Lesson 10

This warm-up prompts students to reason in monetary terms, preparing them for subsequent tasks in the lesson. It also provides insight into students’ understanding of dollars and cents as well as their ability to reason mentally.

Display questions for all to see. Ask students to solve them mentally.

In response to “how many dollars are in one cent,” students might say there are no dollars at all in one cent. Ask them what fraction of a dollar one cent represents.

After students solved all problems mentally, for each problem, ask 1–2 students to share their thinking. Pause between problems to give everyone time to reflect on the shared answers.

In this activity, students learn the definition of a percentage as a rate per 100 and apply this definition in the context of money. They label various coin amounts as percentages of 100 cents or 1 dollar.

Students are likely able to name the values of each coin and their individual percentages (in the first two questions) fairly quickly. Assigning a percentage to a group of coins (in the last two questions) adds complexity and should be the focus of the activity as students may use a variety of strategies. One possible strategy is to reason in terms of ratios. For example, a student may think that if a dime is 10% of a dollar, then 6 dimes is 60% of a dollar. This type of ratio thinking is a robust way for dealing with percent problems and should be encouraged early.

As students work, notice the strategies being used to solve the two problems and identify those with effective approaches so they can share later.

Remind students that previously they have learned that a “rate per 1” tells us the amount of one quantity for 1 of another quantity. Explain that in this task, they will explore “rates per 100.”

Solicit a couple of ideas on what “rates per 100” might mean. Students are likely to suggest a description along the lines of “the amount of something for 100 of something else.” Tell students that a rate per 100 is called a percentage and that they will explore percentages in the context of money. Point out the half-dollar and dollar coins in the task, as some students may not be familiar with them.

Arrange students in groups of 2. Give students 3 minutes of quiet think time to begin work on the task. After that time, ask students to share their responses with a partner and complete the remaining questions together.

Students may notice a pattern particular to this activity—that the percent value is the same as that for cents—and carry that assumption forward and apply it incorrectly to situations in which 100% does not correspond to 100. This conversation is addressed in the Activity Synthesis.

Focus the discussion on the ways students approached the last two questions and on precise use of language and notation (MP6). For example, in the first two problems students can write only a number or matched a coin to a pre-written phrase. In the last two problems, however, expressing a percentage with only a number and without the % symbol should be considered an incomplete answer.

Select students with successful strategies to share their thinking with the class. Display a concise version of their reasoning for all to see. Invite others to express support, disagreement, or questions (MP3).

If no one reasoned about percentages in terms of ratios (e.g., If a quarter is 25% of a dollar, 6 quarters are 150% of a dollar), illustrate it.

Many students may reason by noticing a pattern—that the number of cents in an amount matches its percentage of a dollar (e.g., 60 cents is 60%)—rather by thinking in terms of ratio or scaling. Since the pattern only holds up in the context of percentages of 100 of a quantity, students will need to be prompted to look more closely at the meaning of “rate per 100.” Conclude the discussion by displaying the following double number line with 100 at the 100%:

Point out that we were finding percentages of 100, so in the double number line, we line up 100% and 100 because 100% of 100 is 100.

Previously, students found percentages of 100 cents. In this activity, they reason about percentages of 1 dollar.

One important question to think about here is how students know or decide how the numbers on the double number line diagram should be aligned. Students build on their extensive work on equivalent ratios and double number lines to make sense of percentages and “per 100” reasoning.

Recap that in the previous activity students found percentages of 100 cents. Tell students they will now find percentages of 1 dollar. Draw their attention to the fact that, on the double number line, the 1 dollar and 100% are lined up vertically to reflect this.

Keep students in the same groups. Give students 2–3 minutes of quiet think time, and then ask them to share their responses with their partner. Display and read aloud the following questions. Ask partners to use them to guide their discussion.

• How did each of you arrive at your answers for the first two questions?
• Where do your answers fall on the double number line diagram? How do you know?
• Are your answers the same for the third question? If they are not, can they both be correct? If they are, can you think of another answer that would also be correct?

Based on previous work with labeling number lines less than 1, students may label the tick marks with fractions instead of the decimal value of the coins. This may not be helpful for answering the first two questions, but provides an opportunity to discuss alternative ways to label the number line given the context of the problem. Consider prompting them to write fractional values as cents or to rewrite the cents as dollar values.

Select students who used the provided double number line to share their reasoning. This is an opportunity to refresh students’ number line reasoning. Some students may see the four equally spaced tick marks from 0 to 1 and conclude that each is worth 0.25, or \(\frac14\). Others may fill in the 0.50 first, as it is half of 1, then the 0.25 for half of 0.50, and then use additive thinking to fill in the other tick mark values along the top.

Some students may reason in terms of equivalent ratios and say, for example, that since 100 divided by 4 is 25, then \(1\div4=0.25\) must be 25% of 1. They would then assign the 0.25 value to the first tick mark and use additive thinking to conclude that 0.75 is 75% of a dollar. Ask students who used such an approach to present last to emphasize that the familiar ratio thinking applies to percentage problems as well, even though the % symbol may be unfamiliar. If no students took this approach, illustrate it to make this point.