Proofing #SubtractionbyHeart! The new cards are looking lovely :-)
We should have these out by early 2026!
#iteachmath #mtbos

I either need ones on the first die to make 2, or 2s on the second die. In the first case, that's going to force die 2 to consider of odd numbers, and die 1 to contain three blank sides and three ones.

Forgot to check the total. But I'm okay - it's 21 dots totay!

There are two other possibilities: a 2 by 3 and 3 by 2 array of ones.

Each of these is going to lead to potentially more choices. Though once one die is complete, it forces the numbers on the other die.

Let's see - let's go with the first image and see the options.

Yep, the rest of this is forced.

Another solution! Still, there's more here.

Interestingly, I could also have the second die consist entirely of ones, and get a 6 by 1 array instead.

Wait a minute... I see where this is going. This is going to force Die 1 to lose one face from every side. I'll put the 1 in on die 1, and I'll get all six 2s. And so on.

Interestingly, those six spots are going to need to be arranged in a rectangular shape, since they're determined by the column and row numbers.

So I could have a 1 by 6 rectangle of ones, which could only happen if the first die was blank. That'll be my trivial solution again.

I forgot to fill in what the total roll is. 0 + 1 = 1. I'll fill that in.

Now there are 36 possible rolls (corresponding to the 36 empty squares in the table). I need 6 of them to be ones, 6 to be twos, and so on, up to 6 sixes.

So what are the options for having 6 ones?

Let's think about this. To get a one, I need a 1 on one die, and a 0 on the other. (No half dots.)

Also, I can't have a 0 on both dice, because then 0 could be an outcome. So without loss of generality, I can assume my table looks like this.

Normally I'd use pen and paper, but I think it'll be easier to take screenshots of a computer visualizations, so I'll do that for the purposes of speed in posting. Unless I get stuck, in which case I'll need to move to pen and paper to let my brain roam a little more.

This image, featuring the purple concave hexagons, blue rhombuses, and gray darts, is one of my favorites.

The 21st Century Pattern Blocks allow for even more opportunities for these kinds of provocations! We were just playing around with them in a teacher session recently.

Having blocks that can't be made from the base green triangle makes this kind of argument really appealing.

Just something to consider.

Newsletter: The Power of Openers

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