Pythagoras Proofs

Why do this problem?

This problem shows three different approaches to Pythagoras' Theorem, and links to a fourth one (A Matter of Scale). It could be used with a group who have recently met the theorem, to provide a variety of ways of thinking about it, or with a group who are familiar with the theorem, to explore different proofs. There are great opportunities for communicating mathematical ideas and debating which method appeals the most and why!

There is also a link to a proof of the Converse of Pythagoras' theorem, which uses proof by contradiction.

Possible approach

This problem featured in an NRICH Secondary webinar in March 2022.

These printable cards for sorting may be useful: Pythagoras Proofs

Divide the class into small groups and assign each group one of the proofs to work on. Explain that once they understand how the proof works they will be asked to produce a poster or presentation to persuade others of the value of their proof.

After allowing plenty of time for exploring their proof and producing their explanation, bring groups together so that each group can present their proof to others who have worked on a different proof. Everyone should get the opportunity to see all three proofs explained. Encourage learners to be (constructively) critical of each other's proofs and to question points that aren't clear.

Finally, bring the whole class back together for a vote on which proof they thought was the clearest and most satisfying, and a discussion on the merits of each proof.

Key questions

What exactly are you trying to prove?

Possible extension

Read more about proof here.

Possible support

Tilted squares can be used to introduce Pythagoras' Theorem.