Dice mining (grades 2 and 3 math)

Notes for Grade 1 will come later

This week, we continued our run of dice games. This is a simple game we found in Marilyn Burn's About Teaching Mathematics. She calls it Two Dice Sum Game, but I like Dice Miner better (to pair with Dice Farmer, of course). Here's how to play:

1. Students make a number line with slots for each integer 2 to 12
2. Everyone gets 11 blocks or other counters (we used unifix cubes which were nice for stacking).
3. Players put their counters on the numbers, distributing them in whatever way they want. In particular, they can put more than one counter on a number or none on a number.
4. Everyone takes turns rolling two dice (normal 6-sided). Players can take one counter off their board for the sum of the two dice. For example, if I have 3 counters on 7 and roll 6+1, then I can take one counter off and now have 2 counters remaining on 7. If I had no counters on 7, then I would not take any action.
5. The first player to have all counters removed is the winner.

Any activity with construction cubes is bound to create opportunities for other mini-conversations:

• using the cubes as a ruler to make the number lines
• comparing how many cubes one person has to another by measuring lengths (before we made sure everyone had exactly 11)
• using the cubes to form a ruler to measure other things (our whiteboard, a table leg, a friend)
• making a pattern with light and dark colors from the cubes

As to the main game, there is some basic arithmetic practice through all the addition, but the real interest lies in trying to figure out how best to arrange your counters at the start. All the kids had their own hypotheses, but it was interesting to see these highlights:

• One student realized that 7 was the single most likely result and put all his cubes on 7 (this did not win that round)
• Most students started with a uniform distribution, one cube on all numbers
• There was a surprising amount of enthusiasm for 2 and 12, with many students initially playing multiple cubes on those extremes
• One student tried to be tricky and put cubes half on 6 and 7, presumably planning to take them off if either number came up.

Toward the end of the class, there were two comments that were really interesting. First, one student suggested rolling the dice many times, recording the results, and seeing how often all the numbers came up. This would then inform his strategy for how to distribute the dice.

We asked the students how many times they would have to roll. Would 1 or 2 rolls be sufficient? No, everyone was sure that wasn't enough. What about 25 times? One student pointed out that there are 11 slots, so 25 times is only an average of 2 per slot, so that didn't feel like enough to get a good sense of the "true" answer. Most students were eager to see for themselves, so this became their homework.

Homework

1. test the distribution by rolling two dice 100 times (or more) and tallying how often each number comes up as the sum of the two dice.
2. Play Dice Miner 5 times with your friends/parents at home. Record your initial starting position for the cubes, how many rolls to take them all off and who won each round.