Revised 2/2/05
Lesson: Potato Chip Can Geometry
Grades 7 and 8
Topic: Deriving the Surface Area Formula of a Cylinder
Skills to Review: (a) area of rectangle; (b) area of circle (c) circumference of circle.
Lesson Components
Lesson Plan Outline
Student Log
SOLVE-IT: Real-world Surface Area Problems
Practice Short-Response Assessment Problem
Notes for the Teacher (with answer keys)
Blank SOLVE-IT: Real World Problems
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Lesson Plan Outline for Potato Chip Can Geometry
Florida DOE Sunshine State Standards
Strand B: Measurement, Strand C: Geometry and Spatial Sense
Grade 7
Benchmark MA.B.1.3.1: The student uses concrete and graphic models to derive
formulas for finding perimeter, area, surface area, circumference, and volume of two- and
three-dimensional shapes, including rectangular solids and cylinders.
Grade Level Expectations (GLE) for Mathematics Grade 7: 2. uses concrete or
graphic models to create formulas for finding surface area of prisms and cylinders.
Grade 7, Benchmark MA.C.1.3.1: The student understands the basic properties of, and
relationships pertaining to, regular and irregular geometric shapes in two-and threedimensions. 6. knows the properties of two-and three-dimensional shapes.
Grade Level Expectations (GLE) for Mathematics Grade 7; 5. knows the attributes of
and draws three-dimensional figures (pyramid, cone, sphere, hemisphere).
Grade 8
Benchmark MA.B.1.3.1: The student uses concrete and graphic models to derive
formulas for finding perimeter, area, surface area, circumference, and volume of two- and
three-dimensional shapes, including rectangular solids and cylinders.
Grade Level Expectations (GLE): 1. uses concrete and graphic models to explore and
derive formulas for surface area and volume of three-dimensional regular shapes,
including pyramids, prisms, and cones. 2. solves and explains real-world problems
involving surface area and volume of three-dimensional shapes.
Benchmark MA.C.1.3.1: The student understands the basic properties of, and
relationships pertaining to, regular and irregular geometric shapes in two-and threedimensions.
Grade Level Expectations (GLE); 3. draws and builds figures from various
perspectives (for example, flat patterns, isometric drawings, nets); 4. knows the
properties of two-and three-dimensional figures.
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Suggested Timeline
(Based on 45-minute classes)
Math Content
Day 1
Derive Formula for Surface Area of Cylinder
Derive formula S.A. = 2πrh + 2πr2 using potato chip cans
SOLVE-IT: Real World Surface Area Problems
Days
2& 3
Apply surface area formula to real-world cylinder problems
(Problems are challenging. Provide many opportunities for
students to re-work problems without penalty.)
Anne C. Patterson 2004
Learning
Mode
Small group;
Hands-on;
Demonstration
Pair /Individual
Pencil-paper
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Class _______ Date _____________Name__________________________________
Student Log for “Potato Chip Can Geometry”
Your Mission: Follow the directions below. Use only the materials your teacher gives
your group to answer the question:
What is the formula for the surface area of a cylinder and how did you figure it out?
Materials
One (1) paper-covered potato chip can with lids on the top and bottom of can
Calculators
Pencils and paper
Overhead transparencies and pens
Step 1: Carefully remove all the wrapping paper from the potato chip can. In the space
below, draw smaller versions of the three (3) figures formed by the wrapping paper.
Label the dimensions on all three shapes, using symbols: π (pi), h (height of the can), r
(radius of a circle). Write in all the area formulas that you know for the shapes you drew.
Were you able to label the longer side of the rectangle? ________
will give you a hint.
If not, your teacher
Step 2: Next, look and listen while your teacher gives a short demonstration to help you
derive the final formula for the surface area of a cylinder. When your group agrees upon
a formula, write it below.
S.A. = _____________________
Step 3: In the space below, explain in writing below how your group discovered the
formula.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Class _______ Date _____________Name(s)__________________________________
__________________________________
__________________________________
SOLVE-IT: Real World Surface Area Problems
Set aside one activity sheet for recording your group’s final answers.
Raise your hands when your group answer sheet has been completed.
All answers must be correct to win.
All group members must be able to explain solutions to all problems.
Use 3.14 for π in all problems below.
1. Find the amount of cardboard, in cm2,
needed to construct a cylinder-shaped salt
container if the radius of the base is 4 cm and
its height is 10 cm.
_________________
2. A can of paint covers 50m2. How many cans
does Pat need to buy to paint the top, bottom
and outside of a cylinder-shaped water tank
that has a height 5m and has a radius of 2 m?
