Where: In school

When: after science and before lunch

**Warm-up patterns**Today, we started skip counting by 4 (first grade) and 6 (second grade). We are extending the damult dice game today and the kids benefit from as much practice as they can get.

# What did we notice (dominoes)?

We split the class into pairs (the extra student went with P) with the following instructions: talk with your partner about what you noticed or discovered about the dominoes puzzle, write down one sentence and be prepared to share with the rest of the class. I assigned the partners and intentionally broke up traditional pairs. I made sure to have some boy-girl mixing and also paired students with different levels of historical engagement. Many kids had questions about what types of ideas they were meant to be discussing, so we gave them the following prompts:

- What were your strategies for solving the puzzle?
- What did you find for the smallest side sum?
- What did you find for the largest side sum?
- How many solutions did you find?
- How did this compare with the triangle puzzle? What was the same, what was different?

With this structure, we had a really good conversation. There were a lot of ideas shared and some good debate about each of the points. Some highlights:

- Some students claimed to solve the puzzle by randomly ordering the paper dominoes we made last week. They were surprised when I praised this approach as they were expecting to be criticized. We discussed that the whole strategy was actually: make manipulatives, create a possible solution, check whether the solution is valid, reorganize into a different configuration. P emphasized that this is widely valid, if they are working on something and get stuck, one thing to try is to use a tool to help them, not to suffer in confusion.
- For each answer we discussed, there were usually multiple ideas. This gave kids an opportunity to talk about what they found and how they found it.
- Comparing with the triangle puzzles was a very rich vein for conversation. The contrast helped them identify structure in the puzzle that, otherwise, would have been either too subtle or too obvious to mention. For examples, the dominoes force the numbers to sit in pairs, the square arrangement has 4 sides vs 3 for the triangle, the domino puzzle had 8 numbers vs 9 (or 6) for the triangle puzzles, and there were some repeated numbers in the domino puzzle while the triangle puzzle had distinct digits.

# What did we notice (damult dice)?

We repeated the discussion process again, this time based on playing damult dice. Again, this was a really good discussion, covering a range of ideas:- the smallest result possible (2= (1+1) x 2)
- the largest result (72 = (6+6) x 6)
- strategies to make the largest result with any given roll
- strategies to make the smallest result with a given roll
- why do we get so many multiples of 3? (more on this below)

# "New" Game: exact damult dice

We turn again to Math4Love's page on Damult Dice. In the comments, there were many suggestions for extensions to the basic game. The one we introduced today was simple: you have to hit the target number of points exactly and you are allowed to subtract the result of your dice roll instead of adding it. The basic calculation remains the same: roll 3 dice, add two, then multiple the sum by the third.This is a very simple extension, but it makes the game much richer. Gone is the simple strategy of accumulating as many points as you can on each roll, replaced by something a lot more subtle. I'm looking forward to hearing what the kids find.

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