This segment begins the collaborative task of adding the measurement models (Set C) to the card sort. On the previous day, students had matched the numerical representations of fractions to area models. Students were given instructions to work with their partner to make the decisions. They used the sentence frame: Card ___ is equivalent to ___ because it shows _____. The partner needed to agree or disagree and explain why.

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 we are going to go ahead and get into our card sort. We're going to do a little bit more. We're going to move on to our card sort. And you did a really good job working with your partners yesterday, and we're going to do some more work today. So we're going to get out our supplies first, and then I'm going to give you the instructions for today. So you can put whiteboards and markers today, and I'm going to need my paper passers to come up and get supplies. Wait. So you're going to get another set of cards, and once you get it, I just want you looking at that set. So we're getting another card set. Don't talk about it yet. I want it to be placed in the center so both you and your partner can have a chance to look at this set. It looks a little different than that card set that we did yesterday. So yesterday, we had a fraction that we were matching to an object that was cut into pieces, which was an area model. And our fraction was representing a certain area of, um, of the object. And now we're going to a different kind of representation of fractions. So we'd use this for a number line or sometimes we'd use it for measurement. Um, and if you're looking at the card, um, right now I want you to start thinking about which of these cards might be equivalent to one of the fractions that you have on your poster. Don't talk about it yet. Please put it in the middle so both partners can see. And you guys are going to end up working up in a group of three because Neri is absent. So.

Okay, so I'm going to remind you about how we're going to do this sort because most of you did a really, really great job yesterday working with your partner and not leaving your partner behind, so I'm going to remind you how we're going to do this sort. So I'm going to tell who's going to start. It will be either the odd or the even person, and they are going to choose one of these cards. And I don't want you to worry about the blank ones yet. So you're going to choose one of the cards that you think is equivalent to the fraction in the area model that you matched yesterday. And you're going to give a reason. So your reason is going to be within your sentence frame. So you're going to say card, and these are cards C. So “Card C,” and then say the number, “is equivalent to,” and then I want you to name the fraction. And then say, “Because it shows...” and explain to your partner what it shows. Make sure you're really clear with your explanation. You may want to use some of the fraction vocabulary when you're explaining to your partner. Now once you've really explained clearly to your partner, your partner's either going to agree or disagree. If you disagree, you need to give a reason. “So I disagree with you because ...” and you can point things out on, on that card. Now if your partner is giving you an explanation and you're really confused about what your partner is saying, remember you can ask for clarification. You can ask them to repeat or you can ask them what they meant by that, have them explain a little bit more, push them to explain so that you can understand their thinking.

And then after you've matched the cards and you're very sure that you have the, um, cards matched that aren't blanks and you have some cards left over and you're pretty sure that, um, you don't, um, that the, the other cards are matched correctly, then you're going to be drawing. And you're not going to draw, um, a diagram like you did yesterday. It's going to need to be, um, a measurement model. It's going to need to have a line, like a number line. And there are a couple different ways that they did it here. So you're going to make your own drawing that goes with the ones that weren't matched. But remember, just like yesterday, you are not cutting any of these out until you agree, and are you going to cut all the cards out first and then do something with them?

STUDENT: No.

MICHELLE KIOUS: No. So it needs to be one at a time. You're going to do one at a time. So you... one partner is going to say their match, agree or disagree. Once you've agreed, cut it out, put it down. Then the other partner will do the same thing. Make sure that you're using that sentence frame so that we can hear your thinking and your reasons why. And I'm going to be coming around to ask you some questions as we're working. And we will be sharing out at the end, um, but I want you to really take your time and focus on why these cards matched. And I am also going to tell you that you're allowed to do things with the cards. So if you need to make a mark on the cards to kind of help you see why it matches, you're allowed to do that. If you need to use any tools that you have, you're allowed to use those, so think about, um, what you can do to help you figure this out. So I'm going to let you go ahead and get started in a minute. Any questions before we start? So the person that's going to start is going to be the even person. So go ahead and begin.

STUDENT: Because I measured it and, uh, I found out that right there, and it's in the middle, between of one and two.

STUDENT: Ah.

STUDENT: You got it?

STUDENT: Yes, I did.

STUDENT: Um, we just matched the card, so, uh...

STUDENT: I'm cutting it.

STUDENT: Uh, she's cutting it up.

STUDENT: C, C8 is equivalent to A6, uh...

STUDENT: Which is one and a half.

STUDENT: C7 is equivalent to one and one-fourth because it's already one whole, and then right here right where the star is, is one-fourth.

STUDENT: It shows that, uh, there are 12 middle lines and 10 shaded.

STUDENT: Malachi, you can cut yours out because you're right. This, on the C4 and the three, C2, um, they're correct. I looked at it on here. You can cut that one out. I agree. Well, you can cut it out now.

STUDENT: No, no. C2 is equivalent to three-tenths because it shows three are shaded in, and there's 10 in total. So it's, it’s, it...that equals three-tenth. Do you agree?

STUDENT: C4 is equivalent to ten-twelfths because there's 10 that's shaded and only 12.

STUDENT: I agree.

STUDENT: So, um, it ... this one is equivalent to, um, ten-twelfth because 10 are shaded in out of the 12.

STUDENT: Um, C8 is equivalent to one-half because it shows five shaded out of the ten, and five is half of ten. And, um, five-tenths is equivalent to one-half also.

STUDENT: Card C4 and ten-twelfths, we put ... is equivalent because, um, there are 10 pieces...

STUDENT: Shaded.

STUDENT: ...shaded over here and two left.

STUDENT: There's the, um, there's one, two, three, four, five, six ... six equal pieces here and six equal pieces here, and that's a half.

I wanted students to look at the cards and have time to make sense of the cards individually before starting to share with a partner. The purpose of the sentence frames was to make sure that students were giving reasons for their matches and that both students had a chance to explain why a card was equivalent. A fraction vocabulary list was posted, and I was hoping to hear correct use of words like equivalent, numerator, denominator, etc. during the conversations. I told students that they could use tools or mark the cards to help them make sense of the representations. On the pre-assessment, many students had trouble with precision when placing fractions on a number line. Some students gave a more clear explanation that showed understanding, and some students probably needed to be pushed a little more.