standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
Students practice using precise wording (MP6) to describe the positive or negative association between two variables given scatter plots of data.
Launch
Display the scatterplot for all to see. Remind students that we investigated the relationship between car weight and fuel efficiency earlier.
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Ask students how they would describe the relationship between weight and fuel efficiency. After 30 seconds of quiet think time, select 1–2 students to share their responses. (The scatter plot shows that for these cars, as the weight of a car increases, its fuel efficiency decreases.)
Give students 3–5 minutes to construct similar sentences to describe scatter plots of three other data sets followed by a whole-class discussion.
Student Facing
For each scatter plot, decide if there is an association between the two variables, and describe the situation using one of these sentences:
For these data, as ________________ increases, ________________ tends to increase.
For these data, as ________________ increases, ________________ tends to decrease.
For these data, ________________ and ________________ do not appear to be related.
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Student Response
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Anticipated Misconceptions
Students may assume they need to use each sentence exactly one time. Let them know it is acceptable to use a sentence more than once and that it is acceptable to not use a sentence.
Activity Synthesis
The purpose of the discussion is to talk about trends in data based on the representations in scatter plots.
Consider asking some of the following questions:
“Is it surprising that the price decreases as mileage increases? Why doesn’t mileage predict price perfectly?” (There are other factors, like whether the car has any damage or has any extra features.)
“Is it surprising that heavier diamonds cost more? Why doesn’t weight predict price perfectly?” (There are other factors, like the cut, the clarity, etc.)
“Is it surprising that energy consumption goes up as the temperature increases?” (No, because of air conditioning.)
For the last scatter plot, highlight the outliers by asking:
“What might cause more energy consumption on a cool day?” (Laundry, using power tools, etc.)
“What might cause less energy consumption on a hot day?” (Being gone from home, etc.)
Students may notice that the association between high temperature and energy consumed is more variable than the other situations. There is still a positive association or positive trend, but we would describe the association as “weaker.”
Speaking: MLR8 Discussion Supports.
Use this routine to amplify mathematical uses of language to communicate about the relationship between the two quantities represented in each scatter plot. As students describe the trends, press for details by requesting that students challenge an idea, elaborate on an idea, or give an example of their process. Revoice student ideas to model mathematical language use in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making; Optimize output (for explanation)


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