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For younger learners

  • Early Years Foundation Stage

Sums, Squares and Substantiation

Creating convincing arguments or "proofs", to show that statements are always true, is a key mathematical skill. The problems in this feature offer you the chance to explore number patterns and create proofs to show that these are always true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs.  There are also a selection of "dodgy proofs" where your challenge is to find out where the logic breaks down.

Plus magazine has a selection of interesting articles about proofs here.

What Does it All Add up To?

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

Impossible Sums

Age 14 to 18
Challenge Level Yellow starYellow star
Which numbers cannot be written as the sum of two or more consecutive numbers?

Difference of Odd Squares

Age 14 to 18
Challenge Level Yellow starYellow star
$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?

Direct Logic

Age 16 to 18
Challenge Level Yellow star
Can you work through these direct proofs, using our interactive proof sorters?

Adding Odd Numbers (part 2)

Age 16 to 18
Challenge Level Yellow star
Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?

KS5 Proof Shorts

Age 16 to 18
Challenge Level Yellow star
Here are a few questions taken from the Test of Mathematics for University Admission (or TMUA).

Dodgy Proofs

Age 16 to 18
Challenge Level Yellow starYellow starYellow star
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.

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The Value of Working Systematically

These problems are best solved using systematic approaches.

Integration as Area

These problems invite you to explore integration as area, and use area integrals to solve problems.

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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