standard_description: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
problem_activity: Activity
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.
Launch
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?
Student Facing
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A
$
1 coin is worth 100% of the value of a dollar. Here is a double number line that shows this.
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The coins in Jada’s pocket are worth 75% of a dollar. How much are they worth (in dollars)?
The coins in Diego’s pocket are worth 150% of a dollar. How much are they worth (in dollars)?
Elena has 3 quarters and 5 dimes. What percentage of a dollar does she have?
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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.
Speaking, Listening: MLR7 Compare and Connect.
As students prepare a visual display of how they made sense of the problem, look for students who labeled the tick marks on the double number line with fractions or cents instead of dollar values. This may result in answers such as
\(¾\)
, or 75 cents is 75% of a dollar rather than 0.75 is 75% of a dollar. Although 75 cents is 75% of a dollar, the number line should be labeled with the decimal value of the coins in dollars. As students investigate each other's work, ask students to share what worked or did not work well in the way they labeled the double number line. Is there a particular advantage to using decimals instead of fractions to label the double number line? Emphasize that although there are several ways to label the double number line given the context of the problem, certain methods are more helpful for answering the question. This will foster students’ meta-awareness and support constructive conversations as they compare the various ways to label a double number line given a context.
Design Principles(s): Cultivate conversation; Maximize meta-awareness


aligns: no