____________
3. Gaston is designing a black leather case to
totally cover his cylindrical drum that is 26
inches high and has a 14 inch radius. His
brother is designing a leather case for his drum
that is also has a 14 inch radius but is 28 inches
high. If both cases are to fit very snuggly, how
much less leather, in square inches, will Gaston
use in his new drum case than his brother will
use? ________________
4. A cylinder-shaped can of paint lost its label,
and Jolene needs to make a replacement label.
The height of the can is 20 cm. The radius is 8
cm. An uncut label was printed that measures 8
cm wide and 60 cm long. If Jolene needs 2.75
cm in extra length for “overlap”, what is the
length, in centimeters, of the strip of paper that
she cuts away from the uncut label?
____________
Anne C. Patterson 2004
5. A p-nut butter jar, in the shape of a cylinder,
has a height of 6.5 in. and a radius of 2 in.
There is no overlap of the label. If the top of
the label around the jar is located 1.5 in. below
the edge of the jar, what is the area of the label
in square inches (in2 )?
___________________
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Class _______ Date _____________Name(s)__________________________________
Practice Short-Response Assessment Problem
Problem: The formula for the surface area of a cylinder is S.A. = 2πrh + 2πr2. A
restaurant supply manufacturer plans to produce boxes of drinking straws that hold 100
straws each. She needs to know how much plastic is needed to produce one box of
straws. She decided that the formula above was not appropriate to determine the surface
area of a straw, so she changed it.
Part A:
What formula should she use to find out how much plastic to use in one (1) straw?
_______________________.
Part B:
Explain in writing how and WHY she changed the formula listed in the problem above
and why it did not make sense to use it the way it was.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Notes for the Teacher
Lesson: Potato Chip Can Geometry
Grades 7 and 8
Topic: Deriving the Surface Area Formula of a Cylinder
Skills to Review:
(a) area of rectangle, textbook pages: __________________________________;
(b) area of circle; textbook pages: _____________________________________;
(c) circumference of circle; textbook pages: _____________________________.
Your Students’ Mission: Using paper-covered potato chip cans (cylinders) with a papercovered lid on the top and bottom of each can, students will discover and justify the
formula for the surface area of a cylinder—with hints and coaching from you.
S.A. = 2πrh + 2πr2
Materials
Empty cylinder-shaped potato chip cans with a lid snapped on the top AND bottom of each can
(1 or 2 cans per group and 1 for you to use)
Colorful, generic wrapping paper
Masking tape (1 roll)
Double-stick tape (1 roll)
Scissors (1)
Marker (1)
Optional: A few full cans of potato chips for a reward. This formula is challenging!
Advance Preparation
Cut wrapping paper to fit exactly around each can (its lateral area). Use a black marker to write
“h” directly on the shorter side of the rectangular pieces of paper. Secure paper around can with 3
small pieces of masking tape on outside of wrapping paper.
Cut circles from wrapping paper to fit on lids (2 per can). Use a black marker to draw a “radius
segment” marked r on each circle. Stick paper circles to top of lids with double stick tape.
Run copies of the Student Log (1 per student).
Run copies of “SOLVE-IT: Real-world Surface Area Problems” (1 per student).
Select interesting real-world problems on surface area of cylinders from student textbook and
enter them on the attached blank template, “SOLVE-IT: Real-world Problems”. Run 1 per group
Run copies of the Practice Extended Response Assessment Problem. Make a scoring rubric to let
your students know what is expected of them.
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
DAY 1
Delivering the Lesson; Deriving Formulas
Distribute prepared cylinders to groups.
Discuss with students the directions printed on their Student Logs and start them working. Do not
offer suggestions—yet.
Circulate as students work. Listen carefully and notice which students emerge as leaders—not
always “A” students.
After a few minutes, stop students and give them the following hint through demonstration.
Wrap and unwrap the wrapping paper from the can with the circular top facing them. Help them
discover that the long dimension of the label is the same as the circumference of the lid (C = 2πr).
Hopefully, “light bulbs” will go on and they will realize that the long side of the rectangular label
wraps around the circular-shaped opening of the cylinder, so the length of the label is the same as the
circumference of the circle. This measurement must then be multiplied by the short side of the
rectangle (h) because the area of a rectangle is length X width. So, the Lateral Area (use the word) is
(2πr)h or 2πrh . Then, they must remember to add in the area of TWO circles (πr2 + πr2 = 2πr2). So
Surface Area = 2πrh + 2πr2
Sample Answer Key for Student Log
TYPICAL—but incomplete—student answer.
Step 1:
?
h
A = lw
“No, we were not given enough information.”
