The surface area of a cylinder “CAN” be fun
when using toilet paper rolls
Sonya Morris
Arellanes Jr. High School
Santa Maria-Bonita School District
smorris@smbsd.net
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Unit Information
Grade Level:
6th-8th
Subject:
Math: Measurement and Geometry
Description:
This series of lessons methodically leads the student into exploring circles, area of
circles, circumference, surface area of a net and finally the surface area of a cylinder
while engaging the student in every instance. The student will see the relationship
between the circumference of the circle bases of the cylinder and the bases that connect
the circles to the net of a cylinder.
Using the net and then relating the toilet paper roll to the net by cutting it into one flat
surface while opening and closing the manipulative will help the student discover how
the length of the base is equal to the circumference and how the formula is derived. It is
truly magic to watch the concept click!
The lessons encourage student access and success for all by providing hands on activities
that promote discovery for students with multiple learning styles.
Educational Value:
This lesson has much value as it accomplishes the job of teaching the standard of
Measurement and Geometry 2.1 and the Common Core Standards for Mathematical
Practice 5-8 in a fun and creative manner.
Content Standards:
Standards: Measurement and Geometry
2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional
figures and the surface area and volume of basic three-dimensional figures, including
rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
Common Core Standards For Mathematical Practice:
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Comon Core Content Standards
Geometry 7.G
Solve real-life and mathematical problems involving angle measure, area, surface area,
and volume.
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4. Know the formulas for the area and circumference of a circle and use them to solve
problems; give an informal derivation of the relationship between the circumference and
area of a circle.
6. Solve real-world and mathematical problems involving area, volume and surface area
of two- and three-dimensional objects.
Geometry 8.G
Solve real-world and mathematical problems involving volume of cylinders, cones, and
spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to
solve real-world and mathematical problems.
Lessons:
Lesson 1: All About Circles
Topic: Information about circles and area of a circle
Standards: Measurement and Geometry
2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional
figures and the surface area and volume of basic three-dimensional figures, including
rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
Common Core Standards For Mathematical Practice:
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Objective:
The students will be able to identify the radius and diameter of a circle as well as be able
to derive them from using information from either the radius or diameter. They will also
be able to calculate the area of a circle. They will be able to do all these tasks with 85%
accuracy.
The students will explore making circles with a compass. We will then use what we have
made to write our own definition of a circle. The students will take Cornell Notes on
information about circles. They then work in groups to come up with valid questions
pertaining to the notes. They also write a summary using D.L.I.Q. (What Did you do?
What did you Learn? What was Interesting? What Questions do you have?)
Required Materials:
Interactive notebook, colored pencils, markers, compass, white paper and a work sheet
for each student, a poster in the room with the formula for the area of a circle and the
circumference of a circle.
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Anticipatory Set:
Let’s draw some circles using compasses. Let’s see the different sizes you can draw.
Please fill the page and draw at least 5.
Let’s see what we can learn about a circle now.
Let’s use what we have learned in the past to come up with some definitions that we
understand.
The center of our circle could be called the center point. It is equal distance from all
sides of the circle.
The distance from the center of the circle to the edge is called what? You are correct. It
is called the radius.
This line that goes across the circle from one side to the other crossing the center point is
called the what? It starts with a ‘D”? You are correct again, it’s the diameter.
What do you notice about the radius and the diameter? Exactly, the radius is twice the
diameter. How about the diameter? Yes you are correct again, half the diameter is equal
to the radius.
Now the distance around the circle is called something as well. Well you are correct in
saying that it is the perimeter but in circle language it is called another term. It starts with
a C. Yes you are correct it is called the circumference.
Just a few more facts about a circle:
Pi, yes pi is very important because we will use it in formulas to find area and
circumference of a circle.
Pi is irrational so it can not be expressed as a fraction or a decimal, it has no repeating
pattern and the number goes on forever. Some may use 22/7 or 3.14 but these are only
estimates of pi. For easier calculations 3.14 is used frequently.
Here are some important formulas that we will use today and in the next few days.
Area of a circle:
r² which means (3.14) (r) (r)
Circumference of a circle: d = (3.14) (d) or 2 r = 2(3.14) (r)
Get out your interactive notebooks and turn to the next two free pages that have no
information on them. The page that you have just done will be your left hand page.
Your right hand page will be your Cornell notes that you will write from your discovery
page.
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Let’s take Cornell notes on information about circles.
In our groups let’s brainstorm some valid questions about the notes we just took to help
us study later on.
Remember to write your summary using D.L.I.Q. (What Did you do? What did you
Learn? What was Interesting? What Questions do you have?
Guided Practice:
The students will be engaged in some guided practice on their individual white boards
that will cover radius, diameter and area of a circle. We will go over what we have
learned from our notes. We will flash our answers up when we are finished to check to
see if everyone has an understanding of the concepts. When there is evidence of
comprehension we will transition to a work sheet on radius, diameter and area of a circle
for homework.
Reflection:
What did we learn today? How can we use this in a real life situation? Let’s name the
parts of a circle together. Can you remember any formulas off hand?
Lesson 2: Discovering Surface Area of a Cylinder with a Net
Topic: finding the surface area of a cylinder without using the whole surface area
formula.
Standards: Measurement and Geometry
2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional
figures and the surface area and volume of basic three-dimensional figures, including
rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
Common Core Standards For Mathematical Practice:
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Objective:
The students will be able to find the area of the cylinder net using only the area of circle
and the length times width. They will be able to do this task with 85% accuracy
Required Materials:
A net of the surface area of a cylinder and red and blue markers
Anticipatory Set:
Let’s recap on what we have learned about circles. Here is what I have been leading up
to.
