Triangular Numbers, More on Pascal's Triangle and Odd/Even Numbers

Remember when we played with squares and square numbers?

I thought maybe it was time to look at triangular numbers.

We took out circles because triangular number is the number of dots in an equilateral triangle uniformly filled with dots. 

Okay, one is the smallest way to make a triangle. How many circles does it take to make the next size larger triangle? We played and built.

Three.

Next size up?

Six.

Can you predict the next size up triangle?

Nine? Is it multiples of three?

Let's see.

No, ten.

Let's build one more.

Fifteen.

Now let's take them and make squares out of them. How many more do you need?

One more to the three.

Three more to the six.

Can you predict the next amount it will take to make the triangles a square?

Six?

Let's try and see.

Why did you say six?

Because it was the same sequence of numbers...1...3...and 6.

The sum of two consecutive triangular numbers is a square number.

Then I told them a bit about Karl Gauss. He lived in the 1800's and he proved that you can make any whole number by adding no more than three triangular numbers and that triangular numbers never end in 2, 4, 7 or 9.

What triangular numbers add up to your age.

Ten is already a triangular number but it can also be made by1 +3+6!

Seven is 6+1!

Thirteen can be made by adding 10 + 3.

What about you? Hmmmm...49 can be made by adding 28+28+3.

Lastly, remember Pascal's Triangle?

Can you find triangular numbers in it?

Then I found this investigation of number patterns and Pascal's Triangle and I had to try it out too.

So, I printed out a copy of Pascal's Triangle and we went to work coloring multiples of two and three.

We tag teamed this activity by starting with Quentin until he wanted to pass the marker to James and then James to Sam.

You 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 colored, but 560 = 5+6 = 11, is not.

You could also look at it in terms of odd and even numbers and if two cells above are of the same type (both odd, both even), you color the cell below even. If the cells above are different (odd + even, or even + odd) you color it odd. Why does this work?

Because of the principle that an odd + an odd always equals an even. Same is true with the sum of two even numbers is itself even. If your student doesn't know this, he will need to play with numbers to prove it to himself. With every step of the way, he needs to see for himself what the rules of math are, and not just take it from you.