# Studying the difference of squares with snap cubes.

The tour through the “Algebra with Integers” section of our number theory book continues.  My instincts are to just jump ahead, but my younger son seems to really be enjoying the problems so we are plodding through them.  Today we looked at the difference of squares:  $A^2 - B^2 = (A - B)*(A + B).$

After a few numerical examples I wasn’t really sure what more to do.  On a whim we decided to try playing around with snap cubes and see if there were any geometric connections.  Turned out to be a lot of fun.

To keep the number of snap cubes from getting out of hand we kept things simple by looking at the example $5^2 - 3^2$.  I asked my son to make a picture with snap cubes showing what $5^2 - 3^2$ should look like and he made the picture below (the white cubes are supposed to be a minus sign and an equals sign).  He described the shape at the bottom of the picture as the yellow square taking a bite out of the blue square.

The next step was to see if we could find a different geometric way to understand how the bottom shape arose from the original  two squares.  The algebraic formula for the difference of squares is so simple, you would hope a geometric interpretation would be simple, too.  I asked him to see if there was a simple shape that we could make out of the bottom blue cubes.  He made this:

Oh, ha ha, my attempt to keep the number of cubes simple accidentally produced a Pythagorean triple 🙂  Oops, but what the heck, let’s not pass up the opportunity:

A few quick numerical examples showed that the difference of squares didn’t always produce a square, so we looked for an alternate way to arrange the 16 blocks and found these two ideas:

and this:

All of the pictures above came from probably 15 to 20 minutes of talking about the geometry.  We finished by talking about the relationship between the two last pictures and the algebraic formula back on the board with two quick discussions:

So, what started out just as a whim turned into to something really fun.  I really enjoy finding the ways to find connections between algebra and geometry.  Turns out to be pretty fun even if you’ve not studied either subject yet 🙂