I have used the official IB Mathematics AA and AI command-term glossaries as the basis for the meanings. Because IB past-paper questions are copyrighted, I have used original IB-style questions rather than reproducing past-paper questions verbatim. This gives you an article you can publish commercially without relying on copied exam content. (International Baccalaureate®)

One of the easiest ways for an IB Mathematics student to lose marks is to solve the mathematics correctly but not answer the question that was actually asked.

In IB Maths: Applications and Interpretation (AI) and Analysis and Approaches (AA), command words tell students exactly what type of response is required. A question asking you to state something is very different from one asking you to show, justify, or prove it.

The good news is that Mathematics AI and AA use the same core glossary of command terms. The IB expects students to become familiar with these terms before sitting their examinations. (International Baccalaureate®)

Understanding them can therefore improve exam performance without learning any additional mathematics.

The most important distinction: what kind of answer is required?

Consider these four questions:

State the value of $x$ .
Find the value of $x$ .
Show that $x = 4$ .
Justify why $x = 4$ .

They might involve exactly the same mathematics, but they do not ask for the same response.

State normally calls for a brief answer.

Find normally requires the relevant working.

Show that gives you the destination and requires you to demonstrate how to get there.

Justify requires mathematical reasons or evidence supporting the conclusion.

Learning these distinctions is one of the simplest ways to make exam technique more precise.

IB Maths command words: a quick reference

The following explanations paraphrase the official command-term glossaries used for both Mathematics AI and AA. (International Baccalaureate®)

IB Maths command words and what they require from a student
Command word	What it means for a student
Calculate	Produce a numerical answer and show the important stages of the calculation.
Comment	Make a reasoned judgment about a result or statement.
Compare	Explain the similarities between two or more things.
Compare and contrast	Explain both similarities and differences.
Construct	Organize or display information in a logical or diagrammatic form.
Contrast	Explain the differences between two or more things.
Deduce	Reach a conclusion using information already provided.
Demonstrate	Make something clear using reasoning, evidence or examples.
Describe	Give a detailed account of what happens or what is observed.
Determine	Obtain the single possible answer.
Differentiate	Find the derivative of a function.
Distinguish	Make the differences between concepts clear.
Draw	Produce an accurate, labelled diagram or graph, normally to scale where appropriate.
Estimate	Obtain an approximate value.
Explain	Give a detailed account including reasons.
Find	Obtain an answer while showing the relevant working.
Hence	Use the result from the previous part to answer the question.
Hence or otherwise	The previous result is suggested, but another valid method may be used.
Identify	Select or recognize the appropriate answer from the possibilities.
Integrate	Find an integral of a function.
Interpret	Use mathematical information to recognize trends and draw conclusions, often in context.
Investigate	Examine something systematically in order to establish facts or conclusions.
Justify	Give valid mathematical reasons or evidence supporting the answer.
Label	Add the required labels to a diagram.
List	Give a sequence of short answers without explanation.
Plot	Mark specified points accurately on a graph or diagram.
Predict	Give an expected result based on the available information.
Prove	Establish a result formally through a logical sequence of mathematical steps.
Show	Display the stages of a calculation or derivation.
Show that	Derive a result that has already been given in the question.
Sketch	Give the general shape of a graph or diagram, including its important features.
Solve	Obtain the solution using an appropriate algebraic, numerical or graphical method.
State	Give a brief answer without explanation or calculation.
Suggest	Propose a reasonable possible answer or hypothesis.
Verify	Provide evidence confirming that a result is valid.
Write down	Give an answer that can usually be read directly from the information provided, with little or no calculation.

Examples of IB Maths command words in questions

The following are original Full Marks Education exam-style examples, rather than reproductions of IB past-paper questions.

1. Calculate

Example — AI/AA SL

A savings account contains $4500 and pays compound interest at a rate of 3.2% per year.

Calculate the value of the investment after 6 years.

What the examiner is asking for

A numerical answer is required, but students should also show the important stage of their calculation.

4500 \times {(1.032)}^{6}

followed by the resulting value.

Simply writing the final calculator result is weaker exam technique than showing how it was obtained.

2. Write down

Example — AI SL

The equation of a regression line relating distance travelled, $x$ , to fuel used, $y$ , is

y = 0.074 x + 1.8

Write down the gradient of the regression line.

What the examiner is asking for

The answer can be taken directly from the equation:

0.074

No derivation is needed. This illustrates why students should not automatically fill the page with calculations. Sometimes the command word is specifically telling them that very little working is required.

3. State

Example — AA SL

The function

f (x) = {(x - 3)}^{2} + 5

has a minimum point.

State the coordinates of the minimum point.

Expected response

(3, 5)

There is no need to expand the quadratic, differentiate it or explain why the vertex occurs there. The command word state tells the student that a concise answer is sufficient.

4. Find

Example — AA SL

Let

f (x) = x^{2} - 6 x + 5

Find the coordinates of the minimum point of the graph of $f$ .

