Chapter 1

Numbers & the Number System

Place value, negative numbers, factors, multiples, primes, squares, cubes, and divisibility rules.

Objectives

Sections

30+

Questions

1.1Place Value

▼

📖 What is Place Value?

Every digit in a number has a value based on its position. In 3,456,789, the 4 is worth 400,000 because it sits in the hundred-thousands place.

🔍 Interactive Place Value Explorer

Billions	H.Millions	T.Millions	Millions	H.Thousands	T.Thousands	Thousands	Hundreds	Tens	Ones
-	-	-	-	-	-	-	-	-	-

💡 Key Fact

1 million = 1,000,000 (6 zeros). 1 billion = 1,000,000,000 (9 zeros).
One place left = ×10 bigger. One place right = ÷10 smaller.

📐 Worked Example 1

Write down the value of the digit 7 in: (a) 3,721,400 (b) 57,000,000 (c) 470,312

3,721,400 → hundred-thousands place

Value = 700,000

57,000,000 → millions place

Value = 7,000,000

470,312 → ten-thousands place

Value = 70,000

✏️ Practice: Place Value

0/3

What is the value of 5 in 2,543,100?

Write in digits: "Forty-two million, three hundred and six thousand, and fifteen"

How many times bigger is the 3 in 30,000 than the 3 in 3,000?

1.2Positive & Negative Numbers

▼

📖 Understanding Negatives

Negative numbers are less than zero: −5, −12. You see them in temperature, altitude, and bank accounts.

💡 Rules

Adding a positive: move RIGHT →
Adding a negative: move LEFT ← (same as subtracting)
Subtracting a positive: move LEFT ←
Subtracting a negative: move RIGHT → (two negatives = positive!)

⚠️ Common Mistake

−3 is NOT bigger than −1. Think temperature: −3°C is colder than −1°C, so −3 < −1.

📐 Worked Example 2

Calculate: (a) −5 + 3 (b) 4 − 7 (c) −2 − 6 (d) −3 − (−5)

Start at −5, move 3 right: −5 → −4 → −3 → −2

−5 + 3 = −2

Start at 4, move 7 left: passes 0, lands on −3

4 − 7 = −3

Start at −2, move 6 left → −8

−2 − 6 = −8

Subtracting negative = adding: −3 + 5 = 2

−3 − (−5) = −3 + 5 = 2

✏️ Practice: Negatives

0/4

Calculate: −8 + 5

Calculate: 3 − 10

Calculate: −4 − (−9)

Moscow is −15°C. London is 18° warmer. What is London's temperature?

1.3Factors, Multiples, HCF & LCM

▼

📖 Factors

Factors divide exactly into a number. Use factor pairs: start from 1 and work up until pairs repeat.

📐 Finding All Factors of 36

Find all factors of 36.

1×36, 2×18, 3×12, 4×9, 6×6 (repeats!)

✅ Answer:

Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}

📖 HCF & LCM

HCF (Highest Common Factor): the biggest factor shared by two numbers.

LCM (Lowest Common Multiple): the smallest number in both multiplication tables.

📐 HCF of 24 and 36

Factors of 24: 1,2,3,4,6,8,12,24

Factors of 36: 1,2,3,4,6,9,12,18,36

Common: 1,2,3,4,6,12

✅

HCF = 12

🧪 Multiple Explorer

Select two numbers to see their multiples highlighted. Common multiples appear in gradient.

Num 1:

Num 2:

✏️ Practice

0/3

What is the HCF of 18 and 30?

What is the LCM of 4 and 6?

What is the LCM of 5 and 8?

1.4Prime Numbers

▼

📖 What is a Prime?

A prime has exactly two factors: 1 and itself. Note: 1 is NOT prime.

💡 Primes to 19

2, 3, 5, 7, 11, 13, 17, 19

2 is the only even prime!

🧪 Sieve of Eratosthenes

Click buttons to cross out multiples and find all primes up to 100.

✏️ Practice

0/2

Is 51 prime?

ANo, 51 = 3×17

BYes

List all primes between 20 and 40.

