AQA GCSE Maths Higher Cheat Sheet 2026

ExpertMinds Editorial·3 March 2026·10 min read

Practice AQA GCSE questions while you read →

Higher tier covers grades 4–9. The content includes everything on Foundation plus significant additional topics. Questions at grades 8 and 9 combine multiple topics in a single question. This cheat sheet focuses on Higher-only content — review the Foundation sheet for baseline rules.

Key fact:Higher grade boundaries are typically: Grade 4 ~15–20%, Grade 7 ~50–60%, Grade 9 ~75–80%. A student who knows all Higher content but is slow will score lower than a student who is fast and accurate on grades 4–7 material.

Quadratics

Method	Use when	Key step
Factorising	Coefficient of x² is 1; integer solutions exist	Find two numbers that multiply to c and add to b in x² + bx + c
Completing the square	Finding vertex; deriving quadratic formula; always works	x² + bx = (x + b/2)² − (b/2)²
Quadratic formula	Always works — use when factorising fails	x = (−b ± √(b²−4ac)) / 2a
Discriminant	Number of real roots: b²−4ac > 0 → 2 roots, = 0 → 1, < 0 → 0	No need to solve — just evaluate b²−4ac

Trigonometry

Rule	Formula	Use when
SOH-CAH-TOA	sin θ = O/H; cos θ = A/H; tan θ = O/A	Right-angled triangles only
Sine rule	a/sinA = b/sinB = c/sinC	Non-right triangle: 2 angles + 1 side, or 2 sides + non-included angle
Cosine rule	a² = b² + c² − 2bc cosA	Non-right triangle: 2 sides + included angle, or 3 sides (find angle)
Area using trig	Area = ½ ab sin C	Any triangle when 2 sides and included angle are known

Tip:Ambiguous case: when using the sine rule with 2 sides and a non-included angle (SSA), there may be two possible triangles. Check if the second solution (180° − θ) gives a valid triangle by confirming angles sum to less than 180°.

Circle Theorems

Practice AQA GCSE questions while you read

Questions graded, hints, and explained.

Angle at centre = 2 × angle at circumference (same arc)
Angles in the same segment are equal
Angle in a semicircle = 90° (angle subtended by diameter)
Opposite angles in a cyclic quadrilateral sum to 180°
Tangent to a circle is perpendicular to the radius at the point of contact
Two tangents from an external point are equal in length
Alternate segment theorem: angle between tangent and chord = angle in alternate segment
Angle between radius and chord (perpendicular from centre bisects chord)

Algebra — Higher Topics

Topic	Method / rule
Algebraic fractions	Factorise numerator and denominator; cancel common factors; find LCM for addition/subtraction
Functions: f(x)	f(3) means substitute x = 3 into f(x)
Composite functions: fg(x)	Apply g first, then f: fg(x) = f(g(x))
Inverse function: f⁻¹(x)	Replace f(x) with y; rearrange for x; replace x with f⁻¹(x)
Iteration	Rearrange to xₙ₊₁ = f(xₙ); substitute repeatedly to converge on root
Nth term of quadratic sequence	Second difference ÷ 2 gives coefficient of n²; adjust for linear and constant terms
Graph transformations	y = f(x+a): left a; y = f(x)+ a: up a; y = f(ax): squeeze horizontal; y = af(x): stretch vertical

Practice GCSE Maths Higher questions

Grade 7–9 questions combine multiple topics. Practice multi-step problems to build the fluency needed for the harder marks.

