Illuminations: Popcorn, Anyone?

# Popcorn, Anyone?

 This lesson can be used for students to discover the relationship between dimension and volume. Students create two rectangular prisms and two cylinders to determine which holds more popcorn. Students then justify their observation by analyzing the formulas and identifying the dimension(s) with the largest impact on the volume.

### Learning Objectives

 Students will: Perform an experiment based on a conjecture Create objects with varying volumes from sheets of paper Compare the volume of similar shaped objects Compare the volume of different shaped objects Discover which dimensions have the largest impact on volume

### Materials

 8.5×11 in. white paper 8.5×11 in. colored paper Popcorn Paper Plates Cups Tape Rulers Popcorn Prisms Anyone? Activity Sheet Popcorn Cylinders Anyone? Activity Sheet Comparing Cylinders Activity Sheet Popcorn Prisms Anyone? Answer Key Popcorn Cylinders Anyone? Answer Key

### Instructional Plan

The goal of this lesson is to have students construct objects and determine the resulting volume. This lesson moves the student from a familiar environment where they substitute values into formulas, into an experiment based on their own conjectures. This activity is based on two shapes, rectangles and cylinders. If time is limited, either part of the activity can be done independently. However, completing all parts of the activity strengthens the connections among shape, dimension, volume, and formula. Consider the following when choosing which parts of the lesson to complete:
• Rectangular prisms are easier for students to measure.
• Students develop an understanding of how the volume formula changes for a square prism (versus a rectangular prism) and the impact on the volume calculation.
• Cylinders are easy to build but it is difficult for many students to understand relationship between the radius and circumference of the circular base and the dimensions of the paper used to create it.
• If both activities are completed, students can transfer the knowledge that a squared dimension has a larger impact on the volume from square prisms to cylinders.

Rectangular Prisms

Students should work in pairs because they will work together to create the objects and in filling the objects with popcorn. Pass out the Popcorn Prisms Anyone?Popcorn Prisms Anyone? activity sheet, a piece of white paper, a piece of colored paper, tape, and a ruler to each pair of students. It is helpful to spend time showing students some model rectangular prisms, reviewing the volume formula, demonstrating the prism construction. Popcorn should not be used for the demonstration, but students should be able to see how the prisms fit inside of each other.

Circulate around the room as students work through the activity sheet. After students finish Question 2, hand each group a bowl of popcorn and a cup for transferring the popcorn. Suggest to students that one hold the rectangular prism as the other fills the tall prism without spilling the popcorn into the shorter one. If availability allows, watch students during this part of the activity to see their reactions.

Question 6 may be difficult for some students. You may choose to guide students by asking them the dimensions of their rectangular prisms. After students conclude that the bases are squares, ask for the formula for finding the area of a square. Students should be able to transfer this knowledge to the volume formula. Ask more advanced students how to relate the side of the rectangular prism to the side of the rectangular piece of paper use to form the prism and create a formula for volume based on this. They should find:

V = (w/4)2 · l
where l and w are the length and width, respectively, of the original rectangular paper

Question 7 can be used as enrichment for students who finish early. Have tactile learners use their original rectangular prisms to determine the length and width by changing the dimension. Encourage students to play with the numbers and explain their methods for solving the problem. At the conclusion of the activity, model the algebraic solution if no students found one.

The beginning of the cylinder activity should closely mimic the prism activity. Distribute the same materials and the Popcorn Cylinders Anyone? activity sheet. Again, model the cylinders and have students follow the same steps as in the rectangular prism activity. Show students how to measure the diameter, stressing it is only an estimate, and the lesson should run smoothly.

Students may struggle with Question 6. Direct them back to the prisms activity. The example in Question 6 is very important for helping students see concrete examples before tackling the remaining questions. If they copy the answer from the prism activity, ask them why they can substitute radius for side-squared. Once most groups have completed the activity, you should write the following formulas on the board:

V = r2h
V = w2h

Provide initial values for the radius and the height and ask students how the volume changes as you increase each by one unit. Duplicate the activity for the volume of a square prism. This is a good place to reinforce what the patterns implied with the activities. For enrichment, provide models of square prisms and ask students to compute the change in volume as the sides and height are increased.

Comparing Cylinders

If time allows, the Comparing Cylinders activity sheet is available to help students understand the concept of calculating radius given circumference and that the circumference of the popcorn cylinder was formed from the side of the rectanglar paper. Have materials available for students who want to recreate the cylinders.

It is suggested that Questions 1–9 be instructor-led with student input. Select students in different groups to help with the answers and question the students as they build the cylinders. The student pairs should be able to complete Questions 10 and 11 based on the prior questions.

To bring closure to this activity, a class discussion of the results is important. Questions for Students can be posted on the board and groups walk around and add their comments for the class discussion. During the discussion, encourage both concrete examples and algebraic reasoning.

### Questions for Students

 If you were buying popcorn at the movie theater and wanted the most popcorn, what type of container would you look for? [Answers may vary before the activity. After, it should be clear that the shorter containers are the better choice.] What is the difference between a rectangular prism and a cylinder? [Rectangular prisms have rectangular bases and cylinders have circular bases.] How can you determine the circumference for a cylinder is you are constructing it from rectangular paper? [The circumference will be formed with a side of the rectangular paper that you roll.] How can you calculate the radius of a cylinder with only the measurement for circumference? [Solve for the radius in the circumference formula: r = C/2] Why did the "squared" dimensions have a bigger impact of the volume? [Squaring a dimension is almost like counting it twice in volume. For example, if the side length of the square base in square prism is 5 and the height is 10, then the volume is 5×5×10. The 5 is multiplied twice and the height is multiplied once. A similar example can be made for the radius and height of a cylinder.]

### Assessment Options

 Provide students different size paper and have them compare the 2 volumes for a square prism and the corresponding cylinders. Ask students to write a journal entry comparing the shapes, dimensions, and volumes of all the containers created during the lesson. What was the most surprising thing they learned?

### Extensions

 Students can create rectangular prisms where the paper is not folded equally. The diagram below provides an example. Students can compare the volume of the rectangular prism to the square prism made during the lesson. Students can create triangular prisms and look for volume impacts based on rotating the side used for the base. Have students try to find the maximum volume for both the prism and cylinder. Since there is no top and bottom to the shapes, there is no maximum. With a top or bottom, however, there is a maximum. Have students experiment with the dimensions while keeping the original paper size constant and explain the situation using physical and mathematical models. Note: The algebraic explanation for the investigation is advanced.

### Teacher Reflection

 How did your lesson address auditory, tactile and visual learning styles? How did the students demonstrate understanding of the materials presented? Did students make the connection between a circumference and the side of a rectangle? Did students make the understand how folding or rolling the rectangular paper in different ways led to different volumes? How could you have brought out this connection more? What were some of the ways that the students illustrated that they were actively engaged in the learning process? Did seeing the outer containers partially full help reinforce retention for this activity.

### NCTM Standards and Expectations

 Geometry 9-12Analyze properties and determine attributes of two- and three-dimensional objects. Visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections. Measurement 6-8Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders. Solve problems involving scale factors, using ratio and proportion. Measurement 9-12Understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders.
 This lesson was prepared by Jamie Chaikin as part of the Illuminations Summer Institute.

2 periods

### NCTM Resources

 More and Better Mathematics for All Students
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