Binomial Multiplication and Factoring Trinomials with The Rectangle

I was reading about Joseph Nebus’s factoring method….

(okay, a brief note. Early in his writeup, Nebus (Joe? Joseph?) writes: “It’s a method for factoring quadratic expressions into binomial expressions, and I must admit, it’s not very good. It’s cumbersome and totally useless once one knows the quadratic equation.”

Many, many math teachers have expelled much breath on the uselessness of factoring, as the skill is completely nullified by the quadratic formula (which I think what he means here). But when they make this comment, they are thinking as mathematicians, not as teachers. Mathematicians work with math to solve problems. Teachers teach math so their kids can demonstrate their knowledge on tests*—not just state tests, but college admissions tests and placement tests. And on these tests, the questions are designed for either factoring or the quadratic formula—and far more the former than the latter. All students must learn to factor trinomials if they are to escape remediation. The quadratic formula is optional. And if the test pragmatism isn’t enough of an argument, please know that students with limited integer operations skills will do better with factoring than the formula because they rarely have squares memorized and please, please believe me when I say that they will ALWAYS subtract 4ac from b squared, even if c is negative. End brief note.)

There are teachers who think this is a science, and teachers who know it’s an art based largely on the audience. And which kind of teacher you are is a religion, or an expression of personality (which I often think is the same thing).

So when I say that the method Nebus describes sounds extremely convoluted, I am simply a Jehovah’s Witness expressing doubt about the utility of the Amish rumspringa.

But many math teachers aren’t even aware of the box/diamond method, and many others who do use it don’t teach it in a fully integrated manner. So for the teachers of the artist mindset looking to find the right method for certain audiences, here’s an outline of the method.

I got the approach from CPM. I don’t know if it originated with CPM, so apologies if the original idea goes back earlier. CPM’s curriculum is insanely irritating: text heavy, lousy examples and insufficient practice. But in many cases, its approach to a topic provides a beautiful, fully integrated, and consistent framework that I steal without shame.

Factoring out common terms

I always introduce the generic rectangle when introducing or reviewing simple factoring (pulling out common terms). The area model uses the fact that the rectangle’s area is both the product of the length and width and the sum of the individual areas. You break up the side of the rectangle into as many different segments as there are terms.

So 8x + 18 is the sum of two areas, both created by a product of length and width. One side is used for both areas, so it must be a factor common to both areas. In other words, what is the greatest common factor of both terms?

Once you find the GCF, work backwards. What do I multiply by 2 in order to get 8x? Most students do well on this, but if they struggle, I show them how to divide in order to find the answer.

I don’t stress its use here, as I do during binomial multiplication; I just want students to be familiar with my use of it. At the same time, I always find a few students who struggle with factoring common terms and find the approach helpful.

Binomial Multiplication

I do not mention FOIL, although most of my students have learned it at one time or another. While I don’t require my students to use any particular method for tests, I require them to use the area model method for binomial multiplication at least for a day or two so they understand the underpinnings of the factoring method.

So obviously, binomial multiplication is the opposite of factoring; the terms go on the outside of the box and generate the individual areas. (x+2) is a segment of length of x and 2, (x+3) a segment of x and 3. I always point out that the lengths don’t need to be accurate or drawn to scale.

I demonstrate this method several times, up front. I explain the area concept again and how multiplication of length times width for each smaller rectangle is the same as the area of the larger rectangle. I don’t really expect my students to be able to repeat it back to me. What I expect, or hope, they will think is “Oh, okay, that makes sense”. Because from this point on, when they think of this method, I want them to remember that the method made sense to them, even if they don’t remember the specifics. That’s also why I don’t write any of this down—most of my students will toss any documentation, anyway. I work a variety of examples (at this point, a=1), picking students to walk through the process.

The kids have a handout with 20-30 problems (this is one of the few topics that kuta software doesn’t have a good handout for), but I don’t have my usual handout online. The original problems would all be a=1, b and c of all signs, because I want them to work dozens of problems and see the pattern, if they are able to. Then, on day 2, 3, and 4, I introduce difference of two squares (what happens to b?), a>1, and 2×3 or 3×3 polynomial multiplication—which works really well with this model, as the kids just make a bigger rectangle.

