# Lesson Part 3C

## lesson part 3c

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As they close the day’s learning, Antoinette Villarin asks the students who returned from the gallery walk and sharing to describe their findings to their partners.

The student pairs work together, discussing and justifying modifications to their recording sheets. Antoinette collects the graph slides, cards, and materials.

She asks students to engage in a brief reflection/exit ticket at the end of the lesson, responding to the prompts “Today I learned… One question I still have about interpreting parts of a graph is… My partner and I left off on… Tomorrow we will continue with….”

Antoinette adjourns the lesson by telling them that their exit ticket responses will help them remember where they will begin the next day’s continued work.

## lesson part 3c

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

Next Up:   Post-Lesson Debrief Part A
Previous:  Lesson Part 3B

STUDENT: How about G4 and G7?

STUDENT: Yes.

STUDENT: G11 and G8?

STUDENT: Yes.

STUDENT: Okay, so turns out, me and Cecilia's...you guys... Me and Cecilia's are, like, the same thing as you guys'.

STUDENT: I'm going to write final slope because you found the differences.

STUDENT: Yeah.

STUDENT: And then how did you get these two?

STUDENT: So I got G12 and G11 as a match because I found the vertical change from two to one, which is negative one, and 0.5 from zero, which is 0.5. So I made 0.5 into one by multiplying that by two, and then I multiply negative one by two to get negative 2. So the slope was negative two divided by one, or over one.

STUDENT: Oh, I get it. Thank you. So G12 is the top and G11 is the bottom?

STUDENT: Yeah.

STUDENT: Thank you.

STUDENT: Ten?

STUDENT: No, I got G4 and seven, or G7.

STUDENT: How?

STUDENT: Because it goes down six centimeters in six seconds, and it goes up six... Or, like, this is the top container and this is the bottom, and it goes down six centimeters and then...in six seconds, and then it goes up six centimeters in six seconds.

STUDENT: G11 and G8?

STUDENT: I kind of just drew it because I don't know how to draw…because it went down...

ANTOINETTE VILLARIN: Don't forget to take notes on the back.

STUDENT: It went down four centimeters in two seconds, and then I just drew... It started at two because you add four, so...in two seconds, so. But it's not really good, so. That's it.

STUDENT: Can you repeat the explanation again?

STUDENT: Oh! There, uh, there...the top is doing the opposite of the bottom, which is... So four...it's the top, it's going from four height to the two seconds. And then from here, this is six, to the six...no, two seconds to the two height, because it's decreasing every one second.

STUDENT: Oh, because one is increasing and one is decreasing?

STUDENT: Yeah. So one centimeter goes off then one second goes off, and then like that. That's the rate that it's going at. Is there any other ones that you wanted [inaudible].

STUDENT: It's going up by six every three seconds. And it's going down by three...then it's going down by three seconds, too.

STUDENT: It's going down by six, right? That's what we got.

ANTOINETTE VILLARIN: You've been collecting notes on the back. Okay, the notes on the back were on different justifications that you were finding, different matches you were finding, and anything that you could record about 4, 5...4, 8, and 9, because those cards had missing information. I want you to share with your original partner any changes that you would recommend for your own, and make those changes on the front. Okay? I'll give you until about 2:55, and then we'll kind of clean up, and then we have a small reflection for you in the last five minutes. Okay? Does everybody understand what you're doing?

STUDENT: The reasoning is they have the same slope, but here, this is like...this both top...

STUDENT: They're both going upwards.

STUDENT: Yeah, they're both going upwards, but the bottom container… so I don't understand why it matches.

STUDENT: I agree with you.

STUDENT: Number four, I mean I kind of get it. Because it goes with six seconds, right? Six seconds. Maybe these are, like, three seconds, maybe. What do you think? I don't think we should make any changes.

STUDENT: I don't either.

STUDENT: So G11 or graph 11, right? Starts at four.

STUDENT: Wait, so starts by four...so starts by four, goes up by two. So it goes up by two...kind of like goes up by four.

STUDENT: Yeah, so don't you add four to six because it's going down four.

STUDENT: Wait, wait. I think you're supposed to fix this. Like, two...I guess two is supposed to be over here.

STUDENT: But this ends up two seconds.

STUDENT: Wait, are you sure?

STUDENT: Do you want me to do 4.5?

STUDENT: No, I think...I think you should erase the two up there. Four...

STUDENT: But that's four seconds. It's two seconds.

STUDENT: Yeah but look. The four height is decreasing by two seconds, but the height going up was going up by four seconds. So. I think you should put the four seconds, like, right over here.

STUDENT: But that wouldn't be even, then.

STUDENT: So...so what are you saying, actually?

STUDENT: Like, okay. So when you go to two seconds, right? It's at two seconds... two, like, you know, both of them, so you kind of, like, don’t you just do that? So it's like even. So it goes down four centimeters on the top and then it increases on the bottom container by four seconds -- four centimeters. Sorry, I cannot… Right?

STUDENT: So...

STUDENT: Do you want me to change this? Should I change this to like...?

STUDENT: No, no. You could leave it like that, actually.

STUDENT: OK. Are you sure you don't want to do 0.5 and then 1, and then 1.5, and then 2, so it goes all...

STUDENT: Oh, you're talking about that? Yeah, I think you should probably change that.

STUDENT: So it'll be, like, 0.5, and then 1, and then 1.5, and then 2. So then it's like the same thing. Right?

STUDENT: Okay, I think you should put 0.0.

STUDENT: Yeah.

STUDENT: I think you should put the...yeah.

STUDENT: And then you just --

ANTOINETTE VILLARIN: You're going to keep only one thing, everything else I'm going to collect from you. So before you clean up I'll tell you what you're keeping. You're just keeping your record sheet and your binder, and bringing this tomorrow so that we can finish this match and get the new set. Okay? I'll be collecting this graph's slide that I've made copies of for everybody, and I'll be collecting your cards all paperclipped together.

Can you put your name on the first card so I can give it back to you? Okay? So if I can take Partner A. Can you paperclip everything and put just your first names on the top card. I'll come around and collect that? While I'm doing that, you're filling out this reflection. This is your exit ticket out, okay? So you can't leave until I get this exit ticket from you. I will stand by the door and this is your ticket out. Okay? Are there questions? No? Okay, I want to honor the quiet think time while people are writing. Okay, that means you're quiet and letting people think. Question three, two people have asked. It's asking just where you left off on so that you remember. "Are you done matching? Are you still matching? And then in the middle of matching." Like, where did you and your partner leave off on?

Yeah, I'm going to stand by the door and you're going to hand it to me on your way out.

I wanted to thank all of you today. Have your reflection ready as your exit ticket, and I want to thank you for working hard, and inviting our guest in today.

I think writing is really important. I think having a lot of reflection time and exit tickets where students are explaining what they've seen or how they can use the math that they learned. I think word problems or MARS tasks are really helpful, where they ask students to explain their thinking of where they see it or give meaning to it. I also think connections and multiple representations of math concepts are also important, so asking students, "Where do you see that in the equation? Where do you see it in the story? Where do you see it in the graph? How can we connect that all?" I think that is helpful, too.