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.
Cylinders
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.
