Earlier this week Fawn Nguyen posted a nice conjecture from one of her students on twitter:
I love Fawn’s approach to teaching kids math and try to
blatantly steal incorporate her work into what I’m doing with my kids. This problem had so many possibilities and I put it in my back pocket for the weekend.
The reason that I didn’t use it right away is the fraction / decimal component seemed like it would be too distracting for my 7 year old, so I wanted to take a day or two to think through how to present it. There was, however, a similar problem that avoided the fractions that we played with right way -> multiplying 3 consecutive integers. Here, instead of being nearly a square, you get a product that is nearly a cube:
We had a lot of fun building these shapes and talking about the pattern. In retrospect, I wish I would have asked about 0 x 1 x 2 as well.
As an aside, building these shapes got me thinking a little more about connecting geometry and arithmetic and the next day we had a really fun talk related to the geometry of imaginary and non-commutative “numbers” :
Anyway, we circled back to Fawn’s problem this morning. I’d toyed around with the idea of modifying the problem to talk about the Arithmetic Mean / Geometric Mean inequality, but thought that would be a little too ambitious. Finally I decided that a better lesson than AM / GM, as well as the easy way to avoid the complexity of decimals, was to talk about multiplying integers that differed by 2 rather than 1. We talked through the original problem, moved on to the slightly modified problem, and then talked through some of the algebra. Made for a really fun morning. Thanks for another great lesson, Fawn!