Chapter 1

Numbers & the Number System

Extend your skills with negative numbers, index notation, prime factorisation, advanced fractions, percentages, and ratio.

18
Objectives
5
Sections
35+
Questions

🎯 Learning Objectives (Chapter 1)

Add, subtract, multiply and divide positive and negative numbers

1

Use index notation and BIDMAS with powers, roots and brackets

2

Write a number as a product of prime factors

3

Find HCF and LCM using prime factorisation

4

Add, subtract, multiply and divide fractions including mixed numbers

5

Calculate percentage increases, decreases, and reverse percentages

6

Calculate simple and compound interest

7

Simplify ratios, divide quantities, and solve proportion problems

8

1.1Operations with Negative Numbers

▼

📖 Multiplying & Dividing Negatives

In Year 7 you added and subtracted negatives. Now we extend to multiplication and division.

  • positive × positive = positive (3 × 4 = 12)
  • negative × negative = positive (−3 × −4 = 12)
  • positive × negative = negative (3 × −4 = −12)
  • negative × positive = negative (−3 × 4 = −12)

The same rules apply for division.

💡 Simple Rule

Same signs → Positive. Different signs → Negative.

📐 Worked Examples
Calculate: (a) −6 × −5   (b) −24 ÷ 3   (c) London is −2°C; Moscow is four times as cold.
a

Same signs (both negative) → positive

−6 × −5 = 30
b

Different signs → negative

−24 ÷ 3 = −8
c

"Four times as cold" = 4 × −2 = −8°C

Moscow = −8°C

✏️ Practice

0/4
Q1
Calculate: −7 × 3
Q2
Calculate: −8 × −4
Q3
Calculate: 36 ÷ −6
Q4
Calculate: −5 + 3 × −2

1.2Index Laws & Prime Factorisation

▼

💡 Index Laws

  • aⁿ × aᵐ = aⁿ⁺ᵐ (multiply → add powers)
  • aⁿ ÷ aᵐ = aⁿ⁻ᵐ (divide → subtract powers)
  • (aⁿ)ᵐ = aⁿˣᵐ (power of a power → multiply)

📖 Prime Factor Decomposition

Every whole number greater than 1 can be written as a product of prime factors. We use a factor tree.

📐 Factor Tree: 60
Write 60 as a product of its prime factors.
1

60 = 2 × 30

2

30 = 2 × 15

3

15 = 3 × 5

✅
60 = 2² × 3 × 5
📐 HCF & LCM via Prime Factors
Find HCF and LCM of 24 and 36.
1

24 = 2³ × 3     36 = 2² × 3²

2

HCF: Take the lowest power of each common prime: 2² × 3 = 12

3

LCM: Take the highest power of each prime: 2³ × 3² = 72

✅

HCF = 12, LCM = 72

✏️ Practice

0/3
Q1
Write 90 as a product of prime factors (use × and powers).
Q2
Simplify: x³ × x²
Q3
Simplify: y⁷ ÷ y³

1.3Fraction Operations

▼

💡 Fraction Rules

  • Adding/Subtracting: Find a common denominator, then add/subtract numerators.
  • Multiplying: Multiply tops together, multiply bottoms together. Simplify.
  • Dividing: Flip the second fraction (reciprocal), then multiply.
  • Mixed numbers: Convert to improper fractions first!
📐 Dividing Fractions
Calculate: 2/3 ÷ 1/6
1
Flip the second fraction and multiply

2/3 ÷ 1/6 = 2/3 × 6/1

2

= 12/3 = 4

✅
2/3 ÷ 1/6 = 4

✏️ Practice

0/4
Q1
Calculate: 2/5 + 1/3
Q2
Calculate: 3/4 × 2/5
Q3
Calculate: 4 ÷ 2/3
Q4
What is the reciprocal of 3/7?

1.4Advanced Percentages

▼

📖 Percentage Increase & Decrease

Use a decimal multiplier: to increase by 15%, multiply by 1.15. To decrease by 20%, multiply by 0.80.

📖 Reverse Percentages

If a price AFTER a 20% reduction is £2.40, find the original: 80% = £2.40, so 1% = 2.40/80, and 100% = 2.40/80 × 100 = £3.00.

📖 Compound Interest

Interest is calculated on the new balance each year: Amount = Principal × (1 + rate)ⁿ

📐 Compound Interest
$4000 invested for 3 years at 2% compound interest per year. Find the total.
1

Multiplier = 1 + 0.02 = 1.02

2

After 3 years: 4000 × 1.02³ = 4000 × 1.061208 = $4244.83

✅

Total after 3 years ≈ $4244.83

✏️ Practice

0/3
Q1
Increase £60 by 15%.
Q2
A sale price is £48 after a 20% reduction. What was the original price?
Q3
£1000 at 5% simple interest for 3 years. How much interest is earned?

