Command words

The verb sets the rules. Maths papers are stricter about this than most: a ‘show that’ marks the steps rather than the answer, a ‘hence’ obliges you to use the previous part, and a ‘prove’ will not accept three examples however convincing they look.

Recall — just state it

State / Write down: Give the fact or the answer with no working and no justification. If a question says write down, it is telling you that no method is expected, so do not invent one. “Write down the period of y = tan x.” — π radians.
Define: Give the precise meaning, in the terms the specification uses. A worked example is not a definition.

Establish it — the proof verbs

Prove: Give a complete, general argument with every step justified. Checking cases is not a proof unless the cases are exhaustive, and a diagram is not an argument. “Prove that the sum of the first n odd numbers is n².” — induction or a direct algebraic argument, not three examples.
Show that: Reach a result you have already been given. The answer is not the mark; the steps are. Because the target is printed, every line has to be visible or there is nothing to credit.
Verify: Substitute the given value or values and confirm the statement holds. Unlike show that, you are not expected to derive anything. “Verify that x = 2 is a root.” — substitute and get zero.
Disprove: Produce a single counterexample and state plainly which claim it defeats. One well-chosen case is a complete answer.
Deduce / Hence: Use the result you have just established. Hence means the previous part is meant to do the work, so a fresh start from first principles risks no marks even when the answer is right. “Hence or otherwise” relaxes that: the intended route is easiest, but any sound method scores.

Work it out — numbers and algebra

Solve: Find every value satisfying the equation, over the interval given. Losing the second root, or the solutions outside the first quadrant, is the standard way to drop marks here.
Find / Calculate / Determine: Work out the value, showing enough method to be followed. Give exact form when the question asks for it, and otherwise round sensibly and say what you rounded to.
Express / Write in the form: Rearrange into exactly the shape specified, and state the constants asked for. Marks are for the named constants, so finish by writing them out. “Express 3sin x + 4cos x in the form Rsin(x + α).” — give R and α, not just the working.
Estimate: Give a value of the right size from sensible assumptions or from a graph, and say what you assumed.

Explain it — in words

Explain / Give a reason: Say why, not what. A restatement of the result in different words scores nothing; the mark is for the cause or the property that forces it.
Interpret: Say what the number means in the context of the question, in the units of that context. A gradient is a rate of something per something, and naming both is the answer. “Interpret the gradient.” — the cost rises by £4 per extra unit produced.
Comment on: Give a short judgement with a reason attached, usually about whether a model or a conclusion is reasonable.
Criticise / Evaluate: Weigh the limitations against the strengths and reach a supported conclusion. For a model, that usually means naming an assumption and saying what it ignores.

Draw it

Sketch: The right shape, with intercepts, asymptotes and turning points marked and the axes labelled. Accuracy of scale is not required; correctness of features is.
Plot: Mark the given points accurately on the axes provided, then draw the line or curve through them.
Draw: Produce an accurate diagram to scale, with instruments where the question implies them.

See also the key ideas for each topic and the revision checklist.