Lesson 12

Percentages and Tape Diagrams

12.1: Notice and Wonder: Tape Diagrams (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that tape diagrams can be used to think about fractions of a whole as percentages of the whole, which will be useful when students interpret and draw tape diagrams in a later activity. While students may notice and wonder many things about these images, the important discussion points are that there are two rectangles of the same length, one of the rectangles is divided into four pieces of equal length, and a percentage is indicated.

Launch

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

Two equivalent diagrams. The top diagram solid orange, labeled 80. Bottom diagram partitioned into 4 parts. Three parts are blue and together labeled question mark %. The fourth part is white.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If the four pieces of equal length do not come up during the conversation, ask students to discuss this idea. It is not necessary to decide what should be used in place of the question mark.

12.2: Revisiting Jada's Puppy (15 minutes)

Activity

The purpose of this activity is for students to study and make sense of tape diagrams that can be used to see benchmark percentages in terms of fractions. The first question shows a percentage as a part of the whole, and the second shows a comparison between two quantities. Both situations can be described in terms of fractions or percentages. The second situation is important for making connections between percentages greater than 100 and fractions greater than 1.

Launch

Give students 1 minute of quiet think time, and then have them turn to a partner to discuss the first question. Poll the class to be sure that everyone can see that the puppy is \(\frac15\) of its adult weight. Ask the students what 100% is in this situation, and label the diagram with 100%. Give students 1 minute of quiet think time, and then have them discuss the second question with a partner.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have had the opportunity to think about the first question, ask students to write a brief explanation for how the puppy’s current weight as a fraction of its adult weight is represented on the tape diagram. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “Can you explain how…”, “You should expand on...”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own explanation and learn about other ways to interpret the tape diagram.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Student Facing

Jada has a new puppy that weighs 9 pounds. It is now at about 20% of its adult weight.

  1. Here is a diagram that Jada drew about the weight of her puppy.

    Tape diagram with five parts, each part 9. One part colored blue and indicated 20%.
    1. The adult weight of the puppy will be 45 pounds. How can you see that in the diagram?

    2. What fraction of its adult weight is the puppy now? How can you see that in the diagram?

  2. Jada’s friend has a dog that weighs 90 pounds. Here is a diagram Jada drew that represents the weight of her friend’s dog and the weight of her puppy.

    Tape diagrams. Top diagram, 10 parts each labeled 9. Bottom diagram, one box labeled 9.
    1. How many times greater is the dog’s weight than the puppy’s?
    2. Compare the weight of the puppy and the dog using fractions.
    3. Compare the weight of the puppy and the dog using percentages.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display the second diagram for all to see.

Tape diagrams. Top diagram, 10 parts each labeled 9. Bottom diagram, one box labeled 9.

Label the dog’s weight with a 1, and ask the students how we should label the puppy's weight if we are comparing using fractions.

Tape diagrams. Top diagram, 10 parts each labeled 9. Bottom diagram, one box labeled 9.

Display another copy of the second diagram. Ask students, “When we compare the puppy’s weight to the dog’s weight, what represents 100%? How should we label the diagram to show it? What should we label the puppy’s weight?” 

12.3: 5 Dollars (15 minutes)

Activity

The purpose of this activity is for students to describe multiplicative comparison problems given in terms of percentages using fractions.

Launch

Give students 3 minutes of quiet work time. Have them turn to a partner to discuss their answer to the first question. Then give them 3 minutes of quiet think time for the second question, followed by a whole-class discussion.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, ask students to use the same color to represent the amount of money Elena has compared to Noah in a tape diagram, written as a value, and represented as a fraction.
Supports accessibility for: Visual-spatial processing
Speaking: MLR8 Discussion Supports. After students have had enough time to work on the first question, and to share their tape diagrams for the first question with a partner, bring the whole class back together. During the discussion, press for details in students’ explanations by asking where they see Elena’s \$2 and Noah’s \$5 represented in the diagram. Use a visual display of the tape diagrams to annotate (or mark) student responses. Since Elena has 40% or \(\frac25\) as much money as Noah, ask students where they see 40% or \(\frac25\) represented in the tape diagram. As an additional challenge, since Noah has 250% or \(\frac52\) as much money as Elena, ask students where they see 250% or \(\frac52\) represented in the tape diagram. This will help students make sense of tape diagrams and see the relationship between percentages and fractions.
Design Principle(s): Support sense-making

Student Facing

Noah has \$5.

