standard_description: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
problem_activity: Warm-up
The purpose of this warm-up is to assess students’ understanding of the volume of a cylinder. Students learned that the volume of either a cylinder or prism is found by multiplying the area of the base by its height. In this warm-up, students are given information to find the area of a cylinder’s base, but they are not given the height. Students propose a volume for the cylinder and explain why it works. Since the diameter of the base is 8, the area of the base is
\(16\pi\)
.
If students have trouble getting started, ask them:
“Do you have enough information to calculate the area of the base?”
“What is the radius?”
Identify students who use these strategies:
find the area of the base first then set up the equation
\(V=16\pi h\)
.
choose a specific value for
\(h\)
then solve for the volume.
Launch
Arrange students in groups of 2. Tell students that in a previous lesson, they learned how to find the volume of a cylinder if they know the cylinder’s radius and height. Draw their attention to where volume formulas are displayed in the classroom as the unit progresses. Give students 1–2 minutes of quiet work time followed by time to explain their reasoning to their partner. Follow this with a whole-class discussion.
Student Facing
What is a possible volume for this cylinder if the diameter is 8 cm? Explain your reasoning.
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Student Response
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Activity Synthesis
The goal of this discussion is for students to communicate how the height of a cylinder is related to its volume. Invite students to share their solutions and their reasoning. Record and display the dimensions and volumes of cylinders that correspond to solutions given by students.


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