Lesson 12

This warm-up reminds students of previous work with tape diagrams and encourages a different way to reason with them. Students are given only a tape diagram and are asked to generate a concrete context to go with the representation.

Students’ stories should have the following components:

• the same unit for both quantities in the ratio
• a ratio of \(7:3\)
• scaling by 3, or 3 units per part
• They may also have a quantity of 30 that represents the sum of the two quantities in the ratio.

As students work, identify a few different students whose stories are clearly described and are consistent with the diagram so that they can share later.

Ask students to share a few things they remember about tape diagrams from the previous lesson. Students may recall that:

• We draw one tape for every quantity in the ratio. Each tape has parts that are the same size.
• We draw as many parts as the numbers in the ratio show (e.g., a \(2:3\) ratio: we draw 2 parts in a tape and 3 parts in another tape).
• Each part represents the same value.
• Tape diagrams can be used to think about a ratio of parts and the total amount.

Tell students their job is to come up with a valid situation to match a given tape diagram. Give students some quiet thinking time, and then time to share their response with a partner.

Students may misunderstand the meaning of the phrase “with two quantities” and simply come up with a situation involving ten identical groups of three. Point out that the phrase means that the row of seven groups of three should represent something different than the row of three groups of three.

Students may also come up with a situation involving different units, for example, quantity purchased and cost, or distance traveled and time elapsed. Remind them that the parts of tape are meant to represent the same value, so we need a situation that uses the same units for both parts of the ratio.

Invite a few students to share their stories with the class. As they share, consider recording key details about each story for all students to see. Then, ask students to notice similarities in the different scenarios.

Guide students to see that they all involve the same units for both quantities of the ratio, a ratio of 7 to 3, and either 3 units per part or scaling by 3. They may also involve an amount of 30 units, representing the sum of the two quantities.

This task prompts students to solve a single problem using a triple number line, a table, or a tape diagram. Either assign each student a representation or allow students to choose the representation they prefer. During the following discussion, they will compare and contrast the three representations and identify the relative merits of the different representations for the different problems.

Students will have varying opinions about which representation they prefer and why. Their views may stem from observations such as:

• Number lines and tables involve scaling up individual quantities to find the total. Tape diagrams involve starting with the total and breaking it down into equal groups.
• It is hard to take shortcuts with number line diagrams.
• We can use the number line diagram or the table efficiently by thinking, “17 times what is 85?” This value tells us how many “batches” of tickets the teacher had to order.
• When using a tape diagram, it was easier to count 17 groups and compute \(85\div17\) to find how many tickets each group needs.

Students may struggle to represent the problem with a tape diagram. As they work, notice any trends that may need to be addressed with the class.

Tell students that they will now solve a ratio problem in one of three different ways. Either assign each student one of the representations or instruct them to choose one representation to use.

Give students quiet think time to complete the activity and then, optionally, time to share their responses with a small group or in pairs. 

The number line and table representations are organized similarly. For example, one could make progress with both of them simply by skip counting and keeping an eye out for a total of 85 people. The tape diagram, though, is organized in a much different way. Equivalent ratios are not listed out, but rather equivalent ratios arise from thinking about how the diagram could represent any number of batches. Students may thus mistakenly treat the tape diagram like a double number line diagram—they may start writing 15, 30, 45, etc. in the “kids” tape, for example. Once this plays out, students may self-regulate once they notice there are only 2 boxes in the chaperones’ row. But they may decide to just draw more boxes! Reorient these students by asking how many parts of tape there are (17), and reminding them each part of tape represents equal numbers of people, and that there are 85 total people. The presentation of correct work during the discussion could be used as an opportunity to remediate, as well. For example, consider asking a student to explain what they understand about another student’s correct work.

Before debriefing as a class, consider arranging students in groups of 3, where a group includes one student who used each representation. Give them time to see if they got the same answer and compare and contrast the representations.

Solicit students’ reactions to the three strategies, encouraging them to identify what is similar and different about the approaches. Consider polling the class for their preferred strategy. Ask 1–2 students favoring each method to explain why. This can also be a discussion about what worked well in this or that approach, and what might make this or that approach more complete or easy to understand. While there is no right or wrong answer with regards to their preferred strategy, look out for unsupported reasoning or misunderstandings.

Some students may prefer the tape diagram because the solution path seems more direct, but caution them against trying to use a tape diagram any time they see a ratio problem. Because tape diagrams involve equal-sized parts, they can only be used to represent quantities with the same unit. If different units are involved, the parts of one tape and those of the other will not represent an identical quantity.

This activity gives students a chance to apply the different strategies they have learned to solve ratio problems and to decide on methods that would make the most sense in given situations. If desired, you can have students complete this activity as an info gap, by instructing them to close their books or devices and distributing slips cut from the blackline master. Alternatively, students can simply complete the two problems that are printed in the task statement.

Each problem may lend itself better to a particular representation than to others. Below are sample arguments in support of a representation for each problem, though options may vary:

• The salad dressing problem can be well represented by all three representations because it involves small numbers and simple multiplication.
• The box-moving problem would be inefficient to represent with a number line diagram, since it would require making 8 half-hour jumps to find the total of 72. Either a tape diagram or a table showing more straightforward multiplication would be more efficient.

The time needed to solve the problems may depend on the representation students choose to use.

Tell students they will continue to practice using the tools at their disposal—number lines, tables, and tape diagrams—to solve ratio problems.

Provide access to graph paper and rulers. Arrange students in groups of 2.

If using the info gap routine: Explain the info gap structure and consider demonstrating the protocol. In each group, distribute a problem card to one student and a data card to the other student. Give students who finish early a different pair of cards and ask them to switch roles.

If not using the info gap routine: Ask students to complete both questions and then compare their strategies with their partner. Partners should work together to resolve any discrepancies.

While thinking through problems, it is common for students to hold the meanings of their representations (numbers, quantities, markings, etc.) in their heads without writing them down. When students are getting their solutions ready for others to look at, though, remind them of the importance of labeling quantities and units of measure and making the steps in their thinking clear.

Some students may choose the same strategy or representation each time. If their answers are accurate, this is fine. However, if time allows, ask them to check if they can verify their answer using an alternative strategy.

If using the info gap routine: Students holding a Problem Card may have trouble thinking of appropriate questions to ask their partner. Encourage them to revisit the problem at hand and think about the kinds of information that might be helpful or relevant. (For example, if the problem is about how long it would take to perform something, ask students how they usually gauge the amount of time needed for something. Ask, “What would the amount of time depend on?”)

After students have had time to work, share the correct answers and ask students for their preferences about representations. Some guiding questions:

• “Is there a particular representation you tend to try first?”
• “Does one seem more efficient than the others?”
• “Did the situation in a problem affect your choice? If so, what features of a problem might steer you toward or away from a particular strategy?”