When I met with my new supervisor in ed school (the second one), I told her that I didn’t feel like I was introducing topics well. She was extremely supportive, and it became one of our favorite discussion points. How do you introduce a math topic? And in my two plus years of teaching, I think I’ve become good at it–particularly in first year algebra, which I’ve taught more than any other. When I taught CPM Geometry, I hated everything about it except the way it introduced tangents (as a slope). I often spend several days mulling a good intro, and have been known to toss in a few days of review just to get my story right.
The story is usually a problem the kids can understand—and understand that they don’t have the tools to solve it. Or sometimes it’s a parallel. Either way, I try to give them an image, a reference, a bucket. Maybe it will help them trigger memories, because retention is a huge issue in teaching math.
It’s weird, the quick descriptors that teachers use. When I say I’m “teaching polynomials” in Algebra II, any math teacher knows I’m teaching everything but quadratics in their binomial/trinomial form, since quadratics is its own unit. Teaching polynomials means the kids are learning polynomial multiplication, polynomial division, synthetic division, maybe some binomial expansion, certainly some brute force factoring In general, the polynomials unit in Algebra II doesn’t have any obvious purpose other than to prepare the kids for pre-calculus. It’s just “let’s learn how to manipulate polynomials to no immediate purpose”. And that makes the intro tough.
Over half my students will not go on to pre-calc next year. Some will be taking Algebra II again, either with or without Trig. Others will be going into remedial college classes. So even leaving aside the intro, how do I help the kids make sense of this? I don’t care if they can expound on function notation or binomial expansion, but I do want to be sure they know the difference between multiplying two trinomials vs. two binomials, and when to factor. And for god’s sakes, I want them to know that they can’t “cancel out” the x2 term when presented with a rational expression.
I started the unit by explaining the preparatory nature of some of this—that they won’t really see how it’s used until pre-calc, that they just need to recognize these equations and know what to do. Multiplication, they’ve been doing for a while. Factoring, too. But then there’s division proper, which most of them won’t use again. I thought about not covering it, until I realized that I could use the lessons as a way to get them to think about division and factoring.
And so, the introduction:
I don’t present this all at once. I start with the first fraction, then ask what could we do. Someone will reliably say “reduce it”, and so we’ll reduce it. I then introduce the term “relatively prime”.
So then, I say, we do the same thing with variables and fractions, and we go through this step by step:
This isn’t the actual whiteboard example; I just wrote it up and took a picture. But it’s the idea.
And it worked. It gave the kids a great point of reference and most of them were able to divide a simple polynomial on a quiz a few days later.
I’m trying to build on that now. Thus far, I’ve taught them two forms of division and factoring. Ideally, they should be able to identify when to factor, when to divide and when, please, synthetic substitution is a good idea. So I went through the pros and cons of each:
- Factoring: the default. Pros: It’s fun to cancel out the common terms. In a test situation, you can pretty much assume that the terms will factor. Cons: Only works with first and second degree polynomials. After that, it’s brute force. Limited: if you can’t factor, you can’t. Nothing to tweak.
- Division: you don’t have to use it much, unless asked. Pros: It’s the easiest to relate to–works just like number division (most know this already. Most). It’s extremely flexible, works in every situation. Cons: you don’t have to use it much.
- Synthetic division: you never really have to use it. Pros: Incredibly useful for evaluating terms, which is what we’ve been using it for. Quickest method to find factors in higher order polynomials. Cons: It’s the most difficult to learn. Unless you use it often, it’s easy to forget. You have to know what it means in order to find it meaningful.
And with one exception, they did pretty well.
Question 2. It got to them. First, they saw the division sign. So they divide, right? No! Don’t they remember? “Division is…..” I prompt. “Oh, yeah, you flip it!” They flipped it. But then they multiplied, which made sense because they were being tested on that too, right?
Still, it was a good quiz. Once I reminded them, they worked it correctly. Few misconceptions. I’ll need another week to beat in the triggers to tell them what to do when. But it’s working.