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Edexcel GCSE Maths (9–1)

Maths Help

A complete set of revision notes, organised by topic, with worked examples and the tips Maths Mummy actually gives in lessons. Use the jump list to find a topic, or scroll through from the top.

1. Number Foundation & Higher

The building blocks: working with numbers, fractions, percentages, standard form and surds. Get this section solid — it underpins everything else.

1.1 Place value, ordering and the four operations

Make sure you can add, subtract, multiply and divide whole numbers and decimals without a calculator. Always line up the decimal points when adding or subtracting.

Example: Work out 4.6 + 0.85.
Line up the decimal points: 4.60 + 0.85 = 5.45
Tip: When multiplying decimals, ignore the decimal points, multiply as whole numbers, then count the total number of decimal places in the question and put the point back in.

1.2 Factors, multiples and primes

  • A factor divides exactly into a number.
  • A multiple is in the times table of a number.
  • A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13…). Note: 1 is not prime.
Example — Prime factorisation: Write 72 as a product of its prime factors.
72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
Tip: Use a factor tree. Keep splitting numbers until every branch ends in a prime.

To find the HCF (highest common factor), list the common prime factors using the lowest power. For the LCM (lowest common multiple), use every prime factor that appears, with the highest power.

1.3 Fractions

  • Adding/subtracting: find a common denominator first, then add or subtract the numerators.
  • Multiplying: multiply the numerators and multiply the denominators. Simplify at the end (or cancel first).
  • Dividing: flip the second fraction (find its reciprocal) and multiply — "flip and times".
Example: Work out 2⅓ ÷ 1¼.
Convert to improper fractions: 7/3 ÷ 5/4.
Flip and times: 7/3 × 4/5 = 28/15 = 1 13/15

1.4 Percentages

  • Percentage of an amount: divide the percentage by 100, then multiply by the amount.
  • Percentage change: (change ÷ original) × 100.
  • Percentage increase/decrease (multiplier method): multiply by (1 ± percentage as a decimal).
  • Reverse percentages: if you're given the amount after a change, divide by the multiplier to find the original.
Example — Multiplier: Increase £80 by 15%.
Multiplier = 1 + 0.15 = 1.15
£80 × 1.15 = £92
Example — Reverse percentage: A jacket costs £51 after a 15% discount. Find the original price.
Multiplier = 1 − 0.15 = 0.85
Original price = £51 ÷ 0.85 = £60
Tip: For compound interest over n years at rate r%, use amount = original × (1 + r/100)ⁿ.

1.5 Standard form

Standard form is A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.

  • For large numbers, n is positive (e.g. 3 400 000 = 3.4 × 10⁶).
  • For small numbers (less than 1), n is negative (e.g. 0.00045 = 4.5 × 10⁻⁴).
Example: Calculate (3 × 10⁵) × (4 × 10³), giving your answer in standard form.
3 × 4 = 12, and 10⁵ × 10³ = 10⁸, so the result is 12 × 10⁸.
Adjust so A is between 1 and 10: 12 × 10⁸ = 1.2 × 10⁹

1.6 Indices (powers and roots)

  • aᵘ × aᵛ = a^(u+v)
  • aᵘ ÷ aᵛ = a^(u−v)
  • (aᵘ)ᵛ = a^(uv)
  • a⁰ = 1
  • a⁻ⁿ = 1/aⁿ
  • a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵘ
Example: Work out 8^(2/3).
Cube root of 8 is 2, then square it: 2² = 4

1.7 Surds Higher

A surd is a root that doesn't simplify to a whole number, e.g. √2, √5.

  • √a × √b = √(ab)
  • √a ÷ √b = √(a/b)
  • To rationalise a denominator like 1/√3, multiply top and bottom by √3 to get √3/3.
Example: Simplify √50.
√50 = √(25 × 2) = √25 × √2 = 5√2

2. Algebra Foundation & Higher

Letters instead of numbers — but the same rules of arithmetic apply. This section covers expanding, factorising, solving equations, sequences, graphs and functions.

