Does (x + y)^2 = x^2 + y^2

In a few projects that we’ve done over the last couple of days my younger son has gotten a little confused on some basic algebra. Not something I’m worried about as ideas like does:

(i) (x + y)^2 = x^2 + y^2, or

(ii) \sqrt(x^2 + y^2) = x + y

(in case that latex isn’t displaying properly, the entire expression is supposed to be under the square root.)

are questions that confuse everyone when learning algebra.

Today we did a short project to talk about these equations. We ended up spending most of the time on (i) just because it was a little easier to talk about. First, though, was just a quick look at both equations:

Now we looked at (x + y)^2 = x^2 + y^2 more carefully. You can see my younger son’s confusion at the beginning. To help get past that confusion we looked at what (x + y)^2 actually means.

As we were talking during the last project I noticed a bunch of snap cubes near by (from one of last week’s projects). Rather than move on to the square root example I thought it would be better (and also fun) to view the square example from a geometric perspective.

This was a fun discussion and I especially enjoyed seeing the boys find a few different geometric approaches to the problem.