Keywords (AQA GCSE Further Maths): Revision Notes

GCSE Mathematics Keywords Guide

Understanding mathematical instruction words is crucial for exam success. These command words tell you exactly what type of response is expected, and knowing their precise meanings can make the difference between gaining full marks and losing valuable points.

Understanding command words

Mathematical exams use specific instruction words that have precise meanings. When you see these words in a question, they're giving you clear guidance about the type of answer required and the level of working you need to show. Let's explore what each of these key terms means and how to respond appropriately.

Calculate	Work out the numerical value (often used after making a substitution)
Draw (a graph)	Draw axes on graph paper, plot points accurately and join with a straight line or smooth curve
Evaluate	Give a numerical value for your answer
Expand	Remove brackets
Expand and simplify	Remove brackets and collect like terms
Explain	Give reasons, either in words, or using mathematical symbols, or both
Expression	One or more terms, for example one side of a formula
Factories	Write as a product
Give your answer in its simplest form	Cancel answers given as ratios or fractions or collect like terms
Hence	Use earlier work to deduce the result
Hence or otherwise	Using previous work to deduce the result is an option
Plot	Mark points (usually on graph paper) and join with a straight line or curve
Prove	Show all relevant steps (include explanations of facts used in geometrical proofs)
Show that	Show all relevant steps to reach a given result
Sketch (a graph)	Do not use graph paper. Draw axes and show the correct shape in each quadrant. Label appropriate points (e.g. intersection with axes, stationary points)
Verify	"Check out a statement or result that you have been given
Work out the exact value of	Give the answer as an integer, fraction, recurring decimal, in terms of n, etc. or a surd
Write down	The answer should be obvious (no working is necessary)

Calculation and numerical work

Basic computational tasks

When you encounter Calculate, you need to determine a numerical result, often after making substitutions into formulas or expressions. This typically involves showing your working steps clearly and arriving at a specific number.

Evaluate requires you to find a numerical value for your final answer. This is similar to calculate but emphasises that you must give a specific numerical result rather than leaving your answer in algebraic form.

Work out the exact value of demands that you provide a precise answer, which might be in the form of a fraction, surd, or exact decimal rather than a rounded approximation. This instruction often appears when dealing with irrational numbers or when precision is particularly important.

Simple answers

Write down indicates that the answer should be immediately obvious from your previous work or from given information. No complex working is necessary - the solution should be straightforward to identify and state.

Algebraic manipulation

Expanding and simplifying

Expand means you need to multiply out brackets to remove them from an expression. For example, expanding $(x + 3)(x - 2)$ would give you $x^2 + x - 6$.

Expand and simplify goes one step further - after removing brackets, you must collect like terms together to create the simplest possible form of the expression.

Give your answer in its simplest form requires you to ensure your final answer cannot be simplified further. This might involve cancelling fractions, collecting like terms, or expressing ratios in their lowest terms.

Working with expressions

An Expression refers to one or more mathematical terms - essentially one side of a formula or equation. When asked to write an expression, you're creating a mathematical phrase that represents a particular situation.

Factorise is the reverse of expanding - you need to write an expression as a product of its factors, typically by taking out common factors or recognising special patterns like difference of squares.

Drawing and graphing

Creating visual representations

Draw (a graph) requires you to use graph paper, plot points with precision, and connect them with either straight lines or smooth curves as appropriate. Accuracy is important, and you should include proper labelling of axes.

Plot specifically means marking points accurately, usually on graph paper, and joining them with straight lines or curves. The emphasis is on precise positioning of coordinates.

Sketch (a graph) has different requirements - you don't need graph paper, but you must show the correct overall shape in each quadrant. Key features like intercepts with axes and stationary points should be clearly labelled and positioned correctly.

Reasoning and proof

Logical demonstration

Prove demands that you demonstrate all relevant steps, including explanations of mathematical facts used, particularly in geometrical proofs. Every statement must be justified logically.

Show that requires you to demonstrate all relevant steps to reach a specific given result. You're working towards a predetermined conclusion and must show the complete logical path.

Verify means you need to check or confirm a statement or result that has been provided to you. This involves substituting values or working backwards to confirm the given information is correct.

Using previous work

Hence tells you to use your earlier work to determine the next result. This creates a logical connection between different parts of a question.

Hence or otherwise gives you the option to use previous work to find the result, but you can choose an alternative method if you prefer.

Communication and explanation

Explain requires you to provide reasons for your answer, either in words, through mathematical symbols, or both. You need to make your reasoning clear and justify your conclusions.