    1. Elena has 40% as much as Noah. How much does Elena have?
    2. Compare Elena’s and Noah’s money using fractions. Draw a diagram to illustrate.
    1. Diego has 150% as much as Noah. How much does Diego have?
    2. Compare Diego’s and Noah’s money using fractions. Draw a diagram to illustrate.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Have students show and explain their diagrams. Then show these if no one has something equivalent:

Tape diagram.
Tape diagram.

12.4: Staying Hydrated (10 minutes)

Optional activity

In this activity, students explore percentages that describe parts of a whole. They find both \(B\) and \(C\), where \(A\%\) of \(B\) is \(C\), in the context of available and consumed water on a hike.

Students who use a double number line may notice that the value of \(B\) is the same in both questions, so the same double number line can be used to solve both parts of the problem. To solve the second question, however, the diagram needs to be partitioned with more tick marks.

As in the previous task, students may solve using other strategies, including by simply multiplying or dividing, i.e., \((1.5) \boldcdot 2 = 3.0\) and \(\frac{80}{100} \boldcdot (3.0) = 2.4\). Encourage them to also explain their reasoning with a double number line, table, or tape diagram. Monitor for at least one student using each of these representations.

Launch

Give students quiet think time to complete the activity and then time to share their explanation with a partner. Follow with a whole-class discussion.

Student Facing

During the first part of a hike, Andre drank 1.5 liters of the water he brought.

  1. If this is 50% of the water he brought, how much water did he bring?
  2. If he drank 80% of his water on his entire hike, how much did he drink?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Decide if each scenario is possible.

  1. Andre plans to bring his dog on his next hike, along with 150% as much water as he brought on this hike.
  2. Andre plans to drink 150% of the water he brought on his hike.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

After students create a double number line diagram with tick marks at 50% and 100%, some may struggle to know how to fit 80% in between. Encourage them to draw and label tick marks at 10% increments or work with a table instead. Some students may think that the second question is asking for the amount of water Andre drank on the second part of the hike. Clarify that it is asking for his total water consumption on the entire hike.

Activity Synthesis

Select 1–2 students who used a double number line, a table of equivalent ratios, and a tape diagram to share their strategies. As students explain, illustrate and display those representations for all to see.

Ask students how they knew what 100% means in the context. At this point it is not necessary for students to formally conceptualize the two ways percentages are used (to describe parts of whole, and to describe comparative relationships). Drawing their attention to concrete and contextualized examples of both, however, serves to build this understanding intuitively.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their answers for the second question, present an incorrect answer and explanation. For example, “If Andre drank 80% of his water on his entire hike, then he drank 1.2 liters of water because 0.8 times 1.5 liters is 1.2 liters.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss in partners, listen for students who identify and clarify the assumption in the statement. For example, the author assumed that Andre brought 1.5 liters for his entire hike; however, the problem states that Andre drank 1.5 liters of the water he brought. This will remind students to carefully identify the quantity they are finding a percentage of.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

If you are comparing two quantities using percentages, you can also compare them using fractions. Drawing a tape diagram can sometimes help us see how to do this more easily.

Questions for discussion:

  • If you have 50% of the money needed to buy a book, what fraction is that?
  • If you run 125% of your goal for the week, what fraction is that?

Seeing percentages in terms of fractions can help us solve percentage problems. 

12.5: Cool-down - Small and Large (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Tape diagrams can help us make sense of percentages.

Consider two problems that we solved earlier using double number lines and tables: “What is 30% of 50 pounds?” and “What is 100% of a number if 140% of it is 28?”

Here is a tape diagram that shows that 30% of 50 pounds is 15 pounds.

A tape diagram divided into 10 parts, each labeled 5. The entire diagram is labeled 100 %. The first part is labeled 10 %. The first three parts are colored orange, the rest are colored white.

This diagram shows that if 140% of some number is 28, then that number must be 20.

A tape diagram with seven parts, each labeled 4. The first part is labeled 20%. Five parts are together labeled 100%.