Students who were done were asked to number the fractions from least to greatest. Students were also asked to think about the warm-up and revise the previous day’s sort. Xita and Dwight were having a hard time moving forward and needed some prompting. Ethan and Jenny were asked to revise their drawing for accuracy.

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

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MICHELLE KIOUS: If you are done on the page that you were recording your information last time, um, you can look at your fractions and write down, number them from least to greatest, what you think because we're going to be doing that at the end if you're completely done and you're waiting for others to be done. I am going to give you a little more time, but there's one other thing that I didn't talk to you about in the beginning. I think, because we had our warm-up, where we were looking at, um, some different kinds of fractions, I think there are probably a few groups that may have changed their mind about something that they put on your poster yesterday. So I just wanted to let you know that I do have extra copies of yesterday's papers up here. So if you did change your minds about something from yesterday, and it won't peel off very easily and you want to change something, you can come up and get an extra copy. So I'm going to give you about seven more minutes to work on this sort and try to get some more sorted. And then we're going to be doing some sharing. So, go ahead and continue. Okay, so you're confused about these.

STUDENT: Those...

MICHELLE KIOUS: So wait, you said you...did you match this or you're not sure about this?

STUDENT: I’m not sure.

MICHELLE KIOUS: So which one are you concerned about? Confused about? Let's, let's move them over so that we can still see.

MICHELLE KIOUS: ...so if you're completely done, remember what I just said? Write numbers from least to greatest. So you said you're concerned, confused about these. So let, which ... let's pick one to work on since you're confused about both of them.

STUDENT: Um, this one.

MICHELLE KIOUS: Okay, so let's pick this one. So I'm going to ask you a question. Dwight, do you see where it's marked?

STUDENT: Yeah.

MICHELLE KIOUS: Do you think it's marked at more or less than one-half?

STUDENT: More.

MICHELLE KIOUS: Do you agree, Cida?

STUDENT: Disagree.

MICHELLE KIOUS: So you think that this mark is at less than one-half.

STUDENT: Oh, I mean ...

MICHELLE KIOUS: You think it's more? Okay and you also think it's more? How do you know? Where would one-half be?

STUDENT: Right here.

MICHELLE KIOUS: Okay. All right. So I noticed that there are some marks that are already here. So what do you think...let’s forget about this right now. What do you think these marks indicate?

STUDENT: I think they indicate one-thirds.

MICHELLE KIOUS: So let's... Thirds, can you point out the thirds? Where would one-third be?

STUDENT: One-third and two-thirds.

MICHELLE KIOUS: So one-third, two-thirds, three-thirds. Is three-thirds the same as one whole?

STUDENT: Yeah.

MICHELLE KIOUS: Hmm. So you said one-third, two-thirds, three-thirds. So what would this line, Dwight, indicate, that she just said?

STUDENT: Two-thirds.

MICHELLE KIOUS: So two-thirds. And so this is one whole. What do you notice about where the star is placed?

STUDENT: It's near the middle.

MICHELLE KIOUS: Near the middle, between what two points?

STUDENT: Between two-thirds and one-third.

MICHELLE KIOUS: Okay. So do you remember that I said it was okay to draw on this diagram? Do you... might you want to draw, a line? Some lines to sho-, that might help you figure out where this is? Do you want to try or do you want...Dwight, do you want to draw some lines? Do you want Cida to try?

STUDENT: Yes.

MICHELLE KIOUS: Okay, so you can agree or disagree. So can you draw some lines that might help you figure out where this place is?

STUDENT: Oh!

MICHELLE KIOUS: Ah! Okay. So now that we're looking at that, can you tell me how many equal pieces it's now divided into, if we consider that a line? So let's go ahead. Ready, count.

STUDENT: One, two, three, four, five, six. Six.

MICHELLE KIOUS: So what ... how many equal parts?

STUDENT: Six?

MICHELLE KIOUS: Six. So one whole would be how many sixths?

STUDENT: Six.

