Every IB Maths command term explained (with worked examples)

Understanding IB Maths Command Terms

For over a decade, I have marked countless IB Maths papers. One pattern emerges consistently: students often lose marks not because they lack the mathematical understanding, but because they fail to answer the question asked. The culprit? Misinterpreting command terms. These specific words at the start of an instruction tell you exactly what kind of response the examiner expects. Ignore them at your peril.

In this article, I will break down the most common IB Maths command terms. I'll explain what each means, give an example, and show you what a correct response looks like. My goal is to equip you with the knowledge to approach every question with precision, ensuring your hard work translates into full marks.

The Essential Command Terms Explained

Let's dive into the core terms you'll encounter across IB Maths AA and AI papers, both SL and HL. Pay close attention; a subtle difference in wording can mean a significant difference in marks.

Calculate / Find / Determine

These terms are often interchangeable and are among the most common. They require you to obtain a numerical answer. You must show sufficient working to justify your result. The final answer should be given to an appropriate degree of accuracy, usually three significant figures unless otherwise specified.

Example (AA SL/HL, AI SL/HL): Calculate the value of $\int_1^3 (x^2 - 1) \, dx$.

Correct Response:
$\int_1^3 (x^2 - 1) \, dx = \left[ \frac{x^3}{3} - x \right]_1^3$
$= \left( \frac{3^3}{3} - 3 \right) - \left( \frac{1^3}{3} - 1 \right)$
$= (9 - 3) - \left( \frac{1}{3} - 1 \right)$
$= 6 - \left( -\frac{2}{3} \right)$
$= 6 + \frac{2}{3} = \frac{20}{3}$ or $6.67$ (3 s.f.)

Show That / Prove

These are critical. When you see "Show that," the given answer is your destination. Your task is to provide a clear, logical, step-by-step derivation from the initial premise to the stated conclusion. You cannot use the conclusion in your working. "Prove" demands a rigorous mathematical argument, often requiring a deeper understanding of definitions and theorems.

Example (AA HL, AI HL): Show that the derivative of $f(x) = \sin(2x)$ is $f'(x) = 2\cos(2x)$.

Correct Response (using chain rule):
Let $u = 2x$. Then $f(x) = \sin u$.
$\frac{du}{dx} = 2$
$\frac{df}{du} = \cos u$
Using the chain rule, $\frac{df}{dx} = \frac{df}{du} \times \frac{du}{dx} = \cos u \times 2 = 2\cos u$.
Substituting $u = 2x$ back, we get $f'(x) = 2\cos(2x)$.

Explain / Justify

These terms require you to provide reasons for your answer. This isn't just about showing working; it's about articulating the mathematical principles, theorems, or logical steps you used. A clear sentence or two is usually sufficient. This is particularly common in Paper 1 questions where calculator use is restricted, or in conceptual questions.

Example (AA SL/HL, AI SL/HL): Explain why the function $f(x) = \sqrt{x-4}$ has a domain of $x \ge 4$.

Correct Response: For $f(x)$ to be defined for real numbers, the expression under the square root must be non-negative. Therefore, $x-4 \ge 0$, which implies $x \ge 4$.

Sketch / Draw

These terms mean you need to produce a graph, usually without the aid of a calculator. Your sketch should capture the key features: intercepts, asymptotes, local maxima/minima, points of inflection, and general shape. Labels for axes and important points are crucial. Accuracy is important, but a perfect scale is not always required unless specified (e.g., "draw on axes with a scale of..."). My students often lose easy marks by not labeling axes or key points.

Example (AA SL/HL, AI SL/HL): Sketch the graph of $y = e^{-x} + 1$, clearly indicating any asymptotes and intercepts.

Correct Response (description for text, imagine a graph):
The graph should show an exponential decay curve.

• Horizontal asymptote: As $x \to \infty$, $e^{-x} \to 0$, so $y \to 1$. Label $y=1$ as an asymptote.
• $y$-intercept: When $x=0$, $y = e^0 + 1 = 1+1=2$. Label the point $(0, 2)$.
• The graph decreases as $x$ increases, approaching $y=1$ from above. There is no $x$-intercept as $e^{-x}+1$ is always positive.

Solve

This command term typically means finding the value(s) of an unknown variable that satisfy an equation or inequality. Your answer should be the solution set or the specific values. Show your algebraic working clearly.

Example (AA SL/HL, AI SL/HL): Solve the equation $2\sin x - 1 = 0$ for $0 \le x \le 2\pi$.

Correct Response:
$2\sin x - 1 = 0$
$2\sin x = 1$
$\sin x = \frac{1}{2}$
The principal value is $x = \frac{\pi}{6}$.
In the interval $0 \le x \le 2\pi$, sine is positive in the first and second quadrants.
So, $x = \frac{\pi}{6}$ or $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.

Less Common, But Equally Important Terms

While the terms above cover the majority, a few others appear less frequently but carry specific instructions.

State / Write Down

These terms mean no working is required. The answer should be direct. This usually applies to standard results, definitions, or values that can be read directly from a diagram or calculator. My students often waste time showing steps when a direct answer is enough.

Example (AA SL/HL, AI SL/HL): State the amplitude of the function $f(x) = 3\cos(x) - 2$.

Correct Response: The amplitude is $3$.

Estimate

This term implies you need to provide an approximate value, often by using a simplified model, rounding, or graphical interpretation. Precision is not the goal here; a reasonable approximation with a clear method is.

Example (AI SL/HL): A population grows according to the model $P(t) = 100e^{0.05t}$, where $t$ is in years. Estimate the population after 10 years without using a calculator.

Correct Response:
$P(10) = 100e^{0.05 \times 10} = 100e^{0.5}$
We know $e \approx 2.7$. So $e^{0.5} = \sqrt{e} \approx \sqrt{2.7}$.
$\sqrt{2.7}$ is between $\sqrt{1}=1$ and $\sqrt{4}=2$. Closer to $\sqrt{2.25}=1.5$. Let's estimate it as $1.6$.
So, $P(10) \approx 100 \times 1.6 = 160$.

Suggest / Comment On

These require you to offer an idea, hypothesis, or observation, often based on a given context or data. There might not be one single "correct" answer, but your suggestion or comment must be mathematically sound and relevant to the information provided. These are more common in AI papers, especially Paper 3 for HL, where modelling and interpretation are key. You might find this similar to tasks covered in our Paper 3 HL AI guide.

Example (AI SL/HL): A student collected data on plant growth and found a strong positive correlation. Suggest a possible causal link.

Correct Response: Increased sunlight exposure (causal factor) could lead to increased photosynthesis, which in turn causes greater plant growth (observed correlation). However, other factors like water or nutrients also contribute.

Final Thoughts and Next Steps

Mastering these command terms is not just about memorizing definitions; it's about developing a strategic approach to every question. When you sit down for an exam, take a moment to identify the command term. This single habit can significantly improve your performance.

My students who consistently score well are those who internalise these distinctions. Make it a part of your revision routine. Practice questions not just for the math, but for the precise way you answer them. Consider checking out my math flashcards for quick recall of definitions and common formulas, as these often feed directly into 'State' or 'Explain' questions. Understanding these terms will clarify expectations and allow your mathematical ability to shine through, not be obscured by misinterpretation.