Command Words in Additional Mathematics | Solve, Show, Prove, Hence, Find, Sketch and Explain

The first mathematical decision often happens before the first equation.

It happens when the student reads the verb.

Find asks for a quantity. Solve asks for the values that satisfy a mathematical condition. Show asks for a visible route to a stated result. Prove asks for a general argument. Hence points backward to a result that is meant to become useful. Sketch asks for structure, not decorative drawing. Explain asks the student to make a relationship explicit.

The command word does not determine the entire marking requirement by itself. The full question, its mathematical context and the instructions on the actual examination paper control what must be shown. But learning to read these verbs as different jobs can prevent a large class of avoidable errors.

This guide is aligned with the 2026 O-Level Additional Mathematics 4049 and 2027 SEC G3 Additional Mathematics K341 assessment architecture. Both syllabuses explicitly assess standard techniques, problem solving in varied contexts, and mathematical reasoning and communication.

Official references: 2026 O-Level Additional Mathematics 4049 syllabus · 2027 SEC G3 Additional Mathematics K341 syllabus.

Wait, What? The Same Mathematics Can Require a Different Answer

Consider the quadratic x²−5x+6.

• Find the roots. The answer is x=2,3, supported by suitable working.
• Show that the roots are integers. The student must exhibit a route such as factorisation and make the integer conclusion visible.
• Prove that the graph crosses the x-axis at two distinct points. The response must connect the two distinct real roots to graph intersections.
• Hence find the distance between the x-intercepts. The earlier roots are now intended inputs, so the distance is 3−2=1.

The algebra overlaps. The communication job changes.

The Core Reading Rule

Before calculating, translate the command word into an evidence question:

What must be visible in my answer for this verb to be satisfied?

Then combine that with the mathematical object and target.

A useful three-part read is:

COMMAND → OBJECT → TARGET.

For example:

“Hence find the maximum value of y.”

Command: use earlier work and produce a value. Object: the function or transformed expression from the preceding part. Target: the maximum value, not merely the x-coordinate where it occurs.

FIND: Produce the Requested Mathematical Object

Find is broad. It can ask for a number, expression, coordinate, parameter, angle, equation, area or set of values.

The important question is not “How much working does ‘find’ require?” There is no universal answer from the verb alone. Ask what mathematical chain is necessary to support the requested result, remembering that the official syllabuses warn that omission of essential working can lose marks.

Original Example

Find the coordinates of the turning point of y=x²−8x+11.

Complete the square:

y=(x−4)²−5.

Therefore the turning point is (4,−5).

If the student gives only x=4, the target has been read incompletely. The question asks for coordinates, so both components matter.

SOLVE: Find All Values That Satisfy the Stated Conditions

Solve usually points to an equation, inequality or system. The task is not only to produce a candidate value but to find the complete solution set within the stated domain or interval.

Original Example: Logarithm

Solve ln(x−1)+ln(x−3)=ln8.

For real logarithms, x>3. Combining gives:

(x−1)(x−3)=8.

So x²−4x−5=0, giving x=5 or −1. The domain removes −1. Therefore x=5.

The word solve includes the conditions of the original problem. Algebraic candidates are not automatically final solutions.

Original Example: Trigonometry

Solve 2sin x=1 for 0°≤x≤360°.

The principal value is 30°, but the interval contains another solution in the second quadrant. The complete answer is x=30°,150°.

Stopping after the inverse-calculator output solves a different, smaller problem.

SHOW THAT: Build a Route to a Result You Have Been Told

A “show that” question supplies the destination. The student’s job is to construct a legitimate path from the given information to that destination.

The supplied answer is not permission to use the result as an unexplained premise.

Original Example

Given y=x²+2x+2, show that the tangent at x=1 has equation y=4x.

Differentiate:

dy/dx=2x+2.

At x=1, the gradient is 4. The point on the curve is (1,5), so the tangent is:

y−5=4(x−1)

which simplifies to y=4x+1.

That means the proposed result y=4x would actually be false for this curve. This is exactly why “show that” should not switch off verification. If your mathematics contradicts the target, check both your work and the stated problem rather than forcing the supplied form.

Now change the curve to y=x²+2x+1. At x=1, the point is (1,4) and the gradient is 4, giving y−4=4(x−1), hence y=4x. The route now supports the target.

A strong student uses the stated result as a destination and consistency check, not as a licence to reverse-engineer unsupported algebra.

PROVE: Establish a General Statement From Accepted Facts

A proof is not a successful numerical example. It is an argument showing why the conclusion follows for every case covered by the stated conditions.