What the examiner is asking for

Unlike state, find normally requires the relevant mathematical stages. A student might complete the square:

f (x) = {(x - 3)}^{2} - 4

and therefore conclude that the minimum point is

(3, - 4)

Alternatively, an appropriate calculus method could be used.

Find versus calculate versus determine

These three words are sometimes treated as interchangeable by students, but there are subtle differences.

Calculate emphasizes obtaining a numerical result through calculation.

Find asks for an answer together with the relevant stages.

Determine indicates that the information leads to a particular, uniquely determined result. (International Baccalaureate®)

The equation $x 2 - k x + 9 = 0$ has exactly one real solution. Determine the value of $k$ , given that $k > 0$ .

Here, the discriminant must be zero:

k^{2} - 36 = 0

k = 6

because the question specifies $k > 0$ .

Show that

This is one of the most important IB Maths command phrases.

In a show that question, the answer has already effectively been supplied. Your job is to demonstrate how the result follows.

Example — AA SL

A geometric sequence has first term 4 and common ratio 1.5.

Show that its $n$ th term can be written as

u_{n} = 4 {(1.5)}^{n - 1}

A response should start from the general geometric sequence formula,

u_{n} = u_{1} r^{n - 1}

then substitute $u 1 = 4$ , $r = 1.5$ , to obtain the stated result. The important point is that the reasoning must be visible.

Students should be particularly careful about using a calculator to work backwards from the supplied result. The official IB glossary notes that show that questions generally do not require calculator use. (International Baccalaureate®)

Hence

A question containing hence is telling you something very specific: use what you have just established.

Continuing the previous example:

u_{n} = 4 {(1.5)}^{n - 1}

Hence, find the first value of $n$ for which $u n > 100$ .

The intention is that the student uses

4 {(1.5)}^{n - 1} > 100

rather than starting the entire sequence problem again.

Hence or otherwise

If the wording instead says hence or otherwise, using the previous result is encouraged but not compulsory. Another valid method may also receive credit. (International Baccalaureate®)

Solve

Example — AA SL

Solve

3^{2 x - 1} = 27

Since $27 = 3 3$ , we have $2 x - 1 = 3$ , giving $x = 2$ .

The command word solve means that the required solution or solutions can be obtained using an appropriate algebraic, numerical or graphical method.

Students should also check whether the question places restrictions on the domain.

For example: Solve $sin x = 0.6$ for $0 \leq x \leq 2π$ .

Finding only one calculator value is not enough because the interval may contain multiple solutions.

Explain

Explain is especially important in contextual questions. A numerical answer alone is often insufficient.

Example — AI SL

A model predicts that the population of a town increases by approximately 4.8% each year.

Explain why an exponential model may be more appropriate than a linear model.

A strong response might say that the increase is proportional to the current population, so the amount added each year changes as the population changes. The important feature is the reason.

Whenever students see explain, they should ask themselves: Have I actually said why?

Interpret

Interpretation is particularly important in Mathematics AI because mathematics is frequently presented in context.

Example — AI SL

A regression model relating advertising expenditure, $x$ , measured in thousands of dollars, to monthly sales, $y$ , measured in thousands of dollars, is

y = 2.6 x + 41

Interpret the value 2.6 in context.

An appropriate response would be: For each additional $1000 spent on advertising, the model predicts an increase of approximately $2600 in monthly sales.

Simply writing The gradient is 2.6 does not fully interpret the parameter. The units and real-world meaning matter.

Comment

Example — AI SL

The Pearson product-moment correlation coefficient between weekly exercise time and resting heart rate for a sample is

r = - 0.81

Comment on the relationship between the two variables.

A suitable response might note that the data show a strong negative linear correlation: people recording more weekly exercise tended to have lower resting heart rates.

A particularly careful student would avoid claiming that exercise has been proven to cause the lower heart rate solely from the correlation. Comment therefore goes beyond repeating a calculator output. It requires a judgment based on the mathematical result.

Justify

Students frequently give an answer when the examiner actually asks them to defend that answer.

Example — AI HL

Two models are proposed for a data set.

R_{2}^{A} = 0.82

and

R_{2}^{B} = 0.96

The residual plot for Model B also shows no obvious pattern. A student chooses Model B.

Justify the student's choice.

A strong response would use the evidence given: Model B explains a greater proportion of the variation in the data and its residuals do not display an obvious systematic pattern. Writing only Model B is better would not answer the command word. A justification needs evidence.

Show that versus prove

These are not the same instruction.

A show that question asks you to derive a stated result.

A prove question requires a more formal logical argument. (International Baccalaureate®)

Example — AA HL

Prove by mathematical induction that

1 + 3 + 5 + \dots + (2 n - 1) = n^{2}

for all positive integers $n$ .

An appropriate proof would require:

Establishing a base case.
Assuming the statement is true for $n = k$ .
Using that assumption to establish the result for $n = k + 1$ .
Giving a valid concluding statement.

Simply substituting several values of $n$ does not constitute a proof.

Verify

Example — AA HL

The function

y = 2 e^{3 x}

is claimed to satisfy

\frac{d y}{d x} = 3 y

Verify this statement.

