After her students’ initial sorting process, Erika reminds them that they’ve been working with bar model representations. She asks her students to make sure the pictures match the stories.

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

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ERIKA ISOMURA: One more time. I'm not so concerned about you getting an answer right now. I really am curious to know if you can identify which problems are like Jesus's and which problems are like Camila's. So one thing that the people at table four are doing is they're saying, "Oh, this is kind of like Camila's," and they're putting a C, or "This is kind of like Jesus's," and they're putting a J. They're just doing that for now as a first step, just to get your brain kind of making connections, and then your next step would be a little bit more into the content of the problems. Okay?

STUDENT: Okay.

STUDENT: This one, I think this one is like...

STUDENT: This one.

STUDENT: Oh, this one is yours but I think this one is like Jesus's because he will need six pieces of string and, like, because like his, he needs pieces of string and you don't because you already have them.

STUDENT: Yeah, I already have the whole.

STUDENT: He needs the whole so he just...yeah. And then...well, yeah, so he needs the whole. So I think this is Jesus's.

STUDENT: So this is the C.

STUDENT: This is the C and this is the J. Yeah.

STUDENT: Do you want to put them?

STUDENT: Yeah.

STUDENT: You put Jesus and I'll put this one.

ERIKA ISOMURA: Question to you is, when I look at these problems and I kind of know some are like Camila's and some are like Jesus's, and that might be helpful for me to think about solving the problem, this could also be helpful to solve the problem. So here's Rosa Linda's drawing. She has a box of eight crayons but she decides to only use half. Do you see any picture that might be a way to represent her eight crayons but she's only using half?

STUDENT: This one.

ERIKA ISOMURA: Okay, that's your opinion. What do you think, Sofia?

STUDENT: I don't know if I'm going with that one or this one.

ERIKA ISOMURA: Okay, so you think...oh, that's interesting because this one also has halves, and there's eight boxes. Hm. That's a good point. So there's eight boxes of half and here's an eight box. So what I like you to do is, I like you to talk about which one better represents Rosa Linda's problem and then if you like to, feel free to get scissors and cut and put them together.

STUDENT: Okay.

ERIKA ISOMURA: I will warn you there are a few word...like two word problems that don't have pictures. So if you find those two that don't have pictures, feel free to try drawing out what that might look like. Okay? So right now I think your discussion needs to be around Rosa Linda, Rosa Linda. Which one is Rosa Linda's? So I'm going to hang out for a little bit and just hear what you guys have to say about it.

STUDENT: So we already know the whole, right?

STUDENT: Yeah.

STUDENT: We just need to, like, cut it in half but then this one is just half, half, half, half, half. It doesn't...we have eight. Half...two halves make one whole, and then two halves, two halves, two halves. There's only four...four wholes. So now... So we already know the whole, so it's this one because this is only four. It says eight over here.

ERIKA ISOMURA: Sofia, what do you think?

STUDENT: I think I agree with Dylan.

ERIKA ISOMURA: Think so? It's okay to disagree with him, he's not always right. You know that, right? Okay. Keep going.

STUDENT: So the problem with this one is, we're trying to find the half.

STUDENT: [Inaudible] because there's like eight whole and you, like, cut it in half [inaudible]. And then this is four and this is the other side of four because that's half of eight.

STUDENT: So we get scissors now, right?

STUDENT: Yeah, we can cut it out.

STUDENT: Okay, I'll get scissors then.

ERIKA ISOMURA: We know the whole and we're trying to find the part of it versus we know Jesus's, we have all these pieces and let's talk about how much it is together. We've been talking about how to represent it as a bar model. So these are bar models, and let's be honest, they're better than the ones I do on the board because I actually used the computer, and they're the right size and everything. So these are the pictures that connect to some of our stories. So your challenge is, is there a way to figure out like Rosa Linda? I just talked to another table about your problem and they said, "Is Rosa Linda's this one, where there's an eight cut in half, or is Rosa Linda's these eight crayons where she cuts them all in half?" Which one is really Rosa Linda's crayons?

STUDENT: Oh, so we have to figure out which one it is?

