KS3 Mathematics Study Guide
Years 7 & 8 Syllabus
Table of Contents
1. Percentages
2. Area of Circle
3. Area of Trapezium
4. Equations with Brackets
5. Multi-Step Equations
6. Significant Figures
7. Enlargement
8. Indices
9. Standard Form
10. Interior & Exterior Angles of Polygon
11. Sum of Angles in a Polygon
12. Transformations
13. Practice Resources
Percentages
Fraction-Decimal-Percentage Conversions
Converting Fractions to Percentages
To convert a fraction to a percentage, multiply by 100:

Example: 3/5 = 3 ÷ 5 × 100 = 60%

Example: 1/4 = 0.25 × 100 = 25%
Converting Decimals to Percentages
To convert a decimal to a percentage, multiply by 100:

Example: 0.75 = 0.75 × 100 = 75%

Example: 0.08 = 0.08 × 100 = 8%
Converting Percentages to Fractions
To convert a percentage to a fraction, divide by 100 and simplify:

Example: 35% = 35/100 = 7/20

Example: 75% = 75/100 = 3/4
Converting Percentages to Decimals
To convert a percentage to a decimal, divide by 100:

Example: 56% = 56 ÷ 100 = 0.56

Example: 8% = 8 ÷ 100 = 0.08
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator:

Example: 3/4 = 3 ÷ 4 = 0.75

Example: 2/5 = 2 ÷ 5 = 0.4
Writing One Number as a Percentage of Another
To express one number as a percentage of another:

Formula: (Value ÷ Total) × 100%

Example: What percentage of 80 is 20? 20 ÷ 80 × 100% = 25%
Percentage of an Amount
Finding a Percentage of a Given Amount
Method 1: Convert the percentage to a decimal and multiply:

Example: Find 15% of £80 15% = 0.15 0.15 × £80 = £12
Method 2: Find 10%, then 5%, then add:

Example: Find 15% of £80 10% of £80 = £8 5% of £80 = £4 15% of £80 = £8 + £4 = £12
Percentage Change
Finding Prices After Percentage Increase/Decrease
Method 1: Find the change, then add or subtract:

Example: Increase £50 by 20% 20% of £50 = £10 £50 + £10 = £60
Method 2: Use a multiplier:

For percentage increase: Multiplier = 1 + percentage/100

For percentage decrease: Multiplier = 1 - percentage/100
Example: Increase £50 by 20%

Multiplier = 1 + 20/100 = 1.2

£50 × 1.2 = £60
Example: Decrease £80 by 15%

Multiplier = 1 - 15/100 = 0.85

£80 × 0.85 = £68
Calculating Original Price Before Percentage Change
To find the original price:

Divide the new price by the appropriate multiplier
Example: After a 20% increase, the price is £60. Find the original price.

Multiplier = 1.2

Original price = £60 ÷ 1.2 = £50
Change as Percentage
Calculating Change as a Percentage
Formula: (Change ÷ Original amount) × 100%
Example: Original price: £80, New price: £92

Change = £92 - £80 = £12

Percentage change = (£12 ÷ £80) × 100% = 15%
Area of Circle
Formulas

Area of circle = πr² (where r is the radius)

Area of semicircle = πr²/2

Area of quarter circle = πr²/4
Key Points

Use π = 3.14 or π = 22/7 as approximations unless a calculator is available

Remember that diameter = 2 × radius
Examples
1. Find the area of a circle with radius 5 cm.
o
Area = πr² = π × 5² = π × 25 = 78.5 cm² (using π = 3.14)
2. Find the area of a semicircle with diameter 10 cm.
o
Radius = 10 ÷ 2 = 5 cm
o
Area of semicircle = πr²/2 = π × 5²/2 = π × 25/2 = 39.25 cm²
3. Find the radius of a circle with area 154 cm².
o
πr² = 154
o
r² = 154 ÷ π
o
r² = 154 ÷ 3.14 = 49.04
o
r = 7 cm
4. Find the area of a compound shape consisting of a square with side length 8 cm and a
semicircle with diameter 8 cm.
o
Area of square = 8² = 64 cm²
o
Radius of semicircle = 8 ÷ 2 = 4 cm
o
Area of semicircle = πr²/2 = π × 4²/2 = π × 16/2 = 25.12 cm²
o
Total area = 64 + 25.12 = 89.12 cm²
Area of Trapezium
Formula
Area of trapezium = ½ × (a + b) × h Where:

