As they move toward the end of the lesson, Molly McNinch reminds her students to represent their thinking on their status posters: “What was your journey. What thinking did you do? Did you come up with a solution?” Students work to prepare their posters for sharing.

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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MOLLY MCNINCH: So I know a lot of you guys are still working on the problem and you're getting very close. So what I want to do ... Oops, sorry. What I want to do is because I want us to all share our -- our ideas, your posters are going to be posted up on the whiteboards.

So, I'm noticing that some posters are very blank, so I want you guys to start -- again, these are status posters, so we're just looking at what was your journey. What thinking did you do? Did you come up with a solution? Now, I don’t mean -- it doesn't need to be super spiffy with all the beautiful colors, but I do need to be able to see it from a distance, so, I would start working on your posters, because I'm going to stop you guys soon so we can go over and look at them together. All right, get going.

STUDENT: We know these are true, but -- we -- but -- if you don't solve -- so -- if you took -- if you took the first numbers you took, so --

STUDENT: Well, since you're solving for R, you don't need to plug it in, because you're assuming that we don't know what the roll radius is.

STUDENT: Yeah, so if we plug in ... Hold on where --

STUDENT: If you plug in everything else it works out so the R equals what we found the roll radius.

STUDENT: Wait, what about RA equals WR minus S, so R -- okay --

STUDENT: Well, remember you have to --

STUDENT: Yeah, that’s where -- here -- so this would be the equation that you need to put in.

STUDENT: Yeah.

STUDENT: You want to put N equals W minus F, what?

STUDENT: No, so ...

STUDENT: I'm reading the sentence, so I can have ...

STUDENT: Okay, I'll flip it.

STUDENT: What? No, this fine, this is fine.

STUDENT: So the roll radius divided by the wide diameter, so the big triangle, is equal to the roll radius minus the slant length, so this length times the narrow diameter. Remember when we were comparing two triangles?

STUDENT: Yeah.

STUDENT: It's that exact thing. You're comparing the big triangle to the smaller triangle. And if you plug in, like, the values from our first -- these values --

STUDENT: I thought when we tried that it didn't work. How -- what's different about that?

STUDENT: What do you mean? I think when -- when you tried it here?

STUDENT: Yeah.

STUDENT: The only difference is you don't plug in the roll radius, because that's what you're solving for, so we're assuming that we don't know.

STUDENT: Okay, I get it.

STUDENT: Yeah, so if you try it -- you can try it again with, like, pretending we don't know what it already is, you can figure out and you solve for this number.

STUDENT: Okay.

STUDENT: Yeah. I think -- can we -- can I rotate it?

A lot of students grasped that concept of modeling, because I think modeling, so many students will think "picture," and I liked that there were still some posters that had no images at all. I think by advancing that student thinking through encouraging the modeling of actually manipulating equations and manipulating the proportions and ratios, that's helpful. I think with regard to, "How did it advance them with modeling as diagrams?," I think by pushing their thinking towards seeing the triangles, that was a big jumping off point/hint for them. Once they saw that diagram, a lot of them, it just clicked.