Yesterday, I spent some time writing down a plausible path for rediscovering the Pythagorean theorem, and the algebra steps at the end got me started illustrating more algebraic identities with area and volume models.

Let’s start with some basic properties of multiplication:

The commutative property, expressed in an area model, just says that the area of a rectangle stays the same when it’s turned sideways!

The associative property, involving the product of three numbers, requires a volume model rather than an area model. We can draw each box like a loaf of sliced bread, and the associative property tells us that the volume stays the same no matter which direction it’s sliced.

The distributive property is the first that involves addition. A sum is represented by just lining up two shapes side-by-side. Viewed this way, the distributive property allows us to cut a rectangle without changing its area.

These are fairly trivial, but the same can be applied to more interesting examples. Perhaps the most common non-trivial area model came up in the last article mentioned earlier. Here it is.

A minor variant on this formula, which is taught separately in some textbooks, is the square of a difference. This is a little trickier to draw, but it’s a great introduction to subtraction.

Even trickier yet is the difference of squares, and I haven’t figured out how to draw it without motion in the picture.

Volume models can also be used for non-trivial identities with 3rd-degree polynomials, such as the cube of a sum pictured below. Unfortunately, I was unable to find a two-dimensional layout that communicates this well in a still drawing, so I’ve instead produced an animation using CodeWorld, the math and computer science learning environment I’ve built (though this is more advanced code than I’d expect from any of my middle school students!) If you click the link for the animated version, you can pick out the eight parts corresponding to the expanded form.

Illustration of (a+b)³ = a³ + 3a²b + 3ab² + b³, as a volume model. Click below for animation

I haven’t had the chance to think about models of more interesting 3rd-degree expressions, such as the sum of cubes: a³ + b³ = (a + b)(a² + b² - ab). If you have ideas for modeling these, I’d love to see them.