As soon as I could I set my 7 year old to work understanding patterns within the dominoes. We spread them across the kitchen table and began lining them up.
Dominoes are made so that each number is matched with every other number just once. Which number is on the left side or the right side doesn’t matter. There’s only one domino with the five on one side and the four on the other. You can see it as 5:4 or you can flip it around to be 4:5.
We lined up all the dominoes that have a 12 on one side, and we put them in order from 12:0 to 12:12. Then we lined up all the dominoes that had an 11. Of course the 12:11 domino was already used in the first row, so the second row has one less domino in it. Then the next row had all the dominos that have a 10, except those two that 10 dominos that are already used, 10:11, and 10:12. Each row ended with the double of the number it focused on. If you keep going with the pattern it makes a big beautiful triangle and since our dominoes are colored, you can see the colored lines.
So how many dominoes are in a Double 12 set? The first column has 13, the last column has 1, and there are 13 columns. You could count 1 + 2 + 3 and so on up until you add 13. Or you could combine some of the rows together so that you have 6 rows of 14 and one of 7. Or you could combine the rows slightly differently, so that you have 7 rows of 13, as shown in my second picture. Either way you get 91, a beautiful triangular and square number.
We set the dominoes up in the big triangle again and my next task for the children was to put little markers on all the dominoes where the dots totaled up to a specific number. The four year old had to make the number 6, the seven year old the number 10. They quickly saw the pattern: diagonal lines.
My second task was to total up the number of dots in each row. Most of the time when I come up with projects like this I’m flying by the seat of my pants trying to hold the two year old with one hand and trying to get the kids into something before all chaos erupts. This was no exception. I had no real idea where I was going with this, except that I thought the seven year old could practice figuring out triangular numbers. This time the triangular number wouldn’t be the number of dominoes but the number of dots.
So he started figuring out the rows by adding the number of dots on the first and last domino and then multiplying that number by the number of pairs of dominoes in the row, and adding any remaining domino’s dots. The rows went like this:
12 x 6 + 6 = 78
15 x 6 = 90
18 x 5 + 9 = 99
As he went around he noticed an abundance of multiples of three. The total number of dots (shown in red) was always a multiple of three. The first and last domino together always added up to a multiple of three (shown in dark blue). Even the number of dots on the extra domino (shown in green) in rows with uneven numbers of dominoes was always a multiple of three. Why, he asked? Each row started with a domino that had 12 on one side and then whatever number was on the other side of that domino, the end of the row would be double that. So in other words the blue number was always 12 + 3n. N always increased by one, so each blue number increased by three. Now it made sense that all the numbers were multiples of three. We were multiplying a number by three, adding it to another multiple of three, and then multiplying it by something and adding the extras. The number for the middle domino (shown in green) was always a multiple of three as well. It was half of the blue number… but not all multiples of three divided in half equal a multiple three…. but then every second one did, and only ever second row had a green number!
We were getting excited. We were noticing patterns. We saw that the number was increasing and my seven year old wanted to find out if it would increase indefinitely. We peaked ahead to the end. The row with just one domino (12:12) was obviously lower than our first. So the numbers must peak at some point. We kept going:
21 x 5 = 105
24 x 4 + 12 = 108
27 x 4 = 108
30 x 3 + 15 = 105
33 x 3 = 99
36 x 2 + 18 = 90
39 x 2 = 78
42 x 1 + 21 = 108
55 x 1 = 55
24 = 24
My seven year old had the idea that he should graph this, so he started to do that, and he thought about the idea that the totals for the rows started at 78, went up to 108 and then down to 24. Would it be possible, he wondered, to get the other side down to 24 too? Perhaps with negative numbers?
Our time for math that day had run out but a few days later we got out the dominoes again. This time I took only the Double 6 set and made the triangle out of that. Then I took pieces of paper and started making paper dominoes to represent dominoes with -1, -2 and -3 dots. (Could I market Double -6 to 6 dominoes, do you think? Probably not, but it was fun to experiment with.)
After we got a feel for what it would look like we moved from the paper dominoes to graph paper writing not the number of dots (and negative dots) on each side but what the total would be with both sides added together.
The row totals went:
12 (just one 6:6 domino)
21 (6:5, 5:5)
27 (6:4, 5:4, 4:4)
21 (this is the last row of the normal dominoes)
12 (this row has all the normal domino numbers combined with -1)
0 (this row has all the normal domino numbers combined with -2)
-15 (this row has has all the normal domino numbers combined with -3)
We stumbled over the -15 first wondering if it should be -12 instead but it wasn’t. So what could we see? The numbers peaked at 30 then went down. How did the totals move? They went up by 9, then 6, 3, and finally 0 (between the two 30 rows) and then down by increasing multiples of three.
What if we added the columns instead of the rows? Before we introduced the negative numbers these created drastically different results than adding the rows. Now with negative numbers in the mix our longest column ended up totally to 72 (whereas our longest row totaled to -72). And our shortest column consisted of just the -6:-6 domino making it the mirror of the shortest row’s 6:6.
There’s just so much possibility for exploration in math. We were both learning, both discovering new things. There was no answer key.