1.5Squares, Cubes & Index Notation

▼

📖 Square & Cube Numbers

Square: n × n = n². E.g., 7² = 49. Square root: √49 = 7.

Cube: n × n × n = n³. E.g., 4³ = 64. Cube root: ∛64 = 4.

💡 Know These!

1²=1 2²=4 3²=9 4²=16 5²=25 6²=36 7²=49 8²=64 9²=81 10²=100 11²=121 12²=144

1³=1 2³=8 3³=27 4³=64 5³=125

🧪 Visualise Squares

n = 5

5² = 25

5³ = 125

5 × 5 × 5 = 125

✏️ Practice

0/4

What is 9²?

What is √121?

What is 4³?

What is ∛125?

1.6Divisibility Rules

▼

💡 Rules

÷2	Last digit even
÷3	Digit sum ÷3
÷4	Last 2 digits ÷4
÷5	Ends in 0 or 5
÷9	Digit sum ÷9
÷10	Ends in 0

✏️ Practice

0/2

Is 4,572 divisible by 3?

Is 8,991 divisible by 9?

📝 Chapter 1 Exam

30 marks | 40 minutes

[2]1.Write in digits: "Three hundred and fifty-two million, seven hundred and four thousand, and nineteen."

352,704,019

[2]2.Oslo is −14°C. Cairo is 38° warmer. Find Cairo's temperature.

−14 + 38 = 24°C

[3]3.Find (a) HCF of 42 and 56. (b) LCM of 12 and 15.

(a) Factors 42:{1,2,3,6,7,14,21,42} 56:{1,2,4,7,8,14,28,56}. Common:{1,2,7,14}. HCF=14
(b) Multiples 12:12,24,36,48,60. 15:15,30,45,60. LCM=60

[3]4.Calculate: (a) 7² + 3² (b) √144 − √64 (c) 2³ + 5²

(a) 49+9=58 (b) 12−8=4 (c) 8+25=33

[4]5.A square pool has area 81m². (a) Side length? (b) Perimeter? (c) A cube tank has side 3m. Volume?

(a) √81=9m (b) 4×9=36m (c) 3³=27m³

Chapter 2

Fractions, Decimals & Percentages

Convert between representations of parts. Calculate with decimals and fractions. Find percentages of amounts.

2.1Understanding Decimals

▼

📖 Decimal Place Value

The decimal point separates whole numbers from fractional parts. Tenths (1/10), hundredths (1/100), thousandths (1/1000).

⚠️ Say It Right

0.451 = "nought point four five one" — NOT "four hundred and fifty-one"!

✏️ Practice

0/2

Which is larger: 0.451 or 0.7?

A0.451

B0.7

Order smallest to largest: 0.8, 0.18, 0.81, 0.08

2.2Rounding & Estimating

▼

📖 How to Round

To round to 1 d.p.: look at the 2nd decimal digit. If 5+, round up. If 4−, keep same.

📐 Estimation

Estimate 5.78 × 7.2

5.78 ≈ 6, 7.2 ≈ 7

✅

6 × 7 = 42 (actual: 41.616)

✏️ Practice

0/1

Round 12.55 to 1 decimal place.

2.3Decimal Calculations

▼

💡 Golden Rule

Line up the decimal points! Add zeros as placeholders.

📐 Column Subtraction

Calculate 82.1 − 0.574

82.100 − 0.574 ──────── 81.526

✏️ Practice

0/2

7.742 + 9.41

Four ribbons, each 0.6m. Total length?

2.4Fractions & Equivalence

▼

📖 Fractions

Numerator (top) = how many parts you have. Denominator (bottom) = how many equal parts total.

Equivalent fractions have the same value: 1/2 = 2/4 = 4/8. Multiply/divide top and bottom by the same number.

📐 Improper → Mixed

Change 7/3 to a mixed number.

7 ÷ 3 = 2 remainder 1

✅

7/3 = 2⅓

✏️ Practice

0/2

Simplify 12/20 to its simplest form.

