In this section, Patty and I begin to move students to the core mathematics of the Pizza Crust problem by looking at the relationship between area and perimeter.
ANTOINETTE VILLARIN: Do these problems look familiar, by the way? Yeah? Okay. All right. This one we’re not gonna write, we’re just gonna talk. Okay? I have here a square and inside, someone wrote, “36 square units.” What does the “36 square units” mean? And how would you prove it? Okay? So think, first, what that means, ‘cause when you see numbers, sometimes you associate them with different things. So what do you think that means? And how would you prove it? And then share with your partner.
ANTOINETTE VILLARIN: (I’m not giving them enough think time by themselves. I think they need more time.)
PATTY FERRANT: That’s good.
ANTOINETTE VILLARIN: You know? Because, okay.
PATTY FERRANT: I had to address the inches squared, and the inches, because I saw kids write 28 with a 2.
ANTOINETTE VILLARIN: So I had to..
PATTY FERRANT: This is good. This won’t take long.
ANTOINETTE VILLARIN: Okay. When you and your partner are done sharing, you can look at me, so I know.
PATTY FERRANT: What’s the 36 square units mean?
ANTOINETTE VILLARIN: Now when you guys share, remember what the giver does, and what the getter does.
STUDENT: shape, and the..
PATTY FERRANT: Measuring what, though?
PATTY FERRANT: Measuring what.
STUDENT: The length and the width.
PATTY FERRANT: So, there’s 36 …
PATTY FERRANT: You think area, you think length and width. So what’s the 36 mean?
ANTOINETTE VILLARIN: So who can tell me what the 36 means? Somebody wrote that in there. What does that mean? And how would you prove it? Ah, David?
STUDENT: Because, if you put squares in, you could fit 36 little squares in.
ANTOINETTE VILLARIN: Very good! You could fit 36 little squares in. So it doesn’t mean perimeter? It means the squares inside? Okay, good! Okay. Is there anyone that disagrees with David? No? Okay. Let’s go on to the next question, then. Here is a square pizza with an area of 36 square inches. What length of stuffed crust will be around the edge? What does “around the edge”—what does that mean? Okay? Think what that means. Julian, do you have a question about this one? Okay, now share with your partner. Stuffed crust around the edge.
PATTY FERRANT: Around the edge…
STUDENT: Yeah, it means perimeter, because perimeter’s around.
PATTY FERRANT: Perimeter’s around?
ANTOINETTE VILLARIN: When you’re ready, look at me, so I know we can move on. Okay. All right. Who would like to tell me what that means? Around the edge. Stuffed crust around the edge. Kenneth?
ANTOINETTE VILLARIN: Perimeter! Why.
STUDENT: Because around the edge is the perimeter.
ANTOINETTE VILLARIN: Around the edge is the perimeter. What color was it when we drew it here?
ANTOINETTE VILLARIN: The green! Okay. So, what information, thank you Kenneth, are you given in this problem? What geometric measurement, what information are you given? Jericho?
STUDENT: The area?
ANTOINETTE VILLARIN: The area? And then in the question, thank you Jericho, what are they asking you to find? What are they asking you to find? Dale?
STUDENT: The perimeter?
ANTOINETTE VILLARIN: The perimeter. Okay?
Here, our goal was to focus on justification, connecting what we had done in the previous problem to problem two; we strategically chose student work that could elicit this. We wanted to reinforce that idea of proof by asking the students "Where does the 36 square units come from?" Little did we realize that our focus would become language-based. The idea of "square units" and "inches squared" came up, and we had not anticipated this coming up in our planning. We had to address that within the lesson, as well as the idea of "crust" and "edge."
So while we intended to focus on justification, we had to address language. This meant a discussion with the class on the terms "inches squared," or what we meant when we say "around the edge." With populations of English language learners like this classroom, developing that academic language became an important part that we needed to address before we could even get to justification.