a and b are the lengths of the parallel sides

h is the perpendicular height between the parallel sides
Examples
1. Find the area of a trapezium with parallel sides 7 cm and 13 cm, and height 5 cm.
o
Area = ½ × (7 + 13) × 5
o
Area = ½ × 20 × 5
o
Area = ½ × 100
o
Area = 50 cm²
2. Find the height of a trapezium with parallel sides 8 cm and 12 cm and area 50 cm².
o
50 = ½ × (8 + 12) × h
o
50 = ½ × 20 × h
o
50 = 10h
o
h = 5 cm
3. Find the length of one parallel side of a trapezium with height 6 cm, area 60 cm², and
the other parallel side measuring 10 cm.
o
60 = ½ × (a + 10) × 6
o
60 = 3 × (a + 10)
o
60 = 3a + 30
o
3a = 30
o
a = 10 cm
Equations with Brackets
Expanding Single Brackets
To expand a bracket multiplied by a number or variable:

Multiply each term inside the bracket by the number outside
Examples:

3(x + 4) = 3x + 12

5(2y - 3) = 10y - 15

-2(4x + 1) = -8x - 2
Expanding Double Brackets
To expand two brackets multiplied together: Use the FOIL method (First, Outer, Inner, Last):
(a + b)(c + d) = ac + ad + bc + bd
Examples:

(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12

(2x - 1)(x + 3) = 2x² + 6x - x - 3 = 2x² + 5x - 3

(3x + 2)(4x - 5) = 12x² - 15x + 8x - 10 = 12x² - 7x - 10
Collecting Like Terms
After expanding brackets, collect like terms:
Example:

3(2x + 1) + 2(x - 4)

= 6x + 3 + 2x - 8

= 8x - 5
Multi-Step Equations
Solving Multi-Step Equations
1. Expand brackets if present
2. Collect like terms on each side
3. Get all variable terms on one side and all number terms on the other
4. Solve for the variable
Examples
1. Solve 3x + 5 = 17
o
3x = 17 - 5
o
3x = 12
o
x=4
2. Solve 4(2y - 3) = 20
o
8y - 12 = 20
o
8y = 20 + 12
o
8y = 32
o
y=4
3. Solve 5(z - 1) + 2 = 3(z + 4)
o
5z - 5 + 2 = 3z + 12
o
5z - 3 = 3z + 12
o
5z - 3z = 12 + 3
o
2z = 15
o
z = 7.5
Equations with Variables on Both Sides
To solve equations with variables on both sides:
1. Collect variable terms on one side
2. Collect number terms on the other side
3. Solve for the variable
Example:

5x + 7 = 3x - 5

5x - 3x = -5 - 7

2x = -12

x = -6
Significant Figures
Understanding Significant Figures

Significant figures (sig figs) are the digits in a number that carry meaningful information

Non-zero digits are always significant

Zeros between non-zero digits are always significant

Leading zeros (zeros before the first non-zero digit) are never significant

Trailing zeros after a decimal point are significant

Trailing zeros in a whole number may or may not be significant (assume they are unless
told otherwise)
Examples of Counting Significant Figures

5.73 has 3 significant figures (5, 7, and 3)

0.0058 has 2 significant figures (5 and 8); leading zeros are not significant

40.50 has 4 significant figures (4, 0, 5, and 0)

23000 has 2 to 5 significant figures depending on context (if written as 2.3 × 10⁴, it has 2
sig figs)
Rounding to Significant Figures
1. Identify the digit in the position you are rounding to
2. Look at the next digit:
o
If it's less than 5, round down
o
If it's 5 or more, round up
Examples for rounding to 2 significant figures:

3.762 rounds to 3.8

0.0451 rounds to 0.045

856 rounds to 860

1049 rounds to 1000 (or 1.0 × 10³)
Enlargement
Understanding Enlargement
Enlargement is a transformation that changes the size of a shape while maintaining its
proportions.
Key components:

Scale factor: Determines how much larger or smaller the image will be

Center of enlargement: The fixed point from which the enlargement is performed
Properties of Enlargement

All lengths in the image are multiplied by the scale factor

Angles remain unchanged

Area is multiplied by (scale factor)²

Volume is multiplied by (scale factor)³

If the scale factor is between 0 and 1, the image is smaller than the original

If the scale factor is negative, the image is inverted through the center of enlargement
Performing an Enlargement
To enlarge a shape from a center of enlargement:
1. Draw lines from the center of enlargement through each vertex of the original shape
2. Measure the distance from the center to each vertex and multiply by the scale factor
3. Mark the new positions along these lines
4. Connect the new vertices to form the enlarged shape
Example
For a shape with vertices at (2,1), (2,3), and (4,3), enlarge by scale factor 2 from the origin (0,0):
Original vertices:

