜𝑝𝑝

ℎ𝑦𝑝

cos θ = 𝑎𝑑𝑗

ℎ𝑦𝑝

tan θ = 𝑜𝑝𝑝

𝑎𝑑𝑗

SOHCAHTOA

Year 10 – Spring 1 – Maths - Proof

Angle facts 𝒚 = 𝒎𝒙 + 𝒄

1 Angles around a point sum to 360° 14 𝑦 = 𝑚𝑥 + 𝑐 M = gradientC = y-intercept

2 Adjacent angles on a straight line sum to 180° 15 Parallel lines Have the same gradient/m is the same

3 Vertically opposite angles are equal 16 Perpendicular lines Gradient is −1

𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡

4 Interior angles in a triangle sum to 180° Algebraic proof

5 Interior angles in a quadrilateral sum to 360 17 2𝑛 Even number

6 All angles in an equilateral triangle are 60° 18 2𝑛 + 1 Odd number

7 Alternate angles are equal 19 2𝑛 + 1, 2𝑛 + 3, 2𝑛 + 5… Consecutive odd numbers

8 Corresponding angles are equal Kinematic formulae

9 Co-interior angles sum to 180 20 Kinematic formulae 𝑣 = 𝑢 + 𝑎𝑡

𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

𝑣2 = 𝑢2 + 2𝑎𝑠

10 Exterior angles of a polygon

add up to 360° Inequalities

11 The interior and exterior angle of any polygon

add up to 180° 21 𝑥 > 𝑦 𝑥 is bigger than 𝑦Represented by a dashed line

12 The sum of the interior angles of a polygon can be found by using the formula

(number of sides-2) x 180º 22 𝑥 ≥ 𝑦 𝑥 is bigger than or the same as 𝑦Represented by a solid line

13 Regular polygons have all sides the same length and all angles the same size

23 −3 ≤ 𝑥 < 2

Year 10 – Maths - Volume, Surface Area, Similarity and Advanced Trig- Spring 2

1 Area ofrectangle

Base x height 9 Volume of a Sphere

4

3𝜋𝑟3

2 Area of triangle

(𝐵𝑎𝑠𝑒 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡)

2

10 Surface area of a sphere

4𝜋𝑟2

3 Area of a Trapezium

1

2𝑎 + 𝑏 𝑥 ℎ

“Half the sum of the parallel sides times the difference between them”

11 Cosine Rule a2 = b2 + c2 - 2bc cos A.

4 Area of a Parallelogram

Base x Perpendicular Height 12 Area formula

Area = 1

2absinC

5 A Prism A 3D solid which has the same 2D shape running all the way through it

13 Sine Rule 𝑎

𝑠𝑖𝑛𝐴=

𝑏

𝑠𝑖𝑛𝐵=

𝑐

𝑠𝑖𝑛𝐶

6 Volume of a prism

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑥 𝑙𝑒𝑛𝑔𝑡ℎ 14 Congruent Shapes that have exactly the same lengths and angles in any rotation

7 Surface Area of a 3D solid

The sum of all the 2D faces 15 Congruency Tests for Triangles

SSS- Side, Side, SideASA-Side, Angle, SideSAS-Side, Angle, SideRASH-Right angles, side, hypotenuse

8 Volume of a cone

1

3𝜋𝑟2ℎ

16 Similar Shapes that have the same angles but are not the same size and whose lengths are in the same ratio.

9 SurfaceArea of a Cone

𝜋𝑟𝑙 + 𝜋𝑟217 Plan View The 2D outline of a shape from

above

10 Volume of a square based pyramid

1

3𝑥 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡

18

The circle song:“Circumference is pi times diameter, pi times diameter, pi times diameter (repeat) and area is pi r squared”


P A

P B

'P A

'P B

'P A B

P A B

Notation Meaning

ξ The Universal Set

∈ Element of

∉ Not an Element of

∩ Intersection (overlap) ‘AND’

U Union (all together) ‘OR’

∅ Empty Set


You must learn this for your

GCSE


Higher Content

OnlyCircumference The distance around the edge of the circle

Radius The distance from the centre of the circle to the edge of the circle

Diameter The distance across the circle from edge to edge, going through the centre

Tangent A straight line that touches the circle

Chord A line that touch each edge of the circle but does not go through the centre

Segment The area create between the circumference and a chord

Arc Part of the circumference

Sector A slice of the circle- looks like pizza!

