Antoinette Villarin’s students continue their partner work, matching graphical representations of rates of change with each other.

In this clip, two different student pairs engage in mathematical discourse, comparing graphs to each other to discern which graph represents the top prism out of which the fluid is draining, and which represents the corresponding receptacle prism. Students offer ideas and defend their thinking to each other.

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

STUDENT: So this is top, right?

STUDENT: Yeah.

STUDENT: So we have to make a bottom. So it has to increase then. Zero. Zero. Should it start here? Or do you think it should start here? How should we do it? Maybe we can do... What should we go by? Like, two's...

STUDENT: We can start at two.

STUDENT: Two? Okay, so this would be zero, 2, 4, 6. Like this?

STUDENT: I guess.

STUDENT: Okay. Then, how about these ones? Do these ones by ones or...? Maybe we should do -- maybe we should go by these ones. We should go by decimals.

STUDENT: Like this?

STUDENT: Yeah. So 1, 2, 3, 4, 5. We can do 0.0, .5, 1, and then 1.5, and then 2. Okay, so now we just draw the line. So maybe we can do... Do you want to start at two?

STUDENT: Yeah.

STUDENT: Okay, so let’s try to do... Line going up to here. And then end at two?

STUDENT: At six.

STUDENT: End at six?

STUDENT: I think it ends at four.

STUDENT: Ending at four? Or should we start at four?

STUDENT: Start at two and end at six.

STUDENT: End at six? So should we do it by ones? Like, 1, 2, 3, 4, 5, 6. Okay, let's try. Do you want to do it by decimals?

STUDENT: [Inaudible].

STUDENT: Here. So should we go by decimals or whole numbers like this?

STUDENT: Go by half a second.

STUDENT: Half seconds? So, these ones?

STUDENT: Yeah.

STUDENT: Okay, let's... Should… What should the ending be? Four?

STUDENT: Six.

STUDENT: G3 starts and then at four centimeters, and goes...two centimeters...

STUDENT: From .5 seconds it would be...

STUDENT: One.

STUDENT: Yeah. And then… so we have to find the --

STUDENT: Slope.

STUDENT: Hm?

STUDENT: Find the rate of change.

STUDENT: So look at two points. Just find two points.

STUDENT: Two points here?

STUDENT: No, like, so this would be one point and then this would be another.

STUDENT: Mm-hm.

STUDENT: So the left, so it goes 1, 2 then over 1, 2. So that's the slope, two over two. So it equals one.

STUDENT: So we don't need to do this one. So this one, we're already done. This one, we still have to do. So this one is correct, right?

STUDENT: Yeah.

STUDENT: Okay, so that one is correct, and then there's these ones we still have to do. These ones are correct, right?

STUDENT: Yeah.

STUDENT: Okay. Yeah. [inaudible] Now... So, let's see this one. This one...this one we already did because...let's see. See if we compare this one, these ones match and then these two match. This is the top, bottom and these are the ones that are top, bottom, top, bottom. And then this one, it goes four and it ends at one second. And then this one starts around, like maybe two, and then this one ends at six or, like, around six. So, let's see. So it stops, like, maybe every...every, it decreases every, like, maybe one second. Or, maybe like, one centimeter...or it decreases one second.

So after one centimeter, it decreases one second. And for these ones, for the same thing. After one...they all have the same constant rate. So after one second...after one centimeter it decreases one second.

To support students in partner work, I make sure they have access to tools to support their learning. It depends on the unit that we’re in or the content that we’re doing.

I would pull out tiles if we were doing growing patterns in middle school; or if we were doing geometry, definitely protractors. Students always have access to calculators.

As long as they ask, they know where to get it. I never offer it to them, but I tell them, "They're available here." I also think highlighters are very important, especially when you're learning about graphs. With the graphs from the lesson, I definitely think, even though it was worksheet-driven, we were strategic about which ones we wanted to photocopy so that students could see. I think allowing them to mark it up was is a really important tool that you would need when you're graphing.