Surface Area and Volume of Cylinders: A Food Approach
Grade level: Eighth grade Resource Room (group of four students)
LESSON RATIONALE
New York State Learning Standards and Key Ideas
Mathematics, Science, and Technology Learning Standard 1 (Analysis, Inquiry, and
Design): Students will use mathematical analysis, scientific inquiry, and
engineering design, as appropriate, to pose questions, seek answers, and develop
solutions.
Key idea 1: Abstraction and symbolic representation are used to communicate
mathematically.
Key idea 2: Deductive and inductive reasoning are used to reach mathematical conclusions.
Key idea 3: Critical thinking skills are used in the solution of mathematical problems.
Mathematics, Science, and Technology Learning Standard 3 (Mathematics): Students
will understand mathematics and become mathematically confident by communicating and
reasoning mathematically, by applying mathematics in real-world settings, and by solving
problems through the integrated study of number systems, geometry, algebra, data analysis,
probability, and trigonometry.
Key idea 1 (Mathematical Reasoning): Students use mathematical reasoning to analyze
mathematical situations, make conjectures, gather evidence, and construct an
argument.
Key idea 2 (Number and Numeration): Students use number sense and numeration to
develop an understanding of the multiple uses of numbers in
the real world, the use of numbers to communicate mathematically, and the use of numbers
in the development of mathematical ideas.
Key idea 3 (Operations): Students use mathematical operations and relationships
among them to understand mathematics.
Key idea 4 (Modeling/Multiple Representation): Students use mathematical
modeling/multiple representation to provide a means of presenting,
interpreting, communicating, and connecting mathematical information and relationships.
Key idea 5 (Measurement): Students use measurement in both metric and English
measure to provide a major link between the abstractions of mathematics and the real world
in order to describe and compare objects and data.
English Language Arts Learning Standard 1 (Language for Information and Understanding): Students will read, write, listen, and speak for information and understanding.
Key idea 1 (Listening and Reading): Listening and reading to acquire information
and understanding involves collecting data, facts, and ideas; discovering relationships,
concepts, and generalizations; and using knowledge from oral, written, and electronic
sources.
Key idea 2 (Speaking and Writing): Speaking and writing to acquire and
transmit information requires asking probing and clarifying questions, interpreting
information in one’s own words, applying information from one content to another, and
presenting the information and interpretation clearly, concisely, and comprehensibly.
Instructional Objectives
Students will recognize and distinguish the difference(s) between formulas for area and
volume of a cylinder. (comprehension)
Students will use manipulatives to compute the surface area of two cylinders and then
compare the results. (application/evaluation)
Adaptations
Based on the learning styles inventory, manipulatives will benefit hands-on learners while
the activity as a whole addresses the needs of other learners as well.
Numerous short steps will be given orally using clear and concise directions that will ideally hold the students’ attention. Open-ended questions will also be posed, ideally guiding the lesson.
Students with attention deficit-hyperactive disorder are allowed to walk around the room during the lesson if they deem necessary as long as they are still engaged in the lesson at hand and are not distracting other students.
Although reading/writing is not implemented into this lesson, students having a learning disability in reading and writing who choose to write out the answers to any questions (i.e. the independent practice), they may have a scribe or access to a word processor.
Materials
Cylindrical food containers of different sizes will be needed for this activity (i.e.
soda/juice cans, Pringles, and/or candy in tubes/rolls).
Colored pencils, blank white paper, scissors, rulers, tape, and a calculator will also be
necessary for this investigation.
LESSON OPENING
Anticipatory Set
"You identified that you wanted to work on surface area of shapes some more, and
cylinders in specific. Today we’re going to do a mini-lab that will hopefully help
you understand the concept better. To make it even more fun, we’re going to use food
and beverage containers in our experiment, and you get to eat and drink the food
afterwards! Let’s concentrate on the process and procedure though. If you have any
questions during the lesson, please feel free to confer with each other or with
myself."