Calculate 2/3 × 15

2.5Percentages

▼

📖 Percent = out of 100

50% = 50/100 = 0.5. Convert: ÷100 for % → decimal. ×100 for decimal → %.

🧪 FDP Converter

Percentage: 40%

📐 Find 35% of 80

10% of 80 = 8

30% = 3 × 8 = 24. 5% = 8 ÷ 2 = 4.

✅

35% = 24 + 4 = 28

✏️ Practice

0/2

Find 40% of £150

Write 3/5 as a percentage.

📝 Chapter 2 Exam

20 marks | 25 minutes

[2]1.Calculate 14.2 − 5.83

14.20 − 5.83 = 8.37

[3]2.Write 12/20 as: (a) simplest fraction (b) decimal (c) percentage

(a) 3/5 (b) 0.6 (c) 60%

[2]3.Change 11/4 to a mixed number.

11÷4 = 2 r 3. 2¾

[2]4.Find 70% of 90.

10%=9. 70%=7×9=63

Chapter 3

Introduction to Algebra

Expressions, simplifying, expanding brackets, substitution, sequences, and graphs.

3.1Writing Expressions

▼

💡 Notation Rules

3 × m → 3m
b ÷ 5 → b/5
a × a → a²
1 × x → just x

✏️ Practice

0/2

Write y × 7 in algebraic notation.

Write d × d × d using index notation.

3.2Collecting Like Terms

▼

📖 Like Terms

Like terms have the same letter part. 3a and 5a are like. 3a and 5b are not.

📐 Simplify 2x + 3y + 4x − y

x-terms: 2x + 4x = 6x

y-terms: 3y − y = 2y

✅

6x + 2y

✏️ Practice

0/3

Simplify: a + 3 + a

Simplify: 2a × 5

Simplify: 2c × 3d

3.3Expanding Brackets

▼

📖 Expanding

Multiply the outside term by every term inside the bracket.

📐 Expand 6(3a + 4)

6 × 3a = 18a

6 × 4 = 24

✅

6(3a + 4) = 18a + 24

✏️ Practice

0/3

Expand 2(x + 3)

Expand 3(y − 5)

Expand 5(2b − 1)

3.4Substitution

▼

📖 Substitution

Replace letters with numbers, then calculate using BIDMAS.

🧪 Formula Calculator: y = 3x + 2

x =4

✏️ Practice

0/3

If x=7, find 3x+5

If a=4, b=6, find 5a−b

P=2l+2w. Find P when l=12, w=5.

3.5Sequences

▼

📖 Sequences

An ordered list following a rule. Term-to-term: how to get the next term. nth term: formula for any position.

🧪 Sequence Generator

First term:

Difference:

📐 Find nth term of: 5, 8, 11, 14, 17...

Difference = 3 → starts with 3n

3n: 3,6,9,12,15. Our seq: 5,8,11,14,17. Each is +2.

✅

nth term = 3n + 2

✏️ Practice

0/3

Next term: 4, 10, 16, 22, ___

Find nth term of: 7, 12, 17, 22...

nth term = 4n−1. What is the 10th term?

3.6Straight-Line Graphs

▼

📖 Plotting Graphs

Make a table of x, y values.
Plot points on a grid.
Draw a straight line through them.

🧪 Graph Plotter

y = x +

📝 Chapter 3 Exam

25 marks | 30 minutes

[2]1.Simplify: 5a + 3b − 2a + b

3a + 4b

[2]2.Expand: 4(3x − 2)

12x − 8

[3]3.If p=5, q=−2: (a) 3p+q (b) p²−q

(a) 15−2=13 (b) 25−(−2)=25+2=27

[3]4.nth term of 1, 6, 11, 16, 21...

Diff=5. 5n:5,10,15,20,25. Ours is −4. 5n − 4

[4]5.Father is 3× son's age. Together=52. Find each age.

Son=x. x+3x=52. 4x=52. x=13. Son=13, Father=39

Chapter 4

Geometry & Measure

Metric conversions, perimeter, area, and essential angle rules.

