Our next goal in the lesson is to have students take what they had learned in the previous discussions and apply it to the core problem.
ANTOINETTE VILLARIN: I now have a new problem for you, okay? You guys are doing really well. This one we want you to write. Okay? So I’m gonna read it to you, and you’re gonna first do it like we’ve been doing. Privately, for a little bit, and then sharing with your partner. And if we have time today, ‘cause I don’t know when we get out of here, we’re gonna share as a class. Okay? So what I’d like you to do on your paper, is you’re gonna design two rectangular pizzas, also with an area of 36. Okay? And it has to have different perimeters. Okay? And your job is, Robbie wants more pizza crust on his pizza, because on the pizza crust on the original problem, there was only 24 inches of pizza crust. He wants a little more. Okay? So you want to come up with different sized pizzas, as long as the area’s 36. In each case, calculate what the perimeter will be. We want you and your partner to come up with at least two. Okay? Now, you can draw it, and we might have some tiles for you, so if you want to build it, you’re also welcome to do that. Okay? So do what you’re comfortable with, and Ms. Furant will come around with some tiles. Before, before you do that, though, are there any questions about it? Does it... okay, all right. So work with your partner, you and your partner can have the same one.
ANTOINETTE VILLARIN: Prove it to me, don’t forget to prove it to me, I don’t just want the length and the width, I want... to prove it to me.
ANTOINETTE VILLARIN: What time does the bell ring?
PATTY FERRANT: 58. You could say they could prove it by building it.
ANTOINETTE VILLARIN: So, eight minutes...
PATTY FERRANT: I think I would just ask the question of, if we give you these tiles...
STUDENT: Um, so, 36? I mean, 13. Wait wait wait, 36. Divided by 2. 18. So, 18 for .... oops.
ANTOINETTE VILLARIN: Sometimes, building it helps. So if you like to build it, you can, as a way to prove it. If you come up with two, just raise your hand and I’ll come and check it.
STUDENT: Need lots more.
ANTOINETTE VILLARIN: Okay, some of you have two, so if you are finished, we have an extra challenge for you, that you and your partner can work on. See if you can come up with ALL the possible rectangular boxes, that have the area of 36.
PATTY FERRANT: But if you can do that, absolutely! But once you do that, there’s also an extra extra challenge.
PATTY FERRANT: What do you have for 9?
STUDENT: I have width and length, then I get 36.
PATTY FERRANT And how did you get that?
STUDENT: I... It’s like, I get it, the rectangle, and think different ways it would be the area.
PATTY FERRANT: Mmm hmm.
ANTOINETTE VILLARIN: And remember, proving to us, when we say prove, or justify, your answer, saying “6 times 6 is 36” isn’t proving it to us. We have to see it in your model. Okay? Let’s see it. Okay, good! 9 times 4. Are you proving it to me on here?
ANTOINETTE VILLARIN: Okay, good job.What was your name?
ANTOINETTE VILLARIN: Lucia?
PATTY FERRANT: There’s only like 3 minutes.
ANTOINETTE VILLARIN: Oh, there is? Okay. Can I tell them that you’ll continue tomorrow? Okay. Okay. Finish up whatever you’re working on right now, and then put your pencil down.
We took this problem from MARS, which was the second to the last problem of the task. We asked students to come up with different sized rectangular pizzas that had an area of 36. We wanted them to justify their work based on an imposed condition. This was not where we wanted to end originally, but this ended up being the last problem in the time given. Later, Patty finished the lesson with the last problem on the MARS task with the same students the following day. Here, students solved the problem on paper (the previous problems having been a discussion). We wanted them to work with each other, using manipulatives if desired. Their work would be collected so that we could assess growth from actual student work. We wanted to see if students had a better understanding of the differences of area and perimeter, thus clarifying their previous misconceptions. Additionally, we also hoped that students would be better at reasoning and justifying their thinking through visual models and verbal explanations. During this clip, you will see Patty and me walk around and ask students "Where does your answer come from?", "Can you show us?", "Can you prove it on your paper?", etc.