Multiples of 10: 10, 20, 30, 40, 50, 60 etc. Factors of 10: 1, 10, 2, 5

Year 9 Foundation – Unit 1

Number

Understand place value

Non-calculator - add and subtract Non-calculator - multiply (short & long) Non-calculator - divide (short & long)

List factors, multiples, squares & cubes systematically Use of calculator including bracket, squares, powers Order of operations – BIDMAS

Prime numbers

Next Steps

• Write a number as a product of prime factors • Find the LCM & HCF from prime factorisation

using Venn diagrams • Estimate roots and powers

Key words

Place Value – The value of a number in

a particular position

Negative – Below zero Positive – Above zero

Factor – a whole number that divides

exactly into another whole number.

Square numbers – the outcome of a

number multiplied by itself twice

Cube numbers – the outcome of a

number multiplied by itself three times

Indices/ Powers- numbers that tell us

how many times a number has to be multiplied by itself

Calculating with Negative Numbers

Adding and subtracting negative numbers:

e.g. 10 + - 3 = 7 10 - - 9 = 19

- 9 + - 3 = -12 - 7 - - 3 = - 4 Multiplying and dividing negative numbers:

e.g.

7 x – 4 = - 28 - 6 x – 4 = 24 - 35 ÷ 5 = - 7 -40 ÷ -8 = 5

Powers and Roots

Powers: 32 _{= 3 x 3 = 9} 43 =_{4 x 4 x 4 = 64} Roots: √121 = 11 or -11 (because 11 x 11 = 121 and -11 x -11 = 121) √64 = 8 or -8 (because 8 x 8 = 64 and -8 x -8 = 64) Square numbers – 1 , 4 , 9 , 16 , 25 , 36 , … Cube numbers – 1 , 8 , 27 , 64 , 125 , …

Multiple or factor?

Multiples of 10: 10, 20, 30, 40, 50, 60 etc. Factors of 10: 1, 10, 2, 5

Highest Common Factor

To find the highest common factor of two numbers, systematically list the factors of both numbers. Find the highest number in both lists. 12: 18 1 x 12 1 x 18 2 x 6 2 x 9 3 x 4 3 x 6 HCF = 6

BIDMAS

BIDMASis the order that operations need to be completed in: Brackets Indices (powers) Division Multiplication Addition Subtraction Example B(3+2)2_{x 2 ÷ 2 + 4 -1 =} I52_{x 2 ÷ 2 + 4 -1 =} D25 x 2 ÷ 2 + 4 -1 = M25 x 1 + 4 – 1 = A25 + 4 – 1 = S29 – 1 Answer: 28

Lowest Common Multiple

To find the lowest common multiple of two

numbers write out a list of their multiples until you spot one that is common in both.

(Hint: Write out five of each at a time) Example

LCM of 12 and 18 12: 12, 24, 36, 48, 60

18: 18, 36, 54, 72, 90 LCM: = 36

Year 9 Foundation – Unit 2

Powers and roots

Recall squares up to 10x10 & square roots

Recall cubes up to 5x5x5 and 10x10x10 & cube roots Recognise powers of 2, 3, 4 and 5

Calculate with squares and cubes and roots Complex calculations with a calculator Use positive integer indices.

Calculate positive integer powers and exact roots Estimate roots e.g. √20

Converting numbers to and from standard form

Next Steps

Using standard form with negative powers

Indices using algebra

Squares up to 10 12_{= 1} 22_{= 4} 32_{= 9} 42_{= 16} 52_{= 25} 62_{= 36} 72_{= 49} 82_{= 64} 92_{= 81} 102_{= 100}

Key words

Powers – The number of times

you use a number or expression in a calculation, its written in a small raised number

Square Roots –A number that

produces a specified quantity when multiplied by itself.

Cube numbers –multiplied by

itself 3 times

Standard form/ Standard form index- A way of writing a very

small or a very large number

Powers

The power tells you how many ‘lots’ of a number to multiply together ; Example

What happens when you multiply numbers that are written as powers of the same number or variable (letter)? Cubes to 5 12_{= 1} 23_{= 8} 33_{= 27} 42_{= 64} 52_{= 125} 102_{= 1000}

Square Roots

The square root of a given number is a number that, when multiplied by itself, produces the given number.

The square root of 9 is 3, since 3 × 3 = 9.

Numbers also have a negative square root, since –3 × –3 also equals 9. A square root is represented by the symbol √ . For example, 16−−√=4.16=4.

Remember: A square root is treated like a power or index, according to BIDMAS

Estimating square roots

I want √31

52_=5x5=25 62_=6x6=36

√26 is bigger than 5 but smaller than 6

Converting numbers to and from standard form

Write a number in standard form

Writing ordinary numbers as standard form. Write these numbers in standard form.

