# I think you can share the surreal numbers with kids

A few months ago I saw a really cool blog post by Jim Propp about the surreal numbers. It inspired me to try explaining the some of the ideas to my kids and it went really well:

Walking down the path to the surreal numbers Part 1

Walking down the path to the surreal numbers Part 2

Then, a few weeks ago, I saw Donald Knuth’s book about the Surreal Numbers:

which inspired me to revisit these prior projects:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

The second time around we explored slightly more complicated ideas -> 1/4 and $\infty + 1$, for example.

These two visits to the surreal numbers made me think that kids would really have fun seeing them, so I’ve decided to do a dive into the surreal numbers at next week’s 4th and 5th grade Family Math night.

My goal will be to make the hour long lesson / discussion look a lot like the two prior excursions with my kids:

(1) Introduce the game of “checker stacks” from Propp’s blog.

(2) Explain that the game positions can be represented by numbers and that the game has special properties when the position is positive, negative, and 0. The kids (and parents) will have to take this part for granted.

(3) Begin to explore the positions by looking at game positions with integer values. I’ll have the kids play 1 on 1 games versus their parents here.

(4) Explore the game positions that have value 1/2 and -1/2. This is where things get fun ðŸ™‚

Now – based on how things go up to this point we can either try to explore the positions that have value 1/4 and -1/4 or just skip ahead to infinity. With only an hour for the activity, I’ll probably skip 1/4 to be sure that we don’t have to skip infinity!

(5) To get to infinity I have to introduce the “deep blue” and “deep red” checkers from Propp’s blog. Hopefully a couple of examples will help, but this part should also be a neat way for the parents and kids to explore the properties of these checkers.

(6) Once we discover that these checkers have values of plus and minus infinity, we can talk about infinity +1 and things like that.

(7) I’ll wrap up with with the sort of inverse of infinity -> the stack whose value is positive, but smaller than any positive number you can think of!

(8) At the end I’ll mention Siobhan Roberts’s biography of John Conway:

and the quote from the book that the surreal numbers sort of break your brain when you learn about them for the first time.

I hope this will be a fun lesson for the kids in addition to breaking their brains – ha ha ðŸ™‚

# What learning math can look like: Guessing vs. Knowing

My older son and I looked at problem 19 from the 2014 AMC 10 b this morning:

He told me that he’d solved the problem correctly this morning and it seemed like a neat problem to talk about, so I wanted to hear his solution. It turned out that he had some good intuition, but not exactly a full solution:

So, after the rough argument that the answer was 1/3 I wanted him to try to find a more complete argument. Finding the right argument proved a little tough, so I asked him to go back to the information that came with the original problem. Using this information helped him find a more complete solution:

It was interesting to me to see the “hand waving” argument that showed the probability was likely to be around 1/3, and also instructive to see how much more work was required to find the complete solution. The whole process today gave me some insight into how kids think about the difference between thinking something is true and knowing something is true.