Last Sunday we had our first lesson of the year at the Golden Key Russian School – an event that inspired me to start blogging again after the long summer break. This year I am teaching two groups, one consisting mostly of first graders and the other one of mostly 4th and 5th graders. That said, it may be surprising that we had the same main activity in both groups. Fortunately, it worked fairly well in both cases, although of course, the level of discussion varied between the two groups.
The activity was the game NIM – more specifically, 2-pile NIM (described below). I started off by asking the kids to list some games they knew. The ones that were familiar to myself and most of the class I wrote on the board. Here is the list from the 1st graders:
We then discussed the difference between deterministic games and games of chance, and labeled our list. Some confusion (in both groups) was caused by the fact that in a deterministic game you don’t control the moves of your opponents and so some kids argued that this meant that there was “chance” involved. Finally, we agreed that deterministic meant that each player is in full control of his/her moves.
Next, I wanted to illustrate for them what it meant for a player to have a winning strategy. The game I chose for that purpose was very simple: Two players take turns saying numbers between 1 and 9, keeping track of the total. Whoever can bring the total to 10 wins. Naturally, they found this game very silly, but no one in either group had any trouble seeing that the second player can always win if she plays perfectly. The next challenge was to figure out the winning strategy when the target sum was 20 instead of 10. And here is where the difference in age between the two groups showed. In both classes I played the game against two kids, going first each time. All the kids played perfectly and won the game, but the younger kids couldn’t explain the winning strategy in words, even though they intuitively figured it out. The older kids, however, were able to explicitly say that “on the first turn the second player makes the sum be 10 and then he can always make it 20 on the second turn.”
After that prelude, we moved on to our main activity: 2-pile NIM. The game is played as follows: There are 2 piles of stones. On your turn, you are allowed to take any number of stones, but only from one of the two piles. The players alternate taking stones and whoever takes the last stone wins. I had the kids play with 5 stones in each pile. (Unfortunately, there were an odd number of kids in both classes so I had to play with someone both times and did not get around to taking any pictures.)
The kids in both groups really enjoyed the game, but it seemed that most of the kids in the younger group were playing randomly and getting very excited when they would win. However, many of them noticed that if on your turn there is either exactly one stone in each pile or exactly two stones in each pile, then you would lose. In the older group, one of the 5th graders figured out the winning strategy for the second player pretty quickly, but the rest of the class needed some help. With the younger group, I had to explain the strategy to them more explicitly: on her turn, the second player always takes as many stones as the first player just took, but from the opposite pile. At the end, I believe that all of the kids in the older group and most of the kids in the younger group understood the winning strategy for the second player.
I was hoping that after they understood who should win and the winning strategy for the game with two 5-stone piles, the older group would be able to figure out the game with one 5-stone and one 7-stone pile. However, this was not the case. Perhaps this was because there was insufficient time left at the end of class for them to really think it through. I told them to play the game with a parent or friend for homework and plan to discuss it at the beginning of class next time.
I hope to come back to the many variations of this game as the year goes on (more piles, more stones in the piles, as well as the one pile version where there is a limit on the number of stones you can take from the pile). But for now, this was a fun start to a new school year!