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
This activity requires students to work with larger numbers, which is intended to encourage students to use an equation and notice the efficiencies of doing so. It also emphasizes the interpretation of the constant of proportionality in the context. In this case, the constant represents the cost of a single ticket, and makes it easy to identify which singer would make more money for similar ticket sales in a concert series. Note that asking students to give the revenues for different ticket sales encourages looking for and expressing regularity in repeated reasoning (MP8). The last set of questions ask students to interpret the constant of proportionality as represented in an equation in terms of the context (MP2).
Monitor for students who solve the problems using the following strategies and invite them to share during the whole-class discussion.
writing many calculations, without any organization
creating a table to organize their results
writing an equation to encapsulate repeated reasoning
Launch
Provide access to calculators. Consider using the names of actual performers to make the task more interesting to students.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
A performer expects to sell 5,000 tickets for an upcoming concert. They want to make a total of
$
311,000 in sales from these tickets.
Assuming that all tickets have the same price, what is the price for one ticket?
How much will they make if they sell 7,000 tickets?
How much will they make if they sell 10,000 tickets? 50,000? 120,000? a million?
\(x\)
tickets?
If they make
$
404,300, how many tickets have they sold?
How many tickets will they have to sell to make
$
5,000,000?
Student Response
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Activity Synthesis
Select student responses to be shared with the whole class in discussion. Sequence their explanations from less efficient and organized to more efficient and organized. Discuss how the solutions are the same and different, and the advantages and disadvantages of each method. An important part of this discussion is correspondences and connections between different approaches.
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each explanation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no