How to Factor Quadratics

After years of teaching how to factor quadratics and then getting in my car and banging my head on the steering wheel, I decided that enough was enough. I was going to spend some time finding a better method. I took to my favorite community of math educators, the #MTBoS.

Several different ideas were thrown my way, but the one that was most attractive was Mary Bourasa’s method, sent to me by Helene Matte.

Last year I had tried the “diamond” method, which worked a bit better than simply guessing and checking, which I’d done in previous years. The first problem I ran into was that I had trouble remembering what went in the top of the x and what went in the bottom. In videos I watch online teachers did it different ways. I think this was because of the second problem I ran into, which is where the hell did this giant “x” thing come from anyway? It’s not a “trick” really, but it does seem to have no connection to other things we do in math.

I might as well have said, “Today we are going to factor quadratics. Draw a random shape, fill it with numbers in the recipe I give you, then get your answer.” And kids weren’t that good at it.

Enter Box Method (or area method, or whatever)

A few years ago I learned about the “box method” for multiply polynomials, binomials included. Put the first polynomial along the top, the second along the left side, multiple rows by columns. Very similar to multiplying actual numbers with this method.

Example with Numbers

Example with Binomials

The approach to factoring using this method is attractive because it feels like working the box method, in reverse. If students are familiar with the box method for multiplying binomials, it’s a natural extension to use this method to factor them (as I often talk about factoring as the reverse of distributing).

Step one

The first step in this process is writing down a*c (M), the coefficient on the middle term (A), and then finding two Numbers that multiply to give you M and add to give you A.

Step Two

Once you have your numbers, fill the box. The upper left corner and lower right corner have to contain the squared term and constant term respectively. Fill the upper right and lower left with the two numbers you found in step one.

Step Three

The last step is to factor out the GCF of each row and column. Then you’re done. You have the factors that multiply to give you the quadratic.

A couple notes

This method fails miserably if you don’t factor out any common factors at the beginning. For instance, if you have a 2 in each term that can be factored out, you have to do that first before using this method.

I still have to grade the quizzes that cover this section but the kids seemed to respond a lot better than they have in previous years. I’ll update this post once I know more.

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