7.3.1B Volume & Surface Area of Cylinders
Overview
7.3.1 Essential Understandings
As the overarching theme of 7th grade mathematics is proportionality, the placement of circle measurement in 7th grade is intentional. Circles provide the perfect vehicle for deepening students' understanding of proportionality. There are at least 2 specific ways proportionality is evident in circle measurement:
 The circumference of a circle is directly proportional to the radius of the circle. That is, there is some constant k such that for all circles, C = kr. This implies, for instance, that if you double the radius of a circle, then you double its circumference.
 The area of a circle is directly proportional to the square of the radius of the circle. That is, there is some constant h such that for all circles, A = hr^{2}. This implies, for instance, that if you double the radius of a circle, then you quadruple its area.
By decomposing two and threedimensional shapes into smaller, component shapes, students ﬁnd surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base × Height). They apply these formulas in problem solving to determine volumes of cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two and threedimensional shapes to model realworld situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of cylinders. (7th Grade NCTM Curriculum Focal Points)
All Standard Benchmarks
7.3.1.1
Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is π. Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts.
7.3.1.2
Calculate the volume and surface area of cylinders and justify the formulas used.
For example: Justify the formula for the surface area of a cylinder by decomposing the surface into two circles and a rectangle.
Benchmark Group B  Volume and Surface Area of Cylinders
7.3.1.2 Calculate the volume and surface area of cylinders and justify the formulas used.
For example: Justify the formula for the surface area of a cylinder by decomposing the surface into two circles and a rectangle.
What students should know and be able to do [at a mastery level] related to these benchmarks
 volume formula is area of base x height
 knowledge and use of area of circle
 calculate the volume of a cylinder
 calculate the surface area of a cylinder
 justify the formula for the volume and surface area of a cylinder.
 generalize, algebraically, how to find the volume of a cylinder (Volume = area of the base times height or V = πr^{2}h).
Work from previous grades that supports this new learning includes:
 using formulas
 know how to find volume of other 3D figures (rectangular prisms)
 finding surface area of rectangular prismsadding up the areas of all the faces
 conceptual understanding of volume and surface area
 finding area of 2 dimensional objects (except circles)
 connecting back to find volume for rectangular prisms and use that same strategy for cylinders
 identifying nets of various 3D figures.
NCTM Standards
Use visualization, spatial reasoning, and geometric modeling to solve problems:
 use twodimensional representations of threedimensional objects to visualize and solve problems such as those involving surface area and volume.
Measurement Standard for Grades 68
Understand measurable attributes of objects and the units, systems, and processes of measurement.
 Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
Apply appropriate techniques, tools, and formulas to determine measurements.
 Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision;
 Develop and use formulas to determine and circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of morecomplex shapes;
 Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.
Common Core State Standards (CCSS)
Geometry 7.G Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.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.
7.G.6 Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Geometry 8.G Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.9 Know the formulas for the volume of cones, cylinders and spheres and use them to solve realworld and mathematical problems.
Misconceptions
Student Misconceptions and Common Errors
 Not understanding that the length of the rectangle (the lateral surface of the cylinder) is the circumference of the circle (base).
 Confusing 'area of base' with 'base' in formulas like V = Bh or A = bh. The formula for volume on Minnesota's MCA3 7th Grade formula sheet is a general formula for any prism. Students may be more comfortable with V = πr^{2}h for finding the volume of a cylinder.
 Confusing "lateral surface area" with "total surface area." It is helpful for students to draw nets of cylinders to calculate surface areas to reinforce the differences.
 The formula from the Minnesota MCA3 7th grade formula sheet for finding surface area is a general formula that works for any prism. Students may find it difficult to apply to a cylinder. If they decompose the cylinder, they will see it is 2 circles (2πr^{2}) added to the area of the lateral rectangle (l × w). The formula uses 'p' for perimeter as opposed to 'c' for circumference used in most textbooks when finding the surface area of cylinders.
 Students do not label their answers with appropriate units (e.g., not using units cubed when referring to the volume of a cylinder).
 If the cylinder is lying on its side, students sometimes misread the height as the vertical distance, which would actually be the diameter in this orientation, when height is actually the distance between the two circular bases.
