This activity builds students’ understanding of “percent of ” in a different context, prompting them to revisit their previous idea. Students begin to see the term “percent” as describing a comparison to 100 more broadly. The activity also introduces the percent symbol (%).

First, students see examples of the phrase “percent of a dollar” being used to describe the values of some U.S. coins. They interpret “percent” in terms of the amount of money in cents relative to the 100 cents in a dollar. Then, students write a definition for the term, accounting for the usage seen in this activity.

The goals of the discussion are to elicit new insights on the idea of percentage, clarify the meaning of “percent,” and introduce the symbol for percent.

Invite students to share their responses to the first two sets of questions. Then, direct their attention to the reference created using Collect and Display. Ask students to share their definitions of “percent,” borrowing language from the display as needed. As they respond, update the reference to include additional phrases.

Discuss questions such as:

• “We used ‘percent’ to talk about the value of coins and the area of a shaded region. What is the same about the use of ‘percent’ in the two situations? What is different?”
• “How would you describe the meaning of ‘25 percent of a dollar’ to someone else? What about ‘25 percent of a large square’?”
• “What does ‘100 percent of a dollar’ mean? What about ‘100 percent of a large square’?”
• “How would you define the word ‘percent’ now?”

Clarify that percent means “per 100” or “for every 100.” It is a way to describe a comparison to 100 or the ratio of a quantity to 100. Use examples to illustrate this idea:

• The value of a nickel is 5 percent of a dollar because there are 5 cents in a nickel to 100 cents in a dollar. (The values of a nickel and a dollar are in the ratio of 5 to 100.)
• The value of a dime is 10 percent of a dollar because there are 10 cents in a dime to 100 cents in a dollar. (The values of a dime and a dollar are in the ratio of 10 to 100.)
• Two dollars is 200 percent of a dollar because there are 200 cents in 2 dollars to 100 cents in a dollar. (The ratio of their values is 200 to 100.)
• In Diagram E of the Warm-up, the area of the shaded regions is 108 percent of the area of the large square because their areas in square units are in the ratio of 108 to 100.

Display the symbol “%” for all to see. Tell students that it is used to express “percent.”

In this activity, students develop their understanding of percentages as rates by reasoning with double number line diagrams and relating 100% to 1 dollar.

Prior to this point, students related “100% of a quantity” to 100 units of that quantity (square units of area, or cents in a dollar). The work here makes it more explicit to students that 100% can correspond to other amounts. Students see “1.00” (dollar)—rather than “100” (cents)—being aligned to 100% on a double number line diagram. They use the diagram to find amounts that correspond to some percentages of a dollar as well as to identify given amounts of money as percentages of a dollar.

The reasoning here encourages students to attend to precision (MP6). In working with partially labeled number lines, students need to attend to the units being used and the meaning of the intervals before locating and labeling other relevant values. In expressing a value as a percentage of another value, they need to include “percent” or the % symbol.

Students may more readily see coin values as percentages of 100 cents than as percentages of 1 dollar. To support this transition, ask students: “What percentage of 100 cents is 6 dimes?” Then, ask how this percentage (60%) can be shown on a double number line diagram such as the one given in the first question. If the idea of expressing the coin values in cents instead of dollars doesn’t come up, ask them about it. Consider inviting students to write “100 cents” above “\$1.00” and “50 cents” above “\$0.50,” and so on.

The purpose of this discussion is to reinforce the idea of percentage as a rate and highlight that double number lines are helpful for reasoning about percentages.

Invite students to explain how the diagrams support or show their reasoning about 6 dimes, 6 quarters, and 8 nickels as percentages. If no students mention the ratios of coin values to 1 dollar, consider asking: “What do we know about the ratios of the coins’ values to 1 dollar? How can these ratios help us find percentages of a dollar?”

Emphasize that a percentage is a rate relating a quantity to 100. One dollar is 100% of a dollar (or 100 per 100), so:

• 6 dimes is 60% of a dollar because the ratio of \$0.60 to \$1.00 is 60 to 100.
• 6 quarters is 150% of a dollar because the ratio of \$1.50 to \$1.00 is 150 to 100.
• 8 nickels is 40% of a dollar because the ratio of \$0.40 to \$1.00 is 40 to 100.

On a double number line diagram, these coin values and percentages—\$0.60 and 60%, \$1.50 and 150%, and \$0.40 and 40%—are aligned vertically, just as \$1.00 and 100% are.