Key Stage 3 maths · Working mathematically

KS3 reasoning in maths: conjectures, proof and proportion

Reasoning is the second of the three threads that run through Key Stage 3 maths. Pupils in Years 7, 8 and 9 are expected to explain why something is true, not just calculate it. They make and test conjectures, look for proofs and counter-examples, decide when a problem needs additive, multiplicative or proportional thinking, and judge what data can and cannot tell us.

Reasoning mathematically, explained

In secondary maths a correct answer is only part of the job. Pupils are asked “how do you know?” and “will that always work?”. The National Curriculum for Key Stage 3 describes this as reasoning, and it is taught through the ordinary topics rather than as separate lessons. The central routine is to make a conjecture, which is a statement that seems to be true, and then test it with examples. A pupil can look for a proof, meaning an argument that shows it is true in every case, or a counter-example, a single case that shows it is false. Many examples that work do not make a proof, but one example that fails settles the matter.

Reasoning connects number, algebra and graphs. Pupils extend their understanding of the number system and see how a number relationship can be written as a rule and drawn as a graph. They identify the variables in a situation, which are the quantities that can change, and express how they are related. If a square has side s, the perimeter P is 4s, and a straight-line graph through the origin shows this too. Algebra is the language for saying that something is true for any number, which is why it is central to proof.

Ratio and proportion are developed in measures and geometry as well as number. Pupils learn to decide whether a situation is additive, where amounts are added or taken away, multiplicative, where they are scaled up or down by a factor, or proportional, where two quantities keep the same ratio. If 6 pancakes need 2 eggs, then 18 pancakes need 6 eggs because everything is multiplied by 3. Adding 12 pancakes would not mean adding 12 eggs. Pupils write proportional relations algebraically, for example cost = 1.5 × number of items, and recognise scale factors in maps and similar shapes.

In geometry, pupils begin to reason deductively. They use known facts, such as the angles of a triangle adding up to 180° or angles on a straight line adding up to 180°, to work out unknown angles step by step, giving a reason for each step instead of measuring. They also carry out constructions with a ruler and compasses, such as the perpendicular bisector of a line, and explain why they work. In statistics and probability they explore what can and cannot be inferred. A graph showing two things rising together does not show that one causes the other, and a small sample may not represent a whole group. Pupils begin to set out arguments like these formally, as a chain of clear statements with a reason for each. The programme also says that pupils who grasp ideas quickly should meet richer and more demanding problems, before being moved on to new content.

Words your child will hear

Conjecture: A statement that seems to be true but has not yet been proved, such as “the sum of three consecutive numbers is a multiple of 3”.
Proof: A logical argument that shows a statement is true in every case, not just in the examples tried.
Counter-example: One example that shows a statement is false.
Variable: A quantity that can change, or a letter that stands for such a quantity.
Direct proportion: Two quantities that stay in the same ratio, so doubling one doubles the other.
Deduce: Work something out logically from facts you already know, with a reason at each step.
Correlation: A link between two sets of data, such as both rising together. It doesn’t show that one causes the other.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Extend their understanding of the number system; make connections between number relationships, and their algebraic and graphical representations
Extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically
Identify variables and express relations between variables algebraically and graphically
Make and test conjectures about patterns and relationships; look for proofs or counter-examples
Begin to 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 begin to express their arguments formally

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

A worked example

Ali says: “The sum of any three consecutive whole numbers is a multiple of 3.” Is he right? Then test: “The sum of any four consecutive numbers is a multiple of 4.”

Test some examples. 1 + 2 + 3 = 6, 4 + 5 + 6 = 15 and 10 + 11 + 12 = 33. All are multiples of 3, and each is 3 times the middle number. This supports the conjecture but doesn’t prove it.
To prove it for every case, let n be the smallest number. The three numbers are n, n + 1 and n + 2.
```
n + (n + 1) + (n + 2)
= 3n + 3
= 3(n + 1)
```
3(n + 1) is 3 multiplied by a whole number, whatever n is. So the sum is always a multiple of 3. Ali is right, and the proof explains the pattern.
Now test four consecutive numbers. 1 + 2 + 3 + 4 = 10, which is not a multiple of 4. One counter-example is enough, so the second conjecture is false.
Algebra shows why. The sum is n + (n + 1) + (n + 2) + (n + 3) = 4n + 6, which is the same as 4(n + 1) + 2. It is always 2 more than a multiple of 4. Check: 10 is 2 more than 8.

Ali is right about three numbers, because the sum is always 3(n + 1). The statement about four numbers is false: 1 + 2 + 3 + 4 = 10 is a counter-example, and the sum is always 2 more than a multiple of 4.

Common mistakes and how to help

Believing that a few examples prove a rule

If a pattern works three or four times, it feels certain. But some rules fail only after many cases. The expression n² + n + 41 gives a prime number for every whole number from 0 to 39, then gives 41² when n is 40.

What helps: Ask “can you find an example where it doesn’t work?” and treat finding one as a success. For a true rule, ask for a reason in words, a diagram or algebra.

Adding when the situation needs scaling

Additive thinking is familiar. If 6 pancakes use 2 eggs and the pancakes go up by 12, a child adds 12 to the eggs and gets 14 instead of 6.

What helps: Use a ratio table and ask “what has everything been multiplied by?”. Sketching a small version of the problem helps to show which type of reasoning is needed.

Assuming a link in data means one thing causes the other

A graph might show ice cream sales and sunburn cases rising together. It is tempting to see a cause, but hot weather could explain both.

What helps: Ask “what else could explain this?” and “how many people were asked?”. Discuss what the data could show and what it couldn’t.

Using what a diagram looks like instead of what is known

Geometry diagrams are often marked “not drawn accurately”. Pupils who say two angles are equal “because they look equal” have skipped the reasoning.

What helps: Make a habit of finishing the sentence “I know this because…”, quoting a fact such as “angles in a triangle add up to 180°”.

How to help at home

Always, sometimes or never? Offer statements such as “multiplying makes a number bigger”, “a square number is even” or “the sum of two odd numbers is even”, and ask which of the three words fits. Finding a counter-example, such as 0.5 × 4 = 2, is the most useful part.
Ask “how do you know?”. When your child gets an answer right, ask for the reason as well. Even a diagram on the back of an envelope counts as reasoning, such as dots arranged in pairs to show why two odd numbers always add up to an even one.
Scale a recipe on paper. A recipe for 4 people needs 300 g of rice. How much for 6? Ask whether they added or multiplied, and why 302 g can’t be right. This is proportional reasoning in an ordinary setting.
Question what a headline claims. When a news item says one thing is linked to another, ask what the evidence is, how many were asked and whether anything else could explain it. This is the statistical reasoning that pupils practise in class.
Flip a coin. Toss a coin 20 times and record the results. Ask whether you would expect exactly 10 heads, and whether tails is more likely after five heads in a row. It isn’t, because each toss is independent.

Leads on to

Key Stage 4: Fluency in number and algebra
Key Stage 4: Reasoning and proof in maths
Key Stage 4: Problem solving and modelling
Key Stage 4: Surds, indices and proof (higher)