Vignette
In the Classroom
T: "We are going to find the volume of a cylinder today. What is a cylinder?"
S: "A can."
S: "A threedimensional object that looks like a can."
S: "A three dimensional object that has a circle for a bottom and top."
T: "Ok. You are all right, but what shape is the lateral surface of the cylinder?"
S: "A rectangle."
T: "Correct. So we can model a cylinder with a sheet of paper. I am giving each group a sheet of paper, some grid paper, some clear transparency film, some cm cubes, markers, and you may use anything else you find in the room. Your task is to find the volume of the cylinder made by your sheet of paper."
S: "Does it matter which way we hold our paper?"
T: "What do you think? Do you think it will matter?"
S: "No, it won't."
S: "I think it will, because if we hold it this way (the horizontal way), the bottom is bigger, so it'll hold more."
S: "But it is shorter, so I think they'll be the same."
T: "That's why we are doing this. We're going to find out. So with that, I do not care which way you have your paper, you just need to find the volume it would hold. Good luck!"
As the teacher walks around the room, she hears many good conversations. There are some disagreements as to how to best do the activity, but all are busy trying to get started. She stops at one group and asks what they are doing.
S: "Well, I think we should tape the cylinder together and fill it with the cubes and just count them."
T: "Would there be any problems with that strategy?"
S: "The cubes are square, and the cylinder is circular."
T: "So what are some other ways? But won't that strategy work? Really work?"
S: "Well, I guess it would work, but it won't be very accurate."
T: "So does that make it wrong?"
S: "I guess not. I guess it would give us a starting point."
T: "Good. Keep going with that thought and I'll come back to your group."
As the teacher again circulates, she comes to a group that made a cylinder, set it on their grid paper, and is attempting to trace it. Since it is made out of paper, it doesn't hold its shape, so it keeps moving.
T: "What are you doing, can you tell me?"
S: "We are trying to trace the bottom of the cylinder, but it's not working!"
T: "Is there any way you can make it easier?"
S: "Yes, have thicker paper that doesn't move so much."
T: "Do you have any materials here that might work better?"
S: "The transparency film seems thicker, it might work."
T: "Ok, try it."
The students try it and it seems to work for them. They trace it and then count the number of blocks in the circle, but struggle with what to do with the partial squares, so they just combine the partials to make whole squares.
Other groups have used similar strategies; one has even used their traced circle and found the area of it by using the formula for the area of a circle.
All the groups share their strategies, and see how no matter which strategy they used, their answers are similar to one another. Some are in decimal form, some are whole numbers. The class has a discussion on why they think their group's answer is more or less accurate than others.
T: "All right, that is just the first step of this lesson. Our task here is to see how much the cylinder will hold or its... what?"
S: "Volume."
T: "Yes. That's correct, volume is the amount of material an object will hold. So what could we do now to find the volume of this cylinder? How does finding the area of the base, or the number of blocks the base will hold, help us?"
S: "Because volume is the area of the base multiplied by the height."
T: "Correct. But why is it that?"
S: "If you find the amount that will fit in one layer, and you know the number of layers, then you can just multiply those two values by one another."
T: "Pretty well said. You are correct."
S: "Can't we think of it like stacking cases of pop. If you know how many you can fit across the floor, and you know 12 will stack on top of one another, you can multiply the number on the floor, which would be like the area of the floor, by 12 to get the number of cases of pop that would fill the room."
T: "Good example. You all found areas of the base, and those were all minutely different and yet similar, so how can we find the height?"
S: "Just measure the paper."
T: "Should our answers be the same or different?"
S: "It depends on which way you did your cylinder. But there should really be only two answers for the height."
T: "Why only 2? We all had different answers before, why can't we now?"
S: "It's the same piece of paper, well, not the SAME piece, but they all have the same dimensions."
T: "So let's do that, measure your height and multiply that by the area your group got for your base. Compare your answers to another group that used the same cylinder."
S: "Our answers are kind of close but not really. If we used the same cylinder, or held our paper the same way, why would they be different?"
T: "Who can answer that?"
S: "I think I can. It's because we used different strategies to find the areas of the bases."
T: "Good. We also wanted to compare the volumes of the two different shapes of cylinders. What do we have for volumes?"