I wish I could say that this method eliminates the problem of (x+2)(x+3) = x2 + 6. Alas. However, when a student makes the mistake and I scowl and draw the rectangle, with no other explanation, 9 out of 10 kids making the mistake go “Oh, yeah.” That’s the win, such as it is.

Factoring Trinomials

So after a week or so of multiplication, I point out something interesting about the completed rectangle:

This is particularly interesting when we consider the two “middle” terms that add up to bx. We now know that they add to bx and multiply to the same product as ax2 and c.

Interesting, yes, but also useful. I remind the kids that distribution is the inverse of factoring, that distribution converts a product into a sum, while factoring turns a sum into a product. So if they are faced with a quadratic equation in ax2 + bx + c—say, for example, x2 + 9x + 14—how could they turn this sum back into a product?

I ask the kids, if I’d multiplied two binomials to get x2 + 9x + 14, what would have been in the box?

Factoring trinomials is the task of finding the numbers for the other half of the rectangle.

And thanks to the properties of the generic rectangle, we know that the terms we are looking for add to 9x and multiply to 14x2, the product of the first kittycorner.

So we use the “diamond” as a visual tool to help find those terms.

No matter what method a teacher chooses to teach, factoring comes down to that question: What do I multiply to get ac that I add to get b?

I teach the students to write out the factors in pairs, starting with 1 and the number itself (otherwise they tend to forget) and working up from there. Remember that I teach students who will often have a tough time remembering all the factors of 24, and pause on each term to remember the pair.

So once you find the terms that meet the requirements, you put them in the box. It doesn’t matter which goes where. I repeat that phrase a lot. I sometimes wonder if I should create a rule for where the terms go, just so I won’t get the question any more.

It’s worth stressing to your students that, while you’ve found the missing terms, you still have one more step! They’ll still forget, and this will bite them back when they start factoring trinomials in which a>1.

The last step involves finding the GCF for each row. This is where I get the payoff for introducing “single row” factoring much earlier. The students are familiar with the process; they’ve seen me explain that the outside terms are the GCF for a month or more, even if they didn’t use it themselves.

Again, I work five or six problems with the class as a group each day. The kids have a page of 20-30 problems they work through; if they finish one page, I give them another with more complex problems. Anyone who can do the work peels off from the class discussion and works independently from the beginning, the rest are “released” after the class discussion. I put worked examples all over the whiteboards to give them models to follow. Many of my struggling students don’t move past a=1. Some of the weakest will only be reasonable competent at c>0 in the first go-round and struggle with finding the difference of two terms for a while. So over the next two-three days, the kids work on factoring at their own pace. The strongest kids are working a>1 by the last day (and their third page of problems), and working problems like x2 – 9x = 10, learning to set it equal to 0 and factor.

Here’s an example with c<0:

Here’s a>1—and this, by the way, is where anyone can benefit from the generic rectangle. Any other method of factoring a>1 trinomials is a pain in comparison:

I return to factoring throughout the year. Every so often I’ll declare it time to build on existing skills, so kids who had just gotten competent at c>0 can get more practice time on c1, and then the strongest kids start to identify patterns—identifying perfect squares, difference of two squares, and so on.

As time goes on, I give fewer worked examples and just the general outline below, to see how they do at moving from general to specific:

Next Steps

I have traditionally gone from this to completing the square and quadratic formula, then onto graphing parabolas. I am going to reverse these two topics this year. Teach factoring trinomials, then graph parabolas. Get that going well, and then move onto completing the square, quadratic formula, and then graphing those cases. See how that goes.

Finally, I can’t stress this enough: a quarter or more of my algebra classes are low ability kids, so if you’re thinking Jesus, two weeks or more for multiplying and factoring quadratics? then you aren’t teaching low ability kids or you’re just ignoring the fact that they’re flunking your class. My top kids are doing in depth work on the topic or, in some cases, moving onto another topic entirely.

I’ve been getting some people lately asking, or complaining, that “low ability” is vague. I’m sorry, but it’s not. Potter Stewart was right: you know it when you see it. If you want a specific metric, it’s a kid with cognitive abilities measured at the 50th percentile (say, IQ from 95-105, but that’s a guess). In other words, kids that are perfectly functional in the real world, but simply don’t have the interest or ability for advanced math. Kids with cognitive abilities any lower than that aren’t, as a rule, going to be able to even fake it in algebra, much less anything past that. There are always exceptions.