These numbers are now written in standard form.

Standard form is often used to write very large Or very small numbers.

Year 9 Foundation –

Unit 3 Expressions

Recognise the difference between an expression, equation, identity and formulae

Substitute into simple expressions (positive integers)

Add & subtract directed numbers Multiply & divide directed numbers

Simplify expressions by collecting like terms Multiply expressions including with indices

Substitute into simple expressions including indices & negative numbers

Expand a single bracket

Factorising (common term factors) e.g. 6x – 4; x2₊ 5x

Next Steps

Expanding with double brackets Solving with two unknown varables

Expressions

Use algebraic notation to create expressions;

We must be able to translate worded problems into numbers to form an expression

. 5(t-7)

Multiplying indices

2n+4

Key words

Expressions-A combination of letters and

numbers

Equations-Two expressions that are

separated by an equals sign with one or more variables.

Inverse-Opposite

Identity-Expressions either side of an =

sign with one or more variables which are true to for all values.

Formula-A rule, using numbers and letters

which shows relationship between variables.

Substitution-Replace a variable with a

number to work out the value.

Like terms-Terms where the variables are

identical but the coefficients maybe different.

Expand Multiply out (terms with brackets) Factorise- The inverse of expanding

brackets

Expression, Equation & Identity

Sometimes the identity will not have a triple equals sign, you may need to spot that both sides will give the same answer for any variable. E.G. y2 _{= y x y is an identity.} A formula is another type you will learn in later units.

Simply expressions/collection

like terms

We can collect together terms that have the same letter in them. We can add up normal numbers as we normally would.

For example; 3x + 4x = 7x 9 + 4 + x = 13 + x 8 + 1 + x + 2x = 9 + 3x

You can use circles and squares around the terms to easily identify the like terms and collect them up more easily.

Expanding brackets

Factorising

This is the opposite of expanding out brackets, to put them back in.

By extracting a highest common factor we can factorise an expression, say 9y + 81.

The biggest number to fit into 9and 81exactly (highest common factor) is 9. This goes on the outside of the bracket.

9 (y + __) To get to 81, it is 9 x 9so: 9y + 81 = 9 (y + 9)

Y9 Foundation – Unit 4 Angles

Use letter notation to describe a point, line & angles Know names, estimate, measure & draw angles Angles on a line/ at a point/ vertically opposite Angle sum of a triangle, using to Calculate missing angles in triangles including exterior.

Identify & name triangles – equilateral, isosceles, right-angled

Angle sum & properties of quadrilateral (Justify as 2 triangles) including exterior angle

Find & use the sum of interior and exterior angles of polygons.

Explain why some polygons tessellate and others do not

Recognise and name pentagons, hexagons,

heptagons, octagons and decagons Irregular polygons with 3,4,5,6,8 sides only

Angles in parallel lines

Next Steps

Include algebra in angle problems.

Work with compound shapes and polygons such as parallelograms or rhombi.

Key Words

Polygon – A 2D shape with any number of

straight sides (they have to be straight, so there isn't a 1 or 2 sided polygon).

Angle – The rotational distance between

two straight lines, i.e. how much of a turn is required.

Sum of angles – Total of all the angles. Sum

can be denoted by the Greek Σ (sigma)

Interior angles – The angles inside a shape

at each vertex (corner).

Exterior angles – If the side was extended

outside the shape the exterior angle is the angle between the extended side and the adjacent side (the side next to it).

Quadrilateral – Special name for 4 sided

polygon (shape)

Tessellate – Fit together with no gaps. NOTE: Only those shapes whose interior

angle (at one vertex/corner) divides exactly into 360 with no remainder will tessellate perfectly. Any other shape will not give a full perfect tessellation on its own.

Angle Facts

The special angle facts are shown here. For the total degrees in a polygon: (n-2) x 180

Where n is the number of sides the polygon has. External = 360/n

E.G. Hexagon – INT = 4x180=720 EXT = 360/6=60 Because an n sided polygon can be split into n-2 triangles. Acute: less than 90° Right: 90

Obtuse: over 90, less than 180 Reflex: over 180, less than 360

Using Properties of Shape

Use the sum of angles to find a missing angle. In the triangle below:

The angle at V or ∠UVW = 51° ∠VUW = 90°

∠UWV = ? °

Sum of all angles in a triangle is 180°. 180 – 51 – 90 = 39 so ∠UWV = 39° We can name the sides

Edges by the start And end letters E.G. VU, UW and VW

Angles in Parallel Lines

Some of these are incorrectly known by the shape they look like, so, in order: Z angles --- (no name) F angles C angles --- (no name) Use these only to remember, do not describe them as ‘letter’ angles, you will get no marks!