Original Geometry Example

In triangle ABC, D and E are the midpoints of AB and AC. Prove that DE is parallel to BC.

Because D and E are midpoints:

AD/AB = AE/AC = 1/2.

Also ∠DAE=∠BAC. Therefore triangles ADE and ABC are similar by SAS similarity. Corresponding angles are equal, so ∠ADE=∠ABC. Hence DE∥BC.

The drawing may make the parallel lines look obvious, but appearance is not proof. The argument transfers the conclusion through established relationships.

HENCE: Reuse What the Question Has Already Built

Hence is a routing signal. It tells you that the previous result is intended to make the next step shorter, clearer or possible.

This is a major examination-reading skill because students sometimes restart from the beginning and miss the dependency the question has deliberately constructed.

Original Example

Part (a): Show that x³−4x²−x+4=(x−4)(x−1)(x+1).

Part (b): Hence solve x³−4x²−x+4=0.

Use the factorisation directly:

(x−4)(x−1)(x+1)=0

so x=4,1,−1.

The earlier part has converted a cubic problem into a zero-product problem. “Hence” tells you to collect that benefit.

SKETCH: Show Structure Without Pretending It Is an Exact Plot

A sketch should communicate the defining features of a graph clearly enough that the mathematical behaviour is visible.

Depending on the function and question, useful features may include:

• axes and relevant labels;
• intercepts;
• turning points;
• asymptotes;
• period or key trigonometric points;
• end behaviour;
• domain restrictions;
• relative position of branches.

A sketch does not require the graphical precision of a plotted data set, but it must not contradict the function.

Original Example: Reciprocal Function

Sketch y=1/(x−2)+3.

The vertical asymptote is x=2. The horizontal asymptote is y=3. At x=0, y=2.5. The two branches approach the asymptotes without crossing the vertical asymptote.

A beautiful curve without labelled asymptotes communicates less mathematics than a simple controlled sketch that shows the correct structure.

EXPLAIN: Name the Relationship, Not Merely the Result

Explain often asks for the bridge between evidence and conclusion.

Suppose a model predicts a negative length. Writing “negative answer” is observation. Writing “the negative solution is rejected because length is nonnegative in this physical context” is explanation.

Suppose a transformed graph is approximately linear. Writing “the graph is straight” is observation. Explaining that the linearity supports the proposed transformed relationship over the observed range makes the inference explicit.

Explanation is not necessarily long. One precise sentence can carry the whole reasoning step.

STATE: Give the Requested Fact or Result Directly

State usually signals that a concise response is appropriate. But the full question still matters.

If asked to “state the period of y=sin 3x”, the response 2π/3 may be enough if no working is requested. If the student is uncertain, writing the short relation 3T=2π can protect the reasoning during practice.

Do not turn every “state” into a page of derivation. Examination control includes knowing when to stop.

WRITE DOWN: Retrieve or Read a Result With Minimal Processing

When a question asks you to write down a value, equation or result, the information may be directly available from a graph, a previous part or a familiar structure.

The danger is overworking. If a graph visibly gives an x-intercept, do not automatically rebuild the entire equation unless the question demands it.

Fast recognition is part of mathematical fluency.

EVALUATE: Produce the Value of an Expression

Evaluate asks for a numerical or exact value of an expression at the stated inputs.

For example, if f(x)=2x²−3x+1, then evaluating f(4) means:

f(4)=2(16)−12+1=21.

It does not mean solving f(x)=4. The direction of substitution matters.

SIMPLIFY: Change Form Without Changing Value

Simplify asks for an equivalent expression in a more useful or conventional form.

Example:

(x²−9)/(x²−3x) = [(x−3)(x+3)]/[x(x−3)] = (x+3)/x, with the original restriction x≠0,3.

Cancelling a factor changes the visible form. It does not erase the domain restriction inherited from the original denominator.

EXPRESS: Rewrite in a Requested Form

Express often specifies the destination form.

Express x²−6x+11 in the form (x−a)²+b.

Completing the square gives:

x²−6x+11=(x−3)²+2.

Stopping at a correct expanded expression does not satisfy the command because the required representation is part of the target.

DETERMINE: Use the Available Evidence to Establish an Unknown

Determine often behaves like “find”, but the wording can emphasise inference from supplied conditions, graphs, models or relationships.

For example, if a line is tangent to a curve and contains an unknown parameter, determining the parameter means using the tangency condition, not merely substituting visible coordinates.

VERIFY: Check That a Claim or Result Survives a Test

Verification can use substitution, differentiation, graph behaviour, units, restrictions, inverse operations or boundary checks.