Differentiate:

\frac{d y}{d x} = 6 e^{3 x}

Since

3 y = 3 (2 e^{3 x}) = 6 e^{3 x}

we have $d y / d x = 3 y$ . The calculation provides evidence confirming the claim.

Sketch

Example — AA SL

Sketch the graph of

y = \frac{1}{x - 2} + 3

showing the equations of any asymptotes.

Students do not need to produce a perfectly scaled engineering drawing. However, the sketch should show the important features:

$x = 2$ as the vertical asymptote;
$y = 3$ as the horizontal asymptote; and
the correct general shape and position of the branches.

The IB explicitly distinguishes between draw and sketch. A sketch communicates the essential shape and relevant features, whereas a drawing is expected to be more accurate. (International Baccalaureate®)

Draw

Consider: Draw the line $y = 2 x - 1$ for $-2 \leq x \leq 3$ .

Here, accuracy matters. Points should be plotted correctly, axes should be labelled appropriately, and straight lines should be represented accurately. This is different from a quick qualitative sketch.

Estimate

Example — AI/AA SL

The graph of $f$ crosses the $x$ -axis between $x = 2.3$ and $x = 2.4$ .

Estimate the root of $f (x) = 0$ .

An approximate value is expected. A student should not interpret the word estimate as requiring an unjustifiably precise answer such as $x = 2.347826194$ .

Predict

Example — AI SL

A linear regression model is

y = 4.2 x + 17.5

Predict the value of $y$ when $x = 8$ .

y = 4.2 (8) + 17.5 = 51.1

Because this is a prediction from a mathematical model, students should recognize that it is a model-based expected value rather than necessarily an exact future observation.

Deduce

Example — AA HL

It has been established that

f^{'} (x) > 0

for all $x > 4$ .

Deduce what this tells you about $f$ on the interval $x > 4$ .

Since the derivative is positive throughout the interval, $f$ is increasing there. The calculation has already been done. The student now has to draw the mathematical conclusion.

Differentiate and integrate

These command words are unusually direct.

Differentiate

f (x) = x^{3} e^{x}

means: obtain $f'(x)$ .

Likewise, integrate

\int (3 x^{2} + 4 x - 1) d x

means: obtain the integral.

Students should still show appropriate mathematical working, particularly when techniques such as the product rule, chain rule, integration by parts or substitution are involved.

Describe versus explain

These two words are easily confused.

Suppose a graph rises rapidly and then begins to level off.

A question asking students to describe the graph might expect:

The function increases throughout the interval, but its rate of increase becomes smaller.

A question asking students to explain this behaviour requires a reason:

The function increases because its derivative is positive, but the decreasing derivative shows that its rate of increase is becoming smaller.

A useful rule is: Describe = what happens. Explain = what happens and why.

Compare, contrast and distinguish

These words also require different approaches.

If two probability distributions are shown:

Compare asks about similarities.

Contrast asks about differences.

Compare and contrast asks for both.

Distinguish asks you to make the important difference between concepts clear.

For example: Distinguish between independent events and mutually exclusive events.

A suitable answer would explain that independent events do not affect each other's probabilities, whereas mutually exclusive events cannot occur together.

Why command words matter for marks

IB Mathematics does not assess only whether students can obtain a final number.

The examinations also assess mathematical communication, reasoning, interpretation and problem solving. The official course guides emphasize these skills in the assessment objectives for both mathematics courses. (International Baccalaureate®)

That is why a student can sometimes have the correct mathematical idea but still fail to earn all available marks.

If the instruction says justify, provide evidence.
If it says interpret, return to the context.
If it says show that, make the derivation visible.
If it says prove, construct a logically complete proof.
If it says state, do not waste five minutes producing unnecessary algebra.

A simple exam strategy

Before starting any IB Maths question:

Circle the command word. Identify what the question actually wants you to do.
Identify the mathematical task. Is this a calculation, proof, interpretation, graph, model or explanation?
Match your response to the command word. Ask yourself whether you need working, a reason, evidence, context, a formal proof, or simply a short answer.
Read your answer against the command word. If it says explain, check that you gave a reason. If it says justify, check that you supplied evidence. If it says interpret, check that you mentioned the variables and their context.

This final check can take only a few seconds and can prevent easily avoidable lost marks.

Final takeaway

IB Maths command words are not decorative language.

They are instructions.

Two students may know exactly the same mathematics, yet the student who understands what state, find, show that, justify, interpret, comment, and prove require is much more likely to communicate that mathematics in the form the examination expects.

Learning the mathematics is essential. Learning how the IB asks you to demonstrate that mathematics is part of preparing for the exam as well.

At Full Marks Education, our IB Maths AI and AA resources integrate command words throughout worked examples and exam-style practice so that students become familiar not only with how to solve a problem, but also with how the IB expects that solution to be communicated.

Examples in this article are original Full Marks Education questions written in IB-style format. They are not reproduced IB examination questions.

IB Maths AI & AA Command Words Explained: What They Mean and How to Answer Them