Jayecelle and Michael were stuck. It was Jayecelle’s turn to pick a card and give a reason for a match. Jayecelle is learning English and Michael is fluent. I asked questions to help them move forward with their thinking.

5th Grade Math - Interpreting Fractions*Michelle Kious, Woodrow Wilson Elementary School, Jefferson Elementary School District, Daly City, California*

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MICHELLE KIOUS: So are you guys working together? Yeah. You're not leaving any big behinds? No? Okay, so whose turn is it?

STUDENT: Me.

MICHELLE KIOUS: Me, you. Okay, so did you...you think you have a match or you're still working on it?

STUDENT: We can't find anymore.

MICHELLE KIOUS: You can't find anymore. Okay. So there are more matches. So I want you keep thinking about it because all of these have a match, and then the ones that don't have a match, you will be drawing your own on the cards. So take a look at what you haven't matched yet, and then let's see if you can find a match. So Jayecelle, I'm going to ask you to look at one, and I'm going to see, ask you what you noticed. Can you look at this number line and tell me what you noticed about the number line. So, Michael, can you pay attention to what Jayecelle's doing right now? So, Jayecelle, can you tell Michael again what you noticed?

STUDENT: Um, if you look, the one, and if you put, uh, a little bit right here, there’s going to be one-half.

MICHELLE KIOUS: So she said if you look at the one and you put a little bit right here, it's going to be one and a half. Do you agree with that statement? That this would be one and a half? Or do you disagree?

STUDENT: I kind of disagree because it's not basically in the middle.

MICHELLE KIOUS: Ah.

STUDENT: It's like...more.

MICHELLE KIOUS: So you're saying it's not quite in the middle. Do you understand what he's saying?

STUDENT: Yeah.

MICHELLE KIOUS: And he...I think you're saying that a half would be in the middle? One and a half? Michael, yes or no?

STUDENT: Yeah.

MICHELLE KIOUS: Yeah. So, but, what she was saying, she said one and a half. But do we know, Jayecelle, is this part on the number line, is it more than one or less than one? Right here. Michael, I want you to think about it too? Is this, is this, um, mark on the number line, is it more than one or less than one? Jayecelle? Hmm?

STUDENT: Less.

MICHELLE KIOUS: She says less. What do you say?

STUDENT: More.

MICHELLE KIOUS: You see more. Okay. So we have a zero here, and then we have a one, and then we have a two. So, take a look at it. If the one is here and the two is here, is this mark more than one or less than one? Well, is two more than one?

STUDENT: Yeah.

MICHELLE KIOUS: Yeah? So this is one, and you said it would... First, she said it was one and a half, but Michael said to move it over. So is one and a half more than one? If you have one and a half pizzas, do you have more than one pizza?

STUDENT: Um...

MICHELLE KIOUS: I have one and a half pizzas. Do I have more than one pizza? You think what?

STUDENT: Yeah.

MICHELLE KIOUS: You think yeah? I think yeah, too. So, Michael? So we were thinking about, she was saying one and a half, and you said it wasn't quite one and a half, but you both agreed that it was more than one. So I want you to look at what you had from yesterday, and if you can see if there are any representations that indicate more than one whole. And then look really carefully, if there's more than one, that is more than one whole, which one do you think would match with this model? Okay?

STUDENT: Um...

MICHELLE KIOUS: So I want you both to look together. So look at your list. Do you see any that are more than one whole?

STUDENT: I do.

STUDENT: This.

MICHELLE KIOUS: That one? Okay. Do you see any others that might be more than one whole? This one.

STUDENT: I think that one.

MICHELLE KIOUS: You think that one is more than one whole? So if this is one whole, what is this part?

STUDENT: Uh, three-thirds.

STUDENT: One-third.

MICHELLE KIOUS: Hmm?

STUDENT: One and third.

MICHELLE KIOUS: Well, what does the shaded part represent? What part of a fraction? Huh?

STUDENT: One-fourth.

MICHELLE KIOUS: Oh, you think it's one-fourth. So what was the shaded part in the, in...is it the one or the four?

STUDENT: The one.

MICHELLE KIOUS: Okay. And Jayecelle, he said one-fourth. Do you see the four equal pieces?

STUDENT: Oh, yeah.

MICHELLE KIOUS: Point them out.

STUDENT: Oh, one, two, three, four.

MICHELLE KIOUS: Okay, so he said it was more than a whole. So whole and one-fourth. And look at this one. What do you see here? Michael?

STUDENT: Uh, I see...

MICHELLE KIOUS: And Jayecelle?

STUDENT: ...one whole, um, four-fifths.

MICHELLE KIOUS: One whole and four-fifths. So take a look at these two and look at this number line, and which one do you think it matches? So think for a minute. You said they're both more than one whole? Do you have a reason?

STUDENT: Because it’s not that far. I mean, it's not that far off to the other part.

MICHELLE KIOUS: Okay.

STUDENT: And it seems like less than ten.

MICHELLE KIOUS: Less than what?

STUDENT: Less than ten.

MICHELLE KIOUS: Less than?

STUDENT: Ten.

MICHELLE KIOUS: Less than ten. Where do you get the ten?

STUDENT: It's not a ten.

MICHELLE KIOUS: This is ten? So you said this part was four-fifths, right? This part, this one whole, where would that be on the number line? Okay. Is it more...? And it's more than one whole. So where would, where do you think the four-fifths part would be on the number line? Right about here. Ah! Okay, so did you guys decide? Is it going to be this representation or this one? What do you think?

STUDENT: This one.

MICHELLE KIOUS: Do you agree? Okay, you can cut that one out. How are we doing over here? Malachi? You're doing okay?

STUDENT: We're doing good, but we haven’t cut out that much.

MICHELLE KIOUS: Well, you know what? I'm really more concerned that you're think...

STUDENT: That C6 was equal to A7 and B6 because there were, um, I split it up into fours. So I count one, two, three, four, then that was one-third. Then another four; that was two-thirds. And then one, two, three, four. That was three-thirds, but there was only two, two out of the three, um, two out of the three column that was shaded in, so.

STUDENT: Equal one-fourth or no?

STUDENT: One-fourth?

STUDENT: Or no?

STUDENT: Um, let's see. So we said three of four lines.

STUDENT: I did three.

STUDENT: I said this one we count as, um, lines.

STUDENT: No, I know.

I picked to work with this group because I knew they were having difficulty and that sometimes Michael will just take over and do the work. I really wanted to push Jayecelle to express what she did know about one of the cards. I picked one card that represented more than a whole to see if she understood the concept of a fraction greater than 1. It was hard to think of questions to help push her thinking. I also wanted Michael to be clear with his explanation because he often has a good understanding but doesn’t always take the time to express it. I often struggle with the kind of questions to ask that help students move forward without being too leading. I was pleased to see how Cornelius gave a justification for 2/3 which included equivalent equal parts.