Had a great morning working with the boys today that included a particularly fun exchange with my older son that I wanted to write about.

It started with a continuation of a brief conversation on Twitter about Khan Academy. I’ve been quite happy to use their problem bank for review exercises with my kids. I’m glad that the resource is around, and I’m glad that it is free. No need to post the whole conversation, but you can get to it from here if you enjoy reading twitter conversations:

This school year I’m walking through Art of Problem Solving’s Geometry book with my older son and today we were covering the section on exterior angles in triangles. I’ve never taught geometry before and I find thinking about how to talk to my son about basic concepts in geometry to be both fun and challenging. Exploring the beginnings of mathematical proof from a few basic axioms has been particularly fun.

The talk about exterior angles got pushed back a tad because of a challenge problem that gave my son a bit of trouble. The problem is from an old AMC 8 exam and is here:

http://www.artofproblemsolving.com/Wiki/index.php/2003_AMC_8_Problems/Problem_22

Two interesting things came up talking about this problem. The first was showing how you could find the area of a square if you knew the length of one of the diagonals. Maybe not the most interesting piece of mathl, but still pretty cool when you see it for the first time.

More interesting from a math point of view was the idea that could never be equal to an expression like , or even something like I explained to my son that was “more than irrational” and couldn’t be written as the sum of a rational number or square (or other) roots of rational numbers. His response was great – “So it is super irrational?” Well, sort of yes, actually!

These are the conversations that I love having. Going back to the Khan Academy point, I’d much rather have the kids spend time doing a few Khan Academy review problems on fractions, or basic facts about prime numbers (to name two things that I’ve used it for so far this year) if it means I can have these conversations with them.

Not wanting to pass up the opportunity to explore this particular conversation a little more, we went to the whiteboard and I explained the difference between algebraic numbers and transcendental numbers. I even wrote down the series for Sin(x) and Cos(x) to show him that did satisfy some polynomial-like equations, just not ones with finitely many terms. Next, almost if the morning was a set up, he asked if satisfied any special equations.

In response to that question I down the series for and showed how these three equations led to the relation that

With this little side track into transcendental numbers behind us, we finally got around to talking about exterior angles in triangles. We started with a couple of simple examples and then moved on to proving the theorem that the measure of the exterior angle in a triangle is equal to sum of the other two angles in the triangle. The proof we did is probably the standard one and uses the fact that the angles in a triangle add up to 180 degrees. After we finished this proof my son turned to me and said that he thought there was another way to prove the same theorem. That was a nice little surprise and I asked him if he wanted to do this new proof for his morning movie. He agreed and I turned on the camera with no idea of what he was going to say. Turned out that he had a really good idea:

Fun morning. Really love having conversations like these.

Have you explored with them the rotating and sliding pencil approach ? Seeing angles as measured by amounts of rotation. No notation required at all.