There is much discussion. They find that the cylinder that was made with the paper going horizontally had more volume.
T: "Why do you think that is? It IS shorter, so shouldn't it have less?"
S: "I think it's because the circle on the bottom is bigger."
T: "So why would that make a difference?
S: "Because each layer holds more, so it adds up faster."
T: "I guess that's one way to say it. You are right, the larger the base, the more it will hold. Nice job on this lesson, and here's a quick recap: To find the volume of a cylinder, take the area of the base times the height. Now you know WHY that is the formula used."
Resources
Teacher Notes
Students may need support in further development of previously studied concepts and skills.
 Make sure students understand what they are doing; e.g., "knowing" or "understanding" vs. "memorizing."
 NCTM's 'Focus on Grade 7: Teaching with the Curriculum Focal Points' (2010, Chapter 3, pp 4576) provides a full discussion on how to conceptually develop students understanding of surface area and volume in 3dimensional shapes. The diagrams from this book are invaluable. See below for examples:
pg. 60. This diagram is useful for connecting circumference with "perimeter of the base." Teachers could start by students accessing prior knowledge by drawing a net of a rectangular prism. They then calculate the surface area. Many students may find the area of each of the 6 faces and calculate the sum of these 6 faces to find total surface area. Teachers can use this prior knowledge to show how students could calculate total surface area a different way using the perimeter of the rectangular base. Helping students see this connection will reinforce the idea of the circumference of the circular base of a cylinder being part of the calculation for total surface area.
pg 6061
 This diagram is useful for showing students the connection between the general formula for the surface area of any prism; they will see on the formula sheet (SA = ph + 2B) and what actual measurements will be used in the calculation.
pg 63
 The best way for students to understand surface area correctly is to have them use a net of the figure; just memorizing the formula will NOT ensure they know how to find surface area.
 Help students see that the volume of a prism can be figured by multiplying the area of base (the amount that will fit in the base) by the height (the number of layers).
 An interactive visual of how to find the surface area of a cylinder. This applet allows students to visualize the circumference as the length of the rectangle represented as the lateral surface area.
 The same site, Geometry 3D shapes, also has a tutorial for finding the volume of a cylinder.
Additional Instructional Resources
 Math video lesson on finding volume of a cylinder.
 Geometry  Math Warehouse provides detailed lessons about angles, triangles, quadrilaterals, circles, similar triangles, parallelograms, polygons, and trapezoids.
 Interactive Geometry at Mathsnet.net. This is by far the most comprehensive & magnificent interactive geometry resource on the Internet.
 Geometry course from Learning Math. This online geometry course includes readings, problems, videos, interactive activities, homework problems and solutions. It is meant for K8 teachers but will work well for middle school students as well.
cylinder: A threedimensional shape with two opposite faces that are congruent circles. The side (lateral surface) is a rectangle that is "wrapped around" the circular faces at the ends
.
surface area: The area required to cover a threedimensional shape.
lateral surface area: The area of all faces of a prism not including the two bases.
Reflection  Critical Questions regarding the teaching and learning of these benchmarks
Do students see the cylinder as a 2 or 3 dimensional figure?
What common mistakes are students consistently making in finding volume and surface area of a cylinder?
Are students understanding the volume formula, or just "going through the motions" by plugging the values into the formula?
Are students clear on the difference between bases of a twodimensional figure and a threedimensional figure?
How do you find surface area?
How do you find volume?
What shapes make up the net of a cylinder?
What is the height of a cylinder?
How is finding volume of a cylinder both different from and similar to finding volume of a rectangular prism?
How is finding surface area of a cylinder both different from and similar to finding surface area of a rectangular prism?
Focus in Grade 7: Teaching with the curriculum focal points. NCTM, 2010. pg. 68.
Geometry wordproblem: The box problem and the goat problem.
http://www.purplemath.com/modules/perimetr5.htm
Minneapolis Public Schools Teacher created quiz to assess benchmark 7.3.1.2.
Minnesota Grade 7 Mathematics MCAIII Item Sampler Item, 2011, Benchmark 7.3.1.2
TAKS (Texas Assessment of Knowledge and Skills) Mathematics, Grade 8, 2010 released items.
Assessment
1.
