Now, I'll be honest, this is NOT something I have done with my students. I have had the tendency to instruct without letting them do a whole lot of exploring. Part of it for me is that there is

**so**much material to teach in Algebra 2 (and I am having to play catch up from Algebra 1) and part of it is my own comfort level. However, as I was thinking about my lesson on Friday about the Remainder and Factor Theorem, inspiration struck me on Wednesday. My Algebra 2 students were having an assessment on Thursday and there would be enough time afterwards for them to do a little noticing and wondering. After their assessment, I asked them to complete a paper with the following questions:

Do (x^3 + 6x ^2 - 3x + 7) / (x + 3) using synthetic division.

Find f(-3) if f(x) = x^3 + 6x ^2 - 3x + 7.

What do you notice?

What do you wonder?

Do (2x^4 + 6x^3 - 15x^2 + 15x - 50) / (x - 2) using synthetic division.

If f(x) = 2x^4 + 6x^3 - 15x^2 + 15x - 50, what is f(2)?

What do you notice?

What do you wonder?

Here are some of the responses I received:

**Notice:**

- I don't remember how to do (the f(__)) problem.
- same problem and you get the remainder
- the remainder is the same as the answer of the function and they have the same number of terms
- The remainder of the synthetic division is the same answer as the 2nd problem I worked out (the f(__) problem).
- I noticed that the numbers are the same in the problems and f(x) is the same number as it is in the box of in synthetic division.

(I had several answers that were similar to the 2nd, 3rd, and 4th notice bullets.)

**Wonder:**

- How to do it (the f(__) problem)
- Could you use synthetic division to find f(x)?
- (Written under the 1st wonder) I got the same remainder for the 2nd set of problems, but not the 1st set of problems. What did I do wrong on the top question?
- Can you use functions to solve synthetic division?
- What do they have in common? Why are they both remainders?
- How does that happen? Why are they the same?
- Are we going to do the same thing as before or is it different?
- I wonder if the two problems are related. I wonder if you can use the 2nd problem to figure out the synthetic division problem.
- Will this be the case every time? (Then the student added on the 2nd one when his answer did not match) Will the answer match the remainder if the number replacing x is negative?
- Why is the answer the same as the remainder?
- Are they connected? Is this another way?

**My observations:**

- Many (TOO MANY!) of my students did not recognize function notation or how to work with it. This is something we did review earlier in the year and it dismays me how many had no clue. Even several of my top kids came up to ask how to deal with f(-3) and once I told them, they remembered. However, that they even had to ask worries me.
- The students who didn't know how to do synthetic division (all the way) correctly obviously did not make the connection at all. Many of these students left the notice/wonder part blank.
- I half expected some smart-aleck responses from some of my students (especially some of my struggling ones). I didn't get any. However, I had a lot of blanks in the spaces asking for their noticings and wonderings.

I had chosen to do it this way, rather than orally, for two reasons:

1) I knew I was going to have extra time after their assessment and this would help keep them focused and quietly working on something.

2) I felt that this would give my students who do not catch on as quickly as my top students the opportunity to think about it and possibly make the connection on their own. In my Algebra 2 classes, I have a range of students from very bright students (who mostly took Algebra 1 in 8th grade) to students who struggle with math. Not all of my high-ability students caught it, and some of my middle ability students did put together the connection rather nicely. I hope that will help them tomorrow when we discuss the Remainder and Factor Theorems.

Originally, I didn't think I would give them their papers back. However, after reviewing them and reflecting some, I think I will give them back to them. Hopefully we'll have some nice discussion about it tomorrow.

*Addendum:**On Friday, my principal came in to observe me during the 1st of my 4 Algebra 2 classes. (We are a Race to the Top school and we are doing the new Ohio Teacher Evaluation System this year.) Although I'm guessing I'll get dinged for this not being as much of a class discussion since they had completed the noticings and wonderings on paper the day before, I feel like it went pretty well. I would have liked some more discussion out of them. Part of it for me is that I haven't done this before and finding the right questions to elicit discussion out of them was a little bit of a challenge for me. I feel like we answered most of their wonderings, which is a good thing. :-)*
## 2 comments:

They made some great connections and conjectures. I liked this wondering a lot:

"Will this be the case every time? (Then the student added on the 2nd one when his answer did not match) Will the answer match the remainder if the number replacing x is negative?" That could be something to try to answer in class together!

I like these ones too:

Could you use synthetic division to find f(x)?

Can you use functions to solve synthetic division?

And this, although it's sad to assume that you did something wrong, when it's just how the math works out:

I got the same remainder for the 2nd set of problems, but not the 1st set of problems. What did I do wrong on the top question?

One thing I thought of if you want to give more students more access to the problem, if you do this again, is to have one worked-out example that they just notice and wonder about it, and a second (and third!) problem that they do themselves. That can serve as their reminder and maybe the act of noticing and wondering about an example will reveal some of what they are getting stuck on?

I'd love to come do Noticing & Wondering with your kids, you've got some deep thinkers in your classroom!

Brilliant post! I can't wait yo get to remainder theorem again now!

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