Students are presented with three possible ways to begin solving the problem, “Solve for x: 2x2-14x+20=0,” and asked to decide together which way(s) of starting the problem are correct.

9th-11th Grade Math - Quadratic Functions*Barbara Shreve, San Lorenzo High School, San Lorenzo Unified School District, San Leandro, California*

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BARBARA SHREVE: Alright. Let's look at one more altogether and then I'm going to have you try some of these in your pairs. Okay? So, next page. You can just put that to the side and use it as scratch paper in a minute. This time we're going to look at three factoring problems and three more people; and you're going to need to decide who's correct in terms of how they start. Can you do me a favor Adam and pass this back? Thank you. So take a look at these. You have, this time, less time; you have ninety seconds to decide who's correct.

STUDENT: I think they're all right. So it'll be 8...8 and then 2. We do this 8x and then 2x. I think this one plus this one equals this one and this one times this one equals this one. So it's going to be 8 and 2.

STUDENT: Alright.

STUDENT: But this can be positive 6 but it's adding.

STUDENT: So all of them are correct.

STUDENT: No. Because if you add this one, it's not going to be equal.

BARBARA SHREVE: Alright ladies and gentlemen, we're going to do this one a little bit faster. Okay? I want to do a quick show of hands and not just one hand per table; everybody needs to take a stand. You can vote more than once. Raise your hand if you think Stefan is starting this factor problem in a strong way. Stefan, we've got some hands for Stefan. Okay, how about Katie? Is she starting to factor correctly? Is Katie starting to factor correctly? Okay, Miguel? Oh, we've got a couple of people voting for Miguel too. Okay. So now we need to talk. Why do we all like Stefan's? Kayla, you going to help us out again? Go for it.

STUDENT: I think most people start like that to figure out where the factors really are because that's how I start.

BARBARA SHREVE: Okay. Start with a generic rectangle? What's it going to help you find?

STUDENT: Uh. Again, I don't know what to call them. It's going to help you find (inaudible).

BARBARA SHREVE: Yeah, it's going to help you find these two parts right here right? Does anybody else know some ways we can describe those with math words? Those are the math words we always forget because we do it so fast.

STUDENT: Length times width.

BARBARA SHREVE: It's like the length times the width of the rectangle. Absolutely. Nice Terrence. Right? It's going to help us find those lengths and widths, or those expressions that are the factors. Factors I think is the word I think you're looking for right? Yeah, so I like Stefan's too. I think this is a great way to start. We're figuring out if this is the area of our rectangle, how can we find the outsides. But now Katie's. I've seen some people here use diamond problems. Don't you like diamond problems Alejandro? So is this an okay way to start? What's this going to help you find?

STUDENT: The same thing.

BARBARA SHREVE: The same thing right? It's a different way of looking at it. So this uses geometry because it uses the length and the width; and this uses a number pattern. What are you going to look for when you use this number pattern Alejandro? Or anyone who wants to help him out? Ronald?

STUDENT: The multiplication and addition.

BARBARA SHREVE: Tell me more.

STUDENT: Um, the numbers that she did on the side, you multiple and you'll get your answer at the top. And then when you add, you'll get your answers at the bottom.

BARBARA SHREVE: So if that's a number and that's a number, this is going to be x*y and then you're saying this is x+y?

STUDENT: Yeah.

BARBARA SHREVE: So those two numbers are going to help us find what would go on the side, so the length and width also. So Katie's is a great way to start. How about Miguel? You two raised your hands and said you liked what Miguel was doing right?

STUDENT: We changed our minds.

BARBARA SHREVE: Why? Because nobody else voted with you?

STUDENT: No.

BARBARA SHREVE: If I voted with you, would you change your mind?

STUDENT: Yes.

BARBARA SHREVE: Okay, so why...how does my hand help convince you?

STUDENT: Uh...

BARBARA SHREVE: Does this get us started?

STUDENT: Yeah. Oh no.

BARBARA SHREVE: Why are you saying no? What's Miguel doing between here and here?

STUDENT: Factoring.

BARBARA SHREVE: Pardon.

STUDENT: He's factoring.

BARBARA SHREVE: He's starting to factor right? So there's part of what he does that is absolutely correct and part that is a little bit...he's added something to the problem that we don't want to add. What's he added on?

STUDENT: Zero.

BARBARA SHREVE: He's added on this equals zero that wasn't part of the original. And I do see sometimes you guys making that mistake of like, oh you know how to find x now, so you want to keep going and show-off everything that you know, when really you get to stop after factoring. Okay? So he doesn't need to add that on but writing this expression like this is great, that part is fine. Okay? Do you guys have questions about any of these three starting places?

STUDENT: No.

BARBARA SHREVE: Kateef is okay but is everybody else? Dolores, you alright? So here's the thing you guys. We've talked about a couple kinds of problems and couple of starting places, and now what I'm asking you to do at your tables is look at a whole bunch.

I put examples of common mistakes in front of students so that they can confront them when they are not part of the work the student has created him or herself. I find that the distance created when students analyze another person’s work – especially when that person is not immediately present – allows students to be more analytical. In this case, it also presents the challenge for students of trying to make sense of someone else’s work when that person is not present to answer questions about it. In pushing students to take a stand or make a choice about who is correct, I am trying to get them invested in either confirming their answer or understanding why someone disagrees with it so that they are less passive listeners. In this case students were reluctant to do so, leaving me unsure whether they had been able to understand the students’ work or were not sharing their opinions for other reasons. It was helpful to hear how students were interpreting factoring as a tool that is used to rewrite an expression when the rewritten expression is the goal, rather than as a tool that can help you work toward a solution to an equation. Through the discussion I recognized that students did have confusion about what question they were answering, and what the answers meant. I was faced with decisions about how much more to explore in a class discussion and what to leave for them to discuss in pairs and teams, and I opted to move students to work in pairs rather than continuing to pursue the discussion whole class, so that more students would have the chance to participate in articulating questions and trying to answer them.