A(2,1)

B(2,3)

C(4,3)
To find new vertices, multiply each coordinate by the scale factor:

A'(4,2)

B'(4,6)

C'(8,6)
Indices
Basic Rules

x⁰ = 1 (Any number raised to the power of 0 equals 1, except 0⁰ which is undefined)

x¹ = x

x⁻ⁿ = 1/xⁿ (Negative indices represent reciprocals)
Index Laws
1. Multiplication: x^a × x^b = x^(a+b)
o
Example: 2³ × 2⁴ = 2⁷ = 128
2. Division: x^a ÷ x^b = x^(a-b)
o
Example: 5⁵ ÷ 5² = 5³ = 125
3. Power of a Power: (x^a)^b = x^(a×b)
o
Example: (2³)² = 2⁶ = 64
4. Power of a Product: (xy)^a = x^a × y^a
o
Example: (3×2)² = 6² = 36 or 3² × 2² = 9 × 4 = 36
5. Power of a Quotient: (x/y)^a = x^a / y^a
o
Example: (4/2)³ = 2³ = 8 or 4³/2³ = 64/8 = 8
Examples
1. Simplify 2³ × 2⁵
o
Using the multiplication law: 2³ × 2⁵ = 2³⁺⁵ = 2⁸ = 256
2. Simplify 3⁴ ÷ 3²
o
Using the division law: 3⁴ ÷ 3² = 3⁴⁻² = 3² = 9
3. Simplify (5²)³
o
Using the power of a power law: (5²)³ = 5²×³ = 5⁶ = 15,625
4. Simplify 2⁻³
o
2⁻³ = 1/2³ = 1/8 = 0.125
Standard Form
Understanding Standard Form
Standard form (also called scientific notation) is a way of writing very large or very small
numbers.
A number in standard form is written as: a × 10ⁿ where:

1 ≤ a < 10 (a is a number between 1 and 10)

n is an integer (positive or negative whole number)
Writing Numbers in Standard Form
1. For large numbers (≥ 10):
o
Move the decimal point left until you have a number between 1 and 10
o
Count how many places you moved the decimal point - this is the positive power
of 10
Example: 45,000 = 4.5 × 10⁴ (decimal point moved 4 places left)
2. For small numbers (< 1):
o
Move the decimal point right until you have a number between 1 and 10
o
Count how many places you moved the decimal point - this is the negative power
of 10
Example: 0.00067 = 6.7 × 10⁻⁴ (decimal point moved 4 places right)
Converting from Standard Form to Ordinary Numbers
1. If the power of 10 is positive, move the decimal point to the right by that number of
places
2. If the power of 10 is negative, move the decimal point to the left by that number of
places
Examples:

3.4 × 10⁵ = 340,000

7.82 × 10⁻³ = 0.00782
Calculations with Standard Form
1. Multiplication:
o
Multiply the numbers and add the powers
o
Example: (5 × 10³) × (2 × 10⁴) = 10 × 10⁷ = 1 × 10⁸
2. Division:
o
Divide the numbers and subtract the powers
o
Example: (6 × 10⁶) ÷ (2 × 10²) = 3 × 10⁴
3. Addition/Subtraction:
o
Convert to the same power of 10, then add/subtract the numbers
o
Example: (5 × 10⁴) + (3 × 10³) = (5 × 10⁴) + (0.3 × 10⁴) = 5.3 × 10⁴
Interior & Exterior Angles of Polygon
Interior Angles
The interior angle is the angle inside a polygon at each vertex.
For a regular polygon (all sides and angles equal):

Interior angle = (n - 2) × 180° ÷ n Where n is the number of sides
Examples:

Triangle (n = 3): Interior angle = (3 - 2) × 180° ÷ 3 = 60°

Square (n = 4): Interior angle = (4 - 2) × 180° ÷ 4 = 90°

Regular pentagon (n = 5): Interior angle = (5 - 2) × 180° ÷ 5 = 108°
Exterior Angles
The exterior angle is the angle formed between any side of a polygon and the extension of its
adjacent side.
For any polygon:

Sum of exterior angles = 360°

For a regular polygon: Exterior angle = 360° ÷ n
Examples:

