In this activity, students continue to explore different representations—now including tape diagrams and tables—for solving problems involving percentages. The goal is for students to see that a relationship can be represented in different ways, make connections, and choose a representation based on the situation and what makes sense to them. Two or more partially completed representations are given to encourage their use by students.

The situations here involve multiplicative comparisons of two quantities. In the first situation, students need to find the value for 100% when the value for 10% is known. In the second, they find a quantity that is 150% as much as another quantity. Students must interpret the quantities being given and sought as well as what 100% represents. They also need to decide how to represent these quantities. In doing so, students practice reasoning quantitatively and abstractly (MP2).

When students work on the second question, monitor for those who make the value for 100% explicit in their chosen representation (if any is used). Also monitor for students who find the weight of the bunny in the following ways:

• Compute 10% of the weight of Jada’s puppy and multiply it by 15.
• Find 50% of the weight of Jada’s puppy and triple it.
• Multiply the weight of Jada’s puppy by 1.5 directly because 150 is 1.5 times 100.

In both situations here, the quantities can be related in terms of fractions. For example, we can think of the weight of Jada’s puppy as of the weight of another dog and the weight of the bunny as of the weight of Jada’s puppy. The latter can help students relate percentages greater than 100 and fractions greater than 1. Monitor for students who make connections to fractions. Reasoning in terms of fractions can serve as a useful preview for the work on benchmark percentages in an upcoming lesson.

One goal of this discussion is to spotlight various ways to reason about 150% of a quantity and how the same multiplicative relationships are shown in different representations. Another goal is to reiterate that we can think about solving percentage problems in terms of finding equivalent ratios.

Display 2–3 approaches from previously selected students for all to see. Invite students to briefly describe their reasoning. Use Compare and Connect to help students compare, contrast, and connect the different approaches and representations. Here are some questions for discussion:

• “How are the approaches the same?” (They all started with thinking about the 9 pounds as 100%. Many of them involved dividing 9 and 100% by 2 and multiplying the result by 3. They involved finding 50%, or 10%, first before finding 150%.)
• “How are they different?” (They used different representations. The 9 pounds and 100% are described differently in each representation. Some approaches are more convenient or quicker than others.)

Highlight that all the strategies involve finding a pounds-to-percentage ratio of that is equivalent to . It makes sense to multiply 9 by (or divide 9 by 2) and then multiply by 3 because multiplying 100 by and then multiplying the result by 3 gives 150.

In this activity, students reason about percentages that describe both multiplicative comparison and parts of a whole. In the first situation, they reason about a quantity described as “120% as much as” another quantity. In the second situation, they find both and , where of is .

Students can use any representation or strategy. As they work, monitor for different approaches being used and connections that can be made across representations and strategies. For instance, students using a table, a double number line diagram, and a tape diagram to answer the first question may all start by reasoning about 20% of 5 cups before finding 120%. Note how they go about finding the 20% and the 120% in each case.

Regardless of how they reason, students need to first interpret the quantities and recognize the value that corresponds to 100% in each situation. Where and how 100% can be seen across representations and strategies is another connection worth making.

Some students may bypass using representations. For instance, they may reason about 50% and 80% of a quantity in terms of the fractions and (or ), and multiply or divide the given values by these fractions. To find 150% of Kiran’s water (in the Are You Ready for More?) they may calculate . Consider asking these students to explain how their reasoning could be shown on one of the representations.

Focus the discussion on how students interpreted the quantities in each situation and how they reasoned accordingly. For instance, ask students how they knew which quantity corresponds to 100% in each situation and how they represented it.

At this point, it is not necessary for students to formally conceptualize the two ways that 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.

If time permits, discuss students’ responses to the last question. Select a student who used each representation—a double number line diagram, a table, and a tape diagram—to share their reasoning. As students explain, display those representations for all to see. If any students reasoned without these representations, select one to share their thinking as well. Point out connections across strategies and highlight that the percentages and their corresponding values in each case represent equivalent ratios.