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
Scaling by 3, or 3 units per part
The stories 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.
Launch
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. (For instance, for a 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 think time and then time to share their response with a partner.
Activity
None
Describe a situation with two quantities that this tape diagram could represent.
Student Response
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Building on Student Thinking
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.
Activity Synthesis
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.
13.2
Activity
Standards Alignment
Building On
Addressing
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
This activity 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 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.
This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Arrange students in groups of 3–4. Use Three Reads to support reading comprehension and sense-making about this problem.
Display only the problem stem (the first two sentences), without revealing the question.
In the first read, students read the problem with the goal of comprehending the situation.
For the first read, read the problem aloud while everyone else reads along, and then ask, “What is this situation about? What is going on here?” Allow 1 minute to discuss with a partner and then share with the whole class. A typical response may be, “Students are going on a field trip with chaperones. Everyone needs a ticket.” Listen for and clarify any questions about the context.
In the second read, students analyze the mathematical structure of the story by naming quantities.
Select a different student to read to the class, and then ask, “What can be counted or measured in this situation?” Give students 30 seconds of quiet think time, followed by another 30 seconds to share with their partner. A typical response may be: number of students and teachers, number of all participants, number of tickets, and cost of tickets.
In the third read, students brainstorm possible starting points for answering the questions.
Invite students to read the displayed information aloud with their partner, or select a different student to read to the class. After the third read, reveal the question. Ask, “What are some ways we might get started on this?” Instruct students to think of ways to approach the questions without actually solving. Give students 1 minute of quiet think time, followed by another minute to discuss with their group. Invite students to name some possible strategies referencing quantities from the second read.
Provide these sentence frames as students discuss:
“To find the number of tickets for chaperones (or students), I can start by . . . .”
“I know that . . . , so I can . . . .”
As students discuss possible solution strategies, select 1–2 students to share their ideas with the whole class. As the selected students present their strategies, create a display that summarizes possible starting points. (Stop students as needed before they share complete solutions or answers.) If no students mention using diagrams or tables, ask them if those might be useful.
Tell students that they will now solve a ratio problem in one of three different ways. Either assign each student one representation or allow them to choose one.
Give students quiet think time to complete the activity and then, optionally, time to share their responses with a small group.
Activity
None
Activity Synthesis
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.
13.3
Activity
Optional
Standards Alignment
Building On
Addressing
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
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, this activity can be completed as an Info Gap. 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. Here are sample arguments in support of a representation for each problem, though options may vary:
The cleaning fluid 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.
Launch
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. Instruct students to close their books or devices. 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.
Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example: “We already know . . .“ “We are trying to find . . .” “We can use _____ to represent . . .“ Supports accessibility for: Language, Organization
Activity
None
Solve each problem and show your thinking. Organize it so it can be followed by others. If you get stuck, consider drawing a double number line diagram, a table, or a tape diagram.
A recipe for a cleaning liquid says to mix 4 parts water with 3 parts vinegar. How much water is needed to make a total of 28 tablespoons of the solution?
Andre and Han are moving boxes. Andre can move 4 boxes every half hour. Han can move 5 boxes every half hour. How long will it take Andre and Han to move all 72 boxes?
Student Response
Activity Synthesis
Share the solutions to the problems or invite students to share theirs. Ask students about their choice(s) of 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?”
Lesson Synthesis
To reinforce the idea that different representations can be used to solve problems involving equivalent ratios, highlight when each representation might be preferred:
Tape diagrams are most likely to be helpful when the parts of the ratio have the same kind of units (such as cups to cups, miles to miles, or boxes moved to boxes moved) and the sum of the quantities is meaningful in the context.
Number lines are a good choice when it helps to visualize how far apart numbers are from each other. They are harder to use with very big or very small numbers.
Tables work well in almost all situations.
To emphasize good problem-solving habits, ask students: “What should we remember to do when using different representations to solve equivalent ratio problems?” (We should remember to:
Label each part of the diagram with what it represents.
Include units in the answer, for example, writing “4 cups” instead of just “4.”
Use brackets to indicate total amounts.
Read what the question is asking and answer it.)
Student Lesson Summary
When solving a problem involving equivalent ratios, it is often helpful to use a diagram. Any diagram is fine as long as it correctly shows the mathematics and you can explain it.
Let’s compare three different ways to solve the same problem: The ratio of adults to kids in a school is . If there is a total of 180 people, how many of them are adults?
Tape diagrams are especially useful for this type of problem because both parts of the ratio have the same units (“number of people") and we can see the total number of parts.
This tape diagram has 9 equal parts, and they need to represent 180 people total. That means each part represents , or 20 people.
Two parts of the tape diagram represent adults. There are 40 adults in the school because .
Double or triple number lines are useful when we want to see how far apart the numbers are from one another. They are harder to use with very big or very small numbers, but they could support our reasoning.
Tables are especially useful when the problem has very large or very small numbers.
We ask ourselves, “9 times what is 180?” The answer is 20. Next, we multiply 2 by 20 to get the total number of adults in the school.
Another reason to make diagrams is to communicate our thinking to others. Here are some good habits when making diagrams:
Label each part of the diagram with what it represents.
Label important amounts.
Make sure you read what the question is asking and answer it.
Make sure you make the answer easy to find.
Include units in your answer. For example, write “4 cups” instead of just “4.”
Double check that your ratio language is correct and matches your diagram.
Standards Alignment
Building On
Addressing
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
A teacher is planning a class trip to the aquarium. The aquarium requires 2 chaperones for every 15 students. The teacher plans accordingly and orders a total of 85 tickets. How many tickets are for chaperones, and how many are for students?
Solve this problem in one of three ways:
Use a triple number line.
Use a table.
(Fill rows as needed.)
kids
chaperones
total
15
2
17
Use a tape diagram.
After discussing all three representations with your class, which representation do you prefer for this problem and why?
Student Response
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Building on Student Thinking
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.
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Building on Student Thinking
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?”)
