In this problem, we engage our students in a consideration of how others might have approached the Pizza Crusts problem, probing them to identify the misconceptions about perimeter and area.
ANTOINETTE VILLARIN: My name’s Ms. Villarin, I don’t know if Ms. Ferrant told you about me, but I planned with Ms. Ferrant last Friday… and I also have a seventh/ eighth grade math class. So today, her and I will be teaching a lesson together, again my name’s Ms. Villarin and that’ll be on the board. So you can treat me the way you treat her, if you ask questions, and you have something that you don’t understand, you can ask me. Okay? So today we’re going to be doing a really cool lesson. We have a lot of people in the classroom, I don’t want you to be nervous at all. If people stand by you, it’s just ‘cause they want to know what you’re thinking. Okay? All they’re doing is listening to what you’re thinking, and looking at what you’re recording. Okay? So make sure you speak loud enough so that they can hear you, and then just pretend like the cameras aren’t here. Okay? So what we’re going to be doing today is… about two weeks ago I think you guys did a problem called “Pizza Crusts?” Ms. Ferrant gave you a problem, she had you do it by yourself. Do you all remember it? Yeah? What was it about? Can anyone tell me what it was about? And I have a seating chart here, so if I don’t know your name, I apologize, I’ll be looking at this. Is it Cassie?
STUDENT: Mmm hmm. We were looking at the circumference and area of the circle, and the difference.
ANTOINETTE VILLARIN: Okay, so the circumference of areas, I mean, the area and perimeters of rectangles, and then you had circumference and you were looking at pizzas? Okay? Do you guys remember that problem? What else… does anybody else remember what else it was about? No? Okay. So anyway, we’re gonna, we’re gonna talk about that today. Okay? And this is a lesson that Ms. Ferrant and I planned, And we want to see what you, what you know about it. Okay? So our first activity that we’re gonna do, she’s gonna pass out paper. And we just want you to think about it. You don’t have to write anything down. And we’ll be doing a lot of thinking privately, by ourselves, and then sharing with a partner, and then sharing out loud. Okay? I know that’s a common thing you guys do in your class. But the first thing that we want you to do is: Robby is confused. He’s a student, and he’s confused about area and perimeter. He thinks they’re the same. He thinks there’s no difference between the two. And we want you to help him. Okay? So right now, think what you would say to him, if you wanted to tell him the difference between area and perimeter. Think privately to yourself. Okay, and when you’re ready, share with your partner what you would tell Robby.
ANTOINETTE VILLARIN: Okay, did everyone have a chance to talk about it? Yeah? Okay. Who would like to explain what you would say to Robby if you wanted to tell him the difference between area and perimeter. What would you say, okay? Um, is it.. Connie? Angelique?
ANTOINETTE VILLARIN: Okay.
STUDENT: We talked about the area is length times width, and perimeter is length plus width plus length plus width.
ANTOINETTE VILLARIN: Ah! Okay. You’re telling me how to get it, so you’re saying, well, what shape are you talking about?
STUDENT: a triangle. I mean… I mean the rectangle.
ANTOINETTE VILLARIN: The rectangle? So you’re saying, for area, it’d be length times width, and then for perimeter it’d be lengths plus widths. Okay. Is there anybody else who’d like to add to what Angelique said? How ‘bout… is it Alex? Alexander.
STUDENT: Well, the area is what’s inside and the perimeter is what’s outside.
ANTOINETTE VILLARIN: Okay, so the area is what’s inside and the perimeter is what’s outside? Okay. Alex, do you want to come up and color in one of those for me? I want us to color in on these shapes… red, what’s area, and green, what the perimeter is. So choose the one that you want to do.
ANTOINETTE VILLARIN: you can choose whichever shape you want, and you can shade it in. Now let’s move this. Oops! Okay. Now. Can you guys see what Alex is doing?
ANTOINETTE VILLARIN: Yeah? Okay. Let’s try to fill in all of it, so it’s all along the sides. Okay? Does anyone want to come up and color in what the perimeter would be? In green? In green? how about, is Brandon? Are you Brandon? No. Yamlen? What’s your name?
ANTOINETTE VILLARIN: Joey? Okay. So Alex colored in the square, and he colored in the area. Okay? Thank you.
ANTOINETTE VILLARIN: Tell your partner really quickly what, is it Julian? What Julian is doing.
STUDENT: He’s getting the perimeter of the circle
TEACHER: What did Alex do?
STUDENT: Alex, he shaded in the middle of the square, ‘cause it’s the area.
TEACHER: And what’s area? What do you think area is?
ANTOINETTE VILLARIN: I would like… raise your hand and tell me what Julian did. What did he do? Um, is it Jessica? Stacey.
STUDENT: He outlined the circle with the perimeter.
ANTOINETTE VILLARIN: He outlined the circle, which is the perimeter? Okay, can anyone come in and color the area? Of this? Come up and color the area of it? How about Shira? Shira? And Ms. Ferrant has the red one and I have the green one.
ANTOINETTE VILLARIN: Yeah, that’s good. Okay. Do you guys agree with Shira? Yeah?
ANTOINETTE VILLARIN: Okay good, thank you Shira. Can someone come up and color in in green the perimeter? Of this square, that Alex colored in? Desiree? Yes.
ANTOINETTE VILLARIN: Okay, thank you Desiree. So she colored the green around it. And that’s the perimeter. And our very last shape, the rectangle. Okay? Is there somebody that wants to come up and color the area of the rectangle. Erica? and then maybe while she’s doing that, in green, does someone want to come up and color alongside of her, the perimeter, in green? Brandon? Yeah.
ANTOINETTE VILLARIN: And I know it takes a while to color, so you can just kind of shade it in really quickly. You’re welcome! So, looking at what you guys just did, it seems like you know a lot about area and perimeter. Area’s on the inside, and perimeter is the distance… or, area is the space on the inside, and perimeter’s the distance around it. Okay?
In this problem, we engage our students in a consideration of how others might have approached the Pizza Crusts problem, probing them to identify the misconceptions about perimeter and area. At the beginning of the class session, Patty Ferrant and I were trying to invoke what students knew about area and perimeter, focusing mainly on what was causing them problems whenever they dealt with the concepts. In the introduction, I feel it's important to not let kids know it's the introduction, being subtle about the content. Our goal was to set an invitational tone parallel with the task. This lent itself nicely to the MARS task, because the first problem is so accessible. We wanted every student to have access to the task and to give us some information about area and perimeter.
The students in this course, while not my own students, are similar to 7th grade students that I have taught. Patty and I teach in the same city, so the student populations are similar. They also share conceptual challenges--struggling with geometry in 7th grade. Many students had not been exposed to geometry in a hands-on way and were unable to understand the challenges with finding either perimeter or area when a constraint was imposed on the other.