If x=ln3 is proposed as a solution to eˣ=3, substitution gives e^ln3=3, confirming the equation.

Verification is especially valuable when a calculator or long transformation chain is involved.

Command Words and AO1, AO2, AO3

There is no one-to-one mapping, but the verbs often reveal which assessment demand is becoming important.

Read Additional Mathematics Assessment Objectives | AO1, AO2 and AO3 for the full distinction.

Command Word Trap 1: Answering a Nearby Question

Question: Find the maximum value of y=−2(x−3)²+7.

Student answer: x=3.

The mathematics identifies where the maximum occurs, but the question asks for the value of y. The correct answer is 7.

Reading failure can survive correct algebra.

Command Word Trap 2: “Show” Becomes Circular

Question: Show that (x−2) is a factor of P(x).

Weak route: begin by writing P(x)=(x−2)Q(x) with no justification.

Stronger route: evaluate P(2). If P(2)=0, the factor theorem establishes that (x−2) is a factor.

The target is reached from a recognised theorem rather than assumed.

Command Word Trap 3: “Hence” Is Ignored

Part (a) gives a completed-square form. Part (b) asks for the minimum value “hence”. A student expands back to standard form, differentiates, solves and substitutes.

The route may be mathematically valid, but it discards information the question has already paid to construct. If the expression is 2(x−4)²−5, the minimum is visible immediately as −5.

Use the previous result unless there is a mathematical reason not to.

Command Word Trap 4: “Sketch” Becomes “Plot”

A student calculates twenty coordinates for a simple quadratic and spends ten minutes drawing them carefully. Another identifies roots, axis, turning point and opening direction in two minutes.

The second student is doing more mathematics with less labour because the command asks for structure.

Command Word Trap 5: “Explain” Becomes “Calculate Again”

If a question asks why one solution is rejected, repeating the quadratic formula does not answer the communication job.

The explanation may be as short as:

“x=−2 is rejected because x represents a length.”

The words connect the mathematical result to the model.

A Command-Word Annotation Routine

During revision, mark the paper before solving:

1. Box the command word.
2. Underline the target quantity or statement.
3. Circle conditions, intervals, units and accuracy instructions.
4. Draw an arrow to any previous result signalled by “hence”.
5. Write one phrase describing the required evidence.

Example:

“Hence prove that the curve has no stationary points for x>0.”

Box: hence prove. Target: no stationary points. Condition: x>0. Previous result: likely derivative form or inequality. Evidence plan: show derivative cannot equal zero in the stated domain.

Original Mixed Command Set

Use the function f(x)=x²−4x+1.

1. Express f(x) in completed-square form.
2. Hence state the minimum value.
3. Find the x-coordinate at which the minimum occurs.
4. Solve f(x)=0.
5. Sketch y=f(x), showing the roots and turning point.
6. Explain how the sign of the discriminant agrees with your sketch.

Answers:

1. f(x)=(x−2)²−3.

2. Minimum value −3.

3. x=2.

4. x=2±√3.

5. Upward-opening parabola with turning point (2,−3) and roots 2±√3.

6. The discriminant is positive, so there are two distinct real roots, matching the two x-axis intersections on the sketch.

One function generated six different command jobs.

The Seven-Question Final Check

• What verb am I answering?
• What exact object must I produce?
• What conditions limit the answer?
• Does “hence” require an earlier result?
• Does “show” or “prove” require visible justification?
• Does “sketch” require key features rather than many coordinates?
• Have I stopped once the requested job is complete?

For Parents and Tutors

When a student repeatedly says, “I knew how to do it after I saw the solution,” inspect the command-reading stage. The learner may not have failed the mathematics itself. They may have answered the wrong job, missed a condition or failed to recognise that the previous part was meant to unlock the next one.

Ask the student to read a question without solving it and say:

“What does the examiner want me to leave on the page?”

If that answer is unclear, more calculation may not be the first intervention.

Evidence Boundary

This page is an eduKate reader’s guide to common mathematical command verbs. It is not presented as an official SEAB glossary assigning a fixed marking rule to each word. The full wording and instructions of the actual question control the required response. The official syllabuses support the broader demands of standard technique, problem solving, justification, explanation, proof and mathematical communication.

Continue the Examination Interface Layer

• Formula Sheet | What Is Given, What Is Not
• Assessment Objectives | AO1, AO2 and AO3
• Essential Working in Additional Mathematics
• Return to the Additional Mathematics Hub

The Quiet Return

Read the verb. Name the object. Identify the evidence. Then do only the mathematics the question actually asks for.