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.

Diff of Squares 1

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:

Diff of Squares 2

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:

Diff of Squares 3

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:

Diff of Squares 5

and this:

Diff of Squares 6

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 🙂

A neat coincidence with place value problems

Over the last few days I’ve been working through some pretty difficult place value problems with my younger son.  These problems come from the “Algebra with Integers” section of Art of Problem Solving’s Introduction to Number Theory book.  As I wrote yesterday part of the difficulty is that we haven’t really covered any algebra yet.

Despite the lack of familiarity with algebra, my son seems to really like these problems.  He is absolutely fascinated when we arrive at the answers and is amazed that some of the problems have more than one answer.  The problems are all like the one in the video below:

Tonight I tried to throw in a little curve ball – how many two digit integers are 11 times the sum of their digits?  I wasn’t sure if he would see that this situation was impossible right away.  He didn’t, which was lucky because I really wanted to see how he would react to the algebra telling him there were no solutions.  Initially that math gave him a little trouble, but I loved seeing him realize that something was wrong without being quite sure what it was:

Definitely a fun couple of days.  I like these problems both for the easy introduction to algebra and for the place value review.   They also serve as a surprise example of a “low entry / high ceiling” problem.   I hadn’t really thought about this feature of these problems until I saw this challenge problem involving digits and decimal representation from Christopher Long on Twitter yesterday:

as well as a few fun follow ups like this one (and check Chris’s twitter feed for plenty more):

Pretty sure that the first time I saw a problem of this type was in the book Lure of the Integers by Joe Roberts which shows this interesting result:


The problems on Twitter are obviously a little harder than the ones I’m covering with my son, but I think older kids would probably like them a lot.   Even just showing some of the solutions might catch a kid’s interest since It is pretty surprising to see how large the solutions are.   I love simple to state problems that turn out to be a little more difficult than you might initially think!

Always happy to see the math I’m covering with the boys overlap in some small way with the math people are sharing on Twitter!