Looking at sums of integers and squares via geometry and algebra

Earlier this week I carelessly asked my younger son to read a section in a statistics book that relied on knowledge of calculus:

My son had some interesting questions about ideas in calculus, so instead of statistics we spent a few days this week talking about calculus (mainly finding the area under a curve).

For our project today I wanted to revisit one of the ideas in arithmetic that we’d relied on in the calculus discussion – sums of integers and sums of squares.

We started with sums of integers and my son gave a geometric proof of the rule for sums of consecutive integers:

Now we moved on to sums of squares – here we talked about the sum (and some of the number theory hiding in the sum formula), but didn’t yet try to prove the result:

Now we looked at a geometric way to understand the sum of squares formula:

Finally, we did a lightning fast review of mathematical induction and showed how we could proved that the sum of squares formula was true in general:

So, a fun week and a bunch of great discussions . . . even if it started with me being pretty careless!