At the beginning of this lesson, I like to take some time to discuss any issues that arose from the previous lesson. Specifically, I want to discuss what to do when there is a "missing" term in the polynomial dividend. I'll ask for a volunteer to show the class how to solve the third problem from the homework. In the best case, this student will not only be able to explain how to do it, but will also explain why. In the worst case, I'll walk the class through it myself, being careful not to tell them how, but to guide the class with Socratic questioning.
The line of reasoning should go something like this:
When working with integers, we must put a digit in every place, so we include "0" in some places. But this isn't usually necessary with polynomials because each term contains a "place value label" - the associated power of the variable!
During division, however, it helps to make all the "places" apparent, so we fill in the missing terms making sure that the coefficients are zero. In this way, while we are careful not to change the dividend, we also make room for terms of all different types in the quotient.
This section of the lesson might go one of two ways, depending on how much individual assistance I think the students need.
Option 1: If it seems like many students are going to need individual help, I'll begin by writing problem #1 from Classwork for Long Division WO Remainders on the board. The students will try to complete the division as a self-assessment. I'll allow 3 - 5 minutes to complete the long division, then I'll ask the students to explain the division process to me while I write out the steps on the board for all to see. Students will assess their own work and correct their own mistakes. Finally, I'll ask for some brave volunteers to share with the class a mistake they made, why they think they made that mistake, and how they might avoid it in the future.
When this is finished, I'll assign #2 and #3 to be completed in the next 10 minutes. As thes students work, I'll circulate and offer assistance.
Option 2: If it seems that most students "get it", but need some more practice, then I'll simply assign three more problems: #1, 2, & 3 (See Classwork for Long Division). None of these has a remainder, but #3 is "missing" one term. I'll ask the students work work individually for at least 5 minutes so that I have a chance to do some quick formative assessment while I move around the room. All three problems should be completed in about 15 minutes, and this will give me time to work individually with the few students who need it.
To help my students understand how to deal with remainders, I'll simply present them with "just one more problem" and wait for them to run into trouble. You never know, some bright student just might "discover" the correct way to hand it. When they do get stumped by the remainder, I'll use Socratic questioning to help them figure it out. This will go well if you act surprised by the remainder, and then pretend to be confused about what to do with it. (I make enough mistakes in class, that I can usually fool my students into thinking this is just another one of them.) The conversation I'm envisioning might go something like this:
Teacher: "Well, class, what do you think: are you getting better at this [i.e. long division w/o remainders]?"
Students: Yes! It's easy now!
"Ok, let's try another one on your own just to make sure. Please take about three minutes to do this problem as a self-assessment."
[The teacher writes problem #1 from Classwork for Long Division W Remainders on the board. Three minutes pass quietly. Some students seem confused.]
"So, how did it go?"
It was harder. It doesn't seem to divide evenly - there's a remainder.
"Hmm. That's never happened before. Let's work it out together so we can all see."
[Students direct teacher through the steps of the algorithm.]
"Ok, so there is certainly a remainder of 2. What do we do with it?"
Well, when we divide integers we just add more places to the right of the decimal, but that would be strange with polynomials. We'd have to add powers like x^(-1) and x(-2).
"You're right; we don't want to add those terms to our polynomial."
Can't we just write "remainder 2"?
"We could, but that seems like we're saying we haven't really finished dividing the polynomial."
"Is there some other way to 'finish dividing' when we're working with integers? Something other than decimal places?"
"For example, if we divide 17 by 3, we get 5 with a remainder of 2. Is there some way to deal with that 2 without adding decimal places to our quotient?
Sure, we can write the remainder as a fraction! It would be 5 and 2/5.
"Excellent! Let's do the same thing here."
[Teacher and students discuss how this should be done. An example with integers helps.]
"There. Now we know how to handle remainders when dividing polynomials. Let's practice with a couple more problems: #2 & #3. You have 10 minutes until class ends, after that these will become homework. Have fun!" (Assign more than two problems if time permits!)