Regular Polygons

Regular means that all the lengths and angles are of the same size.

Irregular is a shape that isn't regular, so the angles and sides not all the same size

3

4

5

6

7

8

9

10

Triangle Types

Triangles can be classed by how many sides or angles are equal. Above are their names. The equilateral is regular as all angles and sides are the same.

Year 9 Foundation-Unit 5

Averages and range

Mode and Median

Mean and Range All averages from a list Compare distributions

Draw & read stem & leaf diagram Averages from ungrouped table

Next Steps

Calculating averages from a grouped frequency table.

Mean Mode median and Range

Key words

Mean – Add all values and divide by the number of values

Median – The middle value when the numbers are put into order first. Mode- Most occurring value

Range – Largest value minus the smallest value.

Comparing Averages

Stem and Leaf diagrams

The tens are called the stem and the units are the leaf. It’s a way to represent data to find the Mode, Median and Range or large amounts of data much quicker.

Averages from ungrouped table.

Mean = 40 divided by 16 = 2.5

Mode = 1 as 1 is the most popular number of handbags

Median = 16 divided by 2 is 8 so when you total up the frequency column the 8th_value lies in the second row so 2 would be the median.

Year 9 Foundation – Unit 6

Decimals

Decimals place value - ordering, read scales Add & subtract decimals including bills & change and negatives

Multiply & divide decimals by 10, 100, 1000 Round to decimal places (including money from a calculator) - use ≈

Decimals into fractions & simple vice versa including simple recurring decimals

Multiply/divide decimals by whole numbers Rounding to significant figures - use ≈

Next Steps

Use knowledge of rounding to specify error bounds on measurements using inequality signs (<, >, ≤, ≥).

Place value

The following

list of numbers has been put in order, from

smallest to biggest:

0.06, 0.6, 0.606, 0.66, 6, 6.06, 6.6.

The reading on this scale is 4.6 kg.

Key words

Recurring decimal – a continually repeating pattern of numbers after the decimal point Decimal Place (d.p.) – any digit after the decimal point

Significant Figure (s.f.) – any digit that is not a leading zero (a leading zero is a 0 which comes before any other digit)

≈

– approximately equal to

Adding and subtracting

Multiplying/dividing by 10, 100,

1000

To multiply by 10, 100, or 1000, move the decimal point 1, 2, or 3 spaces to the right.

To divide by 10, 100 or 1000, move the decimal point to the left.

For example, 7459.26 ÷ 100 = 74.5926

Multiplying/dividing by whole

numbers

Do the multiplication as normal and put the decimal point in at the end.

Do the division as normal, but don’t forget the decimal point!

Rounding decimals

If the next decimal place after the one you want is 5 or more, round up. π = 3.1415926….

We can round π in a number of ways. π ≈ 3.1 (1 d.p.) π ≈ 3.14 (2 d.p.) π ≈ 3.142 (3 d.p.) π ≈ 3 (1 s.f.) π ≈ 3.1 (2 s.f.) π ≈ 3.14 (3 s.f.) 0.004275 ≈ 0.0043 (2 s.f.) This is because leading zeros are not significant figures.

Converting decimals to fractions

Write down the decimal divided by 1

Multiply the numerator and denominator by 10 until they are both whole numbers Simplify the fraction

Example: Convert 0.75 to a fraction.

Converting fractions to decimals

To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number), adding zeros after the decimal point.

Y9 Foundation – Unit 7

2D Shapes

Calculate the perimeters and areas of rectilinear shapes

Find Area of triangle & parallelogram & trapezium

Convert units of area

Describe and identify the parts of a circle including: centre, radius, diameter, chord, circumference, tangent, sector, segment and arc

Find the circumference and area of a circles to an amount of decimal places or in terms of π.

Next Steps

Work with compound shapes made from several 2D shapes and working with algebra.

Key Words

Perimeter – Literally the distance all the way around the outside of a shape, as if you had to walk around the outside. It is measured in DISTANCE units (m, cm, km, miles, feet etc.) Area – The amount of space trapped inside the edges/lines that make up a shape. It is

measured in AREA or SQUARED units cm2, _m2_,

km2_{, miles}2_{(sometimes square miles) or feet}2

(usually sq. ft.). You may have heard of acres and hectares too.

Circumference – A special version of perimeter that we call the outside edge or perimeter of a circle.

Centre of a circle – The centre is the point exactly in the middle of the circle .

The centre is exactly the same distance from anywhere on the circumference.

