### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

# Pythagoras Proofs

### Why do this problem?

This problem shows three different approaches to Pythagoras' Theorem. 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 methods of proof. There are great opportunities for communicating mathematical ideas and debating which method appeals the most and why!

### Possible approach

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?
What do you know? What can you work out from what you know?
Does each step you have written down follow on directly from the last one?
Are all the assumptions you have made valid ones?