In this clip, Hillary Lewis-Wolfsen invites students to relate equivalent fractions back to a diagram.

5th Grade Math - Proportions & Ratios*Hillary Lewis-Wolfsen, Forest Park Elementary School, Fremont Unified School District, Fremont, California*

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HILLARY LEWIS-WOLFSEN: Okay, All right, let’s jump right in. We’re gonna look at the first problem that was on that candies assessment. This is Amy’s candy box, and what it told you on the assessment was that Amy ate six of the candies. And it asked the question, “What fraction of the candies has Amy eaten?” Do you remember that one? Okay. And a lot of you (peeling off paper) gave us this response. Okay?

I want you to think about this, 6/9ths . I want you to think about the 6 right here, and think about, where you see the 6 in the drawing. All right? Just think. No hands yet. Just think about it. Just private think time. Okay? Where do you see the 6 in that illustration? And…. When you and your partner are ready, I want you to share with your partner where you think you see that 6.

Okay? Where—you know what? Before I ask for that answer, I want you to now think, private think time, where is the 9? Just private think time. And when you and your partner are ready, go ahead and share that.

Yeah, okay. Okay! So who would like to share—where do you see the 6, here? Who would like to share. And… Samuel?

SAMUEL: The empty spaces, or boxes?

HILLARY LEWIS-WOLFSEN: These?

Yeah? Is that what I heard a few of you saying, too, the empty spaces? Okay. Somebody else. Tell me, where do you see the 9 in this illustration? And I don’t remember your name. That’s little, I can barely see – is it, Osi? Osi, where do you see the 9?

OSI: All of them.

HILLARY LEWIS-WOLFSEN: All of them! Yeah, and I think I heard some of you saying the whole thing, or all of the boxes. Okay. Now, there was another answer that we saw on a number of your papers …

It’s this one. 2/3. I want you to think about these numbers, too. Where do you see the 2 in this illustration, and where do you see the 3. Give yourself private think time first, because this one may take a moment longer for some of you. Where is the 2 and where is the 3? Okay. Go ahead and share that with your partner.

Okay. So what do you think about this? Who wants to tell me what you think? And… you wrote your name so small, I can’t see it. What’s your name, Claire? Claire.

CLAIRE: In simplest form.

HILLARY LEWIS-WOLFSEN: In simplest form, you mean, related to this fraction? Yeah. Okay. So are these equivalent fractions? Okay? So how do we relate this back to the illustration? How do we see that here? And what’s your….Charlotte? What’s your…

CHARLOTTE: There are two empty columns, and three columns together.

HILLARY LEWIS-WOLFSEN: 1,2,3 columns? And what… are you telling me that’s the 3? And 2 are the columns that she’s eaten. Can somebody else explain that to me? Because I don’t think everybody heard Charlotte (Charlotte?) I don’t think everybody heard Charlotte’s voice. Can somebody else tell me what she was thinking? Did anyone else hear her? Oh, not many people did.What’s your name? Saurabh? I’m going to ask everybody to write your names really big on the top of that paper. If you wrote it tiny, or if you have your name tag that’s fine. It helps me if I can see your name, too. I can’t see that, it’s this little. Can you say that… tell me what she was saying about the, the 2 and the 3?

SAURABH: There’s 3 columns…

HILLARY LEWIS-WOLFSEN: 3 columns. So, 1, 2, 3 ….

SAURABH: And two of them have been eaten.. so that makes the fraction, which is eaten 2/3 .

HILLARY LEWIS-WOLFSEN: So, 2 of the columns out of 3 have been eaten. Okay, thank you Saurabh. Okay.

With each problem, I asked the students for “private think time.” I wanted them to think through each problem before discussing it with a partner. Even as adult learners we need quiet time to process what is being asked of us. Only once the partners were through discussing did I ask for a whole group discussion. The partner time is especially good for students who are unsure of themselves and can work out their thoughts with the partner without having the eyes of the whole class yet. Only the students who are ready, and comfortable doing so, will be asked to share with the whole group.

When Charlotte answered the question about where 2/3 came from, I asked someone else to explain her answer. I said that I did that because not everyone could hear her. The truth is I wanted another explanation from a different student. She was correct, but students often understand more if they hear the explanation in a variety of ways, and not just from the teacher.