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.
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? 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
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
V = πr2h
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.
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
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
1. 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.]
2. What is the difference between a rectangular prism and a cylinder?
[Rectangular prisms have rectangular bases and cylinders have circular bases.]
3. 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.]
4. 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/(2π).]
5. 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.]
- 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.