# A surprise square root of 2 discussion

I’m having a bit of a roller coaster ride through square roots with my younger son. Sometimes things that I think will be hard about square roots are easy for him, and sometimes things I think will be easy are hard. Today was the latter case as some initial discussions about the square root of 2 led to more confusion than clarity. So I decided to scrap the overall plans for today and just talk about about $\sqrt{2}$. The new goal was to see why it was not rational.

So, we started off by discussion why it wasn’t an integer:

Next we tried to see if the square root of 2 was a fraction. I intended to talk about the standard proof by contradiction here, but about 30 seconds in my son remembered our continued fraction approximation for $\sqrt{2}$. That surprise memory led to a quick review of the first couple of convergents in the continued fraction expansion. We saw some fractions that were nearly equal to $\sqrt{2}$ but none of them were exactly equal.

He understood that if we could write $\sqrt{2}$ as a fraction, the continued fraction expansion would eventually stop (it may be a stretch to say that he understands this, but he at least has the intuition that this fact would be true). Since the continued fraction expansion goes on forever, there must be no rational number that is exactly equal to $\sqrt{2}$.

Finally, we covered what I intended to cover in the last little talk – the usual proof by contradiction that $\sqrt{2}$ is not rational. We end the conversation by mentioning some other numbers that are not rational – some for the same reason as $\sqrt{2}$ and some for other reasons.

So a fun and unplanned discussion about the square root of 2. Hopefully these little side discussions end up building up his number sense a little and help him gain a better understanding of square roots.

# A great problem for kids from the NY Times Wordplay blog

[sorry for what is likley a sloppy write up – had to do it at 5:00 am because of some internet problems last night]

Saw a great problem for kids yesterday via a Steven Strogatz tweet:

Thought it would be a fun one to go through with the kids, and it turned out to be even better than I was expecting as the kids approached the problem completely differently than I did.

I started off with my younger son. We spent the first half of this year going slowly through Art of Problem Solving’s Introduction to Number Theory book, so he has seen problems similar to this one. Another reason that I thought this would be a great problem for him is that we just began talking about square roots last week. So, well-timed Wordplay blog!

His reasoning for how to make the number 256 out of 5’s and 7’s around 4:00 in the movie made me really happy.

Next up was my older son. The problem gave him quite a bit of trouble in the beginning, and in fact he told me at one point he was stuck. We’ve learned an extremely important lesson from James Tanton about what to do when you are stuck – try something.

When he tried something, he found the path to the solution. I think his struggle through this problem was really productive.

Next up we moved to a little “broken calculator” computer program I wrote on Khan Academy’s site to illustrate the problem:

The NY Times Wordplay Blog’s Broken Calculator Problem

Using this calculator we talked through an approach to solving the problem that didn’t require you to find 256 first. Sorry this part is broken into two videos, I accidentally turned off the camera when I was moving it – oops (and the noise in the background is our dishwasher).

Finally, we wrapped up by quickly talking through why you could make every positive integer other than 1. I was much more interested in having the kids think through the first part of this problem rather than this part, so that’s why we didn’t go into all of the details here. Still, they were pretty surprised to see (even informally) that you could make every positive integer other than one.

So, a great problem from the Wordplay blog led to some great math conversations with the kids. The only little issue was some internet problems at our house last night that meant I had to write this one up at 5:00 am – so sorry for the likely sloppy writing.