Summary
These activities ask students to find the surface area and volume of a cylinder.
Materials
Invitation to Learn
Instructional Procedures
Additional Resources
Books
Ancient Egyptians and Their Neighbors, An Activity Guide, by Marian Broida; ISBN 1-55652-
360-2
An Ancient Greek Temple, by John Malam; ISBN 1904194680
Cubes, Cones, Cylinders, & Spheres, by Tana Hoban; ISBN 0-688-15326-7
How To Draw What You See, by Rudy de Reyna; ISBN 0-8230-2375-3
Background for Teachers
To find the surface area and volume of a cylinder, students must
first be able to look at the parts of this geometric solid. A cylinder
has two circular faces known as bases that are connected by a curved
surface known as the lateral surface. The two circular bases are
parallel and have the same area.
The volume of the cylinder is the amount of space inside the
cylinder. To find the volume, students will first need to know the area
of the circular base. The formula for the area of the circle is the same
as that used in surface area or π r2. They next must know the height
of the cylinder. To find the volume, they must multiply the area of the
circle by the height of the cylinder. The formula is V = Bh where B =
the area of the circular base or π r2 and h = the height of the cylinder.
Thus, the final formula is V = π r2h. Volume is measured in cubic
units.
It is helpful to consider the net of the cylinder to see how surface
area is determined. When a cylinder is taken apart and looked at as
a net, there are two congruent circles and a rectangle. The surface
area includes the sum of the areas of each circle (the bases) and the
rectangle (the lateral surface). Thus, students must have previously
learned how to find the areas of circles and rectangles. The formula for
the area of a circle is π r2. The formula for the area of a rectangle is bh.
When looking at the net, students will need to see that the dimensions of the rectangle are equal to the circumference of the circle and the
height of the cylinder. The formula is developed as follows: S = 2πr2 +
2πrh where S is the surface area, r is the radius of the base, and h is the
height of the cylinder. Surface area is measured in square units.
Intended Learning Outcomes
1. Develop a positive learning attitude toward mathematics.
2. Become effective problem solvers by selecting appropriate methods,
employing a variety of strategies, and exploring alternative approaches to
solve problems.
3. Reason logically, using inductive and deductive strategies and justify
conclusions.
Instructional Procedures
Invitation to Learn
Hold up a cylinder. Ask students to identify the geometric solid.
Review the parts of a cylinder with the students, focusing on the
two circular faces called bases and the curved surface. Note that
the two circular bases are parallel and congruent. Point out the
circumference of the circular bases and the height of the cylinder.
Tell students we are going to play a game with some cylinders.
Each student will need a copy of the handout Which Is Larger?
on which to mark their answers. The teacher will need to have a
collection of ten cylindrical objects hidden from student view in a
tub. Have each cylinder numbered one through ten. One at a time,
hold up a cylinder and have students predict through their powers
of observation which is larger, the height of the cylinder or the
circumference of the base. After students have recorded their estimate
for all ten objects, it is time to check their answers. One at a time,
measure the height and circumference of each cylinder in front of the
group. A possible discovery is that the circumference is often larger
than the height. It is a common misconception that the reverse is true.
Instructional Procedures
(NOTE: The activities outlined in Instructional Procedures are
intended to be taught sequentially. They will take several lessons/
days to complete with students.)
Surface Area
- Give each student a roll of Lifesavers® candy. Discuss with
students what the package/label of this cylindrically-shaped
candy would look like if it were opened up and laid out flat.
This would be called a net. Have students carefully open up
the package and lay out the net. Discuss with students what
the net of a closed cylinder (top and bottom included) looks
like. Have students attach the wrapper/net of the cylinder in
their journals and write about its parts.
- Provide students with a copy of Net of Cylinder. Cut out the
net and construct the cylinder, but do not tape it together.
It is important for students to see how the construction
works and how to lay it flat to see the components. Students
may need to put the cylinder together and lay it flat several
times as they are learning. Note how the circumference
of the circle is actually the base of the rectangle. Write
circumference along the base of the rectangle. Next, point
out that the height of the cylinder is also the height of the
rectangle. Write height along the height of the rectangle.
