The Box Method for Factoring Trinomials

I love using the box method (area model without appropriately sized side lengths) to help students learn how to multiply polynomials. I love, love, love to use the box method to divide polynomials. But I only started using it to factor non-monic trinomials last year and I did a horrible job of it. Really horrible. I'm sure none of my students understood it because I didn't really get it. I am happy to report that I now LOVE using the box method for factoring non-monic trinomials. I shared this with the other math teachers at my school (I send out a weekly "Math Minute" - a link to a cool activity, a blog post, an idea that is worth sharing... and this was what I sent this week). I tried to colour code it to make it easier to follow and made a second box to better show the steps.

Here is an attempt at an explanation in case the example is not clear. Start by finding two numbers that multiply to the product of 'a' and 'c' (here -120) and add to 'b' (here -2). In this case the number are 10 and -12. The box represents the area (trinomial) and we are looking for the length and width (binomials). I always put the x^2-term top left, the constant term bottom right and the x-terms along the remaining diagonal. The number we found are used as the coefficients of x so 10x and -12x go in the boxes along the diagonal. Then I common factor the first row and the first column (that's where the 2x and 4x come from). This would normally all happen with one box, but now jump down to the second box. Figure out what multiplied by 2x will produce 10x and what multiplied by 4x will produce -12x. Those complete the factors and you can check that it all works out with the constant term (does -3 times 5 equal -15?). There you go. I love this because it is not a trick - it makes sense and has built-in error checking. I find it really fast, too.

I should note that I also show my students how to factor using decomposition and give them the choice of which method to use. So far more are choosing to use the box method. I can't wait to show this crew how to divide polynomials in a couple of years!