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IGCSE Maths Command Words Explained

Command words tell you exactly what the examiner wants. Understanding the difference between "calculate" and "estimate", or "show that" and "prove", can be the difference between full marks and zero.

Calculate Show that Explain Determine Construct Sketch State Estimate Simplify Solve Factorise Prove Evaluate Describe Write down

Calculate

What it means

Work out a numerical answer, showing your working clearly. You must use the information given in the question.

What the examiner expects

A clear sequence of mathematical steps leading to a correct numerical answer. Working must be shown — even if you use a calculator.

Common mistakes

Writing only the final answer without any working
Forgetting units (e.g., cm, kg, seconds)
Rounding too early in multi-step calculations

Example question

Calculate the area of a circle with radius 7 cm. Give your answer correct to 3 significant figures.

Model answer

Area = pi x r squared
= pi x 7 squared
= pi x 49
= 153.938...
= 154 cm squared (3 s.f.)

Method marks (M1) are awarded for correct working even if the final answer is wrong. Always show your steps.

Show that

What it means

Prove a given result is correct by showing every step of your working. The answer is already given — you must demonstrate how to reach it.

What the examiner expects

ALL intermediate steps clearly shown. You cannot skip any steps because the examiner needs to see the complete logical chain from start to finish.

Common mistakes

Skipping steps because the answer is already known
Working backwards from the given answer
Not showing enough intermediate calculation steps

Example question

Show that the equation 2x squared + 5x - 3 = 0 can be written as (2x - 1)(x + 3) = 0.

Model answer

Expand (2x - 1)(x + 3):
= 2x(x + 3) - 1(x + 3)
= 2x squared + 6x - x - 3
= 2x squared + 5x - 3
Therefore (2x - 1)(x + 3) = 0 is equivalent to 2x squared + 5x - 3 = 0.

"Show that" questions require EVERY step. If you miss even one line of working, you lose marks. These are often worth 3-4 marks specifically for the working.

Explain

What it means

Give clear mathematical reasons for your answer. Use correct mathematical vocabulary and terminology.

What the examiner expects

A written explanation using proper mathematical language, not just a calculation. Link your reasoning to mathematical facts or rules.

Common mistakes

Giving a vague answer like 'because it's bigger'
Using informal language instead of mathematical terms
Not connecting the explanation to a mathematical rule or property

Example question

Explain why the sum of the angles in a quadrilateral is 360 degrees.

Model answer

A quadrilateral can be divided into two triangles by drawing one diagonal. The sum of angles in each triangle is 180 degrees. Therefore the sum of angles in a quadrilateral is 2 x 180 = 360 degrees.

Examiners want mathematical vocabulary: 'because alternate angles are equal' not 'because they look the same'. One clear reason with correct terminology is better than a long vague answer.

Determine

What it means

Work out the answer using the information provided. Similar to 'calculate' but may involve reasoning or interpreting data as well as computation.

What the examiner expects

A logical approach using given information to reach a justified answer. Show how you extracted and used the data.

Common mistakes

Not using all the information given in the question
Guessing instead of working through the data systematically
Confusing 'determine' with 'estimate' — determine requires exact working

Example question

The nth term of a sequence is 3n + 2. Determine the 20th term.

Model answer

Substitute n = 20:
3(20) + 2 = 60 + 2 = 62
The 20th term is 62.

Show your substitution or reasoning clearly. The examiner wants to see that you understood how to use the given information.

Construct

What it means

Draw accurately using mathematical instruments — ruler, compass, and protractor. Construction lines must be visible.

What the examiner expects

Accurate construction with compass arcs clearly visible. Do NOT erase construction lines — they are part of the marks.

Common mistakes

Erasing compass arcs (they must be visible for marks)
Drawing freehand instead of using instruments
Not being accurate to within 1mm or 1 degree

Example question

Construct an equilateral triangle with sides of 6 cm.

