why polynomial long division is awesome

A week ago I had plans for Precalculus. I liked them alright, but I felt they could be better so I did what I always do when I need my brain to percolate over ideas: I start catching up on my reader. I ran across Jason Buell’s post about the IMPROVE model (you should go over there to read the details of that model if you’ve never heard of it). I then scrapped my plans and started re-writing them. That was Sunday night. *shakes fist at Jason and his inspirational posts* 

Before we get into the week, let me be the first to say that I have no grandeous dreams of students going forth into the world and using their ability to long divide polynomials or sketch them from factored form by hand. Polynomial Long Division (henceforth referred to as PLD) is a way for me to do the following:

1. Check on and improve student ability to add/subtract positive and negative numbers as well as distribute them
2. Improve mathematical endurance (why yes, those last two problems may take up a whole page each)
3. Improve student understanding of how the equation, the factors and the graph work together with higher order polynomials
4. Create an opportunity for my studens to feel like mathematical Rock Stars

Let me explain that last one. It goes back to my drive to help students see themselves as being ‘good at math’. PLD is impressive looking.  I mean, really impressive. Especially when you start with a 5th degree, have students figure out the zeros, create the factors and then do multiple rounds of long division in order to come up with the factored form which they then must graph by hand labeling all intercepts. Then they can check their work with the calculator. Yay for instant feedback.

At the start of last week the kids had only worked a day or two with PLD. By then end of the week, with some group assistance, they were all working through these massive problems. They’ll whine that the problem’s huge and takes up too much space, but they’ll do the whole thing and then just kind of sit back and stare at it with this look of ‘did i actually just do all that!?’ on their faces. There were lots of “got it!” mumblings while they worked. It’s not the crowing cheers you get with, say, WCYDWT problems. I liken it to the satisfaction of a job well done. Because what is the true hold up for a lot of students in math? The concepts, or the grammar? Is it that the kid can’t understand what’s going on, or that they are so hung up on the symbols and the difference between coefficients and exponents and plus’ and minus’ that their brain is spinning? I think it’s the latter much of the time that shuts off the student mind.

So here’s the worksheet given out to each table group. I put them in a plastic slip so the four students in each group would have to share. Maybe even, I dunno, read a problem to the group. Work together.

Monday
I started by laying the ground rules. Students were to work with their tablemates to solve the groups of problems: sketching a polynomial, identifying rational zeros, and writing factors given zeros. For each group a set of 5 side-problems needed to be answered using short sentences. I told them I was worried they were not reading problems in full and that the 5 questions were to help give them things to ponder when they approach a problem. The 5 side problems were:

(a) What is the problem asking?
(b) What is the problem giving you?
(c) What are the essential features?
(d) How are the first and second problems in this group similar? Different?
(e) How do you know your right?

Once a table finished the first three problem groups they are to call me over to get checked off. More on how I did that later. I also noted to the classes that I didn’t expect any group to finish the first page today. The problems are about bringing together several of the ideas we’d been working with (factors, zeros, rational zero theorem, PLD, sketching from factored form).

They set off Monday and most tables got through the first group. I was able to meander around the class and ask questions, rephrase problems, and, my personal favorite, listen to students argue about problems.

Tuesday & Wednesday
Set the students straight to work this day. I should also note that it was AP testing week, so I had gaps in many tables. I ended up taking tables of two and splitting them into other groups. This was actually a really effective way to share ideas between groups because when they were put back together on Wednesday they spent time comparing answers. For me, Wednesday was more listening and giving little kicks when they would get stuck. Often it would be pulling the page of one of their tablemembers to the middle for all to see and pointing at what they had written. Then a reminder that they should always ask their table first. Yeah, I’m one of those teachers that if you call me over with a question I will ask the person next to you what the question is. I’m not perfect at it, but it’s something I work on and the kids expect.

By the end of the day I had about a quarter of the tables in each class checked off or ready to be checked off first thing the next day. The check off worked like this: table calls me over. I randomly pick a student’s work to look through. While checking through I make comments about little stuff (“your graph is naked–you should give it labels to cover up” or “I like how you _______” or “that’s a great way to phrase what’s going on there. Thanks.”). I also directed follow up questions towards different members of the table to poke at what I know are common misunderstandings. I learned that side question (b) is badly phrased and students didn’t get that it was meant to be very general. (c) was asking for specifics. Will need to clarify that for later.

I really like the check up since I got face time with every single student. With classes hovering around 32, I often feel like I can go over a week without talking one-on-one with some of my quiet ones. The randomness of the check also keeps groups at the same pace. Yes, there was definitely some copying of a neighbors’ work going on, but I heard far more demands for explanations than I saw copying of work.

Thursday
By this day almost all tables were working on the last two problems (I had a few tables disproportionately affected by AP testing and one that was slacking a bit I had to crack the whip with). This is where the dividends of the previous week and the first three day’s payed off. More times than I can count a table would call me over to ask a question about the last problem group only to end up talking me through each step making connections to the problems on the front page while I just stood there with a carefully blank expression. They would then wave me off. I love that.

So much good practice on fundamentals. Precalculus students would never accept me giving them a quiz on addition/subtraction/multiplication/division, but man, if they didn’t get one while working on PLD. Some confusion over what factor to divide by first, to which I would always ask “what’s 24 divided by 2? then by 4? what about 24 divided by 4? then by 2?” They were able to work from that to figure out that it didn’t matter and then set upon the polynomials using PLD until all that was left were a set of factors they then used to make a sketch. It was awesome to watch.

Probably my biggest role that day was helping tables find the little errors in the basics. I’d have tables get stuck on why they have a remainder and that they know something is wrong but none of them could find it. Heck, sometimes took me over a minute to spot the error. Stupid negatives. It’s always a negative.

Tables that finished earlier were set up with skill quiz questions to work on. I also sent a few off to other tables to help out. Mmm, peer tutoring.

Follow up Thoughts:
For those wondering why I havn’t said anything about synthetic division: I have nothing against it, but I like having students divide by x-squared and the like. I am also unconvinced that students understand what it’s doing.

For bringing together a bunch of ideas, I think this type of table-work/discussion/check-in is great. The side questions had the students using the vocabulary of the unit and focusing on what’s important. I’d like to do more of this in some of my other units, though I’d also like to incorperate more of the PCMI flare of important stuff/neat stuff/tough stuff.

I think my classes are more comfortable working with larger equations now and dealing with longer problems. I really hope that this work pays off when they move on to AP Calculus or AP Statistics. Or just when they are confronted with large problems in general. Break it down! What do you need to do first? Are you double-checking your work? How do you know you’re right?

And how awesome is it that at the beginning of this week you looked at the back problems with wide-eyed fear and now they lay defeated on the page in front of you? Rock on, kiddo.