Lesson 12

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.

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.

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.

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.

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.

Display the second diagram for all to see.

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.

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?” 

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

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.

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

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.

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.

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.

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.