NOTICE Lab - Video Club 5: Lemonade

Video Club 5: Lemonade

In this video, an 8th grade student works on a proportional reasoning problem using manipulatives. The student initially uses additive reasoning to look at the problem, but as she starts to use the manipulatives her thinking changes.

Video Club Overview

Video Club Goals:

- Teachers will attend to and reason about the mathematical resources that the student uses to compare the amounts of water and lemonade in the problem.
- Teachers will leverage the resources to connect to productive ways of using them
- Teachers will think about moves they can make to support the student in connecting her ideas to multiplicative reasoning

Materials:

- Video loaded on Anotemos and shared with all participants
- If watching in person, each participant should have a device on which they can view the video and headphones.

Get ready (20 mins). To prepare participants for viewing the video, review the problem. Give participants a few minutes to reason about the problem themselves. Then have a whole group discussion that gives participants the opportunity to share their explanations. The problem:

Maria and Janelle are making lemonade. Maria mixes 2 cups of lemon juice with 3 cups of water. Janelle mixes 3 cups of lemon juice with 4 cups of water. Whose mixture tastes more lemony?”

Exploring the Math

Exploring ratios and proportions

Part-to-part comparison:

One way to compare two ratios is to examine their component parts and determine what happens when one of the same components between two ratios is equal. This can be represented in several ways with this problem:

Tables: A student might make a table to compare the two different ratios:

In each row of the tables the lemon juice and water are multiplied by the same constant to make equivalent ratios to the original. When both Maria and Janelle have 6 cups of lemon juice Maria has 9 cups of water, while Janelle has 8 cups of water (the blue circles). Since Janelle has less water for the same amount of lemon juice, her mixture is more lemony. Students could also compare the amount of lemon juice when Maria and Janelle have the same amount of water (the red circles). In this case, Maria has 8 cups of water, and Janelle has 9 cups. Since Janelle has more lemon juice for the same amount of water, then her mixture is more lemony.

Other representations: Students might also represent the part-to-part situation using manipulatives, double number lines, tape diagrams, etc.

Part-to -whole comparison:

Another strategy that students might use is to determine the fraction of lemon juice in each of Maria’s and Janelle’s total lemonade mixtures. Maria has 2 cups of lemon juice and 3 cups of water, meaning that the total mixture is 5 cups. This means that the mixture is ⅖ lemon juice. Janelle has 3 cups of lemon juice and 4 cups of water, meaning that the total mixture is 7 cups. This means that the mixture is 3/7 lemon juice.

Then, students might find the decimal values of the two fractions. Since Maria’s mixture is .4 lemon juice and Janelle’s mixture is about .43 lemon juice, Janelle’s mixture is more lemony.

Students might also try to compare the fractions. ⅖ and 3/7 have a common denominator of 35. Maria’s lemon juice would be equivalent to 14/35 of the total mixture and Janelle’s lemon juice would be equivalent to 15/35 of the total mixture.

Algebraic solution

Another solution strategy is to consider how the variables, lemon juice and water, change together. Students might write proportions using variables:

Students can then determine how L and W relate to each other in each equation by multiplying by W:

Students could then compare by plugging in the same value for W and notice that Janelle’s mixture will always have more lemon juice, regardless of the input. They could also interpret the slope. For Maria’s mixture, for every 1 cup of water added, there is ⅔ cups of lemon juice added. For Janelle’s mixture, for every 1 cup of water, there is ¾ cup of lemon juice added. Students could also solve for W instead and then compare those equations.

There are a number of Seeds of Algebraic Thinking that could be activated during this task:

Unit Rate- Another way that students might think about this problem is “How many cups of water are there per one cup of lemon juice for each mixture?” They could also think about it in terms of cups of lemon juice per one cup of water. This thinking might come from children's past experiences of recognizing rules or patterns of how one quantity changes when another goes up by one.

Same Direction Change- Same direction change is a seed related to covariation. Students might explore how one variable changes in relation to the other. In both Maria’s and Janelle’s mixtures as one variable increases the other also increases.

Balance- As students do part-to-part or part-to-whole comparisons, they need to multiply various components of ratios and fractions by constants. For example, when doing the part-to-part comparisons, students multiply both the amounts of lemon juice and water by the same constant to find equivalent ratios. This idea of performing the same operation on all components relates to keeping equations balanced when performing operations to both sides.

Viewing and Discussing the Video

Have participants watch the 6:00 video clip individually on the Anotemos platform. This video has different episodes that aren’t continuous. Here is some context that will be helpful to participants:
1. Robin part I: 0:00-1:52 This clip begins right at the beginning of the interview just after the interviewer has finished reading the question aloud to Robin.
2. Robin part II: 1:53-5:13 This clip begins a little while after the interviewer has asked Robin to use some blocks to show the recipes for two mixtures that would taste the same as Maria’s mixture. As this clip begins, you can see that in response to this question Robin has created three piles of bricks on the table in front of her that represent mixtures that she thinks would all taste the same. The three piles contain: (1) 2 yellow, 3 white, (2) 4 yellow, 6 white (3) 8 yellow, 12 white.
3. Robin part III: 5:13- STOP AT 6:00 This clip begins after Robin has made one more pile of blocks representing the mixture with 9 cups lemon juice and 12 cups water.

Discussion: In discussing the video, focus on the resources that the student uses to think about the problem. What is she thinking about the relationship between the two ratios? What does this tell us about what she knows about ratios/proportions? How does she use the manipulatives? What does that tell us about her resources?

Description of Goal:

Teachers will attend to and reason about the mathematical resources that the student uses to compare the amounts of water and lemonade in the problem.
Teachers will consider how they might respond to the student’s additive reasoning when trying to use a part-to-part solution strategy.

Table - Video Key Moments

VC4 table

Potential Discussion Questions:

What does the student seem to understand about proportional relationships at the beginning of the video?
How does the student’s thinking about the relationship between the lemonade mixtures change when she starts to use the manipulatives?
What role do the student’s gestures play in supporting her understanding? Do her gestures help her in different ways at different times?
What questions might you ask this student? Are there different ways to respond than what the interviewer did?

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