1.5Ratio & Proportion

▼
📐 Divide in Ratio 2:3:6
Share 462g in the ratio 2:3:6
1

Total parts = 2+3+6 = 11

2

One part = 462 ÷ 11 = 42g

3

2 parts = 84g, 3 parts = 126g, 6 parts = 252g

✅

84g : 126g : 252g   Check: 84+126+252 = 462 ✓

✏️ Practice

0/2
Q1
Simplify the ratio 3.25 : 12 (give as whole numbers).
Q2
Share £120 in the ratio 3:5.

📝 Chapter 1 Exam

30 marks | 40 minutes

[3]1.Calculate: (a) −5 × −8 (b) −36 ÷ 4 (c) (−3)²
(a) 40 (b) −9 (c) (−3)² = −3 × −3 = 9
[3]2.Write 120 as a product of primes. Use your answer to find HCF of 120 and 90.
120 = 2³×3×5. 90 = 2×3²×5. HCF: 2×3×5 = 30
[3]3.Calculate: (a) 3/4 − 1/6 (b) 2/5 × 3/7
(a) 9/12 − 2/12 = 7/12 (b) 6/35 = 6/35
[4]4.A tree is pruned to 2.4m, which is 80% of its original height. Find the original height.
80% = 2.4m. 1% = 0.03m. 100% = 0.03m. 100% = 3m
[3]5.Share 462g in ratio 2:3:6.
11 parts. 462÷11=42. 84g, 126g, 252g
Chapter 2

Algebra

Algebraic index laws, factorising, solving equations with unknowns on both sides, and interpreting graphs.

2.1Algebraic Index Laws

▼

💡 Laws

  • m × m = m²   |   3b × b = 3b²
  • x³ × x² = x⁵ (add powers)
  • y⁷ ÷ y³ = y⁴ (subtract powers)

✏️ Practice

0/3
Q1
Simplify: 4a × 3a
Q2
Simplify: 6t⁵ ÷ 2t²
Q3
Simplify: 2d³ × 5d

2.2Expanding & Factorising

▼

📖 Expanding with Negatives & Variables

In Year 8, the term outside the bracket can be negative or contain a variable:

−2(3 − x) = −6 + 2x     4c(c + 2) = 4c² + 8c

📖 Factorising

The reverse of expanding. Find the highest common factor of all terms and put it outside a bracket.

📐 Factorising Examples
1

2x + 4 → HCF is 2 → 2(x + 2)

2

2y² − y → HCF is y → y(2y − 1)

3

8z + 2z → HCF is 2z → 2z(4 + z)

✏️ Practice

0/4
Q1
Expand: −3(x − 5)
Q2
Expand: 5a(a + 3)
Q3
Factorise: 6x + 9
Q4
Factorise: 4m² + 6m

2.3Solving Equations

▼

📖 Equations with Brackets

Expand brackets first, then solve as normal. Always do the same operation to both sides.

📐 Solve: 2(x + 3) = 12
1

Expand: 2x + 6 = 12

2

Subtract 6: 2x = 6

✅

x = 3. Check: 2(3+3) = 2(6) = 12 ✓

📐 Solve: 5x − 2 = 23
1

Add 2: 5x = 25

2

Divide by 5: x = 5

✏️ Practice

0/4
Q1
Solve: x + 3 = 9
Q2
Solve: 3x − 7 = 14
Q3
Solve: 4(x − 1) = 20
Q4
A father is 3× his son's age. Together = 52. Son's age?

2.4Graphs: y = mx + c

▼

💡 y = mx + c

m = gradient (steepness). c = y-intercept (where line crosses y-axis).

Positive m → line goes up ↗. Negative m → line goes down ↘. m=0 → horizontal.

🧪 Graph Plotter

✏️ Practice

0/2
Q1
What is the gradient and y-intercept of y = 3x − 2?
Q2
A line has gradient 4 and passes through (0, 5). Write its equation.

📝 Chapter 2 Exam

25 marks | 30 minutes

[2]1.Simplify: 3x² × 4x³
12x⁵
[2]2.Factorise: 10a − 15a²
5a(2 − 3a)
[3]3.Solve: 3(2x + 1) = 4x + 7
6x+3=4x+7. 2x=4. x=2
[3]4.Write the equation of a line with gradient −2 and y-intercept 7.
y = −2x + 7
Chapter 3

Geometry & Measure

Area formulae for triangles, parallelograms and trapeziums. Volume of cuboids. Polygon angles and parallel line angles.

