In this problem, Hillary Lewis-Wolfsen invites the students to consider several different strategies for answering a proportions and ratios task.

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: There we go. Private think time, again. What do you think of this? Private thumb, when you’re ready. And go ahead and pair, share with your partners. Wow! They were ready!

ANDREW: The 1 and 2, that equals the 3. And then 2 and 4, that’s 6, and then he did 3 cream and 6 chocolate, that’s already 9, but since he kept going, 4 cream and 8 chocolates, it equals the 12, and it’s supposed to be only 9 cups of ingredients. And you?

PARTNER: I don’t really get this one.

ANDREW: It’s kind of like what I did? Um. For every 1 cup of cream, he did 2 cups of chocolate, and he did it again, 1 cup of cream and 2 cups of chocolate, and that would equal to, um, 2 cups of cream, 4 cups of chocolate. And then he kept doing that, but then he kept going. So at the end, he did it wrong. Because he should have stopped when he had 3 cream and 6 cups of chocolate.

PARTNER: Oh.

ANDREW: Because if he kept going on, he would have 12, and it’s not 9.

HILLARY LEWIS-WOLFSEN: So, what do you think about this one? What do you think about this one?

HILLARY LEWIS-WOLFSEN: What do you think?

STUDENT: The answer is logical. On the chart, it says 8 cups of chocolate with the 4 cups of cream, but if you add them together, it makes 12 cups in total. But the problem says Anthony made 9 cups in total for the ingredients.

HILLARY LEWIS-WOLFSEN: 9 cups in total. Okay. What do you think, Saurabh?

SAURABH: He was heading in the right direction, but then he went up to 4, which was wrong because he should have reached up to 3, because there’s 9 cups in total, and…

HILLARY LEWIS-WOLFSEN: Oh. So you’re saying this student should have stopped here?

SAURABH: Yeah.

HILLARY LEWIS-WOLFSEN: And that would have given them their 9 cups for total? Why do you think they went to 8 cups? Did you have something to say?

JAKE: I think he wrote 8 cups because he misinterpreted the sentence. He thought that you have to use 9 cups of the 2, of chocolate only, in the ingredients, so he got 8 cups which is the closest. But then, and he didn’t want to go over the limit, so he just wrote 8.

HILLARY LEWIS-WOLFSEN: Could be, could be! What did you want to add, Timothy?

TIMOTHY: I disagree with Jake, because if he thought that only 1 cup would go for, 1 cup of cream would go for every, 1 cup of cream would go in the whole recipe, then he wouldn’t have had 1 cup of cream and 2 cups of chocolate, 2 cups of cream and 4, he would have just had 1 cup of cream and 2 of chocolate, 1 cup of cream and 4 of chocolate. If he thought that 1 cup, only 1 cup of cream goes into the whole recipe.

HILLARY LEWIS-WOLFSEN: I’m sorry, you’re saying that…Can you say that again? I wasn’t understanding. You said it would go 1 and 4?

TIMOTHY: If he thought only 1 cup of cream goes into the whole recipe…

HILLARY LEWIS-WOLFSEN: Okay.

TIMOTHY: Then he wouldn’t have had 2 cups of cream for 4 cups of chocolate.

HILLARY LEWIS-WOLFSEN: Okay, so you’re saying this 1, because they had the solution of 8 cups, that they were only counting 1 cup of cream in the whole recipe?

TIMOTHY: Yeah. But if they had it up to the 4, that doesn’t make sense that there’s only 1.

HILLARY LEWIS-WOLFSEN: Right, okay. Did you want to respond?

JAKE: I don’t think that Tim understanded me, because I mean like, he thought that all the ingredients combined is 9 cups, he thought it was just the chocolate that needed 9 cups.

I love the supportive language this class uses, i.e. “logical,” “heading in the right direction.” The discussion between the kids is respectful of one another too.