Key Stage 4 maths · Working mathematically

GCSE maths reasoning: proof, counter-examples and arguments

At GCSE, working mathematically means more than getting an answer. Pupils in Years 10 and 11 are expected to explain why something is true, test a pattern to see whether it always holds, prove results with algebra and with reasoned steps in geometry, and judge whether someone else’s argument, graph or statistic can be trusted. This reasoning is built into questions across the whole course rather than taught as a separate unit.

Reasoning and proof in maths, explained

Many GCSE questions are worded to make a pupil justify something: “show that”, “prove that”, “explain why”, “is Sam correct?”. A right answer on its own earns little on these. The marks are for the chain of reasons that leads to it, written clearly enough that a stranger could follow it. For a parent this is often the biggest change from earlier years, when the answer was usually the point.

A big idea here is the difference between a conjecture and a proof. A conjecture is a claim that looks true, perhaps because it worked for several examples. Testing examples is a sensible way to form one, but it never proves it. One counter-example, a single case where the claim fails, is enough to disprove it. The expression n² + n + 41 gives a prime number for every whole number n from 0 to 39, yet n = 40 gives 1,681, which is 41 × 41. To prove a claim for every number, pupils use algebra: an even number is written 2n, an odd number 2n + 1, and consecutive whole numbers n, n + 1, n + 2.

In geometry the reasoning is deductive: each step follows from facts already established, and the reason is named. Pupils write things like “angle ABC = 62° because alternate angles are equal” or “the angles in a triangle add up to 180°, so angle x = 180° − 90° − 35°”. They must not rely on how a diagram looks, since diagrams are often not to scale. They also use constructions with a ruler and a pair of compasses, such as bisecting an angle or drawing a perpendicular bisector, and explain why each construction works, for example because every point on a perpendicular bisector is the same distance from both ends of the line.

Another strand is choosing the right kind of thinking for a problem. Some situations are additive, where the same amount is added each time. Others are multiplicative, where quantities scale together in ratio or direct proportion. Pupils extend ratio and proportion into algebra and graphs: direct proportion gives a straight line through the origin, written y = kx, and inverse proportion gives a curve. They also meet trigonometric ratios (sine, cosine and tangent), which are fixed ratios of sides in right-angled triangles with the same angles. Expressing how variables relate, in words, as a formula and as a graph, is part of the same skill.

Reasoning also covers data. Pupils decide what can and cannot be concluded from statistics and probabilities: a correlation does not prove one thing causes another, a small or biased sample may not represent a whole population, and relative frequency from an experiment only estimates a probability. They also assess whether a graph or statement is presented accurately, for instance noticing a vertical axis that starts at 50 instead of 0, or percentages compared across groups of very different sizes.

Words your child will hear

Conjecture: A claim that seems true but has not yet been proved, often noticed from examples.
Counter-example: One case that shows a claim is false. 2 is a counter-example to “all prime numbers are odd”.
Proof: A logical argument showing a statement is true for every case, not just the ones tested.
Deduce: Work out a new fact from facts you already know, giving the reason for each step.
Consecutive: Following one after another in order, such as 14, 15, 16.
Proportional: Two quantities that stay in the same ratio, so doubling one doubles the other.
Correlation: A link between two sets of data. It shows they change together, not that one causes the other.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically
Extend their ability to identify variables and express relations between variables algebraically and graphically
Make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments
Reason deductively in geometry, number and algebra, including using geometrical constructions
Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning
Explore what can and cannot be inferred in statistical and probabilistic settings, and express their arguments formally
Assess the validity of an argument and the accuracy of a given way of presenting information

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Prove that the sum of any three consecutive whole numbers is a multiple of 3.

Try some examples to see the pattern. 4 + 5 + 6 = 15 and 10 + 11 + 12 = 33. Both are multiples of 3. This suggests the claim is true, but it does not prove it.
```
4 + 5 + 6 = 15 = 3 × 5
10 + 11 + 12 = 33 = 3 × 11
```
To cover every case, let the first whole number be n. The next two consecutive numbers are n + 1 and n + 2.
Add the three numbers and collect like terms.
```
n + (n + 1) + (n + 2)
= 3n + 3
```
Factorise. 3n + 3 has a common factor of 3.
```
3n + 3 = 3(n + 1)
```
Since n is a whole number, n + 1 is also a whole number. So the sum is 3 multiplied by a whole number, which is a multiple of 3, whatever n is.

The sum is 3(n + 1), a multiple of 3 for every whole number n, so the statement is proved.

Common mistakes and how to help

Believing that checking several examples proves a rule

A pattern that holds five or ten times feels convincing, and in earlier years finding a pattern was often the whole task.

What helps: Show a claim that works for many cases and then fails, such as n² + n + 41 giving primes until n = 40. Then ask what would convince someone about every number, which leads naturally to algebra.

Using the same letter for two different numbers in a proof

Writing two even numbers as 2n and 2n looks tidy, but it forces both numbers to be equal, so the proof only covers numbers like 6 + 6.

What helps: Use different letters for different numbers: 2m and 2n for two even numbers, or 2n and 2n + 2 for consecutive even numbers. Ask “could these be different numbers?” before moving on.

Adding when the situation needs scaling

If a recipe uses 3 eggs for 2 people, some pupils reason that 6 people is 4 more people, so 4 more eggs. Adding the same amount feels natural because it works in many simple problems.

What helps: Ask what the scale factor is. 6 people is three times as many, so 3 × 3 = 9 eggs. A ratio table with the two quantities side by side makes the scaling visible.

Treating a correlation as proof of cause

When two graphs rise together, the link seems to explain itself. Ice cream sales and sunburn both go up in summer, but the weather drives both.

What helps: Always ask what else could explain the pattern, and whether the sample is large and fair enough to trust. A good answer says what the data suggests and what it cannot show.

How to help at home

Always, sometimes or never true? Offer a claim such as “the sum of two odd numbers is even” or “a multiple of 4 is a multiple of 8”. Ask your child to decide and explain, then look for a counter-example for any that fail.
Hunt for the counter-example. Try claims that sound reasonable: “squaring a number makes it bigger”, “multiplying always increases a number”. Numbers like 1, 0 and ½ break them, and spotting why is good practice.
Question what you read. When a news story or social media post quotes a statistic or graph, ask what the axes show, how many people were surveyed and whether the claim goes beyond the evidence. Ask whether the conclusion would still hold if the chart were drawn fairly.
Scale a recipe or a journey. Change a recipe from four portions to ten, or work out how long a trip takes if the average speed changes. Ask whether the numbers should be added or multiplied, and why.
Ask “how do you know?”. When your child gets an answer right, ask them to say in one sentence why it must be right. Writing that sentence is what earns marks on “show that” questions.

Builds on

Key Stage 3: Developing fluency
Key Stage 3: Reasoning mathematically
Key Stage 3: Solving problems