# One problem with four lessons

A few days ago I wrote this post:

Pythagorean Triangles and Differences of Squares

Problem #7 from the 2008 AMC 10 A

Yesterday I used the problem to show 4 different things that we could learn from it. This morning we looked at a 5th. Have to write up those two projects quickly – I’m at a Starbucks and have to be back home in a hour!

The first piece of the project was reviewing the problem itself and how we solved it. My son chose the solution that used similar triangles:

Having talked through the similar triangles solution, we moved on to talking through an algebraic solution. Although I didn’t film it, my son’s original attempt at a solution (the one that gave him trouble) was an algebraic solution that resulted in the need to find the solutions of a 4th degree polynomial. Our algebraic solution here is a little less complicated, thankfully!

Next up, an interesting lesson – without really realizing it, in solving this problem we’ve learned how to construct square roots with a compass and straight edge (assuming we are given two lengths). This lesson is a nice little surprise!

Finally, connecting the constructing of square roots back to algebra allows us to see a picture for the “arithmetic mean / geometric mean” inequality. We approach this inequality geometrically first, and then algebraically.

It was not at all obvious to me that there were so many lessons hiding in this problem. Thinking about it now, it really is amazing how much this one little problem has to offer, and the biggest surprise is the next blog piece 🙂