2.1 Simplifying and substituting

Collect like terms by adding/subtracting their coefficients. Only terms with exactly the same letters and powers can be combined.

Example: Simplify 5a + 3b − 2a + 7b.
(5a − 2a) + (3b + 7b) = 3a + 10b
Example — Substitution: If a = 3 and b = −2, work out 2a² − b.
2(3)² − (−2) = 2(9) + 2 = 20

2.2 Expanding brackets

Multiply every term inside the bracket by the term outside.

Example — Single bracket: Expand 4(2x − 3).
8x − 12
Example — Double brackets (FOIL): Expand (x + 3)(x − 5).
x² − 5x + 3x − 15 = x² − 2x − 15
Tip: FOIL = First, Outer, Inner, Last. Multiply each pair, then collect like terms.

2.3 Factorising

  • Common factor: take out the highest common factor, e.g. 6x² + 9x = 3x(2x + 3).
  • Quadratics (x² + bx + c): find two numbers that multiply to c and add to b.
  • Difference of two squares: a² − b² = (a + b)(a − b).
Example: Factorise x² + 5x + 6.
Find two numbers that multiply to 6 and add to 5: 2 and 3.
(x + 2)(x + 3)
Example — Difference of two squares: Factorise x² − 49.
(x + 7)(x − 7)

2.4 Solving linear equations

Whatever you do to one side, do to the other. Work towards getting the unknown on its own.

Example: Solve 3(x − 4) = 2x + 1.
Expand: 3x − 12 = 2x + 1
Subtract 2x: x − 12 = 1
Add 12: x = 13

2.5 Solving quadratic equations

  • Factorising: rearrange to "= 0", factorise, then set each bracket to zero.
  • Quadratic formula: for ax² + bx + c = 0,
    x = (−b ± √(b² − 4ac)) / (2a)
  • Completing the square (Higher): write ax² + bx + c in the form a(x + p)² + q.
Example — Factorising: Solve x² − 2x − 15 = 0.
Factorise: (x − 5)(x + 3) = 0
So x = 5 or x = −3
Example — Quadratic formula: Solve 2x² + 3x − 4 = 0 to 2 d.p.
a = 2, b = 3, c = −4
x = (−3 ± √(9 + 32)) / 4 = (−3 ± √41) / 4
x ≈ 0.85 or x ≈ −2.35

2.6 Rearranging formulae

Treat the formula like an equation — isolate the required letter by undoing operations in reverse order (reverse BIDMAS).

Example: Make r the subject of A = πr².
Divide by π: A/π = r²
Square root: r = √(A/π)

2.7 Simultaneous equations

For linear pairs, add or subtract the equations to eliminate one variable, after scaling so the coefficients match.

Example: Solve 3x + 2y = 16 and 5x − 2y = 8.
Add the equations (the y-terms cancel): 8x = 24, so x = 3.
Substitute: 3(3) + 2y = 16 → 2y = 7 → y = 3.5
Higher: for one linear and one quadratic equation, substitute the linear equation into the quadratic one and solve.

2.8 Inequalities

Solve like an equation, but flip the inequality sign if you multiply or divide by a negative number.

Example: Solve 4 − 2x > 10.
Subtract 4: −2x > 6
Divide by −2 (flip the sign): x < −3
Number lines: use an open circle (○) for < or >, and a filled circle (●) for ≤ or ≥.

2.9 Sequences

  • Linear (arithmetic) sequences: nth term = dn + (a − d), where d is the common difference and a is the first term.
  • Quadratic sequences: look at the second differences — if they're constant, the nth term includes an n² term.
  • Geometric sequences: each term is multiplied by a common ratio.
Example: Find the nth term of 5, 8, 11, 14, …
Common difference d = 3. a − d = 5 − 3 = 2.
nth term = 3n + 2

2.10 Straight-line graphs

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.