Names of the other parts are shown here. Pi

π

– A special Greek letter we use that is the link between the circumference and the diameter. Approximately 3.141…

Area and Perimeter

For a triangle:

Area = (base × height)

/

₂ (Remember the height is the perpendicular or straight up height)

Perimeter = all three sides added together. For a square or rectangle

Area = base x height

(Sometimes called length x width)

Perimeter= 2xbase + 2xheight

[So in the example Perimeter= 6x2+12x2=36cm]

Compound shapes: Area = split up the shape

and add up the parts.

Perimeter = Total distance

around the shape.

A= 6x12 = 72cm2

Reason for Triangle Area

For a triangle:

Area = (base × height)

/

₂

This is because a triangle is exactly half of the rectangle with the same base and height. If you draw the full

dotted rectangle, then cut it in half. It leaves the yellow shape. It is a triangle that

Is exactly half the size of the rectangle.

Other Areas

(Parallelogram & Trapezium)

For a trapezium:

Area =

h x

(a+b)

/

₂

H = perpendicular height A and B are the parallel sides. Example: Area = h x (a+b)

/

₂ =4 x (3+5)

/

₂ =4 x 8/2 = 4 x 4 = 16cm2 For a parallelogram:

Area = base x height

(NOTE: the height needs to be the perpendicular height)

So here Area = 2.5 x 1.8 = 4.5m2

a

b h

Converting Area Units

You need to remember the normal conversions. So 1cm = 10mm.

Since the units are squared, the conversion factor is also squared. E.G 1cm=10mm, so then 102_=100.

1x1cm

2

1cm

2

10x10

100mm

2

Circle Area and Circumference For Area:

We use the formula A = π r2

(Area= π x radius x radius) So here radius = 5

Area = π x 5 x 5 = π x 25

= 78.5cm2_(1dp)

We may be given the wrong one (diameter not radius or vice versa). But look at the radius of the circle and the diameter of the same circle.

Diameter = 2 x radius (or D/2 = r)

For Circumference: We use the formula C=πd

Circumference = π x diameter

Here Circumference = π x 10 = 31.4 cm (1dp) NOTE: it is a distance so cm, not cm2

In terms of π (pi) simply means do the calculation, but work out the numbers only not the pi. So C = 10π cm would be the answer. So would A = 25π cm2_.

Y9 Foundation – Unit 8

Solving Equations

Inverse operations

Solve simple one step equations

Solving simple equations by the balancing method e.g. 3x – 1 = 9

Revise expansion of single bracket

Solving equations with brackets e.g. 2(x + 1) = 5 Solve with unknowns and variables on both sides.

Next Steps

Set up and solve equations

Use equations to solve real life problems Check your solution every time

Key Words

Solve – Find out what the variable (the letter) is worth.

1 Step – Using one inverse operation will find out the answer (what the variable is worth) 2 Step – The equation will need to have one inverse operation, then another to be applied to find out the variable value. You can have as many steps as you wish.

Balancing Method – When we work with the equations, we must keep them balanced like scales, so if we use the inverse we have to do it to both sides of the =

Inverse – The opposite, we do this to get rid of and move the terms we are not interested in. Applying the opposite cancels it out.

Expand - To remove the brackets from an expression or equation.

Brackets – These state that that part of the calculation needs doing first. We can use special rules to remove the brackets.

Unknown/Variable – a letter standing in place of any number.

Alternative Method

If you find it hard to see why the balancing method works, it might help to start off using the

function machine method.

3j – 6 = 9

We can see that here, we have j, times it by 3, then take off 6. We can show this in a flow chart.

If we then follow the chart back from 9, we see that to undo the take away 6 we have to add 6.

If we follow this chart back we can find what j is worth. 9+6 = 15 Then 15/3 = 5. So j = 5!

Solving One Step Equations

These are called one step equations as, to find x we are 1 step away from having x = ___

To solve the question, we use the inverse operation to get the variable

(letter) on its own. In the above example, y has 14 added to it. So the inverse is to subtract 14.

We have to keep a balance, so we do it to both sides The simple inverse operations are:

Add -> Subtract Subtract -> Add Multiply -> Divide Divide -> Multiply To solve 6t=54, remember: 6t means 6 x t.

The inverse of multiply is divide so we ÷6 both sides. We can test the answer, 6 x t = 54, 6 x 9 = 54. Right!

Expanding Brackets

To remove the brackets we have to multiply them out. There are 2 ways: 2(x + 6), grid/box:

So 3(a+4) = 3a + 12

Arrows: Both ways get the same answer. Just use whichever you like the most. x a +4 3 3a +12

Solving Equations with Brackets In Them

If an equation has brackets, the first thing to do is expand out the brackets.

