Tuesday, June 9, 2015

We learn to simplify and Dotty Dice (math class 2 Y2)

who: Baan PathomTham grades 1, 2, 3
when: Tuesday morning
where: in school
what did we use: a pack of playing cards, some cube dice

While our second time with the older kids, this was actually our first time with the new 1st grade students since both were out sick last week.

Warm-up
To get started, we did some skip counting by 2 and counting backward from 20. For counting backward, we got a funny reaction when it came time to say 0. At this stage, neither student was willing/able to go below 0, but we did get to talk a little about whether there are numbers below zero. Next time, we will make this a more explicit part of the class.

Dice sums and re-roll game
Our first game is a simple one. All players have 2 dice (regular, 6-sided, for now). Players roll their dice, identify the numbers that are showing and say the addition number sentence. For example, if someone rolls a 2 and 3, they would say "this is 2, this is 3, 2+3 is 5." Next, they choose how many dice to re-roll, either all, one, or none of their dice. Once everyone has re-rolled, they again talk about the numbers showing, the addition fact, and then figure out which player got the largest sum.

We played a sample game to show them how it works and then split up, one student playing with Pooh and another playing with Josh. As we played, we would ask various questions about what they noticed and why they made certain choices. For example:

• How many sides are on the dice
• What numbers are on the dice? What is the largest, what is the smallest?
• What is the largest sum they could get? What is the smallest?
• What do they think of our choices? Did we play well or badly?

Variation
While we were playing, we noticed there are actually two slightly different versions. In one game (Version A), all players choose which dice to re-roll and then roll at the same time (or, equivalently, in secret). In Version B, players take turns deciding what to re-roll.

We only played as a 2 player game, but there are additional variations if we include more players. Again, they can re-roll simultaneously or in sequence. However, you could also add a points system based on which player has the largest sum, the second largest, etc.

Homework
We asked the students to play this game at home. They should play Version A and Version B at least 5 times (play at least 10 total rounds). Are the games the same or different? Do they use different strategies? Why or why not?

Warm-ups
As with grade 1, we began the older kids with skip counting warm-ups. We did two things slightly differently that I wanted to note. First, when we said that we would skip count by 7, several of the kids grabbed their notebooks with the times table on it. Instead of starting with 0, though, we started with 2. Second, I sat next to one of the students and had him secretly point to the numbers of the sequence on our 100 board while waiting for his turn. That also made it possible for him to predict what was coming next and to figure out the number he would say, which he thought was really cool.

31 strategy
The homework from last week involved playing a variation of the 31 game and investigating strategy. Almost none of the students ended up playing the original 31 version. Instead they generally played with 1 (ace) through 10 with a target of 71. When we talked about strategy, almost everyone had figured out that there was something interesting happening in the 60s, with several ideas that you wanted to make 60 on your turn, though there were some thinking 61 was an interim target. Everyone acknowledged there was complexity if you had already used up all of a particular number and there were some ideas about explicitly forcing that outcome, mostly looking at using all of the aces or other small values.

No one felt they had a clear strategy and no one was sure whether it was best to be first player or second.

I told them this wasn't surprising because the game is actually pretty complex. One of the ways we can get a better understanding is to make it simpler. We then talked about ways to make the game simpler (use fewer card values, have a smaller target) and then began to investigate.

• Ace only: the simplest game we wanted to study was to use only 4 aces. What is the largest target we could use? I joked in each class about playing to 1000. For all the reasonable targets (1 to 4) who wins, first player or second player? This was very easy, but I wanted to emphasize that it is great to start with something we really understand well as preparation for jumping into a more complex version.
• Ace and 2: an obvious next step is to add the next value card. Again, we talked about what the largest reasonable target is and then stepped through strategy for different targets. First player has a winning strategy if we target 1, 2, or 4, while the second player can win targeting 3. Are there any values larger than 3 for which the second player has a winning strategy? We didn't talk through all strategies for all targets, so this is part of the homework.
• Ace, 2, and 3: stepping up again. This time, we played a couple rounds with a target of 24. I was slightly surprised that this held their interest, even on the second round. Again, we worked a bit on strategies for different targets, but we didn't talk through it completely, so this is part of their homework. One student guessed that 17 might have a different strategy to smaller targets.

Dotty Six (another dice game)
The basic idea for this game family comes from NRich, so I won't repeat the basic rules. In addition to the dotty six version, we played a dotty ten version where each cell of the board is only filled once it has ten dots (or tally marks) in it.