This morning I stumbled on an interesting review problem in our

Introduction to Geometry book:

“If all 3 altitudes of a triangle are equal, is the triangle necessarily equilateral?”

This is a fascinating question for kids studying geometry, and since we’ve been focusing on basic proofs lately, I thought it would be fun to film my son working through it “live.”

The whole discussion took about 7 minutes, but I broke it into two pieces just for processing purposes. The first 3 minute video goes like this:

(1) About 1 minute introducing the problem.

(2) About 1 minute with my son checking out a 45-45-90 right triangle to understand that situation, and

(3) About 1 minute looking at an equilateral triangle.

I was really happy to see him looking at some simple cases to get his arms around the problem:

The next 4 minute video roughly breaks down as follows:

(1) About 1 minute of being stuck and wondering what to do,

(2) Then a little bit of wondering if inscribed or circumscribed circles might come into play. This might seem like an unusual thing to wonder about, but it was the last topic that we covered in the book.

(3) Eventually in working down the list of things that he knows about a triangle he arrives at the area formula A = (1/2)*Base*Height. That’s the key idea for this problem.

To me the entire 7 minutes is a great example of what learning math looks like. Nothing to do with speed, just looking at some examples that are easy to understand and then wondering what sorts of properties that we already know that could help us with the new problem. Really fun for me to see these ideas play out this morning.