Monday, May 30, 2016

Improv Math and Division Dice follow-up

We had a really good experience playing Division Dice, the game that we introduced a couple of posts ago.  Mainly, I want to illustrate something fun that came out of really listening and paying attention to what the kids are doing and saying. I like to think of this as "improv math," as a way to credit my improv comedy experiences for heightening my awareness of how important this is.

Division Dice for number sense

I was really pleased about the quality of thinking stimulated by the game. We played with the most loose rules (1s are wild, the components of the 2 digit value can be flipped to their 7s complement). That gave a lot of opportunity for the kids to think through options to (a) make whole number divisions and (b) maximize values.

For example, rolling 3, 4, 6:
  • what are the allowed groupings that give a whole number division? Remember, in the 2 digit number, we can use any of the values 1, 3, 4, 6, and it is possible for us to use two 3s or two 4s in our calculation.
  • What is the highest scoring choice?

Division Dice for arithmetic exercises

As a way to create virtual worksheets, this game is mediocre. The basic structure means that students are never dividing by a divisor larger than 6. This leave out a lot of fact families. However, because the kids are trying to maximize their scores, they quickly realize that they can almost always get away with division by 2, occasionally must divide by 3, and rarely get stuck dividing by 4 or 5. I haven't yet seen a case in a live game where division by 6 was necessary.

Fun exploration: what scenarios will require division by 6?

Using playing cards or other dice shapes allows us to extend the possible values and reduce the likelihood of dividing by 2 or 3. However, it also increases the number of cases that don't have a whole number division relationship. We are thinking about ways to incorporate division with remainder and will try out a variant tomorrow.

Improv Extension

Playing at home, the 3, 4, 6, case led J1 to consider: how do 63 ÷ 3 and 64 ÷ 4 compare?
As he contemplated that, I realized that we had a nice sequence of multiples, meaning all of these are whole numbers:

There were several cool things for J1 to observe here:

  • 4 of the 6 quotients end in 1
  • The quotients are all decreasing
  • The drops between successive quotients are themselves decreasing
  • the dividends are equal to the divisors + 60

We pursued this in two ways:
Extension 1: what if we add something else to the dividends?
We tried three versions.

  1. starting with 60 and adding 6 at each step
  2. Starting with 60 and adding 60 at each step.
  3. starting wit 1 and adding 7 at each step

You can see our notes mid-discussion below:

Later, when J2 was also involved, I offered them another sequence: starting with 66 and adding 6 for each increment:
66 ÷ 1
 72 ÷ 2 
78 ÷ 3
84 ÷ 4
90 ÷ 5
96 ÷ 6
120 ÷ 10
132 ÷ 12
150 ÷ 15
180 ÷ 20
240 ÷ 30
420 ÷ 60
3660 ÷ 600
36060 ÷ 6000
We're breaking the rule about the dividends being multiples of the divisors, but the last two calculations are still easy and nicely illustrate the limiting behavior.

Extension 2: can we find other chains of whole number division equations?
We started this by thinking more simply: for chains shorter than 6. For example, what are the smallest K, L, M, N larger than 1 such that all of the following are whole numbers:

K ÷ 1
 (K+1) ÷ 2 

L ÷ 1
(L+1) ÷ 2
(L+2) ÷ 3

M ÷ 1
(M+1) ÷ 2
(M+2) ÷ 3
(M+3) ÷ 4

N ÷ 1
(N+1) ÷ 2
(N+2) ÷ 3
(N+3) ÷ 4
(N+4) ÷ 5

After getting the shorter cases under our belt, we then went for a chain of length 7. J2 worked by himself for a while, then came back and announced that no chain with dividends smaller than 100 would work.  He went away and then came back quickly with the idea that maybe we could add 7! to each divisor.

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