Command terms

Each question part in an IB maths paper begins with a command term: a verb such as "write down", "find", "show that" or "sketch". Each term has a fixed meaning, defined by the IB, and that meaning sets how much working is expected, what form the answer takes and where the marks are. Reading the command term correctly is the quickest way to avoid two costly errors: writing a page of working for a 1-mark "write down" part, and giving a bare number for a 5-mark "find" part.

The table gives every command term used in IB mathematics, with its meaning in close paraphrase. The sections after it group the terms by the kind of answer they ask for, with a short example of a full answer to each.

Terms that need little or no working

These terms ask for a result that can be read off, recalled or seen directly. Working is not expected, and a long method wastes time.

Write down. The answer can be extracted from information already given or from the previous part, with little or no calculation. "Write down the coordinates of the vertex of $y=2(x-3{)}^2+5$": full answer $(3,5)$, read straight from the vertex form.

State. A name, value or brief fact, given without explanation. "State the range of $f(x)=e^x+1$": full answer $f(x)>1$.

List. Several brief answers, with no explanation. "List the possible outcomes when two coins are tossed": full answer HH, HT, TH, TT.

Identify. Choose from a number of possibilities. "Identify which of the graphs A, B and C could be the graph of $y=\ln (x-2)$": full answer "Graph B", the one with a vertical asymptote at $x=2$ that crosses the $x$-axis at $(3,0)$.

Terms that ask for an answer with working

These are the most common terms on every paper. The marks for them are mostly method marks, so the working must be visible.

Calculate and find

Calculate asks for a numerical answer with the relevant stages shown. "Calculate the mean of $4,7,7,9,12,15$":

Find asks for an answer (numerical or algebraic) with the relevant stages shown. It is the most common command term in IB maths.

Determine and solve

Determine asks for the only possible answer, so the working should show why no other answer is possible. "Determine the value of $k>0$ for which $x^2+kx+9=0$ has exactly one real root": the discriminant must be zero, so $k^2-36=0$, giving $k=\pm 6$, and since $k>0$, $k=6$.

Solve asks for every solution of an equation or inequality, by any suitable method: algebraic, numerical or graphical. "Solve $2\sin x=1$ for $0\le x\le 2\pi$":

Both solutions in the interval are needed; giving only $\frac{\pi }{6}$ loses a mark.

Estimate, differentiate and integrate

Estimate asks for an approximate value: a value from a regression line, a median read from a cumulative frequency graph, or an approximation from a model. "Use the regression line $y=2.1x+3.4$ to estimate $y$ when $x=10$": full answer $y=2.1(10)+3.4=24.4$.

Differentiate asks for the derivative. "Differentiate $y=x^2\sin x$": by the product rule, $\frac{dy}{dx}=2x\sin x+x^2\cos x$.

Integrate asks for the integral. "Integrate $3x^2+4$ with respect to $x$": $\int (3x^2+4)dx=x^3+4x+C$. The constant of integration is part of a complete answer to an indefinite integral.

Terms about a given or earlier result

These terms either give the answer and ask for the reasoning, or tie a part to the work before it.

Show and show that

Show asks for the steps of a calculation or derivation. "For the arithmetic sequence with $u_1=2$ and $d=3$, show that $S_{10}=155$":

Show that gives the result, so every mark is for the steps that lead to it. Start from the given information or from one side of an identity, write each step, and end with the printed result exactly. Using the result to work backwards earns nothing.

Verify and prove

Verify asks for evidence that a result holds, usually by substitution. "Verify that $x=2$ is a solution of $x^3-3x-2=0$": $2^3-3(2)-2=8-6-2=0$, so $x=2$ is a solution. The conclusion sentence is part of the answer.

Prove asks for a formal, logical argument that covers every case. "Prove that the sum of any three consecutive integers is divisible by $3$": let the integers be $n$, $n+1$ and $n+2$, where $n\in \mathbb{Z}$; their sum is $3n+3=3(n+1)$, and since $n+1\in \mathbb{Z}$, the sum is a multiple of $3$. Checking a few cases does not prove the statement (see 1.6).

Hence, and hence or otherwise

Hence means the result must come from the preceding work. A different method, however correct, may earn no marks.

Hence or otherwise suggests using the preceding work, which is usually the quickest route, but any correct method earns the marks.

