Functions and graphs
Modulus and graph reflections
Keep track of which part of an expression changes sign, then use a sketch to organise the cases.
Open the guide: Modulus and graph reflectionsBuild the skill behind the answer
Maths practice by topic.
Work on the step that keeps costing marks: a sign, an unfamiliar identity, a missing case or a diagram that never became an equation.
Start with a question you could not finish
Identify the mathematical decision you missed. Work through the relevant example below, try the practice question before revealing its solution, then return to the original paper. A correct answer you guessed deserves a review too.
Algebra
Factorisation and useful identities
Look for structure before expanding. An identity can reveal a cancellation or turn a long calculation into a short argument.
Open the guide: Factorisation and useful identitiesSequences
Sequences, recurrence and sums
Separate the term number from the value of the term. Write the first few terms before choosing a formula.
Open the guide: Sequences, recurrence and sumsCalculus
Calculus: sketch before calculating
A graph can reveal a cancellation before you integrate. Keep signed area, total area and differentiation rules distinct.
Open the guide: Calculus: sketch before calculatingGeometry
Turn a diagram into equations
Label the quantities you know and translate each condition separately. A useful diagram helps you choose an equation.
Open the guide: Turn a diagram into equationsCounting
Counting without double-counting
Define exactly what is being counted. Check overlap and whether order matters before reaching for a formula.
Open the guide: Counting without double-countingReasoning
Logic and proof
Separate an implication from its converse, test counterexamples and distinguish necessary from sufficient conditions.
Open the logic guidePut the skill back into a paper
Use these guides for focused practice, then return to unfamiliar problems. Historical MAT questions are supplementary practice; the MAT is no longer held from 2026. Use current official specifications when preparing for TMUA, ESAT or STEP.