Antoinette Villarin asks her student pairs to share their discussions with the whole group. She models academic language — constraints, rate of change, initial value, starting situation — that she expects her students to use.

Once a student pair has shared, Antoinette asks the larger group to add on additional details that helped them identify a matched pair of graphs that show the flow of liquid between a given pair of containers.

Antoinette refers back to her anchor chart of the lesson vocabulary and sentence frames that she expects the students to use, and she names and reinforces students’ use of academic language.

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: In your solution or in your sharing out, you could tell me which is the top card and which is the...I'm sorry...the card that represents the top prism and the card that represents the bottom prism. Is there anybody willing to share? Okay, Keiko?

STUDENT: The two cards...I think that the two cards that represent the top and the bottom prism are G2 and G6.

ANTOINETTE VILLARIN: Okay, can you tell the class why? How do you know?

STUDENT: If we're comparing it to your prism example, then there is in total six centimeters in height of the liquid. And if there's four in the top prism, then there's going to be two in the bottom.

ANTOINETTE VILLARIN: Okay, so you're saying in this prism, if I look at it, or this model, if I...my starting situation -- and I did hear some people using these sentence stems -- if I have four centimeters here, you're saying the bottom prism will have a starting situation of two centimeters down here. Is that why you chose G2 and G6, because that one starts at four and this one starts at two?

STUDENT: Yeah.

ANTOINETTE VILLARIN: Okay, good. Is there anybody that could add to that? Was there anything else that told you it was a match? Okay, Sean?

STUDENT: In G2, it decreased by four centimeters in a second, and in G6, it increased four centimeters in one second.

ANTOINETTE VILLARIN: Okay, can you say that again because I got distracted a little bit by the sound...the P.E. [physical education] class outside? Okay, so...

STUDENT: In G2 it decreased four centimeters in a second.

ANTOINETTE VILLARIN: Uh-huh.

STUDENT: And in G6, it increased four centimeters in a second.

ANTOINETTE VILLARIN: Okay, so this is going down four centimeters in a second. Can you picture the water going down?

STUDENTS: Mm-hm.

ANTOINETTE VILLARIN: And then this one, why do we want that one to be going up four centimeters in a second? Sean?

STUDENT: Because it is to be the opposite of the one going down, and so G6 needs to go up.

ANTOINETTE VILLARIN: Okay, it has to go up because what's happening to the liquid in the bottom? What's happening to it as it goes up? It went from two...it's going to...?

STUDENTS: Six.

ANTOINETTE VILLARIN: To six. So what's happening to it?

STUDENTS: Increasing.

ANTOINTETTE VILLARIN: It's increasing. Okay. All right. Is there anybody else that could add to it? So I love that you're talking about the starting situation. I love that you're talking about the rate of change, and I love that you're also considering the constraint of knowing that we're always starting with six, and that's going to stay constant throughout this lesson.

My students learn a lot through stories. When you connect it to the concrete, they retain more and understand it from an applied level: For example, in 8th grade, when you're looking at a graph, asking students what story could it be describing gives it context. I like to teach in context whenever I can. It can be hard to do. It's a lot of research and finding resources. I was a pure math teacher and I always joke with other teachers that I know the pure math, but sometimes I have to research where it gets applied.

With graphs, it's such a perfect place to be connecting it to a story: what could be happening when you're looking at the rate of change increasing, when it's constant, or when it's decreasing?

It's just a very symbolic way to look at a situation in the real world that middle school kids are learning about. I think the more you do that, I think the more connections are made in student learning. I think they can easily connect it to something that they've seen before, whether it's a soda bottle or water flowing out. They can hopefully transfer those connections to other things when they see other graphs in the future, which is also good.