3.1Area Formulae

▼

💡 Area Formulae

  • Triangle: A = ½ × base × height
  • Parallelogram: A = base × perpendicular height
  • Trapezium: A = ½(a + b) × h   (a, b are parallel sides)

✏️ Practice

0/3
Q1
Triangle: base 10cm, height 6cm. Area?
Q2
Parallelogram: base 8cm, height 5cm. Area?
Q3
Trapezium: parallel sides 6cm and 10cm, height 4cm. Area?

3.2Volume & Surface Area

▼

💡 Cuboid

Volume = length × width × height
Surface Area = 2(lw + lh + wh)

✏️ Practice

0/2
Q1
Cuboid: 5cm × 3cm × 4cm. Volume?
Q2
Same cuboid. Surface area?

3.3Interior & Exterior Angles of Polygons

▼

💡 Angle Formulae

  • Sum of interior angles = (n − 2) × 180°   (n = number of sides)
  • Each interior angle (regular polygon) = (n − 2) × 180° ÷ n
  • Sum of exterior angles = 360° (always!)
  • Each exterior angle (regular polygon) = 360° ÷ n

✏️ Practice

0/3
Q1
Sum of interior angles of a pentagon (5 sides)?
Q2
Each exterior angle of a regular hexagon?
Q3
Each interior angle of a regular octagon?

3.4Angles in Parallel Lines

▼

💡 Angle Rules

  • Alternate angles (Z-angles) are equal
  • Corresponding angles (F-angles) are equal
  • Co-interior angles (C/U-angles) add to 180°

✏️ Practice

0/2
Q1
Two parallel lines cut by a transversal. One alternate angle is 107°. What is the other?
Q2
A co-interior angle is 115°. Find the other co-interior angle.

📝 Chapter 3 Exam

25 marks | 30 minutes

[3]1.Triangle: base 12cm, height 7cm. Find area.
½×12×7=42cm²
[3]2.Cuboid: 2.8cm × 1.5cm. Volume = 21cm³. Find height, then surface area.
h=21÷(2.8×1.5)=5cm. SA=2(4.2+14+7.5)=51.4cm²
[3]3.A pentagon has angles 95°, 129°, 97°, 111° and x°. Find x.
Sum=(5−2)×180=540. x=540−95−129−97−111=108°
Chapter 4

Statistics & Probability

Mean from frequency tables, pie charts, scatter graphs, and experimental probability.

4.1Mean from Frequency Tables

▼

📖 Mean from a Frequency Table

Mean = Total of (value × frequency) ÷ Total frequency

📐 Example
Score: 1(freq 3), 2(freq 5), 3(freq 7), 4(freq 4), 5(freq 1). Find mean.
1

Total = 1×3 + 2×5 + 3×7 + 4×4 + 5×1 = 55

2

Total frequency = 3+5+7+4+1 = 20

✅

Mean = 55 ÷ 20 = 2.75

4.2Pie Charts & Scatter Graphs

▼

💡 Pie Charts

Each sector angle = (frequency ÷ total) × 360°

💡 Scatter Graphs

  • Positive correlation: Both variables increase together
  • Negative correlation: One increases, other decreases
  • No correlation: No clear relationship
  • A line of best fit follows the trend of the data

✏️ Practice

0/2
Q1
30 students: 12 like football. What angle on a pie chart?
Q2
As temperature increases, ice cream sales increase. What type of correlation?

4.3Experimental Probability

▼

📖 Relative Frequency

Experimental probability = number of times event occurs ÷ total trials. More trials = more reliable estimate.

🎲 Dice Experiment
🎲

Rolls: 0

✏️ Practice

0/1
Q1
A coin is flipped 200 times and lands on heads 112 times. Estimate P(heads).

📝 Chapter 4 Exam

20 marks | 25 minutes

[3]1.Score: 2(freq 4), 3(freq 6), 4(freq 5), 5(freq 3), 6(freq 2). Find the mean score.
Total=8+18+20+15+12=73. Count=20. Mean=73÷20=3.65
[2]2.40 students: 15 walk to school. What angle on a pie chart?
15/40 × 360 = 135°
[3]3.A spinner is spun 50 times. It lands on red 18 times. (a) Estimate P(red). (b) If spun 200 times, estimate how many times it lands on red.
(a) 18/50 = 0.36 (b) 0.36 × 200 = 72 times

🎉 Year 8 Complete!

Great progress! Review any sections, then continue to Year 9.