4.1Metric Conversions

▼

💡 Key Conversions

Length: 10mm=1cm, 100cm=1m, 1000m=1km
Mass: 1000g=1kg, 1000kg=1 tonne
Capacity: 1000ml=1 litre

Smaller unit → ×. Larger unit → ÷.

✏️ Practice

0/3

0.123m in mm?

2500g in kg?

3.5km in metres?

4.2Perimeter & Area

▼

💡 Formulae

Perimeter = sum of all sides.
Area of Rectangle = length × width.
Area of Square = side².

✏️ Practice

0/2

Rectangle: length 8cm, width 5cm. (a) Perimeter? (b) Area?

Area of square = 49cm². Perimeter?

4.3Angle Rules

▼

💡 Angle Rules

Straight line: 180°
Around a point: 360°
Vertically opposite: equal
Triangle: 180°
Quadrilateral: 360°

🧪 Angle Explorer

Angle:55°

✏️ Practice

0/4

On a straight line with 130°, find angle x.

Vertically opposite to 65° is?

Triangle angles: 58°, 67°, find third.

Quadrilateral: 88°, 102°, 67°. Find fourth.

📝 Chapter 4 Exam

20 marks | 25 minutes

[2]1.Convert 4.5km to m, and 850g to kg.

4500m and 0.85kg

[3]2.Area of square = 25cm². (a) Side? (b) Perimeter?

(a) √25=5cm (b) 4×5=20cm

[3]3.Triangle: x°, x°, 56°. Find x.

2x+56=180. 2x=124. x=62°

Chapter 5

Statistics & Probability

Averages, charts, the probability scale, and probability experiments.

5.1Mean, Median, Mode & Range

▼

📖 Three Averages

Mean: Total ÷ Count
Median: Middle value (data must be ordered!)
Mode: Most frequent value
Range: Highest − Lowest

🧪 Live Averages Calculator

Enter data and click Calculate.

⚠️ Median Mistake

You must order the data first! For an even count, average the two middle values.

✏️ Practice

0/4

Mean of: 6, 8, 4, 10, 7

Median of: 12, 3, 8, 15, 6

Mode of: 5, 3, 5, 8, 3, 5, 9

The mean of 4, 5, 7, n is 5. Find n.

5.2Charts & Data

▼

💡 Discrete vs Continuous

Discrete: Counted, specific values (shoe size, pets). Bars have gaps.
Continuous: Measured, any value (height, temperature). Bars have no gaps.

🧪 Bar Chart Builder

5.3Probability

▼

💡 The Probability Formula

P(event) = Favourable outcomes ÷ Total outcomes

Always between 0 (impossible) and 1 (certain). P(NOT A) = 1 − P(A).

📐 Spinner with sections 1-12

(a) P(3) (b) P(4 or 5) (c) P(NOT 3)

P(3) = 1/12

P(4 or 5) = 2/12 = 1/6

P(NOT 3) = 1 − 1/12 = 11/12

✏️ Practice

0/3

Bag: 3 red, 5 blue, 2 green. P(blue)?

P(rain) = 0.3. P(NOT rain)?

Fair dice. P(even number)?

5.4Probability Experiments

▼

🎲 Dice Roller Experiment

Roll dice and compare experimental vs theoretical probability (1/6 each).

🎲

Rolls: 0

🪙 Coin Flip Experiment

🪙

Heads

Tails

Total

📝 Chapter 5 Exam

25 marks | 30 minutes

[4]1.Ages: 8,11,9,8,10,12,8. Find mean, median, mode, range.

Mean: 66÷7≈9.43. Ordered: 8,8,8,9,10,11,12. Median=9. Mode=8. Range=12−8=4.

[3]2.Mean of 4,5,7,n is 5. Find n.

Total=20. 4+5+7=16. n=4.

[4]3.Bag: 4 red, 3 blue, 1 white. (a) P(red) (b) P(blue or white) (c) P(NOT red)

Total=8. (a) 4/8=1/2 (b) 4/8=1/2 (c) 1−1/2=1/2

🎉 Year 7 Complete!

Well done! Review any tricky sections, then move on to Year 8.