  • Gradient = (change in y) ÷ (change in x).
  • Parallel lines have the same gradient.
  • Perpendicular lines (Higher) have gradients that multiply to give −1.
Example: Find the equation of the line through (0, 3) and (2, 7).
Gradient = (7 − 3) / (2 − 0) = 2. The line crosses the y-axis at 3.
y = 2x + 3

2.11 Quadratic and other graphs

  • y = x² gives a parabola (U-shape).
  • y = 1/x gives a reciprocal graph with two curved branches.
  • y = aˣ gives an exponential growth/decay curve.
Tip: When sketching, find where the graph crosses the axes (set x = 0 and y = 0), and identify any turning points.

3. Ratio, Proportion & Rates of Change Foundation & Higher

Comparing quantities, sharing in a ratio, scaling recipes, speed, density and pressure, and direct/inverse proportion.

3.1 Ratio

To share an amount in a given ratio, add the parts of the ratio to find the total number of parts, then work out the value of one part.

Example: Share £180 in the ratio 2:3:4.
Total parts = 2 + 3 + 4 = 9. One part = £180 ÷ 9 = £20.
Shares: £40 : £60 : £80
Tip: To simplify a ratio, divide every part by their HCF — just like simplifying a fraction.

3.2 Direct and inverse proportion

  • Direct proportion: y = kx — as x increases, y increases at the same rate. Doubling x doubles y.
  • Inverse proportion: y = k/x — as x increases, y decreases. Doubling x halves y.
Example: y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
y = kx → 20 = k × 4 → k = 5.
y = 5 × 7 = 35

3.3 Compound measures: speed, density, pressure

  • Speed = Distance ÷ Time
  • Density = Mass ÷ Volume
  • Pressure = Force ÷ Area
Example: A car travels 150 km in 2.5 hours. Find its average speed.
Speed = 150 ÷ 2.5 = 60 km/h
Tip: Cover up the quantity you want with your finger in a "formula triangle" — the other two tell you whether to multiply or divide.

3.4 Growth and decay

Repeated percentage change uses a multiplier raised to a power: final = initial × (multiplier)ⁿ, where n is the number of time periods.

Example: A car worth £18,000 depreciates by 12% per year. Find its value after 3 years.
Multiplier = 1 − 0.12 = 0.88
£18,000 × 0.88³ ≈ £12,277

4. Geometry & Measures Foundation & Higher

Angles, shapes, area, volume, Pythagoras, trigonometry, transformations and vectors.

4.1 Angle facts

  • Angles on a straight line add up to 180°.
  • Angles around a point add up to 360°.
  • Angles in a triangle add up to 180°.
  • Angles in a quadrilateral add up to 360°.
  • Parallel lines: alternate angles are equal (Z-shape), corresponding angles are equal (F-shape), and co-interior angles add to 180° (C-shape).
Tip: Always state the angle rule you're using — "angles on a straight line = 180°" — as you can get marks for the reason, not just the answer.

4.2 Perimeter and area

  • Rectangle: A = length × width
  • Triangle: A = ½ × base × height
  • Parallelogram: A = base × height
  • Trapezium: A = ½(a + b) × h (a and b are the parallel sides)
  • Circle: A = πr², Circumference = 2πr (or πd)
Example: Find the area of a circle with radius 6 cm (to 1 d.p.).
A = π × 6² = π × 36 ≈ 113.1 cm²

4.3 Volume and surface area

  • Cuboid: V = length × width × height
  • Prism: V = cross-sectional area × length
  • Cylinder: V = πr²h
  • Cone: V = ⅓πr²h
  • Sphere: V = 4/3 πr³
Example: Find the volume of a cylinder with radius 4 cm and height 10 cm (to 1 d.p.).
V = π × 4² × 10 = π × 160 ≈ 502.7 cm³

4.4 Pythagoras' theorem

For a right-angled triangle with hypotenuse c and shorter sides a and b: a² + b² = c²

Example: A ladder of length 5 m leans against a wall, reaching 4 m up. How far is the base from the wall?
5² = 4² + b² → 25 = 16 + b² → b² = 9 → b = 3 m
Tip: The hypotenuse is always the longest side and is opposite the right angle.

