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This page sets out the method for exam technique: “show that” questions in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Exam technique: “show that” questions topic page.
When you see 'Show that …' followed by an expression, a value or a statement.
When the target is a statement such as 'the triangle is right-angled' or 'the offer is cheaper'.
Expanding and simplifying 3(x+4)-x gives 2x+12. Multiply out, then collect the x terms.
Expanding and simplifying (x+2)(x+3)-x^2 gives 5x+6. The x squared terms cancel.
Expanding and simplifying (x+1)^2-x^2 gives 2x+1. The x squared terms cancel and a linear term remains.
Expanding (x+4)(x+1) gives x^2+5x+4. Here nothing is subtracted, so the square stays.
In a show-that question the result is printed for you, so the marks are for the working. To show that 3(x+4)-x = 2x+12, write 3x+12-x, then collect to 2x+12. End on the exact expression you were asked to reach.
After expanding, add the x terms and add the number terms. Do not leave two separate x terms.
Read the printed result before you write. It tells you the form you must finish in, such as an expression in x, a perimeter, or a named term of a sequence. Count the marks. A 3-mark show-that needs about three lines that a marker can see. One line that jumps from the start to the target is not those three lines.
Do not begin with the target and work backwards. If you write the result first and then rearrange, the marker cannot tell that you reached it rather than assumed it. Start from the bracket, the formula, or the definition the question gave you.
Method: 1. Read the printed result and note the form it uses. 2. Count the marks and plan about one visible step per mark. 3. Start from the given expression, not from the target. 4. Leave the words “as required” for the last line.
An expansion show-that wants every product. For a single bracket, multiply the outside number by each term inside. For two brackets, write the four products, then collect. If a bracket is subtracted, subtract every term of that expansion. The minus changes the sign of each of them.
A perimeter show-that starts from the shape. Add every side, or use 2×(length+width) for a rectangle, and simplify until the expression matches the printed one. Then, if the question gives a value, substitute it on a new line and show the arithmetic.
Method: 1. Write the given expression on the first line. 2. Expand one bracket at a time and keep the signs. 3. Collect like terms on their own line. 4. Substitute a given number only after the algebra matches.
Show-that working stays in whole numbers, fractions or exact algebra. Replacing 1/2 with 0.5 can be harmless, but replacing a fraction such as 5/6 with a rounded decimal and then claiming equality is not a proof. If the target is an integer expression, your lines should be integer expressions too.
When you show that a number is a term of a sequence, solve the nth-term equation. The position must be a positive whole number. Name that position: “the 25th term” is a complete answer, and “n=25” on its own leaves the reader to guess what you are claiming.
Method: 1. Keep fractions and whole numbers exact. 2. Do not round a decimal and call it equal to the target. 3. For a sequence, solve for the position n. 4. Say which term it is when n is a positive whole number.
The last algebraic line should be identical to the statement in the question. If the question says 8x+32, do not stop at 4(2x+8) unless you then expand or factor to the printed form. If it says the gradient is 4, the last calculation should show the fraction that equals 4.
A coordinate check and an algebraic argument can sit together. Substitute the x-coordinate into the equation of the line, and also work the gradient between the two given points. Both results should agree with the line. Write “as required” only when both parts are on the page.
Method: 1. Compare your last line with the printed statement. 2. Rewrite it if the form does not match yet. 3. Add a gradient or a substitution when the question asks for a point on a line. 4. Close with the target and the words “as required”.
Number-based show-that questions ask you to prove a claim such as "in total they have less than £250" or "the real distance is 1.5 km". The answer is not a single number, it is a short chain of working that ends in a sentence. Work out each quantity separately on its own line, then put the two numbers you are comparing side by side. For example 84× 1.5=126, then 84+126=210, and 210<250.
The final comparison is where many marks are lost. Writing 210 and stopping gives a calculation but no argument. Write "210<250, so they have less than £250" and the mark is safe. If the claim is "at least double" or "more than half", state the number you are measuring against, such as 2× 2500=5000, so the reader sees both sides of the comparison in numbers.
Method: 1. Write down each amount the claim depends on and work it out on its own line. 2. Keep full accuracy until the last line, then round only to compare. 3. Put the two values being compared next to each other with the correct sign. 4. End with a sentence that repeats the claim in words.
Tip: If a calculator answer is very close to the target, write more figures than the question needs so the comparison is unmistakable.
Some show-that questions give you the answer and ask you to prove it fits. To show that x=4 is a solution of x^2+2x-24=0, substitute it and work out the left-hand side in stages: 4^2+2× 4-24=16+8-24=0. The right-hand side is 0, so both sides agree. Writing =0 straight after the substitution without the middle step does not show anything, because the marker cannot see that you did the arithmetic.
This also covers showing a point lies on a line, a number is a term of a sequence, or a value gives a stated result in a formula. Work with one side at a time, show the intermediate numbers, and finish by saying the two sides are equal. If the value has to satisfy two things, such as both equations of a pair, test it in each one separately and say so for each.
Method: 1. Write the expression or equation first, with the given value in brackets. 2. Work out the left-hand side in at least two lines of arithmetic. 3. Write what the right-hand side is and that the two sides are equal. 4. If there are two conditions, check each one separately.
Higher show-that questions often have an exact target such as 4√(5) or 8(2n+1). The target tells you the form to aim for, so simplify each part into that form. For surds, break each root into a square number times a surd: √(12)=√(4)×√(3)=2√(3). Combine like surds only when the surd part matches. A decimal from the calculator does not count as a proof here, because 8.944ldots only looks like 4√(5).
For algebraic targets, expand fully, collect like terms, and then factorise if the target is factorised. A multiple of 8 is shown by writing the result as 8×(an integer expression). Say what the bracket stands for, for instance "2n+1 is an integer, so the result is a multiple of 8". That closing sentence is usually worth a mark on its own.
Method: 1. Look at the target and decide which form you are heading for. 2. Simplify surds first, then combine like surds, or expand brackets then collect. 3. Factorise if the target is a factorised expression. 4. State the conclusion in words, such as "a multiple of 8".
Tip: A numerical check with a small value of n is a good way to catch a slip, but it is never the proof itself.
A claim of the form "this is true for all values" can be shown to be false with a single example where it fails. Suppose Aisha says that n^2+n+11 is prime for every whole number n from 1 to 10. Testing n=1,2,3 gives 13, 17 and 23, which are all prime, but that does not prove her right. Testing n=10 gives 100+10+11=121=11× 11, which is not prime, so she is wrong.
The question usually says "you must show that your value is not prime", so state the factors. Do the same for other claims: an even number times an even number, a fraction that always gets smaller, and so on. Choose your counter-example to be easy to check, write the full calculation, and end with "so the statement is not true". Showing that something is right needs a proof, but showing it is wrong needs only one failure.
Method: 1. Try small values, but look for one where the claim could fail. 2. Calculate the counter-example in full and write down the result. 3. Say why it breaks the claim, for example by giving its factors. 4. Finish by stating that the claim is not true.
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