# A continued fraction experiment

I’m a big fan of continued fractions – especially the many different ways that you can use them to help kids learn elementary math. Right now I’m studying square roots with my younger son and he’s taken quite a liking to continued fractions, too. See yesterday’s project, for example:

A surprise square root of 2 discussion

I intended for the focus of that discussion to be the standard proof of why $\sqrt{2}$ is irrational. Instead, though, a large part of the discussion was about how you could use the continued fraction for $\sqrt{2}$ to prove that it was irrational.

Having not learned my lesson already, I asked my son to sketch a proof of why $\sqrt{5}$ is irrational, and he went down the continued fraction path again.

Even though this project is pretty difficult and many of the parts are really over my son’s head, I think this was a useful exercise. I also think that it all pretty much stands on its own, so I’ll present the four steps below without much comment.

Following this project, my son asked me if we could study more about continued fractions this week rather than just studying the current chapter in our book about square roots. Something about this topic has really caught his attention!

The continued fraction calculator we are using in the last video is here:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html