When I was homeschooling my daughter for first and second grade we used a LOT of cards, dice, and coins to play math. Our favorite card game was Uno. I’d hand her my handful of lost points and say, “How big a win was it for you?!” giving her an opportunity to be *very* specific about the magnitude of her victory. I’d casually say, “How many different tens can you find?” or “What’s fifty plus twenty [the value of the Wild Draw Four + Skip cards]?” As a result she’d be skip counting by tens and sometimes fives, easily finding different combinations of numbers to make ten, adding numbers to sums way past twenty, and, in the process, improving her capacity for mental arithmetic.

Now she’s in 6th grade. In November we decided, at her request, to homeschool again. We’re doing part time at school and part time at home which her school is fine about. I wrote in an earlier post how and why I decided that we would be Gameschooling our math time and our success with the card game Twelves. I ended that post musing about wanting to find other card or dice games about integers. Well, not a day or two later I ran across a super easy, fun, but by no means trivial, dice game in the Table Talk Math newsletter (**subscribe!!**) created by Molly Babcock that takes things into integer land.

We’ve done some work with adding and subtracting positive and negative numbers on a number line. I used some supplemental material on the Bridges in Mathematics site; the Integer tug-of-war game was a little boring but it set the stage for conceptualizing integer operations, and integers themselves, as processes and objects that have a location, a direction, and a magnitude.

The basics of the integers dice game is to throw the first dice, throw the second dice, subtract the second amount from the first and then roll one dice to determine whether the score goes to zero or whether you get to double your score. (There’s more to it, but this is the gist.)

Today I decided to change things up a little. Instead of throwing single dice, I wondered what would happen if we threw two dice and multiplied them to get the positive number, then do it again to get the second number. What happened is that my kid got in some much needed multiplication review and…the scores got BIGGER, by which I think I mean, “more magnanimous” which probably means their magnitude had gone beyond the single digits in both directions. This is all to say that double digit dice throws got us to some really, REALLY interesting territory.

All was going well until my kid started adding up *my* score. Notice that the -8 comes first in the the tally list of scores for each of our rounds. I forget what her first answer was, but I wasn’t so sure about it myself so I drew out the number line you see in the bottom right of the picture. Essentially my approach was to visualize 12 then “go back” 8 which got me to 4.

My kid, on the other hand was not amused. Her number line is in the middle of the page. She said you don’t need to write out your number line all the way to 12 if you start at -8.

I ended the ~~debate~~ conversation by casually mentioning that it was interesting that we had used different strategies but had ended up with the same result.

In the back of my mind I’m thinking there’s something related to the commutative property going on, but I would love some feedback and/or context. I am strong in some areas of math, but love playing math games with my kid because I can bolster my own number sense along with hers. So, I am SUPER open to hearing your thoughts and insights about both integer operations and our reasoning.

Sounds like the commutative property to me. As when adding positive numbers, you can start with either one, so you pick the order that seems to you the easiest to think about. Just be careful to keep the negative sign with its own number — which should be less of a problem in the context of a game, where the numbers have meaning, than it would be on a prealgebra worksheet.

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Yes, I would say that’s the commutative property, because essentially what you’re doing is (-8) + 12 or 12 + (-8). Either way will give you the same solution. 🙂

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