options: Visualize relationships between two-dimensional and three-dimensional objects.

Apply trigonometry to general triangles.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Prove geometric theorems.

Understand similarity in terms of similarity transformations.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Experiment with transformations in the plane.

Classify two-dimensional figures into categories based on their properties.

Understand and apply the Pythagorean Theorem.

Reason with shapes and their attributes.

Understand and apply theorems about circles.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Translate between the geometric description and the equation for a conic section.

Solve real-world and mathematical problems involving area, surface area, and volume.

Explain volume formulas and use them to solve problems.

Apply geometric concepts in modeling situations.

Use coordinates to prove simple geometric theorems algebraically.

Find arc lengths and areas of sectors of circles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Prove theorems involving similarity.

Make geometric constructions.

Define trigonometric ratios and solve problems involving right triangles.

Understand congruence in terms of rigid motions.

Analyze, compare, create, and compose shapes.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
problem_activity: Activity
The purpose of this activity is for students to reason about the volume of popcorn that a cone- and cylinder-shaped popcorn cup holds and the price they pay for it. Students start by picking the popcorn container they would buy (without doing any calculations) and then work with a partner to answer the question of which container is a better value. This activity is designed for the discussion to happen around a few different concepts:
The volume is a lower estimate because there is still some popcorn coming out the top of the containers.
The fact that one container (the cone) looks like it has more coming out of the top might sway students to think that the cone is a better deal. However, the difference in the volume amounts should support the fact that even with a taller and wider diameter the cone still holds less volume than the cylinder.
Why would the movie theater purposefully sell a container that has less volume for more price?
Identify groups who see these connections while they are working on the task, and ask them to share their arguments during the discussion.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time to decide which popcorn container they would purchase. Students are not asked to use any written calculations to determine this answer. Poll the class and display the results for all to see which containers students chose. Keep this information displayed to refer back to during the discussion.
Give students 2–3 minutes of time to work with their partner to determine which container is a better value. Follow with a whole-class discussion.
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “First, I _____ because. . .,” “How did you get. . .?” and “Can you say more about . . .?”
Supports accessibility for: Language; Organization
Writing, Conversing: MLR5 Co-Craft Questions.
Display only the opening statement (i.e., “A movie theater offers two containers.”) and the two images of the popcorn containers. Ask pairs of students to write possible questions that could be answered with mathematics. Invite 2–3 groups to share their questions with the class. Look for questions that ask students to compare the volumes of the two different containers. Next, reveal the question of the activity. This routine helps students consider the context of this problem and increases awareness about language used to talk about volume of cones and cylinders.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
A movie theater offers two containers:
[IMG]
Which container is the better value? Use 3.14 as an approximation for
\(\pi\)
.
Student Response
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Anticipated Misconceptions
Students may try to use just the image to reason about which is a better deal. Make sure they understand that the image of the two containers is not to scale.
Activity Synthesis
Poll the class again for which container they would purchase. Display the new results next to the original results for all to see during the discussion.
Select previously identified groups to share their arguments for which container has a better value. Encourage groups to share details of their calculations and record these for all to see in order to mark any similarities and differences between the groups’ arguments. Consider asking some of the following questions to help students think deeper about the situation:
“Do you think your volume calculations overestimate or underestimate the amount of popcorn each container can hold?” (I think the calculations underestimate because the popcorn piles higher than the lip of the container.)
“Why do you think movie theaters charged more for the cone?” (It may look like it has more volume to some people since it has a larger diameter and height. It may be easier to place into a cup holder.)
“Do you think a lot of people would buy the cone over the cylinder?” (Yes. The 50 cent difference is not a lot, and since it is taller and wider, people might think the cone is bigger. Although the cylinder is a better value, there may be other considerations like how easy the cup is to hold or put in the cup holder in the seat, or whether you have time in line to calculate the value.)


correct_option_index: 20