What Command Words in Maths Questions Actually Require

Show that, prove, verify: three different obligations

These three get treated as one instruction and they are not. Each demands something different on the page, and each has a characteristic way of being answered wrongly by a student who knows the mathematics perfectly well.

Show that hands you the destination. The answer is printed in the question, so copying it out earns nothing - every mark is for the steps in between. Two rules follow. Your working has to arrive at the stated value exactly, digit for digit, rather than at something near it. And you must not start from the answer and work backwards, because that assumes the very thing you were asked to establish.

Prove asks for an argument covering every case. One example is not a proof however convincing it looks: checking that 3² - 1 is a multiple of 8 tells you nothing about odd numbers in general. Algebra with a general term is what proof looks like at this level - let the number be 2n + 1, then (2n + 1)² - 1 = 4n² + 4n = 4n(n + 1), and since n and n + 1 cannot both be odd, n(n + 1) is even, so the whole thing is a multiple of 8. The one place a single example is enough is a disproof, where one counterexample finishes the job.

Verify is the easy one and is regularly mistaken for the hard one. Verify that x = 3 is a solution asks you to substitute 3 and show that both sides come to the same number. It does not ask you to solve the equation, and solving it spends minutes the paper does not have.

Hence, and hence or otherwise

Hence means: use what you have just done. The previous part was not a warm-up, it was the tool, and a student who ignores it and starts a fresh method may reach the right answer while losing the marks - because what was being tested was whether the connection was seen.

Hence or otherwise is a different instruction with two words added. It tells you the previous part is the intended route and that another method is permitted. Take the hint anyway; the intended route is nearly always the shorter one, and on a timed paper that matters more than being right in an interesting way.

The practical consequence shapes how you should attack a multi-part question. Part (a) is usually a gift, and part (c) is usually unreachable without it. Which is also why a student stuck on part (a) should still attempt (b) and (c) with the most sensible value they can produce - the later marks are for the method applied, not for the number that was carried in.

Estimate, exact, write down, state, explain

This group tells you what form the answer has to take, and each one has an unambiguous meaning that students routinely improve on to their cost.

Estimate deserves a moment because it is the one people think they are being generous by ignoring. Asked to estimate 4.87 × 19.6 ÷ 0.51, the work intended is 5 × 20 ÷ 0.5 = 200. A calculator returns 187.16..., which is the right answer to a different question and scores nothing. The mark is for rounding to 1 significant figure and showing the rounded calculation.

• Estimate - round each number to 1 significant figure, then calculate. An exact answer is not an estimate.
• Give an exact answer, or leave your answer in terms of π, or in surd form - a decimal loses the mark. Write 12π, not 37.7; write 3√2, not 4.24.
• Write down - the answer can be read off a graph, a table or a previous part. Usually one mark, no working expected, and time spent calculating is time stolen from questions that need it.
• State - as with write down, but for a fact or a name rather than a value. No justification required.
• Explain, or give a reason - a sentence is required, and at GCSE the reason usually has to name the rule: alternate angles, angles in the same segment, congruent because two sides and the included angle are equal.

Draw, plot, sketch, construct

The four drawing instructions differ in how much accuracy is being demanded, and reading them correctly decides how long a question should take you.

The gap between sketch and draw is the widest. A sketch of y = x² - 4 needs a parabola crossing the x-axis at -2 and 2 and the y-axis at -4, with those three points labelled, and it takes about fifteen seconds. Drawing the same curve needs a table of values, plotted points and a smooth curve through them, and takes several minutes. Doing the second when the first was asked for is not rewarded, and doing the first when the second was asked for loses the accuracy marks.

• Plot - mark the given points accurately on the grid provided.
• Draw - produce an accurate drawing, to scale, using the grid or the measurements given.
• Sketch - a freehand shape showing the important features, with intercepts and turning points labelled. Accuracy is not the point; the shape and the labels are.
• Construct - use compasses and a straight edge, and leave the arcs on the page. Arcs rubbed out are marks rubbed out, because the arcs are the evidence that a construction was used.

The habit that fixes most of this

Underline the command word before reading the rest of the question. It costs a second and it changes what you are looking for while you read the rest, which is the whole point - a student who has registered the word estimate reads the numbers differently from one who has not.

Then, when marking your own past-paper work, check the command word against what you wrote before you check the answer. Estimate answered exactly, show that answered by quoting the printed value, sketch answered with a table of values - all of those are completely correct mathematics scoring zero, and all of them are invisible if you only compare final answers.

We mark past papers this way in our GCSE and IGCSE sessions, command word first and mathematics second, because they are two separate skills and getting better at one does nothing for the other. A student losing marks to command words has a problem that can be fixed in a fortnight, and it is worth finding out whether that is what is happening before starting on the topics.