Vectors

Vector addition: a + b — head to tail; result is the diagonal of the parallelogram
Scalar multiple: ka — same direction, scaled magnitude
AB⃗ = b − a (position vectors: from origin)
Midpoint M of AB: m = ½(a + b)
To prove collinearity: show vectors are parallel (scalar multiples) and share a common point
To prove a point divides a line in ratio m:n: express the vector and confirm the ratio

Probability — Higher Topics

Topic	Rule
Conditional probability	P(A\|B) = P(A∩B) / P(B)
Multiplication rule (dependent events)	P(A and B) = P(A) × P(B\|A)
Venn diagrams — with algebra	Set up expressions for each region; use total = 1 to solve for unknowns
Histograms	Frequency density = Frequency ÷ Class width; area of bar = frequency (not height)
Cumulative frequency	Plot upper class boundary vs cumulative frequency; median at n/2; IQR = UQ − LQ

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Quadratics

Method	Use when	Key step
Factorising	Coefficient of x² is 1; integer solutions exist	Find two numbers that multiply to c and add to b in x² + bx + c
Completing the square	Finding vertex; deriving quadratic formula; always works	x² + bx = (x + b/2)² − (b/2)²
Quadratic formula	Always works — use when factorising fails	x = (−b ± √(b²−4ac)) / 2a
Discriminant	Number of real roots: b²−4ac > 0 → 2 roots, = 0 → 1, < 0 → 0	No need to solve — just evaluate b²−4ac

Trigonometry

Rule	Formula	Use when
SOH-CAH-TOA	sin θ = O/H; cos θ = A/H; tan θ = O/A	Right-angled triangles only
Sine rule	a/sinA = b/sinB = c/sinC	Non-right triangle: 2 angles + 1 side, or 2 sides + non-included angle
Cosine rule	a² = b² + c² − 2bc cosA	Non-right triangle: 2 sides + included angle, or 3 sides (find angle)
Area using trig	Area = ½ ab sin C	Any triangle when 2 sides and included angle are known

Circle Theorems

Practice AQA GCSE questions while you read

Questions graded, hints, and explained.

Angle at centre = 2 × angle at circumference (same arc)

Angles in the same segment are equal

Angle in a semicircle = 90° (angle subtended by diameter)

Opposite angles in a cyclic quadrilateral sum to 180°

Tangent to a circle is perpendicular to the radius at the point of contact

Two tangents from an external point are equal in length

Alternate segment theorem: angle between tangent and chord = angle in alternate segment

Angle between radius and chord (perpendicular from centre bisects chord)

Algebra — Higher Topics

Topic	Method / rule
Algebraic fractions	Factorise numerator and denominator; cancel common factors; find LCM for addition/subtraction
Functions: f(x)	f(3) means substitute x = 3 into f(x)
Composite functions: fg(x)	Apply g first, then f: fg(x) = f(g(x))
Inverse function: f⁻¹(x)	Replace f(x) with y; rearrange for x; replace x with f⁻¹(x)
Iteration	Rearrange to xₙ₊₁ = f(xₙ); substitute repeatedly to converge on root
Nth term of quadratic sequence	Second difference ÷ 2 gives coefficient of n²; adjust for linear and constant terms
Graph transformations	y = f(x+a): left a; y = f(x)+ a: up a; y = f(ax): squeeze horizontal; y = af(x): stretch vertical

Practice GCSE Maths Higher questions

Grade 7–9 questions combine multiple topics. Practice multi-step problems to build the fluency needed for the harder marks.

Vectors

Vector addition: a + b — head to tail; result is the diagonal of the parallelogram

Scalar multiple: ka — same direction, scaled magnitude

AB⃗ = b − a (position vectors: from origin)

Midpoint M of AB: m = ½(a + b)

To prove collinearity: show vectors are parallel (scalar multiples) and share a common point

To prove a point divides a line in ratio m:n: express the vector and confirm the ratio

Probability — Higher Topics

Topic	Rule
Conditional probability	P(A\|B) = P(A∩B) / P(B)
Multiplication rule (dependent events)	P(A and B) = P(A) × P(B\|A)
Venn diagrams — with algebra	Set up expressions for each region; use total = 1 to solve for unknowns
Histograms	Frequency density = Frequency ÷ Class width; area of bar = frequency (not height)
Cumulative frequency	Plot upper class boundary vs cumulative frequency; median at n/2; IQR = UQ − LQ