Until now, students have worked with ratios of quantities given in terms of specific units, such as milliliters, cups, and teaspoons. This task introduces students to the use of the more generic “parts” as a unit in ratios, and the use of tape diagrams to represent such ratios. In addition to thinking about the ratio between two quantities, students also begin to think about the ratio between the two quantities and their total.

Two important ideas to make explicit through the task and discussion:

• A ratio can associate quantities given in terms of a specific unit (as in 4 teaspoons of this to 3 teaspoons of that). A ratio can also associate quantities of the same kind without specifying particular units, in terms of “parts” (as in 4 parts of this to 3 parts of that). Any appropriate unit can be used in place of “parts” without changing the 4 to 3 ratio.
• A ratio can tell us about how two or more quantities relate to one another, but it can also tell us about the combined quantity (when that makes sense) and allow us to solve problems.

As students work, notice in particular how they approach the last two questions. Identify students who add snap cubes to represent the larger amount of paint, and those who use the original number of snap cubes but adjust their reasoning about what each cube represents. Be sure to leave enough time to debrief as a class and introduce tape diagrams afterward.

Class discussion should center around how students used snap cubes to answer the questions and their approach to the last two questions. Invite some students to share their group’s approach. Ask:

• “How did the snap cubes help you solve the first few problems?”
• “In one of the problems, you were only given the total amount of maroon paint. How did you find out the amounts of red and blue paint needed to produce 80 ml of maroon?”
• “How did you approach the last question?” (Add more cubes, or use the same representation of 5 red cubes and 3 blue ones.)

Discuss how the same 5 red cubes and 3 blue ones can be used to represent a total of 80 ml of blue paint. Explain that this situation can be represented with a tape diagram. A tape diagram is a horizontal strip that is partitioned into parts. Each part (like each snap cube) represents a value. It can be any value, as long as the same value is used throughout.

Show a tape diagram representing a ratio of red paint to blue paint yielding 80 ml of maroon paint. Ask students where they see the 5, the 3, and the 80 being represented in the diagram. Discuss how many batches of paint are represented.

Show the tape diagram for green paint mixture discussed earlier. Students should be able to say that the ratio of blue to yellow paint is . Ask: “What value each part of the diagram would have to take to show a 20 ml mixture of green paint? How do you know?”

Guide students to see that, if each of the 4 total parts must be equal in value and amount to 20 ml, we could divide 20 by 5 to find out what each part represents. , so each part represents 5 ml of paint.

This activity allows students to practice reasoning about situations involving ratios of two quantities and their sum. It also introduces students to using “parts” in recipes (for instance, 4 parts chili powder, 1 part garlic powder, and 2 parts ground cumin), instead of more familiar units, such as cups, teaspoons, milliliters, and so on. Students may use tape diagrams to support their reasoning, or they may use other representations learned so far—discrete diagrams, number lines, tables, or equations. All approaches are welcome as long as students use them to represent the situations appropriately to support their reasoning.

As students work, monitor for different ways students reason about the problems, with or without using tape diagrams.

When solving the last problem, students may say that more than 21 gallons of gravel, 14 gallons of sand, and 7 gallons of cement will be needed to make 42 gallons of concrete. They may reason that the finer ingredients—sand and cement—will fill the spaces between the coarser pieces of gravel. Acknowledge the validity of their reasoning. Clarify that we can make an assumption that the gravel is also on the smaller side, so there are minimal empty spaces.

Select students to share their responses and reasoning. Discuss the ways in which tape diagrams are used to represent the quantities in the problems. Ask questions such as: 

• “For the sneakers and boots problem, where do we see the 6 to 5 ratio in the tape diagram? How do we show the total number of students?”
• “For the taco seasoning problem, how do we represent the 20 teaspoons of ground cumin?”

Highlight that the ratio of quantities expressed using “parts” and the ratio expressed using specific units, such as “gallons” or “teaspoons,” are equivalent. For example, the ratio of parts of gravel to sand to cement in the concrete recipe is 3 to 2 to 1. We can make concrete in a ratio of 3 gallons to 2 gallons to 1 gallon or 21 gallons to 14 gallons to 7 gallons because these ratios are equivalent. Number line diagrams, tables, and tape diagrams can all help us find these equivalent ratios, regardless of the units being used. 

In this activity, students have an opportunity to create their own equivalent ratio problem.

Have each group share the peer-generated question it was assigned and the solution. Though the group that wrote the question will be responsible for confirming the answer, encourage all to listen to the reasoning each group used.