As her students use physical objects and a roll radius calculator to test their ideas for possible inclusion on their group status posters, Molly McNinch circulates among the groups, asking them to consider different quantities and engage each other in a discussion of their ideas.

9th Grade Math - Modeling through Geometry: Circumference of a Cup’s Roll*Molly McNinch, San Lorenzo Unified School District, Woodside High School, Woodside, California*

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STUDENT: And that's 12 inches?

STUDENT: Yeah? Well, if we change the slant length to 5 ...

STUDENT: 3 ... to ... 5.

STUDENT: 15.

STUDENT: So I changed the slant length.

STUDENT: For ... What was the 12 --

STUDENT: 12. Then when we changed the slant length to 5, it went to 15.

STUDENT: You can see that by doing this, the wide slant equals that ... I don't know how that affects ... how 2 affects that. So this one is 15 ... So try, like, 6 or something.

STUDENT: Okay. So I'll reduce that to 4 ... Oh, try this at 6?

STUDENT: Yeah.

STUDENT: Okay it's following a pattern.

STUDENT: 18.

STUDENT: Okay I'll change this back to 4. If we want to change the narrow diameter to --

STUDENT: 1.

STUDENT: 1.

STUDENT: 4 ... So that was 6 ... Well, we had this one up here, and this one ... And this one is double that one.

STUDENT: You're right. You're right.

STUDENT: Yeah.

STUDENT: Okay let's try it at 3 then.

STUDENT: 3, 3, 4 ...

STUDENT: Oh, it's the wide and narrow --

STUDENT: Oh, of course. Okay. Okay let's do one and a half.

STUDENT: Wait, one and a half for the narrow diameter? And wide is 3, then 4, and that's 8. So that is in between ... not in between ...

STUDENT: Okay, so narrow diameter is definitely affecting it.

STUDENT: Yeah.

STUDENT: Yeah.

STUDENT: So ... Hold on, hold on, hold on. So ... if the difference between this is one and a half ... one and a half… no. One… no.

STUDENT: Oh, I see what you mean.

STUDENT: So this is, like, this is ... If you continue this, they'll meet up after another one and a half centimeters -- one and a half inches.

STUDENT: Yeah.

STUDENT: So then a slant length of 8 then ... Would it be ... No, no, that wouldn't change. So -- but if we continue this slant, if we continue it another one and a half inches, if would meet up at the center.

STUDENT: And be a cone.

STUDENT: Oh, I want to try something. Can you do three and a half on this one and then 3 on this one? And then make the slant length, like, 1 inch. Does that make it smaller? Okay, so the slant length needs to be larger than both of those.

STUDENT: So slant length 3.5. Let's just see that. It's the same. I just want to try this ... So wait, is there any correlation between this, this, and this?

STUDENT: Yeah, that's what I was trying to figure out.

STUDENT: Because it's, like 3.5 times 2 times 100.

STUDENT: OK, now do it with, like, 50.

STUDENT: 350. [inaudible] Exactly.

STUDENT: Okay, wait.

STUDENT: 25. And that would be, that would be -- 100 ... 175.

STUDENT: Okay. So it's --

STUDENT: So we know that --

STUDENT: -- 2 times wide --

STUDENT: -- times slant equals --

STUDENT: -- equals roll radius. Okay now let's try it --

STUDENT: -- equals the roll radius.

STUDENT: Okay let's try it with --

STUDENT: Let's try 2 point --

STUDENT: Okay, I'm going to do it with this one. Two and a half times 2 ... times 5 …

STUDENT: It's 500. Yeah. 2.5 times 2. Five times 100 is 500.

STUDENT: Okay.

STUDENT: So wait ... Here let's try --

STUDENT: Yeah let's try it with these ones.

STUDENT: So 2.5 ...

STUDENT: So 5 ...

STUDENT: Times 2 ... So, and then times -- oh, and then slant length is what? 5.75? So ...

STUDENT: It should be --

STUDENT: 5 times --

STUDENT: 28.75. Okay now we have to see ... Make that 1 or something and see if that affects it. Okay --

STUDENT: 9 --

STUDENT: So how do these correlate?

STUDENT: So the difference is --

STUDENT: As long as the wide diameter is .5 larger than the narrow diameter, I'm pretty sure that equation works every time. But --

STUDENT: Here, like, let's try 4.5 and ... Let's try 4 and 3.5.

STUDENT: Yeah. No, wait. Yeah.

STUDENT: Is there a ... And let's try, like, 10. Easy. So, 80, yeah, 4 times 2.

STUDENT: Okay --

STUDENT: Times 10 equals 80.

STUDENT: So this only works if the wide diameter is .5 larger --

STUDENT: Yeah. Okay. So --

STUDENT: And so --

STUDENT: Okay --

STUDENT: So how do these --

STUDENT: So let's try it with an inch and see how those correlate.

STUDENT: Okay.

STUDENT: So 4 and 3? Okay. So 4 times 10. So it was 3 times 2? That's ...

In order to do work well in groups, the students need to know each other and be able to collaborate. In the beginning of the school year, they’re still trying to meet each other, and so those little tiny breaks in teaching where it's like, "Oh I have to go put this thing up on the smart board," or "Oh, I forgot to take roll," instead of letting that be dead time I say, "All right, so you guys are sitting with three new people." Any time I change seats even now I say, "Okay, you guys have 30 seconds. Find out what everybody at your table's favorite color is." Or something just super random where they have to find out something. Then the next few questions that I ask I say, "Okay, let's solve this problem." I use my cards and I'll say, "Okay, Lesley. Tell me the person to your left's favorite color, and then tell me this." So it holds them accountable, too because Lesley needs to know what Danny's favorite color is. I think just those little things force them to talk to each other and hold them accountable. It's like now I have to figure out everyone's favorite color, but you get to introduce yourself and get to know each other. I’m constantly trying to find ways for the students to feel at ease in the classroom. If the classroom is a safe place for students, then they feel comfortable learning AND making mistakes. Never underestimate the impact of your classroom culture.