I recent made a tweet of a pic from a quiz from one of my students on the topic of multiplying polynomials, more specifically multiplying binomials (see pic). I was stoked because the students actually used all 4 methods I modeled for students during class. As a math teacher, when possible, I try to model all of the different strategies for different problems. When I posted the pic, I got a lot of likes and comments, and some people weren’t familiar with all of the techniques, which led to this post.
Below I will talk quickly about teach technique:
Box Method (see problems 10 and 13) – this to me is the most important/useful method with great connections. Later we will explore factoring trinomials, where I will use this model again. It’s a nice tool for visualizing the idea of all 4 multiplications, and some students like the organization that goes along with it. Also works well with more complex polynomial multiplication.
Distributive Property (see problem 12) – Probably the method I saw used least by my students. I’m trying to figure out how to make students value this method more, we do talk about it again when we discuss factoring by grouping. I think the first step shown by the students is something the students don’t think is useful.
FOIL (see problem 9) – I know, I know, I probably shouldn’t use it. I HATE showing students a method that does not work in all cases. My fear is that if I don’t at least acknowledge its existence, it cause issue with their higher level math teachers depending on that teacher’s method. I over emphasize that it only work for binomial x binomial, and relate it back to the distributive property.
Stack and Multiply (see number 11) – I had a bunch of people on twitter mention that they never saw this method, which sort of shocked me. I am not sure where or when I learned this method, but is one that a bunch of students LOVE. I also think it is useful, because it expands upon their prior knowledge of multiplying 2 digit numbers. Usually when I present this method to students I first show them the problem on the left, and students mention that their most common mistake was the “carrying”. I mention that this method is similar, with no “carrying”. I like this method because the students are familiar with the process. I also like the idea that, similar to place value, like terms are lined up vertically. 
I also included an example below that might answer a couple of questions you might have about this method for more involved problems. 
As a teacher I try to give my students as many methods as possible, and let them choose which one works best for them. I also try to make my students aware of the pros/cons of each technique. I teach 8th Grade Algebra 1 in a K-8 district, I rarely get time to articulate with the teachers these students will have in high school. By presenting my students various techniques I hope to give them the skills and knowledge to succeed in mathematics in the future.
Mr . Corley Mathematics 