Use whichever method You like the most. Once you have expanded Solve it the same way As before (inverses). Check the answer, here 3(2p+5)

3(2x5 +5)

3(10+5) = 3 x 15 = 45. That is the same so it is correct!

Solving Equations (Variable on Both Sides)

We can sometimes have equations where the letter appears on both sides. We need a method to solve these.

The method is to still use inverse operations, but we have to get all of the letter on one side, all of the numbers on the other.

The first thing to do is to subtract the smaller letters from both sides. One side has 3n, the other 2n, so -2n both sides. If we follow this

method, then we will have at the end an equation that looks like the simple 1 or 2 step equations.

So at this step -5 (both sides) we get n = 2. CHECK – 3x2+5 = 11. 2x2+7 = 11. Both = 11, so CORRECT!

Year 9 Foundation – Unit 9 Fractions

To understand the meaning of a fraction Fractions - equivalent fractions,

cancelling, ordering, mixed and improper Fraction of a quantity e.g. 5/8 of £20 Fraction of a quantity e.g. 17/8 of £20 Express one quantity as a fraction of another

Improper fraction to mixed numbers and vice versa

Add & subtract fractions same

denominator - including multiplying by an integer and negative fractions

Next Steps

Adding and subtracting fractions with different denominators

Finding reciprocals Dividing by fractions

Multi-step problems using BIDMAS which involve fractions

Adding, subtracting, multiplying and dividing with mixed numbers Equivalent fractions

To find an equivalent fraction, multiply or divide the numerator and denominator by the same number.

To cancel down a fraction, we divide the numerator and denominator by the same number to get them as small as possible.

Key words

Fraction – a way of writing numbers which are not integers.

Integer – a whole number.

Numerator – The top number of a fraction Denominator – the bottom number of a fraction

Equivalent fractions – the same fraction written in different ways.

Improper fraction – a fraction where the numerator is bigger than the denominator Mixed number – a number with a whole number part and a fraction part.

Adding and subtracting fractions – same denominator To add or subtract fractions with the same denominator, keep the denominator the same and add or subtract the numerators.

So "_#+%_#='_#. Also, "_#−%_#=%_#.

Improper fractions and mixed numbers

What is a fraction? Fractions are a way of writing numbers which are not integers (whole numbers). The fraction '_#means 3 out of 4, or 3 divided by 4.The number at the top is called the numerator, and the number at the bottom is called the denominator.

Multiplying with fractions To multiply two fractions together, simply multiply the numerators, multiply the denominators, then simplify. To multiply a fraction by an integer, write the integer as a fraction over 1, then multiply as above. Fractions of a quantity To find a fraction of a quantity, divide by the denominator (to find one part) and then multiply by the numerator (to find the required fraction).

Example: Find '_*of £20. 1 5 𝑜𝑓 £20 = 20 ÷ 5 = £4. 3 5 𝑜𝑓 £20 = 3×4 = £12. ×2 ×2 ÷2 ÷2 Ordering fractions To order fractions, change them to equivalent fractions which all have the same denominator, and then compare the numerators. After that, convert them back to the original fractions.

Quantities as fractions To write a quantity as a fraction of another quantity, put the first quantity over the second and simplify.

For example £45 as a fraction of £50 is #*

*6= 7

Year 9 Foundation – Unit 10

Transformations

Identify Symmetry reflectional and rotational

Identify congruent & similar shapes Translation –draw & describe Translation using Vector notation

Rotation about any point/measure rotation – find the point

Reflect in a vertical or horizontal line Draw & reflect in a diagonal line Reflect in y = ± x

Enlargement simple shape from a given centre using a positive whole number scale factor

Identify the scale factor of an enlargement

Next Steps

Combined Transformations Factional enlargement and describing enlargements

Congruent shapes

Congruent shape have the same size. Reflection, rotations and translations produce congruent shapes. For example:

Key words

Horizontal – a straight line going

from left to right

Vertical – a straight line going

from top to bottom

Congruent-Identical in size and

shape

Reflection – the mirror image of

something

Rotation – turning something

round a single point

Translation – where a shape is

moved but the direction of it stays the same.

Transformation - changing the

direction or position of a shape Vectors- The direction a shape as moved

Enlargement- The process of making something smaller or larger

Scale Factor- The amount a shape as been enlarged by

Translation using vectors

Reflection

Enlargement

Scale factor is the amount the shape as been enlarged by. This diagram shows the blue shaded triangle enlarged by a scale factor of 2. The centre of enlargement is (0,0)

Year 9 Foundation –Unit 11

Formulae

Substitute positive integers into a simple algebraic formula

Substitute negative numbers into a formula to find the value of the subject

Use kinematics formulae

Know what s, u, v, a, t represents Rearrange a linear formula

Next Steps

Rearranging formula with more than 1 variable.

