We are spending a few weeks working through this amazing book:

Currently we are looking at the second on the Euclidean Algorithm, and last night I had a chance to talk through some of the ideas with my older son.

Here are his initial thoughts on the Euclidean Algorithm after reading through a few pages of chapter 1. We worked through the example of finding the greatest common divisor of 85 and 133:

Next we moved on to trying to solve the Diophantine equation 133*x + 85*y = 1. We had already looked at this equation on Mathematica, but had not discussed how to use the ideas from the Euclidean algorithm to solve it.

In this video you’ll see how my son begins to think through some of the ideas about how the Euclidean algorithm helps you solve this equation.

By the end of the last video my son had found some ideas that would help him solve the equation 133*x + 85*y = 1. In this video we finish up the computation and (luckily!) find a solution that was different than then one Mathematica found.

Comparing those two solutions helps to show why there are infinitely many solutions.

I’m on the road today, but hope to be able to talk through some of the ideas from the Euclidean Algorithm with my younger son tonight. The topic is a great one for kids – there are lots of neat math ideas to think about (and to review!). Hopefully we’ll get to explore some of the connections from geometry, too.