Connecting Polynomial Multiplication and Numerical Bases through the Fundamental Counting Principle

I’ve been tutoring a Grade 9 student recently. I love seeing the worksheets and questions her teacher (who I don’t know) assigns. It’s deeply educational to see how another teacher approaches topics and to see how that approach is filtered through the understanding of a student. I feel like I’m learning as much as her each time we meet for a tutoring session.

Two weeks ago, she was learning how to multiply polynomials. Unsurprisingly, she knew an acronym, FOIL, but didn’t know why she was using it, and only had a basic understanding of how to distribute terms. I don’t like to teach FOIL, although I don’t mind the strategy of pairing and multiplying each term. When I teach distribution, I use the following method:

This technique builds on what students already know (distributing a monomial over a binomial)

This technique builds on what students already know (distributing a monomial over a binomial). Once they become fluent with this technique, shortening the approach (and skipping that first step) is great.

Intriguingly, the teacher asked the students to practice multiplying both “horizontally” and “vertically.” The same problem, solved vertically, looks like this (I’ve written out the steps in an exaggerated way, so you can see the progress. When solving, you wouldn’t need to write the question more than once):

I've circled the terms I've multiplied and connected them with their product. Step five is simply summing the different terms.

I’ve circled the terms I’ve multiplied and connected them with their product. Step five is simply summing the different terms. Conveniently, I’ve organized terms in columns according to powers of x, which makes this step particularly simple.

This technique works pretty well, and isn’t fundamentally different from the more common horizontal work. Interesting things happen when you do multiply multiple polynomials though. Using the horizontal technique, we get something like this:

Obviously, once you're proficient with this approach, you can consolidate a few steps.

Obviously, once you’re proficient with this approach, you can consolidate a few steps. That’s what happened with the messy 4 in the third step – my mind was fighting with me, combining terms automatically while I was trying to clearly write one step per line.

In contrast, doing it vertically takes significantly less space:


If you can’t see what I did there, I’ve made you a gif:


Does that pattern look familiar to you at all? Think on it before you read more. What if I turn it sideways?


Ringing any bells yet? Lets change the symbols from Xs and 2s to 1s and 0s:


Anything? One more gif to round out your thinking:


Yessiree Bob. The pattern that you follow when multiplying multiple binomials is the same pattern that you use to count in binary. For higher order polynomials, you’re in a higher base (Trinomials count in Base-3, four-term polynomials in Base-4, etc.). Similarly, the more polynomials you multiply, the more digits your different-base number has ( 3 polynomials yields a three digit number, 4 polynomials yields a four digit number, etc.).

The reason for this connection is the Fundamental Counting Principle (If you have A ways of doing one thing, and B ways of doing another, there are A*B ways to do both things). Distribution requires each term to multiply by all the terms in the other polynomial.  It’s the same combinatoric principle as counting: each position needs to be filled with all possible digits for that base, and unique arrangements of those digits signify a unique number.

I found this realization worth sharing for three reasons:

  1. It’s neat. That’s a pretty good reason to share it.
  2. Vertical multiplication, once I started playing around with it, is way easier for multiplying multiple, multi-termed polynomials. It saves space, effort, and reduces mistakes.
  3. I fight against the siloing of math concepts in education. Seeing deep connections between different topics is pedagogically interesting because drawing students’ attention to those connections can help with both understanding and procedural fluency. It reduces the amount of memorization required by showing how mathematical principles can be applied in a various situations.

Any thoughts here? Further connections that I’ve missed, or mistakes in my work? Lemme know below, yo.


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