This activity prompts students to generalize the process of determining what percentage one number is of another number (or finding as a percentage of , where of is ).

Students begin by using familiar strategies to reason about what percentages some numbers are of 20. They record their responses in a table. Then they express the numbers as fractions of 20 and write the decimal equivalents in the same table.

Next, students analyze the relationships between these numbers. They observe that the percentages and decimals are related by a factor of 100: multiplying the decimals by 100 gives the percentages, and dividing the percentages by 100 gives the decimals. Students then think about how to determine the percentage that any number, , is of 20 and write an expression to describe it.

As students work, monitor for the ways in which they complete the last column of the table. Select students who divide each jumping duration by 20 to share later.

As they analyze the values in the table and describe the relationships they observe in general terms, students practice looking for and making use of structure (MP7).

A key takeaway from the discussion is that we can determine what percentage one number is of another number by dividing the former by the latter and multiplying the result by 100.

First, invite students to share how they completed the last column, ending with students who divide time in minutes by 20. If no students found the decimal equivalents by dividing, remind students that a fraction can also be interpreted as division. For example, can be seen as , which is 1.2.

Next, display the completed table for all to see. Ask students the following questions, and record the responses for all to see:

• “What relationship do you see between the percentages and the decimals?” (The percentages are each 100 times the decimals.)
• “Suppose a student jumped for 1 minute. What fraction of the goal is that?” () “How can this fraction help us find what percentage of 20 that is?” (Divide 1 by 20, which gives 0.05, and multiply 0.05 by 100, which gives 5%.)
• “Someone else jumped for minutes. How can we write that amount as a fraction of the goal?” () “How can we find out what percentage of 20 that is?” (Divide by 20 and multiply the result by 100, or calculate .)
• “If the goal is 30 minutes instead of 20, how would we write each jumping time as a fraction of the goal?” (The time would be the numerator and 30 the denominator.) “How would we write each fraction in decimal form?” (Divide each numerator by 30.) “How would we calculate the percentages?” (Multiply either the fractions or the decimals by 100.)
• “Suppose the goal is minutes instead of 20 minutes. How do we calculate what percentage of the goal is 7 minutes?” (Divide 7 by and multiply by 100, or .) “What percentage of the goal is minutes?” ()

This activity gives students a chance to apply the general process for calculating a value as a percentage of another value in a new context. It also offers students another opportunity to look for and make use of structure in the process of describing as a percentage of , allowing them to better make sense of the expression .

Note that at this point it is expected and perfectly acceptable for students to use strategies or representations that may be more familiar but less efficient. Monitor for the different methods being used so that connections can be made between the methods students know well and those that are newer but more efficient. For instance, monitor for students who reason about what percentage 9 is of 75 in the following ways:

• Using a table to find what percentage 3 people or 1 person is of 75, and then find the percentage that corresponds to 9 people.

  number of people	percentage
  75	100
  3	4
  9	12

  number of people	percentage
  75	100
  1	or
  9	12

• Expressing 9 as a fraction of 75, , rewriting it as and , and multiplying by 100 to get 12.
• Dividing 9 by 75 to get 0.12, and then multiplying 0.12 by 100 to get 12.

The goal of this discussion is to highlight connections between various strategies for finding what percentage a number is of 75, allowing students to further make sense of the general method from an earlier activity.

Display 2–3 previously selected strategies for finding what percentage 9 is of 75. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the strategies have in common?” (They either involve dividing 100 by 25 and multiplying the result by 3, which is multiplying 100 by , or involve dividing 100 by 75 and multiplying the result by 9, which is multiplying by .)
• How are they different?” (Some involve only computations and no tables or diagrams. Some ways are quicker than others.)
• “If some strategies involve multiplying 100 by and others involve multiplying 100 by , why are the results the same?” ( is equivalent to .)

To highlight the structure that leads to the generalized expression, consider using tables to illustrate how the same reasoning is applied to all three questions, as shown. Point out that in each case, the calculations involve dividing the occupancy by 75 and multiplying the result by 100.