It’s the delusion of eduformers and progressives, one and all, that if teachers find the right approach, a low ability kid is transformed into a competent high ability kid. In reality, success in teaching low ability kids comes when they start to feel a sense of competence at some level of math. When a kid goes from staring blankly at a trinomial to thinking “Oh, yeah” when I draw the rectangle, that’s a big goddamn win. I believe a lot of kids in this category could learn specific high level math in the context of a concrete task, although I have no evidence of this. But we’d have to sort kids into different groups and sorting’s just one big no-no.

However, this method is helpful for kids of all abilities. High ability kids get a real kick out of seeing the link between the area model and the algebra, and I’ve rarely met a kid who didn’t appreciate the utility of this method for a>1.

I don’t have a handout per se for this whole method; what I’ve just laid out is 8-10 days of practice, followed by days interspersed here and there throughout the year. However, if I get a kid who comes in late, or who wants a specific tutorial, I have a document that I really need to rework, which is why I spent some time creating images for this writeup. But remember, all of this is religion and on factoring, I’m in a state of epistemic closure. Convert or live life as a heretic. I was going to say “Die, infidel”, but really, the current insanity in the mideast takes all the hyperbole out of that statement and thus all the fun.

*If you are a mathematician who is also a teacher, stop hyperventilating. It’s true. You know it is. Embrace the reality we live.


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38 responses to “Binomial Multiplication and Factoring Trinomials with The Rectangle

  • maiseylou

    Wow. So what is the practical application of this?

    • educationrealist

      Ha. I don’t know from practical. I’m a math teacher!

      Factoring is a HUGE test area, so teachers spend a lot of time on it.

      However, the generic rectangle is a very useful way to teach probability, too. I use it for that as well.

    • EH

      Other than solving equations, there isn’t one in the real world – and when you really have to solve equations, there is always an easier way. There is an application to future studies, though, which I have found makes middle schoolers a lot happier about the drudgery of factoring polynomials: showing how it can be used for optimization problems.

      Teaching them to understand the concept of an instantaneous slope is no problem if they understand slopes at all. Teaching them to compute the derivative of a polynomial cookbook-style takes only a few minutes. Then I show them that solving that equation for zero gives the locations on the graph where the slope is zero, and they can see graphically that that has to be a maximum or a minimum. Pause for a little digression to think of some things that students might want to maximize or minimize. Then I show them how to tell which it is, max or min, without needing the graph. While I’m at it I point out that, if the original equation represented position, the rate of change represents velocity, and taking the derivative of that equals acceleration. If I have another hour, I cover a bit about integrals as well to show how the notation works (long-s integral symbol meaning sum up a bunch of rectangles of height f(x) and width dx) and how it can be used to get an equation for velocity given one for acceleration, or to get position given velocity. Lots of hand-waving, no rigor, but giving them a reason to want to know, a glimpse of where this is going, makes a huge difference. The smart ones also feel like “wow, I actually vaguely understood most of that calculus stuff, I must be good at math”, which also helps their motivation to try to maintain that feeling.

  • Florida resident

    Dear EducationRealist !

    Somehow you have forgotten to declare your definition of factoring. Apparently, you meant the use of integer numbers only.
    Otherwise how about factoring
    (x^2 – 2) ?

    I think it is imperative to tell the sudents, and to your readers as well, what you mean by “factoring”. Other example: for factoring
    (x^2 + 1)
    you need complex numbers,

    On the other hand, later (much later) you should explain to the students, that there is such an object: “Field of all complex numbers”.
    Its importance is confirmed by the Fundamental Theorem of Algebra:

    Over the Field of all complex numbers,
    any polynomial has a root, and thus can be factorized into product of binomials (x – root#1)*(x – root#2)*(x – root#3)*… *(x – root#n).

    Meanwhile, it was the achievement by Evariste Galois and by Niels Henrik Abel, to make this statement:

    Starting from the polynomial of 5-th power and for all higher powers, there can NOT be found explicit formula (however complicated) for the roots of that polynomila in general case.

    Respectfully yours, Florida resident.

    • AllanInPortland

      Is this meant as a parody? I can’t tell. You seem earnest, but really, do you believe kids that need to write down an X to remember what they’re doing would benefit from a (fear-not, I won’t call it a…) digression into imaginary and complex numbers?

