Erika begins her lesson by engaging her students in a conversation about the terms whole and part, activating their prior knowledge with mentor “string” problems, and asking the students to identify the parts and wholes in each scenario.

Her students then work with new problems, sorting and describing differences and similarities between the new problems and the ones they’d done before: “Are any of these problems a lot like Jesus's problem, where he already knows his pieces or his parts, but he needs the whole amount? And are any of these problems like Camila’s, where she needs the pieces because she already has the whole?”

5th Grade Math - Fraction Multiplication Situations*Erika Isomura, Glassbrook Elementary School, Hayward Unified School District, Hayward, California*

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ERIKA ISOMURA: Okay, so we have, as you know, been working on fraction multiplication, and we've been working with these two mentor problems--Camila and Jesus's problems with the string. Right?

STUDENTS: Yes.

ERIKA ISOMURA: Okay. So what I wanted to do before we go on with today's lesson is just do a quick check-in about what we talked about on Wednesday. Okay? So on Wednesday we were talking about these two words: whole and part. We don't really know 100% what we want to say yet, but we had some ideas around whole and part, and how they're reflected in these problems. So Jesus somehow has something about a whole, W-H-O-L-E, and something about parts--and same thing with Camila. So take a moment, turn to your neighbor. What do you remember about whole things and part things in these problems?

STUDENT: Well, what I think a whole is, like, a whole is when you have a fraction, the denominator [inaudible] and then, like, the [inaudible] numerator is [inaudible] and it makes one whole.

ERIKA ISOMURA: All right. If we can come back together. So I heard some ideas from quite a bit of the fraction work we've been doing. Some of the stuff that went to these problems, some of your conversations were about wholes--just in fractions in general.

So let's see if we can recall. This will be helpful for the work you'll be doing today. Jesus is inventing a math game. He needs "mm" pieces of string. Each piece needs to measure "mm" feet long. How much string will he need in total? Where would we find whole--W-H-O-L-E--and parts in this problem? You guys told me this on Wednesday. Diego?

STUDENT: He needed the whole and the...

ERIKA ISOMURA: So he needs the whole.

STUDENT: And he knows the pieces.

ERIKA ISOMURA: And he already knows the pieces. Okay. And was Camila's problem the same way?

STUDENTS: No.

ERIKA ISOMURA: Camila is also inventing a math game. She has "mm" pieces of string. She decides that she only needs "mm" of that length for her project. How long will that piece of string be? So what was going on for wholes and parts for Camila's string problem? Rosa Linda?

STUDENT: She already had the whole, and she needed the pieces.

ERIKA ISOMURA: So she already had the whole and she's trying to figure out her pieces. Needs to figure out her pieces. Maybe one piece, or maybe multiple pieces. Okay, good. I think that was what we talked about. Thank you. So what I have for you today is basically same thing.

So here are eight more problems and what you're going to do first is read them. That's very helpful.

And then the second thing you're going to do is you're going to think about: Are any of these problems a lot like Jesus's problem where he already knows his pieces or his parts, but he needs the whole amount?

And are any of these problems like Camila, where she needs the pieces because she already has the whole?

Okay? So that's step one.

Step two is if you think you've had that conversation, you've read and you've talked to your partner and you agree, maybe cut and sort them into two different groups.

You'll notice there's two blanks. Leave them alone--you'll need them later.

Okay? Any questions?

After you do that, if you're waiting for us to come back together, please try to solve them. Maybe draw a picture. Maybe just talk it out.

What would the answers be to any of these?

ERIKA ISOMURA: Here, I asked my students to recall two mentor problems they’d worked with before:

“Jesus is inventing a math game. He needs ___ pieces of string. Each piece needs to measure ___ feet long. How much string will he need in total?” Where would we find whole and parts in this problem?

“Camila is also inventing a math game. She has ___ feet of string. She decides that she only needs ___ of that length for her project. How long will that piece of string be?” What was going on with wholes and parts for Camila's string problem?”

I then asked my students to work with novel problems, identifying how these novel problems were like the problems we’d worked with previously, sorting the problems into groups.

When we’d worked with these problems before (with Jesus having 7 lengths of string that were each 2/3 feet long, and Camila needing 2/3 of a total length of 7 feet, for example) my students found that somehow these were coming out the same.

Which was really provoking because, one student for example kept saying, "I don't understand why that's happening, I'm dividing.” He didn’t see that as a multiplication, so he was really agitated by the fact that he feels like this is a division problem and yet, it's producing the same result with the same numbers as a multiplication problem.

In his view, Jesus' problem of the repeated 2/3's, that’s clearly multiplication, but the other one seems to be giving the same answer and yet it's clearly, in his brain, not a multiplication problem.