Math by the Mountain

If you’ve followed me for a while, you might already know that I’ve taught just about every factoring method under the sun over the years.

Through many trials and errors (along with teaching progressively more advanced classes such as pre-calculus), I have come to the conclusion that factoring by grouping is the best and most flexible method for students to learn.

Not only is it easy to catch onto and has a built-in check halfway through the process, but it’s flexible enough to keep being used in any future math course (something that is certainly not true for all factoring methods).

Here’s how I structure my factoring unit:

1. I start by teaching how to factor out a GCF
2. From there, we learn how to factor expressions with 4 terms by grouping. This is an easy transition because we’re essentially GCF-ing the GCF when we factor by grouping. I affectionately refer to it as GCF Inception.
3. Next, I spend a day doing side-by-side comparisons between distributing and factoring to help students understand exactly what they’re doing when they’re factoring a quadratic trinomial. There’s just one issue…students don’t know how to “un-combine” like terms yet. If they’re reversing the process after distributing (or expanding, as you might call it), that’s not an issue because students can see what terms were combined because they did it during the distribution process. If they wanted to start from scratch with a trinomial they’ve never seen before, they’d be stuck. This brings us to…
4. I do an investigation to help students discover how to “un-combine” like terms and pick their factors. Students will discover that their two factors need to add to b while multiplying to a*c. Once they know how to find their factors, they can piggy-back that onto their knowledge of factoring by grouping and voila! They can factor any quadratic trinomial!

Let’s take a look at the investigation:

In case your students don’t already know how to factor by grouping, I included a quick mini-lesson on factoring by grouping when there are 4 terms. This is a great warmup, refresher, or quick intro!

Now let’s dig into the investigation! Students are going to distribute and simplify several quadratics that are in factored form and start looking for patterns. I LOVE pattern-based activities. It makes the most sense for my brain, and I’ve found it works really well for my students, too.

Students will be asked to expand several factored quadratics, and then make comparisons. They will be comparing the coefficients of the two middle terms (prior to combining like terms) and how they add and multiply together alongside the values of b and a*c.

Students should be able to generalize their findings at this point. The coefficients of the two middle terms will always add to b and multiply to ac! Next, we take a moment to pause and do some thinking that will help students strategize when it comes to making a list of combinations to find their factor pairs. It is most efficient for students to list out factors of their a*c value instead of b, because there will be fewer options making it more efficient. Students will have a chance to dig into this idea before they use it on the next page!

Okay, now we’re in business! Students have figured out the pattern, and now it’s time to finally un-combine those like terms!

Last, but not least, it’s time to put it all together! Students have learned how to un-combine those middle terms, which means they can turn any quadratic trinomial into an expression with 4 terms that they can factor by grouping. Students will wrap up this investigation by factoring three expressions completely!

After doing this investigation, we spend another couple days doing additional practice to make sure students have the skill locked down.

Do your students struggle with finding factor pairs?

I find that providing students with a factor pairs chart really helps to level the playing field if you have any struggling learners. This is such a simple tool, but it’s one of my most treasured because my students actually refer back to it over and over and over!