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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.

problem

Favourite

Three neighbours

Age

7 to 14

Challenge level

Take three consecutive numbers and add them together. What do you notice?

problem

Three consecutive odd numbers

Age

11 to 16

Challenge level

How many sets of three consecutive odd numbers can you find, in which all three numbers are prime?

problem

Adding odd numbers

Age

11 to 16

Challenge level

Is there a quick and easy way to calculate the sum of the first 100 odd numbers?

problem

Where are the primes?

Age

11 to 16

Challenge level

What can we say about all the primes which are greater than 3?

problem

What does it all add up to?

Age

11 to 18

Challenge level

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

problem

Different Products

Age

14 to 16

Challenge level

Take four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

problem

Impossible sums

Age

14 to 18

Challenge level

Which numbers cannot be written as the sum of two or more consecutive numbers?

problem

Difference of odd squares

Age

14 to 18

Challenge level

$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?

problem

Adding odd numbers (part 2)

Age

16 to 18

Challenge level

Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?

list

KS5 Proof shorts

Age

16 to 18

Challenge level

Here are a few questions taken from the Test of Mathematics for University Admission (or TMUA).

problem

Favourite

Direct logic

Age

16 to 18

Challenge level

Can you work through these direct proofs, using our interactive proof sorters?

problem

Dodgy proofs

Age

16 to 18

Challenge level

These proofs are wrong. Can you see why?

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.