Last week, I had to brush up on my math magic for our unit on Polynomial Division. I had completely forgotten how to factor polynomials bigger than quadratics. It looks daunting when you first look at it. But once I found a good refresher course online, the concept came rushing back to me. Polynomial Division is just like long division except your finding factors, or dividing the bigger polynomial with a factor that's in the form of (x+4).
Starting out with a larger polynomial to factor like x^5+7x^4-3x^3-79x^2-46x+120 didn't faze me. However, it might faze my future students. So, to help my students establish a connection between finding factors for quadratic equations and polynomial division, I would use polynomial long division on a quadratic equation as my introduction to the concept. That way, they can see that polynomial division and factoring quadratic equations are about the same thing: finding factors. It's just using a different method to find them. Also, they can connect what they already know about factoring quadratics and apply it to polynomial division, which might help their understanding.
To begin the lesson, I would pick an easy quadratic, one that is easily recognizable and easy to factor, like x^2+4x+4. The factor of this equation is (x+2)(x+2) or (x+2)^2. The students might say this immediately, but the point of this is to get them familiar with polynomial division. So, we would go through it together step-by-step.
After the first one is done, I would give the students a cubic expression to factor, like x^3+6x^2+11x+6. Now, the students might ask "What do we divide it by?" and my answer would be, just like a quadratic, find and try the factors of the last number (6 in this case), and see if they work. I would also tell the students to check their work by graphing it.
Next, I would throw them a curve-ball, and give them x^3-7x-6. This expression, might give them pause. The x^2 is completely missing, vanished, disappeared! I would let the students offer suggestions or try to think it out at first. Then, I would show them how to factor it with an easier quadratic. Again, one that the students can recognize, so they can make the connection between the two and become less daunted by the process. That quadratic will be: x^2 -1. Here the x^1 part of the equation is missing; it has been canceled out. Again, the students will say almost immediately that the factors are (x+1)(x-1). As a hint, instead of writing the quadratic in the form of x^2-1, I would make the x^1 spot reappear or reveal itself by adding 0x. Then the quadratic would be: x^2+0x-1. I could also reveal more of what I did and why I did it by factoring out the factors for my students: (x-1)(x+1) = x^2-x+x-1=x^2+0x-1. Now, students can see the connection that 0x serves as a placeholder for the missing x^1, since it was canceled out in steps above. This might be the only step the students need to see to feel comfortable about dividing polynomials with missing spots or terms in the expression. If not, I would walk them through it step-by-step, and let them know that we would drop the 0x just like any other polynomial with all it's terms. Then I would give them x^3-7x-6 again to divide by themselves.
Good post, thinking through how you would sequence a lesson. Nice job with sharing your teacher reasoning, and I agree with most. (The missing terms part especially.)
ReplyDeleteOnly thing I'd want to see added is consolidation. This might be a good one to consider 'so what?' - what's important here? (In the math or the teacher thinking.)