I bought a copy of Donald Knuth’s Surreal Numbers recently:
Reading it yesterday gave me the idea to revisit the surreal numbers with the boys today. Our previous look at the surreal numbers was inspired by Jim Propp’s off-the-charts-excellent post:
Although I hope today’s project stands on its own, our previous projects inspired by that post are here if you are interested:
Walking down the path to the surreal numbers Part 1
Walking down the path to the surreal numbers Part 2
Today I started by reminding the kids about the game “checker stacks” that Propp describes in his post and then playing through a few simple examples:
Next up was our first challenge – finding the value of a red + blue checker stack. The boys determine that the value is negative fairly quickly. After a bit of investigating they determine that the value is -1/2. I think this is a wonderful example of what kids doing math looks like:
Now we moved to a problem that is a bit more challenging – a blue – red – red stack. This is a more difficult investigation, but the boys spend about 10 minutes exploring and experimenting. Over the course of the next two videos they determine that the value is 1/4.
Again, I think this is a really nice example of what kids doing math looks like. I’d love to try out this investigation with a larger group of kids:
There are lots of directions to go with this investigation – tomorrow I’d like to explore the “deep blue,” “deep red,” and “deep purple” checkers discussed at the end of Propp’s post. I’m excited to see if kids can understand the ideas that come up with these special checkers.
Mike's Math Page 
The third video made me realize a difference in how I guided my kids in investigating the games: I would frequently ask them about the value of intermediate positions as they analyzed possible moves. This was partly based on my own habits analyzing the games, but also was an intentional decision to give them some practice with some simple fraction calculations and inequalities.
I don’t have a strong opinion about which approach to cuing and amount of scaffolding is better, just wanted to note this difference.
Also, I did see that your older son was getting this when he recognized positions that had value 0, but there were other times they could have known whether the position was positive or negative if they had considered the values of intermediate positions.