Michelle introduces a new card, a contextualized or word-problem representation. She begins by having students create a visual representation matching a contextualized representation, and a verbal representation identifying the mathematical quantities. She emphasizes the importance of the discussion between students that proves that the cards match. She circulates around the classroom, engaging pairs in conversation about their discussion and their representations. Students use tape loops on the back of their cards so that they can rearrange them as needed. She explains the importance of students debating and discussing their representations. Students engage in identifying patterns, structures, and connections between the representations.

4th grade math - Understanding Fractions*Michelle Makinson, Bagby Elementary School, Cambrian School District, San José, California*

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- Clip Transcript PDF
- Understanding and Interpreting Fractions PDF: lesson plan, student pages, pre- and post-activity assessments, and supporting instructional materials
- Sentence Stems PDF
- Discussion Process PDF
- Justification Cards PDF
- Quick Write Representations PDF
- Poster Discussion PDF
- Why It Matches student work PDF
- Which Representation student work PDF
- I Selected This Group Because student work PDF

MICHELLE MAKINSON: We have a new set of cards to match to the green and the white cards, but before we do that, we want to use a blue card and think about if we were going to create a green card for it and a white card for it, what will it look like.

[Speaking to coach Sally Keyes] Like, if I'm looking at a new blue card that's a word problem representation, if I were making the green and the white to go with them, what would I make? So that solved two problems. One, to reacquaint people with what they did a day or so ago because it just sort of flies out of their head. And also causes them to think about it in another way. It's not given to you so the idea of making a card becomes solidified and just getting them refreshed into that. And then once they made all those connections by drawing it out and writing it out themselves, it's that much easier to find it in the existing cards. And it might also prompt them to think about things in a different way.

So this is the new card, one of the new cards that you'll get. Two of Becca's four closest friends live in California. Two of Becca's four closest friends live, should be in California, you're right. What part of these friends live in California? So the job that we're going to do now is we're going to create a visual representation, a picture that goes with that. As if you were making your own green card for that. And we're also going to make our own white card for that. What would be the words that represent what is being said in this word problem? So for these I'm going to put them in the state of California. I've drawn a picture that goes with that. That's where they live. And then if I...so this would've been my green card, right? Does that make sense? Any questions on what I did there?

And then if I were to make a white card, I would want it to be parts of language. So two equal parts out of four. Does that make sense? And that would be my white card, and this is my new blue card. Any questions about what I did? So we're going to become acquainted with the blue card, which I haven't passed out yet, and you're going to do the same A/B partnership process. Partner A will pick a blue card, and then you will both create those types of representations on your white board, and then we'll share out. Does that make sense?

MICHELLE MAKINSON: It's about the discussion that proved that they in fact match.

MICHELLE MAKINSON: So let me make sure you understand what you are doing. So eyes on me. Okay. Partner A is going to pick one blue card. You're both going to work with that one blue card. Separately, you're going to do a green version of it that represents the same information as the blue card and a white version of it in words "parts of" and a picture that makes sense.

Then you're going to share with each other what you did and come to an agreement about how it should be represented. Does that make sense? Any questions about what I want you to do? And so...and your...then match the card to what's on the chart. This is a new way of creating a discussion mat, like we use the purple sheet for, but instead you're drawing the pieces. It's forcing you to have a conversation. Does that make sense? Because otherwise people could jump in and start scanning and matching, right? What's the most important thing in this? The...

STUDENT: Discussion.

MICHELLE MAKINSON: Discussion. So this is forcing you to have that discussion. Okay, I'm coming around to see what people have. I'm looking for people to say "I have part in words," using "parts of" language and a picture that represents what's on that blue card. And when you're done, share with each other, with your partner, and see if you agree.

MICHELLE MAKINSON: Is that what you talked about before?

STUDENT: Oh, this!

MICHELLE MAKINSON: Why?

STUDENT: Because it's also, uh, like there's eight equal parts and there's two whites.

MICHELLE MAKINSON: So take that there and I'm going to give you some tape loops. And then what do you need? Do a...justification card.

MICHELLE MAKINSON: [Speaking to coach Sally Keyes] And there would be no purpose to having the discussion if everything matched up in only one way. There was a reason why there were multiple ways to see it even within the same type of card and so those...that gives you that rich conversation, forcing them to have a conversation, forcing them to debate about things and hash it out. When, if they knew there's a one-to-one-to-one correspondence, they would simply line them up and be done with it. It would have been a more flat activity. And this forces them to think outside the box.

STUDENT: I, um, we're doing (inaudible). So the answer is twenty-four out of twelve because there is three groups and the three groups have...there's twelve three groups. So the answer is twenty-four out of twelve.

STUDENT: No. Oh, wait.

STUDENT: Uh, we're doing this one where Joe bought, um, twenty-two oranges and twelve are in each box. And, um, its question is how many box did he buy. And the answer is two because, um, two boxes will equal twenty-four oranges. And it can't fit twenty-two into one box, so that it can only fit twelve into one. So you need an extra box for the final ten.

STUDENT: So for the green card there is, um, two boxes and twenty-four oranges. You only needed twenty-two oranges but you have two extras because it comes in box of twelve only, so twenty-two twelfths. And Joe bought oranges that came in boxes of twelve oranges. He bought twenty-two oranges. How many boxes of oranges did he buy? Well, he bought two oranges because...oh, yeah!

STUDENT: Twenty-two twelfths. There's twenty-two twelfths...there's twenty-two and twelve oranges (inaudible). And there's twenty-two boxes, so there's twenty-two twelfths.

MICHELLE MAKINSON: If I'm looking at a blue card that's a word problem representation, and if I were making the green and the white cards to go with them, what would I make? That solved two problems, one to reacquaint people with what they did a day or so ago, because it just sort of flies out of their head, and it also causes them to think about it in another way. It's not given to you, so the idea of making a card becomes solidified and getting them refreshed into that. Then once they've made all those connections by drawing it out, and writing it out themselves, it's that much easier to find it in the existing cards, and it might also prompt them to think about things in a different way.