Step 2:
A = πr2
A = πr2
S.A. = 2πrh + 2πr2
Step 3: “First, we flattened the paper we took off the can and found it was not rounded but was a
rectangle. The h was written on it so we copied it on this paper. We wrote down "A = law" because
we knew that was the area of a rectangle. Then we drew 2 circles (the top and bottom) and drew the
radius in, wrote in the r because it was on the wrapping paper and wrote down the area formula for a
circle (A = πr2) two times. We knew that was 2πr2. When we where helped to see that the long side
of the label was equal to the circumference of the lid, we knew it was 2πr times h, so the answer is =
2πrh + 2πr2. Then, we ate potato chips and we really deserved them because this was not easy!!!”
Next, choose a group to present its work to the class on the overhead projector. Point out positive
aspects of their written explanation.
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Days 2 & 3
Applying Formulas
As a class, review yesterday’s lesson and practice applying the formulas to several cylinders
shown in your textbook. Be sure that a few problems state the diameter of the circular base of a
cylinder and several problems state the radius of the circle Also, pick a couple of problems that
ask students to find the total surface area of a cylinder and a couple of problems that ask them to
find the lateral area of a cylinder.
Then, distribute copies of “SOLVE-IT: Real-world Surface Area Problems” Allow students to
work in pairs or in cooperative groups. Encourage them to draw and label figures on notebook
paper and show work. The problems are challenging, so give students sufficient time to work,
probably 1/5 class periods. Encourage re-tries until all problems are correct.
Distribute copies of the Practice Short-Response Assessment Problem at the end of this lesson.
Discuss and post the scoring rubric you will be using and give students plenty of time to work.
Answer Key for “SOLVE-IT: Real World Surface Area Problems”
Answers: # 1.) 351.68 cm2; # 2.) 2 cans of paint; # 3.) 175.84 cm2; # 4.) 7 cm; # 5.) 62.8cm2
Solutions: #1.) Using total surface area formula, 2 X 4 X 3.14 X 10 + 2 X 16x 3.14 = 351.68 cm2 ; # 2.)
Using the total surface area formula, 2 X 2 X 3.14 X 5 + 2 X 2 X 3.14 X 2 is 87.92 m2. Since 1 can of
paint covers 50m2, Pat will need 2 cans of paint because 2 X 50 = 100 m2, which is enough to cover 87.92
m2 #3.) METHOD I: The area of the top and bottom of both boys’ cases are identical. Only their lateral
areas are different, so find lateral area of both boys’ drum cases and subtract areas. The lateral area of
Gaston’s brother’s case = 14 X 2 X 3.14 X 28= 2461.76. The lateral area of Gaston’s case is 14 X 2 X 3.14
X 26 = 2285.92 cm2. The difference is 175.84 cm2. METHOD 2: Gaston’s brother’s drum case is 2 cm
higher than Gaston’s, so the difference in areas is 2 cm X circumference, or 2 X 14 X 2 X 3.14 = 175.84
cm2. METHOD 3: The total surface area of Gaston’s brother’s drum case is 2 X 14 X 3.14 X 28 + 14 X 14
X 3.14 X 2 = 3692.64 and the total surface area of Gaston’s drum case is 2 X 14 X 3.14 X 26 + 14 X 14 X
3.14 X 2 = 3516.8 cm2; the difference is 175.84 cm2 # 4.) Label length needs to be 2.75 cm longer than
the circumference of the can; (16 X 3.14 = 50.24 + 2.75 = 52.99 or almost 53 cm. If the uncut label is 60
cm long, then 60-53 = about 7 cm: # 5.) The length of label with no overlap is the circumference of the
circle (2 X 2 X 3.14 = 12.56); since the label is 1.5 inches below top of jar, the height of label is 6.0 – 1.5 =
5.0 cm. So the area of the label is width (12.56) X height (5) = 62.8 cm2.
Sample Answer Key for the Practice Short-Response Assessment Problem:
Part A: Typical Student Answer: S.A.= 2πrh or S.A.= πdh
Part B: Typical Student Answer: “She changed the first formula because it also finds the area of the top
and bottom of a cylinder (+ 2πr2), total surface area. A drinking straw does not have a top or bottom
because the soda could not get through it and that would be silly. So she took the last part of the formula
off and made it S.A. = 2πrh, lateral area.”
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8
Revised 2/2/05
Class _______ Date _____________Name(s)__________________________________
__________________________________
__________________________________
__________________________________
SOLVE-IT: Real World Problems
Set aside one activity sheet for recording your group’s final answers.
Raise your hands when your group answer sheet has been completed.
All answers must be correct to win.
All group members must be able to explain solutions to all problems.
Anne C. Patterson 2004
Surface Areas of Cylinders, Grades 7 and 8