Teacher Tune Up:What's the difference between a ratio and a fraction, and why should I care?The Intensity of Chocolate Milk
Sixth Grade Poster Problem
Ratios and Proportional Relationships
This problem is designed to help students use ratio language and concepts by focusing on a single compelling problem context – making chocolate milk. Questions in this problem use both part-to-part ratios and part-to-whole ratios.
Common Core State Standards for Mathematics:
Show Slide #1.
Ask students: “Have you ever made chocolate milk at home? How did you do it?” Allow one or two students to describe how they make chocolate milk by mixing the chocolate syrup and milk.
Engage students in a discussion about making chocolate milk. Lead the discussion to elicit ideas about the following.
Tell students that they are about to watch a video showing two different recipes used to make chocolate milk.
Before you start the video, set up the discussion that will follow by asking:
Show Slide #2 (video).
Show Slide #3.
We know that the main ingredients of chocolate milk are chocolate syrup and milk. But how much of each ingredient should you use? Steven is an “expert” at making chocolate milk that he thinks is “just right.”
Discuss Steven’s recipe.
Show Slide #4 as you ask these questions and check the appropriate boxes to highlight the parts and whole.
Show Slide #5.
Distribute Handout #1.
Show Slide #6.
In this workshop, your group will create a recipe for chocolate milk - you can choose the intensity. The intensity of your recipe should be different from Steven’s.
You will then make a poster that describes how much chocolate syrup and how much milk to use for three different situations.
For Situation A and Situation B, use the same recipe. Situation C asks you to modify your recipe for someone who likes less intense chocolate milk.
Situation A: Chocolate milk for just yourself: Use your recipe to make one 8-ounce glass of chocolate milk.
Situation B: Chocolate milk for a group: Use your recipe to make four 6-ounce glasses chocolate milk.
Situation C: Lastly, answer the following question: your grandma likes very weak (not intense) chocolate milk. How would you change your recipe to make a glass of chocolate milk for your grandma? Show how you would change your recipe to make 8 ounces of chocolate milk for your grandma.
Distribute Handout #2.
After you choose your recipe, make a poster.
Your poster should include mathematical representations (mathematical drawings, diagrams, tables, equations, or graphs) to show how you used your recipe in each situation. Include both the ratio of chocolate syrup and milk, and the number of ounces of each ingredient for your recipe. Show all your steps and consider your quantities carefully—some numbers will be easier to with than others.
Have students post their posters around the classroom.
Encourage students to travel around to view the posters created by other groups. Students should be encouraged to write questions for other groups by attaching a small adhesive notes.
During this time, teachers should be reviewing all the posters and considering which to highlight during Phase 5 (Strategic Teacher-led Discussion)
Poster A shows a group whose recipe was 2 parts chocolate syrup to 6 parts milk. These numbers are “friendly” because the parts add to 8. The group’s poster shows a bar diagram for each situation. This representation is very useful for showing how the ratio is constant even as the recipe “scales” up and down.
Poster B shows the work of a group whose recipe was 1 part chocolate syrup and 7 parts milk. That ratio works easily for the 8-ounce glass, but the group had to do some calculations for the four 6-ounce glasses. The 1:7 ratio is not “friendly” for a whole of 6 ounces. However, it does scale up nicely for the total of 4 × 6 = 24 ounces.
Poster C used a ratio of 3:2 (VERY intense chocolate milk!). The numbers in this ratio are more difficult to work with because the whole is 5 parts. The totals of 8 or 24 are not evenly divisible by 5, so you cannot use additive solutions like in Poster 1. These students used a ratio table to tame this ratio.
Facilitate a discussion that uses the students’ work to highlight some critical mathematical concepts and the representations that can be used. Here we have gathered three possible samples of student work, showing different levels of sophistication. Some students who pick “unfriendly” numbers will struggle. Rather than hide this difficulty, take the time to show students examples that didn't work perfectly, and why.
What to look for in terms of numbers chosen for the ratios:
Situations A and B are easier to solve when the parts in the ratio add to 2, 4, or 8. Some students may figure this out and intentionally choose numbers that make the computations easy. Choosing numbers that simplify calculations (when that is an option) is an important practice to highlight. Here is how some easy ratios will scale for Situation A.
1:3 → 2 oz chocolate and 6 oz milk
1:1 → 4 oz chocolate and 4 oz milk
3:5 → 3 oz chocolate and 5 oz milk
Solving for the number of ounces is much more challenging when the parts in the ratio do not add to 2, 4, or 8. For example, if the ratio is 2:3 then you have to multiply each part by 8/5 to make a total of 8 ounces of chocolate milk. The results are
2:3 → 16/5 = 3.2 oz chocolate and 24/5 = 4.8 oz milk (see sample poster 3).
What to look for in terms of diagrams:
The types of diagrams will also be uneven in their level of sophistication. The least sophisticated diagrams do not use a scale. More sophisticated diagrams might not even look like cups anymore, but will preserve the relative sizes of the parts.
Least sophisticated: Drawings of cups not to scale
More sophisticated: Diagrams correctly showing proportions
Most sophisticated: Stacked bar graphs showing part:part or part:whole (as in Poster A)
What to look for in terms of solution strategies:
Less sophisticated: Repeated adding
More sophisticated: Scaling
Most sophisticated: Setting up equations and solving. But check for understanding; some students get good at symbols but lose their connection to the situation.
Distribute Handout #3.
The Focus Problem challenges students to use the ratio reasoning developed in "The Intensity of Chocolate Milk" to solve a problem involving a ratio with four parts.
Answers to Handout #3:
Strategic Education Research Partnership
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Project funding provided by The William and Flora Hewlett Foundation and S.D. Bechtel Jr. Foundation
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