𝜃r

𝑥2 + 𝑦2= 𝑟2

For circles where the centre is not at the origin, the following formula is used:

𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟²

e

Year 11 – The Maths Knowledge – Autumn 2

Average How to find it Advantage Disadvantage1 Mean Add up all the numbers and

divide by how many there are

Includes all values Can be distorted by extreme values

2 Median Order the numbers and find the middle number. If two numbers are in the middle, find the middle of those two.

Not affected by extreme values

Does not include all values

3 Mode Most common value Can be used for non-numerical data

Does not always exist

4 Range Biggest value subtract smallest value

The range tells us how consistent the data is. If there is a large range to data is not as consistent as when there is a smaller range

The range only tells us how spread the data is, it can not be used on its own as an average,

12.Correlation“The relationship between two data sets”

13. line of best fit: a straight line drawn with a ruler that goes through the data with roughly the

same number of points on each side of the line

5 Interpolate To estimate a value within the given data set.

6 Extrapolate To estimate a value outside a given date set by assuming a trend

7 Upper Quartile The middle number between the median and highest value in a data set

8 Lower Quartile The middle number between the lowest value and the median

9 Interquartile Range

Upper Quartile subtract lower quartile

e

11.Need to know Formulae

“Mean is average, mean is averageMode is most, mode is mostMedian’s in the middle, median’s in the middleRange: high take low, range: high take low”

Year 11– Spring 1 - Maths – Functions and Vectors

1 Function 𝑓 𝑥 𝑜𝑟 𝑥:→ 𝑜𝑟 𝑦 = A function is a special relationship where each input has a single output.

It is often written as "f(x)" where x is the input value.

11 a𝑓(𝑥) Shrink or stretch graph vertically by a factor of a.

(Multiply y-coordinates of f(x) by

a)

2 Composite Function A function put inside another

function e.g. fg(x)

12 𝑓(𝑎𝑥) Shrink or stretch graph horizontally by a factor of a.

(Divide x-coordinates f(x) by a)

3 Inverse Function 𝑓−1(𝑥) An inverse function goes the other way.

e.g. if f(x) = 2x + 3 then

𝑓−1 𝑥 =𝑥−3

2

18 Gradient (m) 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑦𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑥

=𝑦2 − 𝑦1𝑥2 − 𝑥1

4 Translation Moves the graph horizontally or

vertically

19 Maximum Point A concave downwards in the curve

5 𝑓(𝑥 + 𝑎) Translate by vector −𝑎0

(Shift in the x-direction by –a)

20 Minimum Point A concave upwards in the curve

6 𝑓(𝑥 − 𝑎) Translate by vector +𝑎0

(Shift in the x-direction by +a)

21 Turning Point At this point the gradient is

zero. Both the minimum and

maximum points are turning

points

7 𝑓(𝑥) + 𝑎 Translate by vector 0+𝑎

(Shift in the y-direction by +a)22 Tangent A straight line touching a curve

8 𝑓(𝑥) − 𝑎 Translate by vector 0−𝑎

(Shift in the y-direction by -a)23 Instantaneous speed Speed at any instant in time

9 −𝑓(𝑥) Reflection in the x-axis 24 Iterative Process Repeated use of the sameformula using the previous

result as the new input.

10 𝑓(−𝑥) Reflection in the y-axis 25 Vector A vector has magnitude (how long it is) and direction.

year 9 - the maths knowledge autumn 1 parallel lines … · 2019. 4. 2. · “same letter, same...

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