This net can also be saved and placed in students journals for
future reference.
- Tell students that knowing the parts of a cylinder can help
to find its surface area. Have a discussion on the definition
of surface area. It is the wrapper or skin of the cylinder
because it includes the top, bottom, and curved side. Discuss
why people might want to find the surface area of a cylinder.
A possible answer might be to know how much tin is needed
to make a tin can in order to calculate the material cost of
production.
- Give each student a copy of Surface Area of Cylinder. Ask
students to determine the area of the cylinder by counting the
squares. For the rectangle, students can multiply base times
height. Have students record their answer for the rectangle on
the net. For the circles, have students count the whole squares
and then estimate the partial squares by putting them together
to create whole squares. The answers will not be perfect, but
should be close. Have students record their answers for each
circle on the net. Have students add together the area of the
rectangle plus the area of each circle to find the total surface
area. They may record their answers on the handout.
- Discuss whether counting squares is the most efficient
method for finding the surface area of a cylinder. Make sure
students see that the surface area is 2 x area of circle + area of
rectangle.
- Guide students to use the formula for each part of the cylinder.
First, remember that the top and bottom bases are both
congruent circles. Review how to find the area of a circle.
The formula for this is π r2. Some students may present this
as π x radius x radius. Find the area of a circle on the Surface
Area of Cylinder net together as a class using the formula and
compare it to the students estimates by counting squares.
Remind students that there are two circles, so the first part of
the formula for finding surface area is 2πr2. Next, find the area
of the rectangle. One edge of the rectangle is actually equal to
the circumference of the circle which is2πr. The other side is
equal to the height of the cylinder. Thus, the formula for the
rectangle is 2πrh. Calculate the area of the rectangle together
as a class using the formula and compare it to the students
estimates by counting squares. Help students to put together
the final formula. The final formula is S = 2πr2 + 2πrh where S
is the total surface area, r is the radius, and h is the height. All
final answers on surface area are measured in square units.
- Have students complete the handout Surface Area Patterns. Set
1 has cylinders that double in radius and height each time.
The pattern students should discover is that the surface area
quadruples each time. Set 2 has cylinders that triple in radius
and height each time. The surface area is nine times larger in
this pattern. Students must then determine the surface area of a
fourth cylinder in each set.
Volume
- Have students work in cooperative groups of four students
to complete this activity. Give each group two sheets of
cardstock paper of the same size. For example, one group will
have two sheets of 8 1/2" x 11", another group will have two
sheets of 8" x 9", another group will have two sheets of 6 1⁄2"
x 10", and so forth. Roll the first sheet into a tall, thin open
cylinder (has no top or bottom) and tape the sides together
with no overlap or gap. Fill the cylinder with popcorn or other
filler. Roll the second sheet the opposite way and tape it into
a shorter, wider open cylinder with no overlap or gap. Place
it around the first cylinder. Have students predict in their
journals if it will have the same volume of the filled cylinder.
Slowly, lift and remove the tall cylinder, allowing the filler
to go into the second cylinder. Students will find that the
contents of the first cylinder do not completely fill the second
cylinder. By completing this activity, students will be able to
determine which cylinder has the larger volume. They should
also discover that cylinders with the same lateral surface area
(rectangle) do not always have the same volume. The size of
the circular base affects the total surface area and, consequently,
affects the volume of the cylinder. Students should discover
that the shorter cylinder of the two has a larger volume.
Discuss each groups findings as a class, and record the findings
on an overhead transparency of Volume Discoveries. Have
students write about their conclusions from this activity in their
journals, including sketches as needed.