Triangle (n = 3): Exterior angle = 360° ÷ 3 = 120°

Square (n = 4): Exterior angle = 360° ÷ 4 = 90°

Regular pentagon (n = 5): Exterior angle = 360° ÷ 5 = 72°
Relationship Between Interior and Exterior Angles
For any vertex in a polygon:

Interior angle + Exterior angle = 180°
Sum of Angles in a Polygon
Formula
The sum of interior angles in a polygon with n sides:

Sum = (n - 2) × 180°
Examples
1. Triangle (n = 3):
o
Sum = (3 - 2) × 180° = 180°
2. Quadrilateral (n = 4):
o
Sum = (4 - 2) × 180° = 360°
3. Pentagon (n = 5):
o
Sum = (5 - 2) × 180° = 540°
4. Hexagon (n = 6):
o
Sum = (6 - 2) × 180° = 720°
Finding the Number of Sides
If you know the sum of interior angles, you can find the number of sides:

n = (Sum ÷ 180°) + 2
Example:

If the sum of interior angles is 1080°, then: n = (1080° ÷ 180°) + 2 = 6 + 2 = 8 sides
(octagon)
Transformations
There are four main types of transformations: translation, reflection, rotation, and enlargement.
Translation
A translation moves every point of a shape the same distance and in the same direction.
Translations are described using column vectors: $\begin{pmatrix} a \ b \end{pmatrix}$

a represents the horizontal movement (positive = right, negative = left)

b represents the vertical movement (positive = up, negative = down)
Example:

Triangle with vertices at (1,1), (3,1), and (2,3) translated by $\begin{pmatrix} 2 \ -1
\end{pmatrix}$ gives a new triangle with vertices at (3,0), (5,0), and (4,2)
Reflection
A reflection produces a mirror image of a shape across a line of reflection.
Common lines of reflection:

y=x

y = -x

x = 0 (y-axis)

y = 0 (x-axis)
To reflect a point (x,y):

In the y-axis: (-x,y)

In the x-axis: (x,-y)

In the line y = x: (y,x)

In the line y = -x: (-y,-x)
Rotation
A rotation turns a shape around a fixed point called the center of rotation.
Key information needed:

Center of rotation

Angle of rotation

Direction (clockwise or counterclockwise)
Example:

A triangle with vertices at (1,1), (3,1), and (2,3) rotated 90° clockwise around the origin
will have new vertices at (1,-1), (1,-3), and (3,-2)
Enlargement
Enlargement changes the size of a shape while maintaining its proportions.
Key components:

Scale factor (k)

Center of enlargement
For a point (x,y) enlarged by scale factor k from center (0,0):

New coordinates: (kx, ky)
For a point (x,y) enlarged by scale factor k from center (a,b):

New coordinates: (a + k(x-a), b + k(y-b))
Practice Resources
Percentages

IXL: Convert between percents, fractions, and decimals

IXL: Percents of numbers and money amounts

IXL: Percent of change

MyiMaths: Fractions, Decimals, and Percentages

MyiMaths: Percentages of Amounts

MyiMaths: Percentage Change

MyiMaths: Change as a Percentage
Area of Circle & Trapezium

MyiMaths: Area of a Circle

IXL: Circles - Word Problems

MyiMaths: Area of a Trapezium

IXL: Area of Trapeziums
Equations

MyiMaths: Single Brackets

IXL: Solve One-Step Equations

MyiMaths: Brackets

MyiMaths: Multi-Step Equations

IXL: Solve Two-Step Equations

MyiMaths: Equations with Variables on Both Sides

IXL: Solve Equations
Significant Figures

MyiMaths: Significant Figures

IXL: Round Decimals
Enlargement & Transformations

IXL: Dilations - Scale Factor and Classification

MyiMaths: Enlarging Shapes

IXL: Identify Reflections, Rotations, and Translations

IXL: Translations - Graph the Image

IXL: Translations - Find the Coordinates

IXL: Rotations - Graph the Image

IXL: Dilations - Graph the Image

MyiMaths: All Transformations
Indices & Standard Form

IXL: Evaluate Indices

MyiMaths: Indices

IXL: Compare Numbers in Standard Form

MyiMaths: Standard Form - Large

IXL: Convert between Ordinary Numbers and Standard Form

MyiMaths: Standard Form - Small
Angles in Polygons

IXL: Exterior Angle Theorem

MyiMaths: Interior & Exterior Angles

IXL: Identify and Classify Polygons

IXL: Sums of Angles in Polygons

MyiMaths: Sum of Angles in a Polygon