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: One quick thing before we move on to the next activity. Okay? Here? Is 28 inches squared. And here, is 22 inches. Okay? Oh, and actually, before we do that, I’m sorry, we didn’t even decide who was correct. Okay? So can you tell me, really quickly, who is correct? Was it Robbie or his friend? Uh, Carl.
ANTOINETTE VILLARIN: Robbie? Why.
STUDENT: Because it’s supposed to be 28.
ANTOINETTE VILLARIN: It’s supposed to be 28, and instead his friend said …
ANTOINETTE VILLARIN: Okay. Before we move on, something that I saw while walking around, is…Robbie said 28 inches squared, whereas his friend said 22 inches. Okay? Can you talk about the difference between that with your partner? What do you think his partner was doing? Or his friend was doing? And what was Robbie doing? So talk about that. About the label.
STUDENT: He didn’t do the…
PATTY FERRANT: What didn’t he do?
STUDENT: The perimeter, and the area…when you measure the area, it has to be square inches.
PATTY FERRANT: mmm!
STUDENT: Instead of just regular inches.
PATTY FERRANT: And how come square inches?
STUDENT: Because there’s squares inside the area.
PATTY FERRANT: Do you agree?
PATTY FERRANT: Do you have anything to add?
PATTY FERRANT: No? Nothing at all? I like how you did that. So how many squares are inside here?
ANTOINETTE VILLARIN: So can someone really quickly, before we go to the next poster, or not really quickly, but …share with the rest of the class, what was his friend thinking, and what was Robbie thinking? Um, Erica?
STUDENT: He’s thinking area is in square inches, and like when you do perimeter you don’t have square inches.
ANTOINETTE VILLARIN: Okay, how… why do you have to add the square inches? Why do you have to tell people “inches squared”? Or “square inches?” Why is that important with area? Lorraine?
STUDENT: So they know it’s the area.
ANTOINETTE VILLARIN: They know it’s the area, and how do we measure area? how we measure area? Alex?
STUDENT: Um, we do it by square inches?
ANTOINETTE VILLARIN: By square inches. Because we physically have squares inside. Okay? So, you ready for the next problem? Okay.
We eventually lead the students to identifying three columns and three rows within the shape to signify the concept of area. That's what we wanted the students to understand. We were trying to get the kids to say it, rather than for us as teachers to just give them the reasoning behind the idea.