Another great problem from Fawn Nguyen (2 of infinity)

Earlier this week Fawn Nguyen posted a nice conjecture from one of her students on twitter:

6th grader made claim: if you multiply 2 consec #s, say 5×6, then the square of the middle # (5.5) is the same product, plus 0.25

— Fawn Nguyen (@fawnpnguyen) January 9, 2014

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!