My older son has been studying in Art of Problem Solving’s Introduction to Geometry book this summer. He’s now in a section about polygons, but the questions in the book are drifting from “solve this” type questions to “prove this” type questions.

He’s struggling a little bit, but seems to be very interested in the proof process.

Today’s question was about octagons -> Prove that if you extend the sides of a regular octagon you get a square.

His initial thoughts about the problem are in the video below. It takes a little while to get going, but eventually some ideas about symmetry help him start getting a proof together:

With some initial ideas about symmetry helping him see that several lines he’s drawn in the picture are parallel, he now starts doing a little angle chasing. Once again an idea about symmetry (this time rotation) helps him see that two triangles are congruent:

Now that we’ve found a few congruent triangles and also a square in the center, we can finally chase a few more angles around and prove that the outside figure is a square:

It is really interesting to see a young kid thinking about what it means to prove something. We’ve spent no time talking about the idea of “proof” in the abstract, but is still fun to see a kid thinking about what an proof should look like.