Summing up Squares

Earlier in the week my son’s math team had a practice problem with this sum:

2 + 5 + 10 + 17 + . . . .

The question was how many terms would it take in order to be above 650?

I though that this problem would make a good starting point for reviewing the formula for the sum of squares:

1^2 + 2^2 + 3^2 + \ldots + n^2 = (n)(n + 1)(2n + 1) / 6

This project didn’t go as well as I’d hoped, but maybe there’s a way to come back to it again later to reinforce some of the ideas. The difficulty the kids had – arranging 6 stacks of 1^2 + 2^2 + 3^2 cubes into a box – comes in the 3rd and 4th videos below.

We started by talking about sums of integers. Of course the infinite sum came up at the end!

With the sums of integers out of the way, we introduced sums of squares. After we wrote down the formula, we started exploring that formula with snap cubes:

Now we studied the slightly more difficult geometric cases of 1^2 + 2^2 and 1^2 + 2^2 + 3^3. The second case gave the kids a lot of trouble.

In the last video we weren’t able to construct the box relating to the sum of the first 3 squares. We kept trying here. I didn’t expect this piece to give them as much trouble as it did, so, unfortunately, I didn’t have any good ideas to help them see what to do.

One thing that was interesting to me is that the pattern with the colors seemed to hinder their progress as much as help them:

As prep for this project yeterday, I 3d printed 6 of the block sets for 1^2 + 2^2 + 3^2. This was a fun 3d printing project all by itself, and came in handy in this project to see if they were able to build the shape without the colors.

Although, as I mentioned above, the color patters seemed to give them as many difficulties as help, my older son thought it was actually harder to build the shape without the colors:

To wrap up the project we took a look at the three pyramids that fold together to make a cube. This idea was a fun project all by itself last year:

A neat geometry project inspired by a James Tanton / James Key tweet

We also looked to see if the sum of squares was approximated well by n^3 / 3 for a few values of n.

So, although this one didn’t go exactly as planned, it was still pretty fun. After we finished I had they try to build the box made out of 6 1^2 + 2^2 + 3^3 + 4^2 pyramids, and they built that shape pretty quickly.

I like the connection with the sum and the 3 dimensional box and am excited to come up with ways to revisit that connection. I’m also trying to think through ways to show a connection with the sum of cubes and a 4D box – I bet that will be fun!