Educating Risa

I left you to figure out why the even/odd (multiples of 2) triangle showed upside-down triangle patterns (when coloured) in a previous post:

Here’s what Daegan and I did to figure it out. We began by investigating what happens when you add two whole numbers. This means you are adding in one of four possible ways:

even + even

odd + odd

odd + even

even + odd

Now the final two ways are in fact the same, as addition is “commutative.” Even little ones can see this, as they can see—using tokens or fingers—that 3+2 = 2+3. The order of the addends does not matter. So we started by investigating what happens when we add:

even + even

Working with number tokens and blank counters, Daegan quickly saw that the sum of two even numbers is itself even. There simply was no way to get an “odd man out” when adding even numbers. We then moved on to odd + even, and Daegan guessed that the answer would vary—sometimes odd, sometimes even. We got some number tokens and made up some odd + even questions at random:

After trying a few sums and seeing that the answer was always odd, we moved on to blank tokens to understand why. When adding odd + even, there is no way to “match up” the “odd token out”:

No matter how many more even-numbered tokens are added, there is no way to “pair up” that odd token out.

We used the tokens and blanks to also determine that odd + odd = even. So now we had our rule for colouring in the blank Pascal’s Triangle: if two cells above are of the same type (both odd, both even), we colour the cell below even. If the cells above are different (odd + even, or even + odd) we colour it odd. Daegan knew that Pascal’s Triangle started with a 1, and had 1s down the sides, so off he went:

A couple tips if you want to try this at home. We got out blank Pascals’ Triangle from here. Older kids can handle colouring it themselves, but for youngers, I recommend colouring it with your child. I suggest you pick a darker colour than your child, as that way it is easy to colour over any mistakes. Once he got the edges done, Daegan and I took turns: he coloured in any odd spots orange, and I did the even ones in green.

At first the colouring seems quite random, but then patterns start to appear—and not just the upside-down triangles! We worked left-to-right, one line at a time down the triangle, and got to a row that alternated perfectly back and forth: odd, even, odd, even. It in fact followed a row that alternated every second cell: odd, odd, even, even, odd, odd, even, even. Daegan found this quite neat:

After the alternating row, I—being the colourer of ‘evens’—got a long break. Since each adjacent cell was of a different type (odd+even, or even+odd), all the sums in the next row are odd. This gave Daegan the giggles as we had been “sing-songing” along as we coloured:  “odd!” (low tone) or “even!” (higher tone). So our constant repeat of “odd! odd! odd! odd!…” on this line of the triangle made us sound like a broken record (or, in my 7-year-old’s world, a CD that kept skipping):

Math can be fun! And silly! And still educational!

The next row was similarly funny, only now our record was broken / CD was skipping on the other word: “even, even, even, even!” Since all the above cells are odd, all the cells to be coloured in the next row (save those outside edges) are even. We pressed on through the triangle, finally completing another bit of math art to add to our wall:

Extension for older kids: recall the other upside-down triangle patterns we saw when we coloured in the multiples of 3, 5 and 7 in my previous post. These patterns happen because when you add two multiples of a given number, the sum itself is a multiple of that number. We saw this with even numbers, or multiples of 2: the sum of any two multiples of 2 (2+4, or 4+6, or 16+ 22, etc.) it itself a multiple of 2 (or even). The sum of multiples of 3 (3+6, 9+12, 24+24) is itself a multiple of 3, and so on. You can use number tokens and blank counters to demonstrate this, just as we did above in exploring odd+even. So in Pascal’s Triangle, whenever multiples of 3 (or 5, 7, etc.) wind up in cells adjacent to one another, when they are summed to yield the numbers of the next row, the resulting cell will again be a multiple of 3 (or 5, 7, etc.). So when you colour in multiples of given numbers on Pascal’s Triangle, you get upside-down triangles.

No posts this weekend. We’re still shaking this stomach bug, and I’ve got much to catch up on. See you Monday!