. . . and the 5th lesson – the Pythagorean Theorem!

[sorry for the rushed feel of this one, as I mentioned in the last post I’m at a Starbucks and have to be home in 20 min!]

The last post talked about one problem with four lessons:

One Problem with 4 Lessons

This morning I talked through a 5th lesson from this problem. That lesson comes courtesy of this amazing video from Numberphile and Harvard math professor Barry Mazur:

 

So, perhaps the greatest surprise of all from the problem we were talking through last week – Problem #7 from the 2008 AMC 10 A – is that it helps you prove the Pythagorean theorem in a clever way.

The first step was reviewing how area scales when you increase the size of a 2 dimensional object:

 

Next we moved on to checking if area seemed to scale the same way for triangles:

 

Finally the punch line – how the combination of the original problem and the idea of how area scales proves the Pythagorean theorem:

 

I’m really happy about these two projects, but the 2nd one makes me really happy for a couple of reasons. First, the ideas are easy enough that I could talk through them with both kids. Yesterday’s lessons, while important, require a bit more background in algebra and geometry than my younger son has right now.

The second reason this lesson makes me happy builds on the first – the combination of the relatively easy ideas and the amazing result – the Pythagorean Theorem! – is a great example of mathematical reasoning. “Hard” theorems do not necessarily require “hard” ideas!

One problem with four lessons

A few days ago I wrote this post:

Pythagorean Triangles and Differences of Squares

about this problem:

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 🙂