Rearrange and substitute

Key words

Substitute – To replace with

Displacement – Distance away from a

fixed point

Velocity- The speed of something in a

given direction

Acceleration- Increase in speed

Time- Time take to travel a given distance

Substituting positive numbers

To substitute positive integers you replace

the letter in a formula the number you are

told it equals.

Example

A =

h

x

w

,

h = 4

and

w = 3

.

So we have

A =

4

x

3

Rearranging Formulae

To change the subject of a formula you must use inverse operations to get the new subject “on its own”.

Example

Make x the subject

y = x + 7

y – 7 = x

So x = y - 7

Kinematics formulae

S,U,V,A, andT represent the different factors in the kinematics formulae or equations of motion Sis displacement U is initial velocity V is final velocity A is acceleration T is time

Using these formulae is very similar to finding the value of the subject. Example

A car is travelling at an initial velocity of 10ms-1_{and accelerates at 2ms}-2_{for 10} seconds, what is the final velocity.

So we have u=10, a=2and t=10and we want to find v. Then substitute these into v =u+at

v = 10 + 2 ×10

v= 30m/s (Don’t forget your units!!!)

Finding the value of the subject

First substitute the values into your

formula.

Next carry out the operations to find the

value of your subject.

Example

D =

E

+

F

, where

E = 6

and

F = -4

So we have

D =

6

+ (

-4

)

D = 2

v = u + at s = ut + %_"at2 _{s =} % "(u+v)2 v2_{= u}2_{+ 2as s = vt -}% "at2

-7

-7

Y9 Foundation – Unit 12

Percentages

Percentage, decimal & fraction equivalence including > 1

Order numbers in percentage, fraction & decimal form (include ratios when ordering)

Percentage of a quantity

Increase/decrease an amount by a % One quantity as % of another

Simple interest (2 or more years) Multipliers

Compound interest

Next Steps

Repeated Change and reverse percentages Comparing simple and compound interest Real life problems using percentages

Key words

Percent – Literally out of 100.

Simple Interest – Interest is only ever paid

on the original investment ONLY.

Compound Interest – Interest is paid on

the investment and on any interest earnt already.

Exponential Growth/Decay– The same as

compound interest but in other contexts (i.e. not invested money). Growth is a percentage added on repeatedly, Decay is a percentage reduction repeatedly.

Reverse Percentage – given the value after

change, what was the original?

Percentages to

Fractions/Decimals/Equivalent

fractions

Percent to Decimal – Divide by 100 Decimal to Percent – Multiply by 100 Percentage to Fraction:

Put percentage over 100, simplify if poss. Fraction to Percentage:

Either-- Try to get the denominator to 100 - Bus stop to divide the fraction (to get a

decimal) and then x100 Equivalent fractions :

Percentage of Amount No Calc/Calc

Convert the percentage to a decimal, eg 10% = 0.1 <= This is your Mulitplier

Do AMOUNT x Multiplier

0.1 x 54 = 5.4 kg. This is easier for random percentages that are not 1,5,10,25,50 etc.

Increase/Decrease by Percentage

Simply follow the above, then (add it on to /subtract it from) the original.

£54 increased by 10% = 54+5.4 = £59.40 £54 decreased by 10% = 54-5.4= £48.60 Find multiplier as before but add (increase) or subtract (decrease). Save that answer Do Ans x AMOUNT E.G. 10%inc –1+0.1 = 1.1 Then 1.1x54=£59.40 10%dec- 1-0.1 = 0.9 Then 0.9x54=£48.60

Amount as % of Another &

Percentage Change

If they are not in the same units, convert so they are

If the amount being expressed is smaller <100% If the amount being expressed is larger >100% This is very similar but involves the change being expressed as a percentage of the original

Interest (Simple & Compound)

3% interest per year on £3000 for 4 years: SIMPLE: Find the amount paid on 1 year:

£3000 @ 3% interest -> 3000x0.03 = £90 per year Multiply amount of interest by years needed. 90 x 4 = 360. Add to original. £3360

COMPOUND: Find the multiplier for the rate 3% increase = 1.03.

Do Original x Ratetime

3000 x 1.034 _{= 3376.52643 = £3376.53.}

Note the different results!

Ordering fractions decimals and percentages

Convert a fraction to a decimal by dividing the top (numerator) by the bottom (denominator) Convert a percentage to a decimal by dividing by 100. When they are all converted to decimals you can then put them in order using place value.

Year 9 Foundation –Unit 13

Presenting data 1

• Collect organise data into two-way tables • Draw & interpret pictograms

• Draw & interpret a bar and line charts (including multiple and composite bar charts)

Next Steps

Construct and interpret pie

charts and scatter graphs

Two way tables

Two-way tables are used for many types of

data, from timetables to data-analysis tables. You read them across and down at the same time.