4.5 Trigonometry (SOH CAH TOA)

For a right-angled triangle, with angle θ:

  • sin θ = Opposite / Hypotenuse
  • cos θ = Adjacent / Hypotenuse
  • tan θ = Opposite / Adjacent
Example: A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the opposite side.
sin 30° = opp / 10 → opp = 10 × sin 30° = 5 cm
Higher: the sine rule (a/sin A = b/sin B) and cosine rule (a² = b² + c² − 2bc cos A) work for any triangle, not just right-angled ones.

4.6 Transformations

  • Translation: described by a vector, e.g. (3, −2) means 3 right, 2 down.
  • Reflection: described by a mirror line, e.g. the line y = x.
  • Rotation: described by an angle, direction, and centre of rotation.
  • Enlargement: described by a scale factor and centre of enlargement. A scale factor between 0 and 1 makes the shape smaller.

4.7 Vectors Higher

A vector has both magnitude (size) and direction, written as a column vector (x, y) or in bold, e.g. a.

  • To add vectors, add the corresponding components.
  • A negative vector reverses the direction.
  • A scalar multiple scales the vector's length but keeps its direction (or reverses it if negative).

5. Probability Foundation & Higher

Working with probabilities, sample spaces, tree diagrams and combining events.

5.1 The basics

  • Probability is always between 0 (impossible) and 1 (certain).
  • P(event) = number of successful outcomes ÷ total number of outcomes
  • The probabilities of all possible outcomes add up to 1.
  • P(not A) = 1 − P(A)

5.2 Combined events

  • "AND" — independent events: multiply the probabilities. P(A and B) = P(A) × P(B)
  • "OR" — mutually exclusive events: add the probabilities. P(A or B) = P(A) + P(B)
Example: A fair coin is flipped and a fair six-sided die is rolled. Find P(heads and a 6).
P(heads) = 1/2, P(6) = 1/6.
P(heads and 6) = 1/2 × 1/6 = 1/12

5.3 Tree diagrams

Tree diagrams show the outcomes of two or more events. Multiply along the branches to find the probability of a combination, and add the relevant end probabilities for an "or" question.

Tip: Check that the probabilities on each pair of branches add up to 1 — a common mistake is forgetting to recalculate probabilities for "without replacement" questions.

5.4 Venn diagrams and set notation Higher

  • A ∪ B means A or B (union).
  • A ∩ B means A and B (intersection).
  • A' means "not A" (everything outside A).
  • P(A | B) means the probability of A given B has happened (conditional probability).

6. Statistics Foundation & Higher

Collecting, displaying and interpreting data — averages, spread, charts and sampling.

6.1 Averages and range

  • Mean: total of all values ÷ number of values.
  • Median: the middle value when data is in order (average of the two middle values if there's an even number of values).
  • Mode: the most frequent value.
  • Range: largest value − smallest value (a measure of spread, not an average).
Example: Find the mean of 4, 7, 7, 9, 13.
Sum = 40, count = 5. Mean = 40 ÷ 5 = 8
Tip: For grouped frequency tables, use the midpoint of each group to estimate the mean: estimated mean = Σ(midpoint × frequency) ÷ Σ(frequency).

6.2 Charts and diagrams

  • Bar charts: compare categories — bars don't touch.
  • Pie charts: show proportions of a whole. Each sector's angle = (frequency ÷ total) × 360°.
  • Scatter graphs: show correlation between two variables. A line of best fit can be used to make estimates.
  • Cumulative frequency graphs: plot running totals — used to estimate the median and interquartile range.
  • Box plots: show the minimum, lower quartile, median, upper quartile and maximum.

6.3 Interquartile range Higher

The interquartile range (IQR) measures the spread of the middle 50% of the data: IQR = Upper Quartile (Q3) − Lower Quartile (Q1). It's a more useful measure of spread than the range when there are outliers.

6.4 Sampling

A sample should be representative of the population it's taken from. Common methods include random sampling (every member has an equal chance of being chosen) and stratified sampling (the sample reflects the proportions of different groups in the population).

Tip: When asked to comment on a sampling method, think about bias — could the way the sample was chosen make certain results more or less likely?