For millennia, humans have loved to operate with mathematical objects. We started with shapes. We moved them around, flipped them over, and even “added” them, laying them side-to-side. Then, we went to town on numbers, and the town grew to a metropolis of operation neighborhoods: addition, subtraction, multiplication, division, and exponentiation. From numbers, a strange sense of manifest destiny took hold, and we expanded our operational influence to conceptual “things” like polynomials, matrices, and even functions themselves.
Polynomials are one of the first abstraction hurdles for young math students. Polynomials are mathematical expressions involving addition (or subtraction) and multiplication. Check out these examples.
You can tell how many terms they have, as the terms are separated by addition.1 The above examples have 3, 4, 2, and 1 terms respectively.
Many high school algebra students learn to “FOIL” when they are learning how to multiply binomials. A binomial is a two-term polynomial. For example, a + 3, x – 7, and 3k2 + m are all binomials.
FOIL is an acronym for First, Outer, Inner, and Last. It is a mnemonic device used to remember what happens when two binomials are multiplied. The “F”, “O”, “I”, and “L” refer to the relative positions of pairs of terms. The pairs get multiplied. FOIL2 tells us to begin by multiplying the two “First” terms. Then, multiply the two “Outer” terms. And, so on.
What is depicted in chalk is another way to think about the multiplication of binomials. Geometry helps to illuminate what is happening. Multiplication can be construed as the calculating of area of a rectangle. Remember how the area of a rectangle is calculated? Simply length times width, i.e. A = lw.
“Squaring” a number, like 3, is how one finds the area of a square with sides with that number as their length. 32 = 9. If you had a square with sides 3 units length, the area of that square would be 9 units2. Squaring a binomial, like with numbers, is simply a matter of multiplying the binomial by itself.
So, consider a square with sides of length (a + b). You can see, in the following photo, how the square is made of an upper and lower rectangle. The upper rectangle is in turn made up of a square (area a2) and a rectangle (area ab). The lower rectangle has another rectangle (area ab ) and the b square (area b2). (I know the “ab” is “upside down” in the photo. It looks better on the bridge.)
Hopefully, the idea is becoming clear. If not, well, let’s try one last time. Multiplying (a + b) by itself results in the area of two squares, a2 and b2 AND two rectangles, technically ab and ba. Of course, ab = ba, so those two rectangles have a combined area of 2ab.3 In which case, when you multiply (a + b) by itself, you get a sum of the squares and two rectangles.
1. Subtraction can be considered addition of a negative. This makes sense if you think of negatives as though they represent being in debt. Having 3 dollars and spending 2 means you have 1 dollar. If you have three dollars and you owe someone 2, you really only have 1 dollar.
3 – 2 = 3 + (–2) = 1
2. FOIL is easy and dangerous. It is a mnemonic device, and it ONLY works for multiplying two binomials. Plus, FOIL doesn’t shed any light on why we multiply the way we do. It simply gives instructions a machine or trained chimp could follow. Since you are neither, we shouldn’t end the story with FOIL. Algebraically, what is happening is an application of the distributive property. Each term of one of the binomials is being distributed to the terms of the other binomial. This post features a geometric interpretation.
3. Multiplication with numbers is commutative. 3 times 5 is fifteen, and so is 5 times 3. If you think about multiplication of numbers as calculation of rectangular area, it makes sense. So, ab = ba. Thus, ab + ba = ab + ab = 2ab.
Note on the equations: In previous posts, I had been copying and pasting the equations as images from a word processing application. I was not fully satisfied with the quality of the images. Then, I remembered
(pronounced la:tek) and figured there would be an online tool. I even went back and redid the equations in Hexagons. I am much happier.
The photographer for this post is Laurie Zimmerman Mann. Check out her site: www.lzmstudio.com.