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Take a look at this figure. What shape do you think this is? Right, it has two circles and
a rectangle. You are right once more, it’s like a can but what is the name of this shape
when it is put all together? Correct once more. It is a cylinder.
Can you please get your blue marker and trace the two circles and the base line that is
connected to the circle as well in blue.
Please get your red marker and trace the lines that are on the sides now.
You will understand why we did this later.
We are going to find the area of this “net”. You can use any method you would like. I
want you to look for “total surface area” that means what? Nice job, yes it means adding
up the area of the two circles and the rectangle to get total surface area.
Great, now that you have one total find another way using what we have learned
yesterday to help you.
What are the two ways you calculated your solution? Yes you can count all the squares
and the other way you can find the area of the two circles and then length times width of
the rectangle to get the area and then add them all together. Which one is more to you
seems more accurate?
Let us draw
+
+
=
On the net write the values in each part and then calculate the total.
Guided Practice:
We are going to do the same for practice. Here are a couple of problems we are going to
do in class and the rest will be for home work. Before we continue we will glue the net
we just worked on into our note book so we can refer to it for our homework. It will be
on the left hand page of the notebook.
Reflection:
What have we learned today and how will it help us with our homework? Can you see
anywhere this would help with real life problems? Let’s brainstorm where this would
apply.
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Lesson 3: Finding the Surface Area of a Cylinder
Topic: Find the surface area of a cylinder relating the width of the rectangle (of the net) to
the circumference of circles (bases of the cylinder).
Standards: Measurement and Geometry
2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional
figures and the surface area and volume of basic three-dimensional figures, including
rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
Common Core Standards For Mathematical Practice:
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Objective: The students will be able to find the area of the cylinder using the area of the
two circles and (diameter) (height). They will be able to do this task with 85%
accuracy
Required Materials:
Blue and red markers, toilet paper rolls, scissors, warm-up, cylinder worksheet (from last
lesson), cylinder with net model, index cards, tape
Anticipatory Set:
Warm-up - (to include circumference, surface area, radius)
Cylinder Pattern - Show model and net (prompt think pair share)
Toilet paper roll activity – Trace exterior of both circles in blue on index card and
around the circumference. Use a ruler to draw a red line down length of the roll. Prompt
discussion of how the blue circumference is related to the rectangle that they get after
they cut the roll. (15min.)
Prompt students to discuss how we can get the blue line information without actually
cutting a cylinder (10 min). Write discoveries on the toilet paper rolls.
Guided Practice – math work sheet
Wrap – up: discuss calculation; prompt discussion about what measurements are needed
to find the surface area of a cylinder. (10 min.)
Reflection:
What did we learn from this unit? Can you tell me what relationships where made with
length of the rectangle that connects with the circle (using the net) and with the side and
the circle and base that connects the
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Extension on lesson:
This problem is a released question from the Common Core Standards Math Site.
Example of Common Core Mathematical Practices
Standards problems using Surface Area of Cylinders.
 In a pottery class Johnny made an 18 in high cylindrical vase with a 6 in radius of
the base. He wants to paint the outside of the vase. How many square inches will
Johnny have to paint?
 A paper cup is a perfect cylinder that is 7 cm tall and 3 cm. across. Find the total
amount of paper needed to make the cup.
 The supermarket around the corner used to sell Grandma’s favorite Mandarin
oranges. For reasons unknown they no longer do and it really upsets Grandma.
Jackson, a thoughtful young seventh grader, remembers this fact as he’s thinking
of what to get his Grandmother for her 80th birthday party. Cans of Geisha
Mandarin Oranges of course! There’s one potential problem, however: Jackson
has two cans to wrap but only has 550cm² of wrapping paper. Does he have
enough paper to wrap the two cans? (Provide students with two cans with the
label sliced and taped back on. Students must measure the cans in order to answer
the question.)
 In an effort to clean up their city, the Parks and Recreation Department is
spending the money to repaint a water tower and 20 trash barrels. They are trying
to fit this into the tight budget, so they need to estimate the cost of the paint job.
Important Information:
 1 water tower and 15 trash barrels need painting
 The hardware store estimates that paint will cost $0.75 per square foot.
 Two painters will be paid $500 each to do the job.
 The water tower is 15 feet tall and 10 feet across.
 The trash barrels are 27 inches tall and 22 inches across.
In total, how much will it cost the city for the renovations?
Helpful website: http://illustrativemathematics.org
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Student Impact:
Students will be able to identify the radius and diameter of a circle as well as be able to
derive them from using information from either the radius or diameter. They will also be
able to calculate the area of a circle.
Students will be able to find the area of the cylinder net using only the area of circle and
the length times width.
Students will also be able to find the surface area of the cylinder using the area of the two
circles and pi (diameter) (height). They will be able to do all of these tasks with 85%
accuracy.
Students will benefit by this lesson for the simple reason that it reaches a variety of
learning styles and is very engaging. It provides the students a time to discover and talk
about what they are learning in a comfortable learning environment. They will lead
viable arguments while proving their points.
Student Assessment:
Students will be given a pre-test and a post-test to assess student learning to check for
understanding. A test in class, Benchmark test and the STAR test will also be indicators
as to how well the lesson was comprehended. In my class all concepts spiral so this
lesson will not be a flash in the pan but something that they will see through out the rest
of the year.
Materials/Budget:
Notebooks: $2 for 200 students = $400
Colored pencil for 200 students = $40
Markers for 200 students = $40
Index Cards for 200 students = $5
Compasses a class set (35) $2 = $70
Scissor a class set (35) $2 = $70
Toilet paper rolls
Use of a smart board
Tape
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