Deduce and demonstrate

Deduce asks for a conclusion that follows from the information given, usually in one or two steps. "Given that ${\int }_0^{2}f(x)dx=5$, deduce the value of ${\int }_0^2(3f(x)+1)dx$":

Demonstrate asks for reasoning or evidence, often with an example. "Demonstrate that $a^2>b^2$ does not imply $a>b$": take $a=-3$ and $b=1$; then $a^2=9>1=b^2$, but $a<b$.

Terms for graphs and diagrams

Sketch

A sketch gives the general shape and the relevant features, labelled. It does not need to be to scale, but intercepts, turning points, asymptotes and end behaviour must be correct and labelled where known.

Draw, construct, plot and label

Draw is stricter than sketch: an accurate diagram or graph, to scale, in pencil, with a ruler for straight lines and plotted points joined by a smooth curve or straight line. It is used when a grid is printed, for example "Draw the box plot on the grid".

Construct asks for information to be displayed in a diagram or logical form: "Construct a tree diagram to show the outcomes of two draws". A constructed tree diagram has every branch labelled with its event and its probability.

Plot asks for points to be marked accurately on a diagram, such as data on a scatter diagram. Label asks for labels to be added: names of points, asymptote equations, axes, or regions of a Venn diagram.

Terms that ask for words

These terms ask for written reasoning. The answer should be a sentence or two, specific to the question, and linked to the mathematics.

Describe and explain

Describe asks for a detailed account of what something is or does, without reasons. "Describe the transformation that maps the graph of $y=f(x)$ to the graph of $y=f(x-3)$": full answer "a horizontal translation of $3$ units in the positive $x$-direction", or "a translation by the vector $\left(\begin{array}{c}3\\ 0\end{array}\right)$". Both the type of transformation and its size and direction are needed.

Explain adds reasons or causes. "Explain why $f(x)=x^2$, $x\in \mathbb{R}$, has no inverse function": full answer "$f$ is not one-to-one; for example $f(-2)=f(2)=4$, so the inverse would not give a single output for the input $4$."

Justify

Justify asks for valid reasons or evidence for an answer. In calculus it usually means a derivative test; in probability, a calculation; in proofs, a stated rule.

Interpret and comment

Interpret asks for a conclusion about the context drawn from a result. "The regression line of exam score $y$ on hours of study $x$ is $y=2.1x+34$. Interpret the value $2.1$": full answer "each extra hour of study is associated with an increase of about $2.1$ in the score, on average".

Comment asks for a judgement on a statement or result. "Comment on the reliability of using this line to estimate the score for 50 hours of study, given that the data ran from 2 to 12 hours": full answer "the estimate is unreliable, because 50 hours is far outside the range of the data (extrapolation)".

Suggest and predict

Suggest asks for a reasonable proposal, often one of several possible answers. "The height of a ball is modelled by $h(t)=20+15t-5t^2$. Suggest why the model is not valid for $t>4$": full answer "for $t>4$ the model gives negative heights, but the ball cannot go below the ground".

Predict asks for an expected result: the next term of a pattern, or the value a model gives. "Predict the value of $u_{20}$ for the sequence $3,7,11,15,\dots$": $u_{20}=3+19(4)=79$.

Compare, contrast and distinguish

Compare asks for similarities, contrast for differences, and compare and contrast for both. In each case the answer must refer to both (or all) items throughout, not describe one and then the other.

"The times, in minutes, taken by two groups to finish a task have medians $12$ and $9$, and interquartile ranges $5$ and $2$. Compare the two distributions": full answer "Group A took longer on average than group B (median $12$ minutes compared with $9$), and the times in group A were more spread out (IQR $5$ minutes compared with $2$)." Each sentence names both groups and uses a statistic for each.

Distinguish asks for the differences between concepts to be made clear: "Distinguish between mutually exclusive events and independent events": mutually exclusive events cannot happen together, so $P(A\cap B)=0$; independent events do not affect each other's probabilities, so $P(A\cap B)=P(A)\times P(B)$.

Investigate

Investigate asks for a systematic study to establish facts and reach new conclusions. It is rare in Papers 1 and 2, more common in Paper 3, and central to the exploration. An investigation works through cases in an organised way, records them, looks for a pattern and states a conclusion.

Reading the command term in context

The same mathematics can be asked with different command terms, and the expected answer changes with it. For the quadratic $y=x^2-4x+3$: