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Proof for All (St)ages

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Creating convincing arguments or "proofs" to show that statements are always true is a key mathematical skill.

The problems in this feature offer students the chance to identify number patterns, make conjectures and create convincing mathematical proofs.

Many of the problems in this feature include proof sorting activities which challenge students to rearrange statements in order to recreate clear, rigorous proofs.  These tasks aim to introduce students to the formality and logic of mathematical proof.

You can watch a recording of the webinar in which we discussed the mathematical thinking which can be prompted by these activities.

The last day for students to send in their solutions to the live problems is Monday 31 January.

Plus magazine has a selection of interesting articles about proofs here

Three neighbours
problem
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Three neighbours

Age
7 to 14
Challenge level
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Take three consecutive numbers and add them together. What do you notice?
Where are the primes?
problem

Where are the primes?

Age
11 to 16
Challenge level
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What can we say about all the primes which are greater than 3?
Three consecutive odd numbers
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Three consecutive odd numbers

Age
11 to 16
Challenge level
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How many sets of three consecutive odd numbers can you find, in which all three numbers are prime?
Adding odd numbers
problem

Adding odd numbers

Age
11 to 16
Challenge level
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Is there a quick and easy way to calculate the sum of the first 100 odd numbers?
What does it all add up to?
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What does it all add up to?

Age
11 to 18
Challenge level
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If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
Different Products
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Different Products

Age
14 to 16
Challenge level
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Take four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Impossible sums
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Impossible sums

Age
14 to 18
Challenge level
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Which numbers cannot be written as the sum of two or more consecutive numbers?
Difference of odd squares
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Difference of odd squares

Age
14 to 18
Challenge level
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$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?
Adding odd numbers (part 2)
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Adding odd numbers (part 2)

Age
16 to 18
Challenge level
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Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?
KS5 Proof shorts
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KS5 Proof shorts

Age
16 to 18
Challenge level
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Here are a few questions taken from the Test of Mathematics for University Admission (or TMUA).
Direct logic
problem
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Direct logic

Age
16 to 18
Challenge level
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Can you work through these direct proofs, using our interactive proof sorters?
Dodgy proofs
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Dodgy proofs

Age
16 to 18
Challenge level
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These proofs are wrong. Can you see why?