In this activity, students encounter percentage problems that are inconvenient to solve by drawing double number line diagrams or by using benchmark percentages, encouraging them to reason differently and look for a more practical approach.

To find 30% of a value in the first question, students can use a familiar percentage, 10%, as a stepping stone. For instance, they may find 10% or of the value and then multiply the result by 3. The next two questions ask students to find 44% and 102% of some values, respectively, prompting students to find different intermediate steps or use alternative strategies.

Some students may opt to use 4% and 2% as stepping stones to answer the two questions. Others may choose to find the value for 1% in each situation and multiply the result by the targeted percentage. Monitor for students who take this path and select them to share later.

The aim of the discussion is to draw attention to the benefits of using 1% as an intermediate step for finding percentages of a number.

Invite a couple of students to share their work on the first question. Then focus the discussion on the last two questions. Select several students who effectively found 44% and 102% of 250 to share their strategies, saving the strategy involving 1% for last. If no one used this method, ask whether finding 1% of 250 would be a helpful stepping stone for finding each of the three percentages.

Highlight that this method of first finding 1% of a quantity can be applied regardless of the percentage involved, much like finding a unit rate is an effective way to solve any ratio problem.

Consider using a table to illustrate this point. Start with the first two rows showing 100% and 1% of 250. Add a new row to show how each percentage can be found using the value for 1%, ending with a table such as shown:

Students may note that it is not possible to have 2.5 people or a non-whole number of people at an event, which is the case here if the percentage is an odd number. Acknowledge that this is true. Clarify that finding 1% of a quantity is still a valid intermediate step even in situations where only even-numbered percentages make sense.

This activity prompts students to generalize the process of finding any percentage of a number. It makes it explicit that, in general, of a number is times that number.

Students are asked to find 1% of a set of values (1, 40, 100, 3,200, and ) and repeat the process to find 15% and 67% of the same set of values. Through repeated reasoning, students notice regularity in their calculations, enabling them to write expressions for finding of any value (MP8).

To find 1% of a number, students may opt to divide the number by 100, multiply it by , or multiply it by 0.01, all of which are valid. When finding of a number, students who divide by 100 or multiply by a fraction can multiply or by the number, respectively. Students who multiply by a decimal, however, may find it more challenging to write a general expression. See Building on Student Thinking for a way to support these students.

As students calculate percentages of different numerical values and describe the process for finding any percentage, , of any value, , they practice expressing regularity through repeated reasoning (MP8).

The goal of the discussion is to highlight the structure in the calculation to find a percentage of a number.

Invite students to share how they found 1% of the different sales amounts in the first table ($1, $40, $100, and $3,200). Record each unique strategy, and ask the students using that strategy for the expression that they wrote to find 1% of . Record the corresponding expression next to each strategy. For instance:

Next, discuss how students completed the table for finding 15% of the same sales amounts. Record each unique approach and the corresponding expression for finding 15% of . For instance:

Discuss with students:

• If different expressions were written for 1% of : “Are the expressions for finding 1% of equivalent?” (Yes) “How do you know?” (They are all equivalent to times .)
• If different expressions were written for 15% of : “Are the expressions for finding 15% of equivalent?” (Yes) “How do you know?” (They are all equivalent to 15-hundredths times or .)
• “How can we find 67% of dollars?” (Use the same strategies for finding 15% but with 67 in place of 15. Multiply 67-hundredths by .) “What expressions can we write?” ( or equivalent)
• “How can we find or any percentage of dollars?” (Multiply -hundredths by or calculate or .)