Difference of Two Cubes (Part 4)

Here’s the formula for the difference of two cubes:

x^3 - y^3 = (x-y)(x^2 + xy + y^2)

The formula isn’t terribly complicated; however, the factoring on the right-hand side is hardly the first thing that a student would guess if only given the left-hand side to simplify. The formula of course can be confirmed by multiplying out the right-hand side, but that’s really cheating. It’d be nice to have a way for students to develop the right-hand side, as opposed to merely confirming that the right-hand side is correct.

To this end, I suggest using base-10 blocks, a common manipulative found in elementary classrooms. The figure below shows a 10×10 cube with a 3×3 cube removed.

difference of two cubes

A (hopefully interesting) challenge for students would be how to build this figure only using the materials found in a typical base-10 kit, and also building it with as few pieces as possible. I think that most high school students, after some thought, can solve this puzzle by using 7 flats (for the bottom 7 layers), 21 rods, and 63 units. This of course provides the correct answer, as

7 \times 100 + 21 \times 10 + 63 \times 1 = 963 = 10^3 - 3^3.

After finding the correct answer, students should give this picture some deeper thought. If we let x = 10 and y = 3, then

7 \times 100 = (x-y) x^2.

This makes sense on physical grounds: the volume of the “base” of 7 layers is 7 \times 10, and the 7 came from the fact that the top 3 layers are incomplete.

Likewise, the 21 rods can be thought of as

21 \times 10 = 7 \times 3 \times 10 = (x-y) y x.

Again, this makes sense just looking at the picture, as the 21 rods makes a solid that is 3 units high (y), 10 units long (x), and 7 units wide (x-y).

Finally, the 63 units can be thought of as

63 = 7 \times 3 \times 3 = (x-y) y^2.

Indeed, these 63 units form a solid with a square base of side 3 and a length of 7.

Adding them together, we find

7 \times 100 + 21 \times 10 + 63 \times 1 = (x-y) x^2 + (x-y) xy + (x-y) y^2 = (x-y) (x^2 + xy+ y^2),

which is indeed the formula for the difference of two cubes. Now that students have discovered the formula for themselves, the formula can then be confirmed using the distributive law.

green line

As a postscript, it should be possible to use two different colors of base-10 blocks (to represent positive and negative numbers) so that students can derive the formula

x^3 + y^3 = (x+y)(x^2 - xy + y^2).

However, I don’t personally own two different colored base 10 kits, so I haven’t had time to think through how to do this.


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