Lesson 11

This warm-up encourages students to use the meaning of fractions and properties of operations to reason about equations. While students may evaluate each side of the equation to determine if it is true or false, encourage students to think about the following ideas in each:

• The first question: Dividing is the same as multiplying by the reciprocal of the divisor.
• The second question: Adjusting the factors adjusts the products. If both factors increase, the resulting product will be greater than the original.
• The third question: The commutative property of multiplication.
• The fourth question: Decomposing a dividend into two numbers and dividing each by the divisor is a way to find the quotient of the original dividend.

Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students if or how the factors in the problem impacted the strategy choice. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

After each true equation, ask students if they could rely on the reasoning used on the given problem to think about or solve other problems that are similar in type. After each false equation, ask students how we could make the equation true.

Up until now, students have worked with ratios of quantities given in terms of specific units such as milliliters, cups, teaspoons, etc. 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 afterwards.

Explain to students that they will explore paint mixtures and use snap cubes to represent them. Say: “To make a particular green paint, we need to mix 1 ml of blue paint to 3 ml of yellow.” Represent this recipe with 1 blue snap cube and 3 yellow ones and display each set horizontally (to mimic the appearance of a tape diagram).

Ask:

• “How much green paint will this recipe yield?” (4 ml of green paint.)
• “If each cube represents 2 ml instead of 1 ml, how much of blue and yellow do the snap cubes represent? How many ml of green paint will we have?” (2 ml of blue, 6 ml of yellow, and 8 ml of green.)
• “Is there another way to represent 2 ml of blue and 6 ml of yellow using snap cubes?” (We could use 2 blue snap cubes and 6 yellow ones.)
• “How do we refer to 2 ml of blue and 6 ml of yellow in terms of ‘batches’?” (2 batches.)

Highlight the fact that they could either represent 2 ml of blue and 6 ml of yellow with 2 blue snap cubes and 6 yellow ones (show this representation, if possible), or with 1 blue snap cube and 3 yellow ones (show representation), with the understanding that each cube stands for 2 ml of paint instead of 1 ml.

Explain to students that, in the past, they had thought about different amounts of ingredients in a recipe in terms of batches, but in this task they will look at another way to mix the right amounts specified by a ratio.

Arrange students in groups of 3–5. Provide 50 red snap cubes and 30 blue snap cubes to each group. Give groups time to complete the activity, and then debrief as a class.

Students may need help interpreting “Suppose each cube represents 2 ml.” If necessary, suggest they keep using one cube to represent 1 ml of paint. So, for example, the second question would be represented by 5 stacks of 2 red cubes and 3 stacks of 2 blue cubes. If they use that strategy, each part of the tape diagram would represent one stack.

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 \(5:3\) 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 \(1:3\). 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. \(20\div4=5\), 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 (e.g., 3 parts oil with 2 parts soy sauce and 1 part orange juice), instead of more familiar units such as cups, teaspoons, milliliters, etc. 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.

Keep students in the same groups. Provide graph paper and snap cubes (any three colors). Explain that they will now practice solving problems involving ratios and their combined quantities (similar to the green and purple paint in the previous task). Draw students to a ratio that uses “parts” as its unit. Ask students what they think “one part” means or amounts to, and how situations expressed in terms of “parts” could be diagrammed.

Before students begin working, make sure they understand that “parts” do not represent specific amounts, that the value of “one part” can vary but the size of all parts is equal, and that a tape diagram can be used to show these parts.

Students may think of each segment of a tape diagram as representing each cube, rather than as a flexible representation of an increment of a quantity. Help them set up the tapes with the correct number of sections and then discuss how many parts there are in all.

Select students to share their reasoning. Help students make connections between different representations, especially any tape diagrams.

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

Keep students in the same groups. Provide graph paper, snap cubes (any three colors), and tools for creating a visual display.

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.