After the time-limited Vivas, it was important to find sufficient time for students to get really immersed in a problem, to feel what 8-month Tom was feeling when he pushed through the mud to the light beyond.
The problem: Proving Pythagoras, using Mathologer’s excellent video, and Cut-the-Knot’s incredible bank of proofs.
300 minutes of lesson time to work together + homework and independent study. Outcome – 5 minute group presentation, and individual write-up. Useful to discuss the similarities and differences of presentations and write-ups. What is the correct level of rigour?
Assessment criteria for presentation:
- Did you attempt an original proof (either of Pythagoras’ Theorem or a related conjecture)
- Did you think about multiple representations of Mathematics?
- Did you convey a sense of wonder and awe at the things you were exploring?
- Did you present well?
Examples of student exploration:
If I just look at the diagrams on the cut-the-knot website, can I work backwards to what the proof must be?
How many 60-triples are there? (These are called Eisenstein triples by other Mathematicians…). This required some really excellent coding. A few patterns began to emerge but nothing concrete – and that is absolutely okay.
Can I prove the following theorem for a right-angled triangle? For any triangle? What if I pretend that I don’t know the cosine rule, can I still prove it?
What if I continue the pattern on and on? Can I prove that there are some parallel lines in this crazy diagram? (Geogebra suggests there are parallel lines, but the group couldn’t prove it…)
Great to encourage students to talk precisely and entertainingly about their mathematical journey, and to provide a space for kind, specific and helpful feedback.