Problem-solving strategies for STEP and MAT

STEP and MAT questions are long because you have to find the way in. These eight habits are the ones our problems use most. Each links to a free problem where you can see it work.

1. Try small cases

Put n = 1, 2, 3, or choose easy numbers, to see what is going on before you try to prove anything. Write the cases down: the pattern you spot is often the key step.

See it in: A squaring recurrence

2. Use the earlier parts

STEP questions are built in layers. If part (iii) looks hard, ask how (i) and (ii) could help. 'Hence' means you must use the previous result.

See it in: A factorisation that telescopes · A cubic in tan

3. Make a substitution

A new variable can turn a messy expression into a familiar one: u = sqrt(x - 3), x = 2cos θ, u = t + 1/t.

See it in: A cubic with cosine roots

4. Sketch it

Counting solutions, finding ranges and checking signs all become easier with a quick sketch showing turning points and asymptotes.

See it in: How many tangents from a point?

5. Look for symmetry

Swapping variables, reflecting an interval, or noticing that every outcome is equally likely can replace pages of algebra.

See it in: Symmetry on a quarter-turn · Waiting for the first red

6. Bound and squeeze

When you cannot compute something exactly, trap it between two things you can compute. Limits and integer parts follow.

See it in: A reduction formula and a limit

7. Solve the general case first

Work with letters before numbers. In mechanics especially, the general solution shows the pattern and catches arithmetic slips.

See it in: When does a cubic have three roots?

8. Check your answer

Substitute back, test a special case, or check units and signs. In a 3-hour paper, one minute of checking can save a whole question.

See it in: A functional equation · A parabolic density

Put them to work

STEP topic guides · MAT-style practice · Competition maths problems

STEP is run by OCR. A Level Math Revision is independent: it is not affiliated with or endorsed by OCR, the University of Cambridge or any university. Every problem here is our own, written in the style of STEP; for real papers use OCR's official past papers.