What a kid learning geometry can look like

My younger son is working through a bit more of Art of Problem Solving’s Introduction to Geometry book this summer. Yesterday he came across a problem that have him a lot of trouble.

The problem asks you to prove that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.

Yesterday he worked through the solution in the book – today I wanted to talk through the problem with him. We started by introducing the problem and having my son talk through a few of the ideas that gave him trouble:

Next he talked through the first part of the solution that he learned from the book. We talked through a few steps of the algebra, but there were still a few things that weren’t clear to him.

Now we dove into some of the algebraic ideas that he was struggling with. One main point for him here, I think, was labeling the important unknowns in the problem.

For the last part, I wrote and he talked. I did this because I wanted him to be able to refer to some of our prior work. The nice thing here was that he was able to recognize the main algebraic connection that allowed him to finish the proof.