The Instruction Behind the Sum: Command Words in GCSE Maths

Every GCSE maths question carries two messages. The first is mathematical: the numbers, the diagram, the relationship to be untangled. The second is procedural, and it is usually delivered in a single word or short phrase — show that, hence, otherwise, justify, explain. Students who attend only to the first message can produce competent algebra and still lose marks, because they have answered a slightly different question from the one set.

Command words are not decoration. They tell a student what kind of response the mark scheme will reward: a derivation, a linked calculation, an independent method, a reason, or a written account of why a result holds. Teaching students to notice them is one of the most efficient forms of exam preparation available, because it changes how a response is planned before any working begins.

The instruction before the mathematics

A useful starting point in the classroom is to separate reading from solving. Ask students to cover the numbers and read only the instruction. What is the question asking them to do with the mathematics? Is the answer already supplied, as in a show that question? Is the method constrained by an earlier part, as with hence? Or is a written justification required, as with explain?

This is not a gimmick. It reflects how mark schemes are constructed. Examiners award method marks for particular stages, communication marks for clarity, and reasoning marks for statements that connect a result to a rule. The command word is the most reliable signal of which of these the question is designed to elicit.

Show that: when the destination is fixed

A show that question supplies the answer and asks the student to demonstrate how it follows. The marks lie in the intermediate steps, not in the final line. Students sometimes treat the given result as something to be assumed, writing a chain of working that quietly relies on the answer it is meant to establish. That is circular, and it rarely earns full credit.

A more secure approach is to treat the given answer as a checkpoint. Work forward from the information in the question, keeping each step visible, and stop when the stated result appears. Where a step involves a non-obvious simplification, say so. The test is whether a reader could follow the working without already knowing the destination.

Teachers can model this by writing a deliberately poor show that response and asking the class to identify where the reasoning assumes what it should prove. The exercise makes the standard explicit without needing a lengthy lecture on proof.

Hence and otherwise: sequence and independence

Hence is a linking word. It requires the student to use the result of the previous part. When a question says hence, find..., a response that starts again from first principles may reach the right answer but can miss the method marks attached to the intended route. The word is asking for a chain, not a fresh start.

Hence or otherwise offers a choice: use the previous result, or use another valid method. Otherwise alone means the previous result should not be used; an independent method is required. Students often misread these distinctions under pressure, particularly when the earlier part was difficult. A sensible habit is to underline the command word and write a brief note beside it — use part (a), different method — before beginning the calculation.

Justify and explain: the standard of evidence

Justify and explain both require written reasoning, but they are not interchangeable with state or write down. A justification should give a mathematical reason: a cited rule, a definition, a calculation that supports the claim. An explanation should communicate why a result is true, often in words, and may refer to a property such as symmetry, proportionality or the behaviour of a graph.

A common weakness is the one-word answer. Asked to explain why two triangles are congruent, a student may write because they are the same. That is a restatement, not a reason. The mark scheme is looking for the specific condition — for example, that the sides correspond. The same principle applies to justification: the response must connect the claim to evidence the reader can check.

Planning a response in three moves

A short routine can be taught across the department and used at home without special resources.

First, circle the command word. Second, say aloud what it demands: derive the given result, use the previous answer, use a different method, give a reason. Third, check the final line against the instruction before moving on. Did the response actually do what was asked?

This takes perhaps twenty seconds. Its value lies in interrupting the habit of starting to calculate before the question has been understood. Students who routinely answer the question they expected rather than the one printed on the page tend to find that the check catches the error while there is still time to correct it.

When the wrong question has been answered

Errors of this kind are diagnostically useful. If a student calculates when asked to estimate, solves when asked to factorise, or gives an answer without justification when asked to explain, the underlying mathematics may be sound. The issue is procedural reading. Feedback should name that specifically, rather than treating the whole response as weak.

Mark schemes can help here. Asking students to compare their response with the published scheme for a show that or explain question makes the standard concrete. They see where the marks sit, and they begin to read command words as instructions rather than as part of the scenery.

The mathematics in a GCSE paper is demanding enough. The small words around it should not be the reason marks are lost.