George Lucas Educational Foundation
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Recently, I applied to a fellowship with Math for America, a program dedicated to improving mathematics education in U.S. public schools by recruiting, training, and retaining highly qualified secondary school math teachers. In my quest to get the fellowship, I started to fiddle around with some math ideas that made me curious. One of those is the idea of factoring a polynomial, and specifically, how we teach it.

For one, I'm not a fan of FOIL (first-outside-inside-last) for a plethora of reasons. While I think it's handy to have an acronym that reminds students of a procedure, it only works in a very special case. In this case, FOIL works only for multiplying a binomial by another binomial. Does FOIL lead students toward understanding multiplication of all types of polynomials, or understanding why the distributive property works even with variables? I'm not so sure.

We do know that there are alternate ways of approaching multiplication of binomials, but I'd like to focus on using the geometric method of multiplication because, well, because I can.

Multiplication and Factoring Using Areas

Multiplying (x + 2) by (x + 3) can be represented like so (see Figure 1):

Figure 1

This makes the following operations look rather simple:

(x + 2)( x + 3)
2 + 2x + 3x + 6
x2 + 5x + 6

Using the area method for multiplying binomials also makes factoring an easy task. We can visualize the squares and rectangles in this shape while thinking to ourselves, "Which two numbers have a sum of 5 (second term) and a product of 6 (third term)?" If we look carefully, students have another method for understanding why we get the terms we do after multiplying the binomials shown. How does this relate to trinomials? Let's see.

Multiplication and Factoring of Cubes

Let's take the last example and multiply it by (x + 4). Like so (see Figure 2):

Figure 2

(x + 2)(x + 3)(x + 4)
(x2 + 5x + 6)(x + 4)
x3 + 9x2 + 26x + 24

Or geometrically (see Figure 2): This has awesome implications for finding both the surface area and volume of this figure. Since we already figured out the "face" of this cube earlier (x2+ 5x + 6), we're basically multiplying that face by the length of x and by the length of 4. This yields:

x(x2 + 5x + 6) + 4(x2 + 5x + 6)
. . .
x3 + 9x2 + 26x + 24

Factoring The Cube

Once we find the quadrinomial, the cube gives us a hint for finding the lengths that created the quadrinomial. One would only need to figure out which three numbers give us a sum of 9 (second term) and a product of 24 (last term). These numbers are 2, 3, and 4, so we'll get (x + 2)(x + 3)(x + 4).

Is this much better than using the cubic formula? Absolutely, especially for our students.

What about a quadrinomial like 2x3 - 11x2 + 12x + 9? We can try to determine all the real roots of this polynomial, or we can take a look at the second and last terms. The simplest combination for a product of 9 is multiplying 3, 3, and 1. The first term's coefficient, 2, makes getting a -11 tricky. We can't get -11 from the set of numbers without considering the first coefficient.

Yet, as we've seen with the other cubes (see Figure 3):

Figure 3

We will see that the term of 2x will multiply with any lengths that aren't associated with it. Thus, (2x - 3) + (2x - 3) will give us 6x + 6x, or 12x. Since we needed to get -11x, this means the remaining 1x is positive and the 12x is actually -12x

Therefore, our quadrinomial gets factored to (x - 3)( x - 3)(2x + 1) or (x - 3)2(2x + 1).

Word of Caution

I'm still exploring this for other cases (looking at x3 + 27, for example), and what imaginary numbers would look like using this model. These methods work great as a way to draw students in, but some special cases will probably require more space than I could lend it here.

Also, this was my rebellion against the cubic formula which, as many mathematicians know, makes little sense to introduce in the classroom.

Let me know what you think in the comments below.

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José Vilson's picture
José Vilson
Middle school math teacher and coach

I feel similarly about SOAP a) because I've never heard of it and b) because I'm not sure if that's going to help students progress upwards ...

José Vilson's picture
José Vilson
Middle school math teacher and coach

Thank you, Bon, and you raise legit questions here. Here's the thing, Bon. At some point, every "model" breaks down for special cases. That's why I'm playing with it right in front of you. At some point, you *must* transition into the abstract to deal with those pesky non-nice trinomials or any other polynomial. Really, it's a way to push our thinking just a bit to examine why we do what we do, or even what made us bring up polynomials at all.

Allison Weisel's picture
Allison Weisel
Math Teacher

A. Men. I was taught not to use FOIL and instead use the area/volume model to teach multiplication and factoring, and I never looked back. I have worked with lower achieving students and I say that this method not only teaches the concept, but allows them to think visually (and kinesthetically, if you use algebra tiles), where their number sense trips them up.

In my mind, you want models to be too cumbersome to serve at some point, to provide impetus for abstraction. Also, doesn't understanding of what is and what is not possible open up whole new worlds of ideas? Teaching with models does not forbid abstraction, or even impede it. Rather, it deepens connections so students know, understand, and remember concepts without (sometimes silly) acronyms.

Nikita Hertz's picture
Nikita Hertz
Passionate about Math & EdTech

I think this a really good way to explain factoring. Geometrical representation helps the student understand the concept much better.

Mrs. J's picture

Have you heard of Algebra Tiles? They are just physical manipulatives for teaching with a geometric model. I've used them in my 9th grade algebra class to teach multiplying all types of polynomials, factoring binomials, and completing the square. It's helpful for kids to have something to physically mess around with, and they think they are toys so it keeps their attention. ;)

Kathy Wolf's picture

This is great for some kids, I think. But I believe there's still danger in believing one method should work for everyone. For me, the symbols really do mean something as they are, without the geometrical representation, which frankly as a 9th grader would have scared me away. Also, it is possible to know an acronym and be able to use it, and at the same time know that the are deeper ideas at work.

Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

Kathy Wolf: in your experience, what percentage (rough estimate) of students ever "think through" the FOIL mnemonic? How many of them (and how many HIGH SCHOOL TEACHERS) will say that to multiply, say, (3x^3 + 2x -7) and (2x^2 +5), you should "FOIL" them? I hear that in so many high school math classrooms, from both teachers and students (and I include teachers and students in AP calculus), that it's a marvel my head hasn't exploded in the last decade. "Deeper ideas at work"? I'd say just the opposite: the suppression of deeper ideas and thinking through mindlessness; what we need in mathematics and other aspects of life is mindFULness.

Kathy Wolf's picture

What I am saying is, it is possible for some people (of course not all) to understand that FOIL is an instance of a broader idea. AND in my experience, which tends to be triage, and unfortunately not as long term in many instances as I would like, I have to choose my battles. If they are familiar with FOIL and they are able to use it, I have to decide whether it is important at that particular moment to mention that that is really just distribution, or whether that will confuse the issue. If we were working on multiplying a trinomial by a binomial, I would of course never call that FOIL, and if they did, I would ask them what they meant by that.

My main point was that mathematics is not one-size-fits-all.

Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

I certainly agree that one size doesn't fit all. I still balk at teaching pointless and potentially counter-productive mnemonics that undermine understanding and make mathematics into something to be memorized rather than understood. I can't see how pointing out that FOIL is a specific case of the distributive property would "confuse the issue." On the contrary, it sheds light on why the mechanical procedure works and is justifiable, and leads to being able to generalize it. And it is incumbent on teachers to ensure that no one confuses "FOIL" with the notion of polynomial multiplication, of which it is a limited, special case.

I think this falls into the category of "dumb stuff math teachers tell kids" that later must be "untaught," like "You can't subtract a bigger number from a smaller number," addition and multiplication always increase," "subtraction and division always decrease," and "multiplication IS repeated addition."

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