For example, in the table below you can see that a total of 8 cars were black, and silver was the most popular colour.

Sometimes you will have to fill in missing values in a two-way table. To do this, look for any row or column with a given total and just one value missing.

Key words

Two-way table / Pictograms / Bar chart / line chart / (multiple bar chart / composite bar charts / frequency / axis / axes

Pictograms

Pictograms show quantities using pictures. The key

tells you how much one picture is worth.

In the above example, regarding mean monthly pocket money, Y11 get £15.

Bar charts

A bar chart presents grouped data with rectangular bars with lengths proportional to the values that they represent.

Multiple and composite bar

charts

Multiple, often ‘duel’, bar charts show

multiple data sets which are represented by drawing the bars side by side , as below.

Example readings include: (1) most books sold in May, (2) January saw the least sales for both CDs and books.

Composite (or ‘Stacked’) bar charts

Composite bar charts display

multiple data points stacked in a

single row or column.

Line charts

A line graph is often used to show a trend over a number of days or hours. It is plotted as a series of points, which are then joined with straight lines.

Year 9 Foundation – Unit 13

Presenting data 2

• Construct & interpret a pie chart • Plot and interpret scatter diagrams. • Identify an outlier.

• Recognise graphical misrepresentations.

Next Steps

Use a scatter graph to make predictions outside the given data range.

Pie charts

A pie chart (or a circle chart) is a circular statistical graphic which is divided into slices (sectors )to illustrate numerical proportion.

To draw a pie chart, we need to represent each part of the data as a proportion of 360, because there are 360 degrees in a circle. For example, if 55 out of 270 vehicles are vans, we will represent this on the circle as a segment with an angle of: (55_/

270) x 360 = 73 degrees.

Key words

Pie chart Scatter diagrams Correlation Misrepresentations Line of best fit Sector

Scatter graphs

Scatter graphs (or ‘scatter plots’) show pairs of numerical (bivariate) data, with one variable on each axis, they show what sort of relationship, if any, there is between the two data sets. There are 3 main kinds of relationship (correlation), as shown below:

A line of ‘best fit’ can be drawn through the plots, from which estimated readings can be taken. The line of ‘best fit’ can also be used to make predictions from the data.

Identify outliers

An outlier is an observation that lies an abnormal distance from all the other values .

Graphical misinterpretations

The “classic” types of misleading graphs include cases where: the vertical scale is too big or too small, or skips numbers, or doesn't start at zero. The graph isn't labelled properly and data is left out. See below for an example:

Also, 3D bar charts give a bar extra height, which is also a way of making a graph misleading.

Key words

Faces – Any of the individual surfaces of a solid object. Vertices – A vertex (plural: vertices) is a point where two or more lines meet. It is a Corner.

Edges – An edge is a line segment that join two vertices Volume - The amount of space that a substance or object occupies, or that is enclosed within a container

Surface area - The total area of the surface of a three-dimensional object.

Net - A pattern that you can cut and fold to make a model of a solid shape.

Plan - A drawing of something as viewed from above Elevation – A view from the side or the front of something

Interpreting plans and elevations

Year 9 Foundation – Unit 14

3D shapes

Types and names of 3-D shapes including real life examples Properties of 3-D shapes (e.g. vertices), cube, cuboid, prism, cylinder, pyramid, cone and sphere

Recognise & draw nets of simple solids, cuboid Nets of solids including triangular pyramids Surface areas and Volume of cuboid Inverse of volume of cuboid

Interpret 2D projections, plans, elevations of 3d solids Volume of right prisms including cylinder including in terms of π

Next Steps

Calculate surface areas and volumes of different shapes.

Volumes of prisms

Volume: Find the cross sectional area and multiply by the depth

Surface Area: Find the area of all faces and add together

1)Find the area of all faces Top circle: πx5² = 78.539..cm²

Bottom circle: 78.539..cm²

Rectangular face = 10 x (πx10) = 314.159…cm² 2)Add together : 78.5..+ 78.5..+ 314.1.. = 471.2cm2 _(1d.p)

1) Find cross sectional area πx5² = 78.539..

2) Multiply by the depth 78.5… x 10 = 785.3cm3_(1d.p)

A cylinder - has 3 faces – two circles and a rectangle. The length of the rectangle in the

length of the circumference of a

circle

Properties of 3D shapes

Some real life examples

Sphere Cylinder Triangular prism _Cone

Nets of 3D shapes

Using plans and elevations given it is possible to draw a 3D shape. You can see how this would be done from the shape on the right.

