Educating Risa

Now that he has addition under his belt, I introduced Daegan to Pascal’s Triangle, one of my favourite devices for playing with math. There are so many fun activities to do with this, so many patterns to find: symmetry, natural numbers, triangle numbers, the Fibonacci sequence, factors and multiples, binomial expansion, etc. We’ll be revisiting this triangle many times in our mathematical journey. But to keep it simple, today we focussed on colouring some fun patterns.

I began with a quick overview of Pascal’s Triangle, showing Daegan how it was constructed: begin with a 1 at the top, and for each successive row, add the (usually) two numbers above to determine what number to write. (Since the edges of the triangle have only one number, and that number is always 1, the edges of the triangle are 1s). Have a look at the first six rows:

We start with 1, and then have 0+1 and 1+0 determine the two 1s in row two (you can think of the space outside the triangle as being 0 (zero)) . In row three, we have a 2 in the middle, as it is the sum of the two numbers above it, 1+1. Skipping to row six at the bottom, we sum the digits in row five above to determine the numbers: 0+1, 1+4, 4+6, 6+4, 4+1, and 1+0. I had Daegan calculate the next row to make sure he understood, and he was able to do so: 1, 6, 15, 20, 15, 6, 1.

We then had a look at a fuller version of Pascal’s Triangle, and I asked him what patterns he saw. He saw several—and I’ll be writing more on one of these in the coming days. I then told him I was going to do some colouring, beginning with the even numbers (or multiples of 2). Here’s what you get (ignore the different colours; I had a hard time finding markers whose caps hadn’t been left off!):

Gareth wanted me to colour in multiples of 5, and Daegan multiples of 7 (their ages). I had Daegan ‘help me out’ by using a calculator to do some dividing:

 

And I wanted to see what multiples of 3 looked like. (Older kids may find it helpful to know that the sum of the digits of any multiple of 3 is itself divisible by 3. For example, 252 = 2+5+2 = 9, and 9 is divisible by 3; 924 = 9+2+4 = 15, and 15 is divisible by 3—so both 252 and 924 are coloured, whereas 560 = 5+6 = 11, is not).

Now this is all fun and neat, but why do these upside-down triangle patterns happen? To try to understand this further, Daegan and I explored the multiples of 2 triangle (i.e., even / odd triangle) at the top. I had Daegan investigate the four possibilities when we add whole numbers. The numbers we are adding are either:

even and even

even and odd

odd and even

odd and odd

We then figured out some rules about adding whole numbers, and used the rules to colour in a large blank Pascal’s Triangle. I coloured the green; Daegan coloured the orange. Pretty, no?

More in my next post; I’ll you to ponder the “why?” of these upside-down triangle patterns for yourselves!