Understanding Keywords in Mathematics Exams

Whether you are approaching your final exams, or you’re at the stage of mid-course exams, you’re probably beginning to put in some practice of past paper questions. Sometimes it isn’t the maths which is hard, but understanding the language of the question; and each question will contain certain keywords which tell you how you should be tackling the question, and what you should be writing down. I can’t emphasise enough how important it is to understand what, in the context of exam questions, these keywords – technically known as “command terms” –  mean. There are over 30 of them, and my list below covers the more common ones.

Write down:   You can obtain the answer with very little, if any, calculation; this may follow on from a previous answer. There will be no marks for showing working.

Hence:   You must use what you have just done to obtain the answer. For example: “Differentiate the function f(x) = 2x² – 8x + 1 and hence find the vertex of the graph y = 2x² – 8x + 1.” You could find the vertex in other ways, but the question here is asking you to find the x value which makes the derivative zero. Sometimes a question will use hence or otherwise; usually, the “hence” method will be simplest, but you are free to use any other method to find the answer.

Show that:  You are being asked to show that a particular statement is true. So, to take a trivial example, “Show that the equation 2x – 6 = 10 has a solution x = 8.” Rather than solving the equation, you could simply substitute
x = 8, and then 16 – 6 = 10, thus showing that x = 8 is the solution. Be careful though: all the marks are for the working, so make sure you have shown enough, and that you aren’t simply stating what the question has asked you to show.

Plot and Sketch:   “Plot” means that you must mark relevant points on a graph; “sketch” means that you should show a graph with roughly the correct shape, marking key points such as axis intercepts. Scales are not necessary on a sketch, although you must give an indication of scale, and also label the axes.

Estimate:   Obtain an approximate value.

Calculate, evaluate:   Obtain a numerical answer, showing working as necessary.

Find:   Obtain an answer, probably algebraic, showing working as necessary.

State:   A brief answer requiring no working.

Prove:   Write out a formal proof. Most likely in HL; for example, “Prove by mathematical induction ….”

Simplify:   Usually requires the simplification of an algebraic expression. For example: “Expand (x – 2)³ and simplify the answer.” The correct solution is
x³ – 6x² + 12x – 8. Anything more complicated will not score full marks.

Exact:   If a question asks you to find a result in an exact form, the answer will be an irrational number which you must not write as a rounded decimal. Most likely it will involve π or a square root. For example, find the hypotenuse of the right-angled triangle with shorter sides 4cm and 6cm, giving your answer in an exact form. The answer is √52, but you will get no marks for 7.211…