Year 9 Foundation- Unit 15

Sequences

Input and output machines including inverse

Generate a sequence by spotting patterns or using the term to term rule

Special sequences of numbers e.g. square, triangular and cube numbers

Recognise simple arithmetic progressions

Next Steps

Negative indices, HCF and LCM with Venn diagrams

Keywords

Linear sequence- a sequence

which increase or decreases by the same amount each time

Power- indicates how many

times a number is to be multiplied by itself

Square numbers- the product of

multiplying a number by itself

Term-to-term rule- how you get

from one term of a sequence to the next

Substitute- to put in place of

another

Inverse- the opposite in effect

Square Numbers

A square number is the product of

multiplying an integer by itself e.g. 4 x 4 = 16, so 14 is a square number. The sequence of square numbers is therefore:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 etc. This is because 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 etc.

Triangle Numbers

Triangle numbers come from a sequence of dots that form an equilateral triangle. The first five triangle numbers are:

1, 3, 6, 10, 15, 21

Finding the nth term of a linear sequence

A linear sequence increases or decreases by the

same amount each time.

The nth term rule is a way of finding any term in a linear sequence, where n is the position in the sequence.

How to find the nth term of a sequence:

1) Find the common difference or term-to-term rule

2) Write out the times tables of the term-to-term rule above you sequence. Find the difference between this and the original sequence.

e.g.

The term-to-term rule is + 5.

The first term of the sequence is 2, the second term is 7 etc.

This is what we put in front of the n 5n 5 10 15 20 25 30 35 -3 -3 -3 -3 -3 -3 -3 We the add or subtract this: 5n - 3 If the term-to-term rule is negative, you would have a negative number in front of n, and your times tables would be negative.

Substitution into the nth term

rule

If you want to find any term of a linear sequence from the nth term rule, you substitute in the term you would like to find e.g.

50th_{term, n = 50}

1000th_{term, n = 1000}

You can then substitute your value of n into the nth term rule

e.g. what is the 25th_{term of a sequence} with an nth term rule of 5n-4

n= 25

5(25) – 4 = 125 – 4 = 121

Function Machines

When we have a function machine with a power, our input is x. If we are given the output and the function, to find the input we do the inverse. The inverse of x2_{is the square root} e.g. which number squared gives 4? 2 or -2.

Squared and cubed numbers

Cubed numbers Squared numbers

Year 9 Foundation – Unit 16

Ratio and Proportion

Begin to understand ratio

Ratio & fraction notation Write and Simplify a ratio,Equivalent ratios

Split a quantity into two parts given the ratio e.g. £2.50 in the ratio 2 : 3

Express the division of a quantity into two parts as a ratio.

Calculate one quantity from another, given the ratio of the two quantities.

Direct proportion including the unitary method, including currency conversion, recipes & better value.

Direct and Inverse proportion

Next Steps

Write & simplify ratio including 1:n or n:1 Share a quantity in a given ratio with three or more parts

Interpreting inverse proportion graphs

Writing and Simplifying

To write a ratio, simply put the two quantities with a colon : between them. Simplify if possible/asked. To simplify, divide by a common factor of both sides.

Key words

Ratio – a set of quantities similar to a fraction, however they are not written out of a total, just the parts written together with a : between them. Variable – the thing that can change. The independent variable is the one you choose to change Proportion- A part or a share of something. Currency – A system of money in general use in a particular country.

Exchange rate - The value of one currency for the

purpose of conversion to another.

Sharing a Quantity

The worked example to the right is with 2 parts, but

The method for more parts is the same.

Share £46.70 in the ratio 1:3:4.

1+3+4 = 8 parts. 1 part = 46.70 divided by 8 = £5.84(2dp) So 3 parts = £17.51 (2dp) and 4 parts = £23.35.

The quantity shared in ratio is: £5.84 : £17.51 : £23.35

Direct and inverse proportion

Currency and exchange rates are always changing.

You have to convert from 1 currency to another.

The exchange rate £1 = 1.21€

If we go from Euros to Pounds we divide and if we go from £ to e you multiply.

There is inverse proportion or inverse variation between two amounts when as one amount increases, the other decreases.

For example, the faster you travel over a given distance, the less time it takes. As the distance is constant the given distance would be the constant of proportionality.

This is shown in the example to the right.

Proportion problems

When making a meal the recipe may not be for the right number of people.

Example

If you can make 18 packed lunches with 8 loaves of bread how many loaves do you need to make 60?

When shopping you want to get the best value. In order to do this you must be able to find the cost per unit mass.

Example

Example

Cars drive around a race track. Driving at an average speed of 180 km/h, a car takes 4%_"minutes to get round the track.