How to Expand Polynomials Using Area Models: The Anti-FOIL method

Like most people working in education today, I was a student before Common Core standards came around. In fact, I graduated high school the same year they were introduced, just missing the boat. It’s been rough seas for these not-so-new-anymore standards in the 10 years since, but I want to celebrate an undeniable gem of Common Core Math: the humble area model.

You may not know it by that name, but area models have countless applications in STEM. For the science nerds out there, the Punnet Square used to breakdown and predict genotypes from cross breeding is a type of area model. In statistics, you might use area models when solving compound probabilities. Or, in Algebra, you might use area models during polynomial division.

All of these applications pre-date the Common Core, but the new standards introduced the area model much earlier and actually named it for what it is, a simple visual tool for deconstructing multiplication or division. Inspired by the simplicity and versatility of the area model, I started looking for more applications. This one actually came to me from a student. While it is by no means new to the world, it was a revelation to me. Let’s take a look.

First, the Basics

Students today are usually introduced to area models in elementary school as a way of deconstructing large numbers for multiplication. Take 102 * 63. We might put it into an area model like this:

Instead of doing one complicated multiplication step, we break it into 4 simple ones, filling in the table like so…

…and then adding up the interior pieces for our solution:

6000 + 120 + 300 + 6 = 6426

You may be more familiar with vertical multiplication. For the pros out there, it’s the faster method when it comes to multiplying by hand. It’s the way I was taught and still favored by many teachers and students today. But multiplying quickly by hand is not a skill of significant value beyond elementary school. It’s much more important to develop number sense, learn how to deconstruct numbers, and use creative reasoning to solve problems. On those fronts, the area model wins.

The FOIL Method of Expanding Quadratics

Most students are introduced to polynomial expansion during their algebra unit on quadratics. Quadratic expressions can be written as factored binomial pairs, and we spend a great deal of time learning how to factor and expand quadratics, over and over and over. For the expansion, the favored method is known simply as FOIL. FOIL is an acronym that stands for First, Outside, Inside, Last. It tells us the order in which we should multiply terms when expanding an expression such as…

(3x + 4)(2x – 5)

We start by multiplying the first terms…

3x * 2x = 6x2

…followed by the outside terms…

3x * -5 = -15x

…the inside terms…

4 * 2x = 8x

…and finally, (quite fittingly) the last terms…

4 * -5 = -20

The products are added together, combining like terms, and voila!

6x2 – 7x – 20

First of all, the order really doesn’t matter here. It could also be LO-FI, which would be cooler. Either way I get it, acronyms are a powerful thing. The adults reading this may not remember much from high school math but I would bet they remember FOIL, or PEMDAS. But, whereas PEMDAS reminds us of something as essential and broadly applicable as Order of Operations, FOIL has one and only one hyper-specific application. What do you do if I throw a trinomial in the mix?!?!

(3x2 – 2x + 7)(6x – 3)

For really anything other than binomial pairs, FOIL is useless. Granted, binomial pairs show up a lot in quadratics, and many students do understand the underlying principle behind FOIL which they can then adapt to different polynomials. Still, the area model is much more inclusive, working with literally every type of polynomial expansion you can imagine. Plus, it organizes the information for us without the requirement of memorizing an acronym.

The Area Model Method of Expanding Polynomials (A.K.A. The Anti-FOIL Method)

Let’s try that first expansion with an area model:

(3x + 4)(2x – 5)

We’ve got 2 terms in each factor, so we set up a 2×2 area model like this…

We’re using the same principle of deconstruction that we use with multiplying large numbers, breaking up the polynomials into their terms. Next, we fill in the model using multiplication…

…then add the products together, combining like terms…

6x2 – 7x – 20

Now what about that expansion of a trinomial and a binomial?

(3x2 – 2x + 7)(6x – 3)

We’ve got 3 terms in the first factor and 2 in the second, so we set up a 3×2 area model like this…

Next, we fill in the model using multiplication…

…then add the products together, combining like terms…

18x3 – 21x2 + 48x – 21

Whereas FOIL works exclusively with binomial pairs, the area model works with any combination of polynomials. You just set the model dimensions according to the number of terms in each factor. No need for keeping track of your steps because it’s all organized in once place. Just fill in the model and combine like terms. Try it out.

Closing Thoughts

You may also recognize that area models are used elsewhere when dealing with polynomials, such as in the box method of polynomial division, or the X-Box method of factoring quadratics. The area model is a real workhorse when it comes to dealing with polynomials because of the way it simplifies multiplication and division through deconstruction. But, for whatever reason, FOIL reigns supreme when it comes to polynomial expansion.

The example of using an area model to multiply large numbers is a lot like how we actually multiply large numbers in our heads. In this way, the area model is not so much a method as it is just a visual aid to our own thinking. That distinction makes it highly adaptable to solving problems of all kinds. With that in mind, I encourage you think about the ways in which you might already be using area models. Look for other applications in this big beautiful world of math. Make your 4th grade teacher proud!