LESSON BODY
"What types of food comes in cylindrical containers? What do you think I brought for you? What other things in the real world are cylindrical?" Record answers on dry erase board. (check for understanding)
Draw an example of a cylinder on the dry erase board. Then set up a table with three
columns underneath this picture. Have the students offer their
interpretations/understanding of the formula for volume of a cylinder (p
r h) and record these ideas in the left-most column. Follow this up with inquiring about
the surface area of a cylinder ((2*(p r )) + (Ch)) and record
their responses in the right column. Have the students identify the common piece between
the two formulas, and list this in the center column. Then have students
identify/distinguish the smaller components/formulas of each of the above formulas and
record these on the board. (check for understanding)
C = p d A = p r A = Ch
Begin the activity by having the students choose a container. The teacher should use a container that is unique to the experiment (one that is not available to the students) to illustrate the steps. "Watch carefully as I start to show you what you are going to do in this activity. Your end result is going to be a paper version of the cylinder you are working with now." Start by tracing one base of the can on the paper and then cut it out. (modeling) "Now it is your turn to trace the bases of your cylinder and cut them out. When everyone is done, we will continue." Use guided practice as necessary to help the students in this process.
"Now we have the two bases. What do we need next? What is the shape of that piece of paper going to be?" Students respond. "How do you think we will be able to get this rectangle?" Students respond. "Now you get to decide how to measure this shape. Make sure you have a rectangle that is the correct size in both width and length." For students who need help, suggest that they find the height of the cylinder first. Then point out that the length of the strip is unknown at this time; they will have to create a long one that can be trimmed following its fitting. (guided practice)
"Now you should have three pieces of paper: two circles and a rectangle." Have each student in the group record all necessary dimensions and formulas on both another sheet of paper and on each individual shape. Then have them compute the surface area of the cylinder under investigation individually, comparing their answers after each has finished. Then allow them to put the pieces of paper together to make the paper version of the cylinder as a group. (guided practice) Do the same with the example you were working with. (modeling)
Using the dimensions recorded by the students, have each student solve for the surface area in yet another way (i.e. if they solved using the formula that does not use the rectangle’s area, encourage that way and vise versa). (check for understanding)
Have each student use the volume formula to derive the volume of the cylinder that they are involved with, again comparing answers after each has found an answer. (check for understanding)
Again using the dimensions recorded by the students, have the students discuss how the circumference of the circles compares with the length of the rectangular strip and what the length of the rectangular strip becomes when taped together. (reversibility of thought) (check for understanding)
Do a quick "drill and practice" example (from the textbook) verbally with the students, making visual recordings on the board as you progress through the problem. (check for understanding)
Enjoy the food!
CLOSURE
"Through today’s experiment I hope you learned more about how to manipulate a
cylinder to find its surface area and distinguish that from its volume. Basically the
length of the rectangle is approximately the circumference of the circular top or bottom
of the cylinder. You "unroll" the cylinder to find the rectangle. Hopefully you
will now be able to remember all of the parts for the volume and surface area of a
cylinder. Next time you have to use these formulas, I hope you can visualize this activity
and can better understand and explain what is happening!
LESSON FOLLOW-UP
Independent Practice
Answer the following questions either verbally in a cooperative group or on paper
(students decide their preference):
Describe in your own words how to find the
surface area of a cylinder.
Give reasons why the surface area of a cylinder
is an approximation and not exact.
Design two new cylindrical containers, one drawn on paper and the other a paper model such
as today’s activity. Remember to record all dimensions. Find the surface area of both
cylinders and the volume for one cylinder. Compare the answers and discuss how you
proceeded through the problems in your mathematics notebook/journal in a short paragraph.
Suggest that the students may want to design one of these cylinders after a common
cylinder, such as a fruit or soup can. Then they can also design a new label for that can.
Evaluation
What did you want the children to learn?
I wanted the students to realize that a cylinder can be broken down into simpler shapes,
namely two circles and a rectangle. The areas of the parts can then be added together to
find the surface area of the cylinder. One can also ideally see that these smaller parts
"mean something," that is, the length of the rectangle is equal to the
circumference of the circle.
Students should now be able to identify the difference between the volume and surface area
formulas for a cylinder and be able to determine what each letter in the equation stands
for and the different parts of each formula.
How will you know that they learned it?
I will be able to determine the students’ level of understanding by listening to
their discussions and answers to questions asked during the activity as well as observing
their work. Some groups/individuals might be more independent than others. The answers to
the questions posed in the independent activity and the results from their own surface
area work will also help me determine the ability level of each student.
LESSON RESOURCES
References for student use:
Agrawal, P. C. (1995). Mathematics:
Applications and connections course 3. New York: McGraw-Hill.
Instructor and any other
teachers/paraprofessionals.
References for teacher use:
Agrawal, P. C. (1995). Mathematics:
Applications and connections course 3. New York: McGraw-Hill.
Burns, M. (1998). About teaching
mathematics: A K-8 resource. Sausalito, CA: Math Solutions.
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