    • educationrealist

      Without Allan’s astonishment, that’s my answer. I did, in fact, declare what factoring was–it is the conversion of a sum into a product. But what you describe is far beyond the capabilities of my students. The students who could understand that passed algebra with an Advanced rating in 7th or 8th grade.

  • AllanInPortland

    I remember my first introduction to factoring polynomials. It was all FOIL all the time. I didn’t do as well at it as I should have because I couldn’t stand that a method as vulgar as brute-forcing an answer warranted any respect or serious effort on my part.

    Don’t get me wrong, I was an 8th grader, so my thoughts we’re that developed by any means, but the attitude was there.

    In retrospect, if the teacher had explained, “Yes, this crummy/vulgar brute-forcing an answer, and there are more elegant solutions, but facility with this style of integer factoring is used others as a proxy for general mathematical aptitude, so it warrants a decent amount of rote practice,” I no doubt would have done much better. A crib table of common factors and their differences would have been handy as well. In fact, that I would have arrogantly memorized, much like Scrabble nuts memorize all the 8 letter words.

  • Florida resident

    Dear “Educationrealist”, dear “AllanInPortland” !

    This was not a parody.

    I do understand your concerns re students at initial stage of math education, at moderate levels of abilities.

    But still, in your terminology, is factoring of

    (x^2 -2) into [x-Sqrt(2)]*[x+Sqrt(2)]

    admissible or not ?

    Your sincirely friendly, Florida resident.

    • educationrealist

      I knew it wasn’t a parody. And yes, I do cover briefly that for practical purposes, we are focusing on integer solutions. When I teach difference of two squares, I do give them some problems like x2 – 3.

      So I mention, but don’t emphasize, a few things that aren’t in this writeup.

  • Florida resident

    You wrote:
    “I do give them some problems like x^2 – 3.”
    But how you solve this problem ?

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  • sergus

    In the prehistoric times in the other country it was taught like this:
    At first, kids were shown the proofs and needed to memorize a couple of formulas, most notably a^2 – b^2 = (a + b)(a – b) and (a + b)^2 = a^2 + 2ab + b^2. Also at that point a^2 + b^2 was taught as non-factorisable. These were considered to be the foundations. After that when kids had something like x^2 + 9x + 14 they were taught to look for a full square, meaning x^2 + 9x + 14 = x^2 + 2*x*(9/2) + (9/2)^2 + 14 – (9/2)^2 =
    (x + (9/2))^2 – ((9/2)^2 – 14) = (x + (9/2))^2 – ((81/4) – 14) =
    (x + (9/2))^2 – (81 – 56)/4 = (x + (9/2))^2 – 25/4 = (x + (9/2))^2 – (5/2)^2 =
    (x + (9/2) + (5/2))(x + (9/2) – (5/2)) = (x + 7)(x + 2).

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  • Tort

    Above the X, I write “Attic” and underline the A and the C. Below the X, I write “basement” and underline the b. You know why one would do this and you also know how to explain it to the kids. Perhaps a picture is worth a thousand words; but I didn’t invent that; somebody taught it to me.

    • educationrealist

      Actually, I have no idea what you’re talking about. And no one would use “attic” and “basement” today; many kids have no understanding of either.

      • Tort

        You don’t know what I am talking about? The attic is above the X and you determine the number for the top of the X by multiplying A and C. The basement is below the X and you determine the number by copying B, the coefficient of x. Well, if you cannot understand that, then I’d better forget about using it.

      • educationrealist

        No, I understood it was a method of learning the quadratic factoring method. But I’ve never seen it before, and today, basement and attic are rare enough for most kids that it wouldn’t carry the same meaning.

      • Tort

        Well, I appreciate the input. I’ve taken it for granted that the kids would know those terms, but you are correct–they might not–and I need to be ready for it in a few weeks. In recent weeks I’ve also picked up on some other new things that I may taken for granted that “they should know.” Thanks for the heads up.

      • educationrealist

        I once used a one and one example (from basketball) to demonstrate probability, and was completely derailed because the kids had no idea what it was.

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  • Heresolong

    Quadratic formula is not particularly useful for discovering replaceable and infinite discontinuities in graphs. Yes, I know, again this only applies to whole numbers, but factoring can be handy to be looking at the limit of a couple polynomials in a fraction and immediately be able to simplify.

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