- Explain to students that knowing the parts of a cylinder can
help to find its volume. Remind students that the volume of
the cylinder is the amount of space inside the cylinder. Show
the students a rectangular prism. Ask them to remember how
to find the volume of a rectangular prism. Have students
discuss in groups how they might be able to find the volume
of a cylinder. Have students share their ideas. If it has not
been brought up in discussion already, remind students that
the formula for volume is base times height or V = Bh. In this
case, the base is a circle so students will need to review how to
find the area of a circle. The formula for the area of the circle
is π r2. To find the volume, students must multiply the base by
the height. Substituting π r2 in the formula for B, the formula
is thus V = π r2h. All final answers on volume are measured in
cubic units.
- Have students answer the following question in their journals:
Which produces a greater effect on the volume of a cylinder
changing the radius or changing the height? After sharing
students ideas, have them complete Effects on Volume handout
to justify their conclusions. Students should be able to use
calculators to complete the computational work.
- Explain the steps to the activity Will You Be Wet or Dry? to the
class. Have each cooperative group of four students select a
can from the teachers collection. Students will need to find the
measurements of the can and fill out the data required on the
handout Will You Be Wet or Dry? as a group. After the volume
is calculated, groups will take turns presenting their data to the
teacher and other class members. The teacher will then fill the
can with the volume of water determined by the group over a
team members head. If the calculation is too high, the water
will overflow onto the students head. If the calculation is too
low, then all of the water will get poured on their head. An
inexpensive rain poncho should be available for students to use
if they would like. Be ready for some fun and have the towels
handy.
Extensions
Curriculum Extensions/Adaptations/
Integration
- Have students look for cylinders in their environment and make
a list of their findings in their math journals.
- Have students compare the parts of cylinders to the parts of
cones and pyramids.
- Have students learn how to draw cylinders and other geometric
shapes.
- Have students learn how the Mesopotamians used the
cylindrical shape to invent cylinder seals, a method of marking
property and signing documents in ancient times. Have
students make cylinder seals from clay and put them on strings
to wear as necklaces as did the Mesopotamians.
- Have students learn how the ancient Greeks used the cylindrical
shape to design columns as part of their architecture. Have
students work in groups of five. Give each member of the group one sheet of 8 1⁄2 x 11 paper. Tell students that their
task is to create cylinders of any size to support a 12 x 12
whiteboard that will balance as many textbooks as possible on
top. Tell students you have also heard that it is possible to
balance a person on the whiteboard placed on paper cylinders
instead of textbooks. Students may use scissors to cut their
paper, and they may use scotch tape to hold the cylinders
closed.
- Have students explore volume and surface area of other
geometric solids.
Family Connections
- Have family members look for cylinder shapes at home. Share
how cylinders are used.
- Have students look for cylindrical food cans. Compare the
volumes listed on the labels. Explain to family members why
certain cans have a greater volume.
Assessment Plan
- Have students design a net of a cylindrical can for a new brand
of peanut butter. After students have completed the writing and
design work, they can put the cylinder together. Have students
compute the surface area and volume of their can.
- Have students place three different cylinders in order from least
to greatest volume using estimation. Have students calculate
the actual volumes of each cylinder to check their work.
- Have students correct the Surface Area Patterns handout for an
assessment grade.
- Have students correct the Effects on Volume handout for an
assessment grade.
Bibliography
Research Basis
Pierce, Rebecca L., & Adams, Cheryll M. Tiered lessons. Gifted Child Today, Spring 2004,
Vol. 27, Number 2, p58-65.
Based on tenets of differentiated instruction supported by the
NCTM, the authors of this article define tiered lessons and outline
eight steps to designing them. The three main ways to differentiate a
lesson are guided by students readiness, interest, or learning profile.
Grouping for differentiated instruction is designed to be flexible from
one lesson to the next.
Pugalee, David K. Writing, mathematics, and metacognition: looking for connections
through students work in mathematical problem solving. School Science and
Mathematics, May 2001, Vol. 101, Number 5, p236-245.
This study looked for evidence of a metacognitive framework
based on students writing about mathematical problem-solving
processes. Students writing was analyzed from the introduction of
a topic through the execution of problems on the same topic. The
findings proved that a metacognitive framework is established through
the process of writing. Furthermore, the author emphasizes the
importance of writing as an integral part of mathematics curriculum.