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.
STUDENT: Okay, so Bryan...
STUDENT: Wait, let's review. Jerry discovered that each multi-link cube...
STUDENT: Ten. Ten. It has to be a ten.
STUDENT: Yeah, that's fine.
STUDENT: Now let's see. Bryan noticed that there was half of a pizza left after the party.
STUDENT: [Inaudible] six pieces of string and two thirds of a foot long. So there's six pieces. This one is supposed to be whole and then pieces.
STUDENT: But we know the pieces.
STUDENT: Oh, yeah, yeah, yeah. Oh yeah, the whole is here. I thought this one was [inaudible]. So this one is right.
STUDENT: And then this one.
STUDENT: Sofia plans to wrap a birthday present for her friend. She has a long ribbon, but doesn't need all of it. In fact, she decides that she wants to use two thirds of [a] six-foot-long ribbon to wrap the gift. How much ribbon will Sofia use?
STUDENT: Yeah, that one's right.
STUDENT: And these three don't match.
STUDENT: Let's just make sure one more time. Uh, Antonio found eight rocks. Each rock...wait.
STUDENT: But all three of these are whole and then these three show a whole, they don't show the pieces.
STUDENT: Maybe we saw it wrong. And then this one doesn't go with any pictures because there's like no, like... Yeah, this one is not a good picture. I think this picture is trying to trick us for this one because this one, it's, like, this is two and then that's the four. I think that picture is trying to trick us, so that picture doesn't match.
STUDENT: It's this one. That's half of a half. This is one. I'm pretty sure it's a typo or something. Let's just make sure. She said we had to draw them, right?
STUDENT: Yeah, she said, like, two of them, like, some of them we have to draw.
STUDENT: So what did we say for Ruchita? Jesus's problem...so knows the pieces, needs the whole. So, okay. So...triple the recipe. So then we have to... It's like a square...it's like a rectangle [inaudible] three pieces.
STUDENT: I'm so bad at drawing [inaudible]. And then I'm trying to make it equal. So bad. One three fourths, one three fourths, one three fourths. And then...how did they draw it again? Now we're going to put this here and you draw the next one.
STUDENT: But how do you know if they...?
STUDENT: It says one, it's not half.
STUDENT: Yeah, so not half.
STUDENT: Your turn. You draw.
STUDENT: But how do you know that it's not half?
STUDENT: So you have to draw like a rectangle or something.
STUDENT: Like a rectangle?
STUDENT: It's, like, we have to make a rectangle but then, like...
STUDENT: Like a rectangle...can I borrow...
STUDENT: Just use mine. Make a rectangle but, like, the opposite of me. Like, he needs the pieces and knows the whole.
STUDENT: He needs the pieces...
STUDENT: So, like, split in the middle, I think.
STUDENT: Like, in the middle, like, it's just one half?
STUDENT: And then you put, like, a question mark at the bottom, I think. Because that's how they did it. He knows the whole, but they need the pieces. So.
STUDENT: They know the whole. What's the whole again?
STUDENT: The whole was a half. So.
STUDENT: Half and then...
STUDENT: Like, you put, like, half but then...
STUDENT: Like that?
STUDENT: I thought you're going to draw it...so you don't need the rectangle at the bottom.
STUDENT: Oh, okay.
STUDENT: Put a question mark here, I think. And then make a line down here. Just to here. Just to here. Put, like, one half. These two don't match.
STUDENT: Let's cut it in half.
STUDENT: You want to cut?
STUDENT: Because we thought that these were the ones that don't have a picture.
ERIKA ISOMURA: No, that would be too easy. Okay, so you feel strongly that neither of these two stories matches either of those pictures?
STUDENT: No, I don't think so. We remember in the past you showed us that each piece in the fraction has to be equal.
ERIKA ISOMURA: Okay.
STUDENT: And right here it says there's two of them that are the length of four feet or four inches, whatever. And right here it's two, and it's smaller. So then it doesn't make sense. So then that's why it doesn't match any of these stories.
ERIKA ISOMURA: We've never actually seen anything like that, have we?
ERIKA ISOMURA: Where we have different parts and the parts are not all the same size. That is a little bit puzzling.
ERIKA ISOMURA: So I think we're going to have to do a little bit more investigating on, would there ever be a time where the parts might be four, four, four, and then something else, and why that might happen.
ERIKA ISOMURA: Okay, so I think that's a really good thing for us to talk about.
STUDENT: Right now I was looking at this, and that's three fifths: 1, 2, 3, 4, 5. Five parts, and it only takes three, and it's ten.
STUDENT: Oh, yeah! That's the Jerry one.
STUDENT: This should be the Jerry one.
STUDENT: So then...
STUDENT: What about this one?
STUDENT: Only the Ruchita one should... Yeah, it's the Ruchita one that doesn't have any...that's blank. So then one right here. And this one is not supposed to be...
STUDENT: Yeah, it's blank. This is blank. This one has no picture. We did it!
STUDENT: Right there.
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.