More fun with counting and geometry

We finished section 3.3 of Art of Problem Solving’s Introduction to Counting and Probability today. One of the problems in the section asked about a few relatively basic integer sums. It made for a fun little morning project.

The first sum I asked the boys to look at was They’d seen this sum already in the examples from the book, but I wanted to hear them talk through it on their own:

The next sum was the first positive even integers The tiny bit of extra difficulty for kids here is keeping track of what is. Sorry about the garbage truck stopping by in the middle of this video . . . .

The next sum was the first positive odd integers. There are a couple of ways to approach this sum, and the answer is a bit of a surprise:

Since we just found an amazingly simple formula for the sum off the first positive odd integers, we brought out the snap cubes to see if we could find the connection to geometry:

So, a fun little project. A few basic sums that we encounter when learning to count give us some nice practice with both arithmetic and basic algebra, and we also get a surprising and pretty cool connection to geometry. Nice morning!