Sum and Difference of Cubes

Teaching the factoring of the sum of cubes and difference of cubes was not a priority for me in my teaching, and the topic does not seem to be part of the Common Core Standards. However, some people do have to include it in their classes, and as a result the subject occasionally comes up in online discussions among math teachers. About four years ago, I came up with a puzzle-like approach to the topic, using the area model of multiplication. Check out the resulting worksheet. If I had to teach this topic, I would start with that.

A few days ago, I saw videos by Jeffrey Smith: sum of cubes, difference of cubes. Those were the inspiration for this blog post.

When I designed the Lab Gear, I made sure it allowed for the representation of (x+y)³:

But seeing Smith’s video made me realize the Lab Gear also allows for a hands-on representation of the sum and difference of cubes — albeit in a non-obvious way.

Sum of Cubes

Let’s start with the sum x³ + y³:

Let’s build on top of those blocks, in order to get two towers of height x + y:

And now, let’s write the total volume of the towers two ways. Block by block, we see that the volume is 

x³ + y³ + x²y + xy² = x³ + y³ + xy(x + y)

As two prisms, we see that the volume is

x² (x + y) + y² (x + y)

Write an equation to show these expressions are equal, and solve for x³ + y³:

x³ + y³ = x² (x + y) + y² (x + y) – xy(x + y)

x³ + y³ = (x + y)(x² – xy + y²)

Admittedly, following this argument requires some facility with algebraic manipulation, and I do not recommend it for students before Algebra 2 or precalculus. Still, as a teacher, I appreciate seeing a geometric interpretation of the formula. It is also interesting that the standard “Make a Box” Lab Gear activity is applied here in a slightly different version: “Make Two Boxes”. The key idea that makes this work is that the boxes have the same height. As always with the Lab Gear, a common dimension represents a common factor.

Difference of Cubes

On to the difference of cubes! This one will require us to make three boxes, all with the same height. However, we must start by choosing new variables. We will use a instead of x+y, and b instead of x. Given that, this building represents a³ – b³:

Take the blocks on the top layer, and rearrange them so we have three towers, all of them with height (a – b):

The volumes of the prisms are ab(a – b), b²(a – b), and a²(a – b) respectively. So:

a³ – b³ = (a – b)(a² + ab + b²)

QED!

— Henri