Model answer

1. Draw a base line of exactly 6 cm with a ruler.
2. Set compass to 6 cm. Place point on one end of the base, draw an arc above.
3. Place compass on the other end, draw another arc to intersect the first.
4. Join the intersection point to both ends of the base.
All construction arcs must be visible.

Leave ALL construction arcs visible. Use a sharp pencil. Marks are lost if lines are more than 1mm out.

Sketch

What it means

Draw a rough diagram showing the key features. Does not need to be perfectly accurate, but must show the correct shape and important points.

What the examiner expects

A clear diagram with key features labelled: intercepts, turning points, asymptotes, or other important values. The general shape must be correct.

Common mistakes

Not labelling key coordinates or features
Getting the general shape wrong (e.g., parabola opening the wrong way)
Spending too long trying to make it perfectly accurate

Example question

Sketch the graph of y = x squared - 4.

Model answer

Draw a U-shaped parabola (opening upward).
Mark the vertex at (0, -4).
Mark the x-intercepts at (-2, 0) and (2, 0).
Label the y-intercept at (0, -4).
Label the axis of symmetry x = 0.

A sketch doesn't need to be exact but MUST show the correct shape and labelled key features. Spend 1-2 minutes, not 10.

State

What it means

Give a direct answer with no working needed. Just write down the answer clearly.

What the examiner expects

A clear, concise answer. No working is required — just the correct value, word, or expression.

Common mistakes

Overthinking it — just write the answer
Wasting time showing unnecessary working
Being imprecise when precision is needed (e.g., not giving exact coordinates)

Example question

State the gradient of the line y = 3x + 7.

Model answer

Gradient = 3

'State' usually means 1 mark. Write a clear answer and move on. Don't spend time on working — there are no method marks here.

Estimate

What it means

Round the numbers to simpler values first, then calculate. Do NOT use exact values — the examiner wants to see sensible rounding.

What the examiner expects

Rounded values shown clearly before the calculation. The rounding must be sensible (usually 1 significant figure).

Common mistakes

Using exact values instead of rounded ones
Not showing what you rounded each number to
Rounding the final answer instead of the inputs

Example question

Estimate the value of (29.7 x 4.03) / 0.489.

Model answer

Round each number:
29.7 is approximately 30
4.03 is approximately 4
0.489 is approximately 0.5

Estimate = (30 x 4) / 0.5
= 120 / 0.5
= 240

You MUST show what you rounded to. If you calculate with exact numbers, you get 0 marks even if the answer is close.

Simplify

What it means

Reduce an expression or fraction to its simplest form. Collect like terms, cancel common factors, or apply index laws.

What the examiner expects

The expression written in its most reduced form. Every possible simplification must be applied.

Common mistakes

Not fully simplifying (stopping too early)
Making sign errors when collecting like terms
Forgetting to simplify coefficients as well as variables

Example question

Simplify 6x squared y / 2xy.

Model answer

6x squared y / 2xy
= (6/2) x (x squared / x) x (y / y)
= 3 x x x 1
= 3x

If the answer can be simplified further, you lose marks. Check: are all common factors cancelled? Are all like terms collected?

Solve

What it means

Find the value(s) of the unknown variable(s). Show your method clearly.

What the examiner expects

Clear algebraic working leading to the value(s) of the unknown. For equations, both sides must balance at every step.

Common mistakes

Not showing the algebraic steps (trial and error alone may not earn method marks)
Forgetting to find all solutions (e.g., quadratics have two roots)
Not checking the answer by substituting back

Example question

Solve 3x + 7 = 22.

Model answer

3x + 7 = 22
3x = 22 - 7
3x = 15
x = 15 / 3
x = 5

Show every algebraic step. Even if you can do it in your head, method marks require visible working.

Factorise

What it means

Write an expression as a product of its factors. This is the reverse of expanding brackets.

What the examiner expects

The expression written as a product of brackets or terms. It must be fully factorised — check for common factors first, then look for quadratic patterns.

