Let's Play Math has a post on Euclid's game. Last night I printed off a hundred chart, grabbed a highlighter and sat down at the kitchen table to play game with Catherine.
Note: Taking a look at the game and the rules might be helpful right now.
Within half a dozen moves Catherine realized the winner had been determined by the very first move. A few moves more and she realized half the board would never be marked and why (the highest of our first two numbers was 47). We talked a bit about the possible outcome and decided that, of course, there were only two outcomes, she'd win or I'd win, but that those two outcomes could be determined by four moves at the beginning.
1. She picked an odd number to start.
2. She picked an even number to start.
3. I picked an odd number to start.
4. I picked an even number to start.
Sure enough, she won through her first move of picking an even number. That let us decide how the other 3 initial moves would end a game.
After that, since this was Euclid's game, I googled Euclid and found this page. We read it and noticed an image of a sheet of Euclid's writing. On it were some shapes so I asked Catherine what kind of math she thought Euclid was interested in. 'Geometry"!' she answered.
That reminded me of an activity a friend (who's also a teacher) had shared with me when I mentioned Catherine was having trouble with the difference between 2 and 3 dimensions. So we grabbed some paper and traced a pattern block. I drew an x and y axis and explained that those were the two dimensions would could measure the shape by. Then I went a little further and showed her how we could number the lines and plot the points of the shape. I thought I was losing her but she said, "I know! Remember when we did maps?" and she drew out a little example of a map with a grid over it and showed me how she'd learned to find a square in the grid by the coordinates she'd been given. Awesome.
That down I stacked up pattern blocks on the tracing and said that the shape was now three dimensional and asked her to show me how that was and where the new axis would be. She understood right away and demonstrated it.
Ta da! Now of course I need to get some activities to reinforce this. I found a little whack-a-mole game on the subject and plan to make up some worksheets where can plot points to come up with some shape or secret message. I know this isn't 4th grade math but she seemed to enjoy it so we'll take a little detour and when she comes back to it in a couple of years there will already be a foundation.
Fifteen minutes of wandering off the path.