STUDENT: Six wholes.

MICHELLE KIOUS: Six-sixths. Hmm. So what do you think that would be?

STUDENT: Five-sixths.

MICHELLE KIOUS: Do we have a, a fraction that is equivalent to five-sixths? Or do we have five-sixths? Look at your fractions.

STUDENT: There’s only five-sixths.

MICHELLE KIOUS: Do we?

STUDENT: Yes.

MICHELLE KIOUS: Okay. Are you ready to cut?

STUDENT: Yes.

MICHELLE KIOUS: Do you both agree?

STUDENT: Yes.

MICHELLE KIOUS: Okay, so I'm looking at these two and this.

STUDENT: Uh-huh.

MICHELLE KIOUS: Do, um, do they represent more or less than one whole?

STUDENT: Um, more.

STUDENT: More.

MICHELLE KIOUS: More? Okay, so does your drawing represent more than one whole?

STUDENT: Uh, yeah.

MICHELLE KIOUS: Where is the one whole marked on your drawing?

STUDENT: There.

MICHELLE KIOUS: Okay. So in this representation, how many parts equal one whole? How many pieces does it take to make one whole?

STUDENT: Five.

MICHELLE KIOUS: Five. So I'm going to move this over so I can see. Um, Jenny, can you show me the five pieces that make up this one whole?

STUDENT: Um, one, two, three, four, five.

MICHELLE KIOUS: Do you agree?

STUDENT: Yeah.

MICHELLE KIOUS: One, two, three ... Oh, from here also? One, two, three, four, five from here? So are you starting here? Or are you starting here?

STUDENT: Um...

MICHELLE KIOUS: You're starting right here? So show me the five. Mm, show...count. Show me the five from here.

STUDENT: One, two, three, four, five.

MICHELLE KIOUS: Mmm. So you said one, two, three, four, five make up one whole, but when I got over to here, I was already past one whole. Hmm. Do you want to change your diagram a little bit?

STUDENT: Yeah.

MICHELLE KIOUS: Okay.

STUDENT: Four and there's eight, right? One, two, three.

STUDENT: It’s upside down.

STUDENT: Really? Oh.

STUDENT: Because of the C.

STUDENT: I’m going to take off my glasses.

STUDENT: Okay, so do you agree with me or disagree?

STUDENT: Agree.

STUDENT: Okay, so this one goes up here.

STUDENT: One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. Twelve, six. Six-twelfths.

STUDENT: We don't have a six-twelfths. We have a six-ninths.

STUDENT: How about this one?

STUDENT: That was eight-twelfths.

STUDENT: One, two ... only a ... we don't have it. How come we don't have these?

STUDENT: I don't know.

STUDENT: Let’s draw one. Wait, which one? Is it mine or yours?

STUDENT: Yours.

STUDENT: Mine?

STUDENT: Yeah.

STUDENT: I'm confused.

STUDENT: Nine-sixths. Okay.

STUDENT: Okay, we can put a star on that.

STUDENT: Let’s put it.

STUDENT: Put a star on six!

STUDENT: Two, three, four, five, six.

STUDENT: Um, where are you putting the six? Put a star. There.

STUDENT: There.

STUDENT: Okay, now, three-tenths C2, um, ten-twelfths.

STUDENT: Three, four. Yes, put this item, item right here.

STUDENT: Uh, you can put other stars to it.

STUDENT: Where's the bottom line?

STUDENT: It's nine fives. It’s nine fives, five nines.

STUDENT: Uh, oh. What number is that? Let me see. C7.

STUDENT: C9.

Xita and Dwight started to get stuck, and I used questioning about benchmarks to move them forward. I wanted them to focus on what they were able to notice about the diagrams. One question was to tell if a diagram represented more or less than ½. After answering that question, I asked them to use the marks on the diagram and add more marks to find the specific fraction that was more than ½. I noticed that Ethan and Jenny had a diagram that was not precise. I asked questions to help them use what they knew to help critique their own diagram.