I continued some direct instruction on factoring today.
I have been prepping the students for a while by having them practice the diamond problems that I have mentioned previously and also by heavily emphasizing area models for multiplying polynomials. My goal for the first was to just get students to practice finding numbers that satisfy a multiplication and addition constraint and the goal for the second was to help them see factoring as backwards multiplication problems. As in, “This polynomial is the answer to a multiplication problem, so what is the question?”
Since we have already done factoring by GCF and Grouping, I started teaching factoring quadratic trinomials. Besides an area model I also drew parallels to factor trees, drawing a factor tree for a number and then creating a factor tree for a polynomial next them. I hit this point hard, repeating that we were trying to pull apart polynomials as much as they could, just like with prime factorizations for numbers.
My original lesson plan was to talk about quadratics with a leading coefficient of 1 using what I called the “Diamond Method” (which was just to set up a diamond problem with the c value in the top and the b value in the bottom) and then introduce Area (Box) and AC methods for quadratics where a is different from 1. But then I felt silly when I realized that AC still works for a=1 and I felt it would be better if I just taught AC method from the beginning since it was essentially the diamond method.
I am not writing this up well at all. It’s coming out jumbled. Anyway.
So I taught all quadratics the following way:
- Set up the AC diamond.
- If a =1 then the solution to the diamond problem is the factorization of your polynomial.
- If a > 1 then the solution to the diamond problem tells you how to split the middle term and you need to either use the Box Method or Grouping from there.
I didn’t bother with “if b is + and c – or if b +…” and so on. I just told them to use the diamond set up for every problem. I really wanted to get away from just having the students remember a bunch of special cases based on how the polynomial is structured and instead just focus on that “backwards multiplication problem” understanding that applies generally to a lot of polynomials they see. I already laid the groundwork for perfect square trinomials and difference of squares when we were in the multiplication unit, but I am curious to see how it those will go later this week.
I thought this strategy worked pretty well but I could tell that I bored my students to tears with how many examples I had them try and then worked through with them.
Oh well. Can’t be exciting every day.
Thanks for reading.