Common mistakes

Not taking out the highest common factor first
Missing the difference of two squares pattern
Leaving the answer partially factorised

Example question

Factorise fully 6x squared - 24.

Model answer

First take out the common factor of 6:
6(x squared - 4)
Recognise the difference of two squares:
6(x + 2)(x - 2)

'Fully factorise' means every factor must be extracted. If the question says 'factorise' (without 'fully'), one level of factorisation is usually enough.

Prove

What it means

Demonstrate mathematically that a statement is always true. Unlike 'show that', a proof must work for all cases, not just one example.

What the examiner expects

A logical chain of mathematical reasoning that demonstrates the statement is universally true. Numerical examples alone are NOT proofs.

Common mistakes

Using a specific numerical example instead of algebraic proof
Not starting from a known fact or definition
Missing logical steps in the argument

Example question

Prove that the sum of two consecutive odd numbers is always even.

Model answer

Let the first odd number be 2n + 1, where n is an integer.
The next consecutive odd number is 2n + 3.
Sum = (2n + 1) + (2n + 3)
= 4n + 4
= 4(n + 1)
Since 4(n + 1) is divisible by 2 for all integers n, the sum is always even.

Use algebra, not examples. Starting with 'Let n be an integer...' is usually the right approach. One numerical example is NOT a proof.

Evaluate

What it means

Substitute given values into an expression or formula and calculate the result.

What the examiner expects

Clear substitution shown before calculation. The examiner wants to see each value placed into the formula.

Common mistakes

Substituting values into the wrong places
Not showing the substitution step (jumping straight to the answer)
Errors with negative numbers when squaring or multiplying

Example question

Evaluate 3a squared - 2b when a = 4 and b = -3.

Model answer

Substitute a = 4 and b = -3:
3(4) squared - 2(-3)
= 3(16) - (-6)
= 48 + 6
= 54

Always show the substitution line — it earns a method mark. Be extra careful with negative numbers and order of operations.

Describe

What it means

Give the key features or properties. Often used for transformations, where you must give the type AND all details.

What the examiner expects

A complete description with ALL required components. For transformations: type, and all defining parameters (e.g., centre + scale factor for enlargement).

Common mistakes

Describing a transformation without giving the centre of rotation
Forgetting to give the scale factor for an enlargement
Not specifying direction and distance for a translation (use column vectors)

Example question

Describe fully the single transformation that maps triangle A onto triangle B.

Model answer

Rotation, 90 degrees clockwise, centre (0, 0).

(All three elements are needed: type, angle + direction, centre.)

'Describe fully' for transformations needs EVERY detail. Rotation needs: type + angle + direction + centre. Enlargement needs: type + scale factor + centre. Translation needs: type + column vector.

Write down

What it means

Give the answer directly. No working is needed or expected. The answer should be clear and obvious from the information given.

What the examiner expects

Just the answer — nothing else is needed. This command word signals that the answer can be found by inspection or simple reading.

Common mistakes

Spending too much time on what should be a quick answer
Showing working when none is needed (wasting time, not marks)
Misreading the data or graph the answer comes from

Example question

From the graph, write down the coordinates of the turning point.

Model answer

(-2, 5)

'Write down' is almost always worth 1 mark. Read the question carefully, give a clear answer, and move on quickly.

Practice by Command Word

Now you know what each command word means, practice with real IGCSE questions filtered by command word.

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1. Draw a base line of exactly 6 cm with a ruler. 2. Set compass to 6 cm. Place point on one end of the base, draw an arc above. 3. Place compass on the other end, draw another arc to intersect the first. 4. Join the intersection point to both ends of the base. All construction arcs must be visible.

Let the first odd number be 2n + 1, where n is an integer. The next consecutive odd number is 2n + 3. Sum = (2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1) Since 4(n + 1) is divisible by 2 for all integers n, the sum is always even.