When I teach the unit on polynomials and factoring in Algebra 1, I start off my first lesson on factoring trinomials with a discussion on which has fewer options: multiplying to a number, or adding to the same number? Students take a couple minutes to list out all pairs of numbers they can think of and then share out to the class. After doing this twice, quite a few students start catching on to the fact that there’s an infinite amount of ways to add to any given number, but only a handful of ways to multiply to the same number. Multiplication gives us fewer options, which will allow us to do less work. This will be really important to what we do in just a moment. (NOTE #1: I provide students with a factor pair chart as an aid to help with identifying factors later on. NOTE #2: We originally began using only whole numbers as a starting point, but students then wanted to branch out further. Could we extend this question to include integers?! Yes-and we did!)
From there, we go back to something students have just learned a few sections earlier: multiplying two linear binomials of the form (x+A)(x+B). We do this a few times and then look at a bunch of already expanded examples and I ask students what they notice. It doesn’t take long before students start realizing that the middle term, the coefficient of the x, always comes from adding the two numbers A+B, and the end term, the constant, always comes from multiplying the two numbers, A⋅B.
Then, I switch the question around. How can we figure out what someone expanded to create a quadratic expression? Is there any easy way to figure this out? Students start to volunteer info that they know: the middle term comes from the addition (A) of the two factors, and the constant term comes from the multiplication of these two numbers (M).
So, then the question is, which number do we look at? The addition number, or the multiplication number? Technically, it doesn’t matter, BUT mathematicians love to be
lazy efficient, so we’ll look at the multiplication number. Students justify looking at the the multiplication number first because, just a few questions prior, they determined that there’ll be fewer options with multiplication than for addition.
From there, I ask students to make further generalizations and predictions about the signs of the terms and the signs of the factors and use that information to work both forward (expanding) and backward (factoring) using some diamond puzzles.
The next day, we practice some more with basic factoring when a=1, using the patterns we found from the investigation the day before:
Then, we kick it up a notch. How the heck do we do this factoring thing when there’s more than one x-squared?! No problem! GCF to the rescue.
After that, we look at what do we do if a GCF alone isn’t enough to get rid of the a-value, or, even worse, there’s no GCF at all?
This brings us to my FAVORITE part of factoring quadratic trinomials: Slide, Divide, Bottoms Up! If you are unfamiliar with this method, let me start off by telling you that it’s awesome. It’s firmly rooted in the same concepts we’ve been using for the last three sections of factoring, and it just makes sense. Another benefit to the Slide, Divide, Bottoms Up method is that it is efficient. Doing guess and check (or the box method) can become very frustrating for students when the a-value is larger than, say, 4 or 5. There’s just too many options and it ends up taking forever, even with a decent intuition about which numbers to test out as factors. Also, this method even works for special factoring cases like difference of squares! Students can certainly utilize the factoring shortcut for difference of squares, but, if they forget, Slide, Divide, Bottoms Up still has their back.
Here’s how Slide, Divide, Bottoms Up works:
Let’s talk through an example:
Like all factoring problems, we check if there is a GCF, first. If we’re lucky, that will remove the a-value and we will be good to do what we normally do. However, in this example, we weren’t that lucky. No GCF, so what to do with the 6? We certainly don’t want more than 1 n-squared, so we’re going to temporarily transfer it to the constant term by multiplication (we “SLIDE” it over). At this point, we discuss what “temporarily” means. It means, “only for a while,” so that tells us that, at some point, we’re going to have to undo it. This should be perfectly “legal” because if we do something but then undo it later, that just cancels out to what we started with. It might also be worth noting that we transfer the a-value through multiplication because we are factoring, which literally means returning an expression back into a product (multiplication) of two factors.
Now that we’ve gotten rid of the a-value of 6 for a moment, we’re left with a standard trinomial that students know how to factor in their sleep with their eyes shut, at this point. The only thing they have to remember after factoring it is that our factored form is for our temporary expression, not the one we started with. So, how to undo what we did?
Well, if we multiplied the a-value into the coefficient, it stands to reason that we should do the inverse operation and just divide it back out (DIVIDE)! Since we’re dividing, make sure to reduce the fractions!
Lastly, we didn’t start out with any fractions. Actually, we started out with a number that was a coefficient (our a-value of 6). To get rid of any fraction(s) that we introduced, we bring the denominator(s) back up in front of the variable to be a coefficient, once again (bring the “BOTTOMS UP“).
Here’s some more examples. Note example 5 where there’s a GCF but we’re still left with an a-value of 4.
Here’s how it works with difference of squares problems.
After using Slide, Divide, Bottoms Up for the past 3 years, I can’t see myself doing factoring any different. I’m pretty smitten with this method, and, hopefully, it’s easy to see why.
After doing all of the different factoring, I give students one last reference sheet to use in their notebooks, which can be used at any time to refresh their memory on how to solve ANY quadratic trinomial.
If you are interested in this flowchart, it is available in three different sizes here.
Let me know how you teach factoring quadratic trinomials in the comments below!