ERIKA ISOMURA: Yeah. So Rosa Linda and Antonio, we were just talking about Rosa Linda's crayons, right? So do we feel like Rosa Linda has a box of eight crayons that we take half of the box, or do we feel like there's eight crayons and we cut each one in half? Because there are two different ways you can think about it and you are right, there are two different ways and they are right there. Which one do you think best matches your story? Just like for all of you, which one best matches Lizzie's story? Which one best matches Randy's story? Okay?

STUDENT: Okay.

ERIKA ISOMURA: All right. So once you do that...and I think you guys all have a story on here, right? Once you do that, if you want to use scissors--I think you already have some scissors--you can start cutting up which ones you think match, where the picture is a way for somebody new maybe to say, "Oh, that's what I'm see in the story." Okay?

STUDENT: Yeah.

STUDENT: So if it says she has a box of eight crayons. So if I have just...

STUDENT: Okay. So you have a box of eight crayons and you have to use half of it.

STUDENT: So a box that has just eight crayons but I have to use just half. So probably, I think it's this one because I have just one box.

STUDENT: Mm-hm.

STUDENT: And then I just have eight crayons but I just need half of one...of each one. And then this one there's eight boxes.

STUDENT: But, like, what if they're just the same if you, like, cut them apart and you use the same colors? But wouldn't it be the same?

STUDENT: Yeah, but it says she has a box of eight crayons. So I just have one box that has eight crayons and then I just, I just need to use one half of each one, but I don't have eight boxes.

STUDENT: No, it says you have a box of eight crayons.

STUDENT: So basically one box has eight crayons and I have to use just one half of each one. Yeah. And this one, like, is just, like, eight boxes of crayons, and then this is just eight crayons, and these are eight boxes. So I think this one...the one that I'm guessing is from mine because it's just, like, eight crayons, that's it. That's not eight...there's no eight boxes of crayons.

STUDENT: Well, I disagree with you. I think it's this one because you have one box of eight crayons and you're using half. Like, if you cut them in half, you'll be using the same color. You get it? Like, if you, like...

STUDENT: Like if I snap this pencil in half?

STUDENT: Yeah. Like, if you're using this half and then you have to cut them in half, you'll still be using the same colors but it will be half. Yeah...wait. It'll be half of the crayon.

STUDENT: Okay. Well, I think you convinced me. So I think we should just, like, uh, just write the letter of the first name so we won't get confused.

ERIKA ISOMURA: Are you guys ready for the next step?

STUDENT: Yeah.

ERIKA ISOMURA: So we've been working on bar models together, and let's be honest, our drawings aren't always fabulous. It's hard to draw perfect [inaudible], right? So I tried to make it easier by drawing them for you on the computer so they're nice, and even, and very clear.

ERIKA ISOMURA: I was hoping that the idea would translate into when they started working with the bar model that, "Oh, these Camila problems, I'm going to start with this amount and then I'm going to fraction it off," versus, "Oh, Jesus's problem, I'm going to draw a bunch of little parts and then count them up."

I was hoping that they'd start seeing that tie-in and start putting it into their own brains. With one group, we ended up talking about they had found a picture that didn't match a problem and it was 1 whole, cut in half, and then it was the half, and so we did a little bit of talking and I asked them, "Whose is this like?" They were able to say, "It's Camila's." Then can we tie it back to Camila's problem? Can I just make it Camila's problem?" I talked them through it and they seem to be ... I thought they were getting, "Oh, yeah. Then, Camila, here's your 1 foot of string and I'm just going to cut it here." I thought that was interesting because I anticipated that that would be a real struggle for that particular group.

Another group was working with the rocks scenario. There was a sticky pad, so I put 8 stickies. I said, "Here's his rocks. Tell me about these rocks." They told me, "Each one is half a pound." I said, "Okay, so I'm going to grab them. How many pounds are there, Diego?” Boom, 4 pounds, because that was making sense of the problem. Both of them are very capable with numbers, which is why I think they jumped to just playing with the numbers.

Then we looked at their card which they had written 1/16 and I said, "You just told me 4 and you wrote 1/16. Which one makes sense just like the rocks?" They both were pretty immediate. "No, it's 4 pounds." Then we had a little discussion about, "Do you think when you did this originally, you were actually working with the problem or the numbers?" They both felt that, yeah, they were just working with the numbers. Then I asked them to go back and stop working with just numbers and trying to match numbers and think about how the stories work.