In this activity, students continue to reason about percentages where 100% corresponds to a value other than 100 or 1. Blank double number line diagrams are given to support students’ thinking, but because the percentages are multiples of 50, students may find it intuitive to reason without diagrams. For instance, to find the amount that Tyler raised (150% of his goal), they may:

• Add the values of 50% of \$40 and 100% of \$40.
• Reason that 150 is and triple the value of 50% of \$40, or find .
• Reason that 150 is and find .

Monitor for students who use these and other ways of reasoning and consider inviting them to share during discussion.

Focus the discussion on how students found 150% and 200% of \$40. Invite students who reasoned with and without double number line diagrams to share.

If not mentioned in students’ explanations, emphasize that we are finding percentages of \$40, since this number—not 100 cents, 1 dollar, or 10 dollars—is the fundraising goal for the three friends. Since 100% of a goal of \$40 is \$40, the 100% and \$40 are lined up on the double number line diagram. To highlight these correspondences, display and annotate them on the double number line diagrams (as shown in the Student Response).

In this activity, students find percentages of a value in the context of distance. They begin by relating a given value to 100% and reasoning about other percentages. Then students reason in the other direction: given a percentage and a corresponding distance, students determine the number of miles that corresponds to 100%. The context does not explicitly state that the value being sought (the distance that Han’s uncle biked) is the value for 100%, so students need to first make that connection.

Double number line diagrams are provided to support students’ thinking and to encourage them to reason in terms of equivalent ratios, but students may also use other representations and strategies.

Monitor for the different approaches students take for each set of questions. Here are some ways in which students may answer the first set of questions, from less direct to more direct:

• Align 8 miles to 100% on the number line, partition the bottom number line into intervals of 25% (), find the corresponding intervals of 2 () on the top line, and skip-count by that value to find distances for 75% and 125%.
• Reason about 75% and 125% of the distance as 25% less and 25% more than 100% of the distance, which means subtracting from 8 and adding to 8, respectively.
• Reason about 75% of 8 directly by multiplying 8 by and 125% of 8 by multiplying 8 by .

Here are some ways in which they may answer the second question:

• Label the top number line with multiples of 3 and the bottom number line with multiples of 50, and see what distance aligns with 100%.
• Double the 3 to find the distance Han’s uncle biked because 100 is .

The diagrams are partially marked and labeled. As students partition and label the diagrams to represent the quantities and percentages in the situation, they practice reasoning abstractly and quantitatively (MP2) and attending to precision (MP6).

One key idea to highlight is that the value assigned to 100% can vary depending on the comparisons being made and the questions being asked in a situation.

Invite previously selected students to share their strategies for finding 75% and 125% of Elena’s biking distance on Saturday and for finding Han’s uncle biking distance. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “In Elena’s case, how did you know what distance corresponds to 100%?”
• “In Han’s case, how did you know that to find the distance that Han’s uncle biked you were looking for a value that corresponds to 100%?”
• “We saw several ways to find 125% of Elena’s Saturday distance. What is the same about the strategies? What is different?”

This activity offers another opportunity for students to find 100% of quantities given other percentages. As in an earlier activity, students are not told that the values being sought correspond to 100%, so they will need to interpret each situation and make that connection.

Double number line diagrams are provided as a reasoning tool, but students may also use a table of equivalent ratios or other methods. Those who use double number line diagrams are likely to find them effective for the first question (find 100% of a quantity given 20%) but less straightforward for the second question (find 100% of a quantity given 30%). Since 100 is not a multiple of 30, students may partition the double number line into intervals of 10% and scale up from there to find the value of 100%. They may also divide 9 by 3 and then multiply by 10, without using the diagram. (Some may multiply 9 by directly, though this is less likely.)

The last question prompts students to make a quick comparison about the two situations. Students’ responses don’t need to be fully formed for class discussions to begin.

One key idea to emphasize here is that reasoning about percentages is like reasoning about equivalent ratios.

Invite students to share their responses and reasoning. For each question, start with strategies that involve double number lines and follow with strategies that might be more efficient. Keep the diagrams displayed for all to see and to refer to throughout discussion.

Highlight that the reasoning involved here is comparable to finding equivalent ratios in an earlier unit. For Jada’s puppy, we are looking for a ratio of weight in pounds to percentage, that is equivalent to . For Andre’s puppy, we are looking for that is equivalent to .

Point out that the same amount (9 pounds, in this case) can represent different percentages depending on the amount to which it is compared or the value for 100%.

• If the value for 100% is 45, then 9 is 20% of that value.
• If the value for 100% is 30, then 9 is 30% of that value.