Antoinette Villarin’s students return to their groups to discuss and share their thinking about the relationship between graphs showing rate of change. She challenges them to try to find the two graphs out of four with the correct combination to describe the same container. As the student partners work, Antoinette circulates around the room engaging pairs in questioning.

8th Grade Math - Representing Constant Rate of Change*Antoinette Villarin, Westborough Middle School, South San Francisco Unified School District, South San Francisco, California*

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- Clip Transcript PDF
- Lesson: Comparing Lines and Linear Equations by the Mathematics Assessment Project Web Link
- Handout: Recording Sheet and Gallery Walk Guide Word Document
- Student Work: The Race PDF
- Student Work: Graphs PDF
- Student Work: Matching Poster PDF
- Student Work: Recording Sheet and Gallery Walk Guide PDF
- Student Work: Exit Ticket PDF

ANTOINETTE VILLARIN: What I just showed you was just a graph of the water flowing out of the top prism. Okay? But there's also a graph of water flowing into the bottom prism. And up here I have four graphs, and these graphs either represent water flowing out of the top prism, or they represent water flowing into the bottom prism. Okay? So they're either a top-prism graph or a bottom-prism graph. Your job is to try to find the two out of these four that have a correct combination, so that you're describing the same container. Okay? The same container.

So, for example, if I pick one of these as my graph to represent the top, then I need to find the other graph out of the three that's going to represent the liquid flowing into the bottom of the same container. Okay? You need to find the correct combination, and there are only two cards that match and do this. When you have it, I'd like you and your partner to circle the two, and on the right side, you're going to tell me why. Are there questions on your task? No? Okay, Partner A, you could take the lead and restate to Partner B what you're doing, and then you can go ahead and get started. Okay, go ahead.

STUDENT: There are only two cards that match, so of... Okay, so this is zero point five seconds. That's why I pinpoint. I can’t do…that’s hard to do…okay…zero, three seconds. And six, this is four. I think these match, maybe. I'm not sure yet, though. I'm not sure yet. Because both have -- it’s kind of both, like, four centimeters. This is four centimeters too. But here, wait...no no no. Maybe I was wrong. Maybe I was wrong. Okay. This is three...no, this is two. Yeah, this is two and this is six, right? Okay, six. Six add one second. Okay, okay. I don't like this. I don't like this at all. Okay, no, this does not match. This does not match because this is like three seconds, this is like one second. Like, immediately. So I don't think so.

ANTOINETTE VILLARIN: And what do you think the bottom is going to start at?

STUDENT: [Inaudible].

ANTOINETTE VILLARIN: Keep going, and then share with Katelynn what you think.

STUDENT: This is six. It can’t be, well, you were right. This is one because this is six, and we're using the measurements from her bottle or prism then the bottom would have to have zero, but these don't start at zero.

STUDENT: I told you.

STUDENT: Yeah. So these are the matching ones.

STUDENT: Why?

STUDENT: Wait, try like -- let’s try six, just… this is like a one right here.

STUDENT: Yeah. That's two and that's four.

STUDENT: Oh, I don't know this one.

STUDENT: This is eight and then this is seven. That's five.

STUDENT: So if we put it at zero point five seconds...if we raise it up, like, kind of like right here, it's like right around the four.

STUDENT: This is going up one, two.

STUDENT: So it's like somewhere right on the four.

STUDENT: That's point five.

STUDENT: So...wait. If we try to match this over here, that's like about six.

STUDENT: That’s like two to the...

STUDENT: Isn't it over [inaudible]?

STUDENT: Two over point five because it's one point zero.

STUDENT: Then how...so it's like two up and then, like...

STUDENT: Yeah and the run is point five because zero to point five.

STUDENT: So it's like two up five across.

STUDENT: Yeah. So don't you divide it? So what's two divided by zero point five? One. No, I’m just kidding. I'm just kidding.

STUDENT: Two divided by five point?

STUDENT: Zero point five.

STUDENT: Um, I think it's... I don't really...

STUDENT: I don't either.

STUDENT: Three point five.

STUDENT: It's not directly in the middle.

STUDENT: Two point six.

STUDENT: Two point five.

STUDENT: It's not two point five.

It's hard when you're walking around and you can’t hear every student pair. I appreciated the videotaping. The videotaping allowed me to observe student thinking from the opposite end of the room. Normally, when I can’t observe every pair, I have to be attuned to what students are saying. Things like “This is too hard” or perceived difficulties remind me to check in with those students eventually.