Minneapolis Public Schools Teacher created quiz to assess benchmark 7.3.1.2
2.
Focus in Grade 7: Teaching with the curriculum focal points. NCTM. 2010. pg. 68
3.
Minnesota Grade 7 Mathematics MCAIII Item Sampler Item, 2011, Benchmark 7.3.1.2
Correct answer: B
4.
Start with four identical sheets of paper with familiar dimensions (e.g., 8 1/2 by 11 in.). Use two of the sheets to make two different cylinders by taping either the long sides or the short sides of the paper together. Imagine that each cylinder has a top and a bottom. Take the other two sheets of paper and fold them to make two different rectangular prisms. Imagine that these rectangular prisms also have a top and a bottom.
a. Which of the four containers has the greatest volume? Explain your reasoning.
b. Which container has the greatest surface area? Explain your reasoning.
c. Take a cylindrical and a rectangular container of the same height. Which one has a greater volume?
Solution:
a. The shorter, wider cylinder has the greatest volume. In the formula for the volume of a cylinder, V = πr^{2}h, notice that the radius is squared and the height is not. Using the larger dimension of the paper as a circumference of the base produces a larger radius, and in turn, an exponentially larger volume, and vice versa: Using a smaller dimension of the paper as a circumference of the base produces a smaller radius, and in turn, an exponentially smaller volume.
b. Again, the shorter, wider cylinder has the greatest surface area. All will have equal lateral surface area (not including the top and bottom), since the same paper is being used. For total surface area, add the area of the bases  so the problem boils down to which has the largest base area. As described in the solution to part (a), the shorter, wider cylinder has the largest base area.
c. The cylindrical container has the greater volume, as long as they both have the same lateral surface. If only the heights are known, then there is no comparison to make  one could have a much larger base area than the other.
http://www.learner.org/courses/learningmath/measurement/session8/part_h/homework.html
5.
TAKS (Texas Assessment of Knowledge and Skills) Mathematics, Grade 8, 2010 released items.
Correct answer: C
6.
TAKS (Texas Assessment of Knowledge and Skills) Mathematics, Grade 8, 2010 released items.
Correct answer: B
7.
A hollow iron pipe is 21 cm long and its external diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weighs 8 g/cm3, then the weight of the pipe is:
A. 3.6 kg B. 3.696 kg C. 36 kg D. 36.0 kg
Answer: B
Differentiation
 Provide a real cylinder/picture/diagram for every cylinder used.
 Students may struggle to see the curved sides of a cylinder as a rectangle when they draw a cylinder's net. To visualize the surface area of a cylinder give groups of students a roll of toilet paper or paper towels. Have students use a marker to draw a line where the toilet paper sheets end. Then have students unroll the toilet paper to see the rectangle that is formed. Have students reroll and unroll several times as they look at the base of the roll of toilet paper to see that the edge of this rectangle is the circumference of the circle.
 netvocabulary term, may confuse with a net used for holding items
 make sure students know the definition of base in this context
 http://illuminations.nctm.org/LessonDetail.aspx?id=L791  In this lesson, students use formulas they have explored for the volume of a cylinder and convert them into the same volume for rectangular prisms while trying to minimize the surface area. Various real world cylindrical objects will be measured and converted into a prism to hold the same volume. As an extension, students may design and create a rectangular prism container according to their dimensions to compare and contrast with the cylinder.
 Ask the following guiding question: Does a double stuff Oreo really have twice as much filling as a regular Oreo? Did the Nabisco Company give a mathematically correct name to the 'Double Stuff Oreo'? Give pairs of students a ruler, one regular Oreo and one doublestuffed Oreo. Allow students time to investigate and present their findings. (Note: If students carefully remove the top cookie portion from the Oreo and measure the volume of the white filling, a regular Oreo cookie has approximately 3 cubic centimeters of filling and a doublestuff Oreo has approximately 6 cubic centimeters of filling.)
● Cylinder
Conceptual Knowledge
 Diameter
 Radius
 Area
 Volume
 Ratio
 Procedural Knowledge
 A = pir^{2}
 V = pir^{2}h
 Problem Solving
 Reasoning
 Communication
 Connections
 Representation
If you compare three different cylinders, how can you tell which one will have the greatest volume?
Students work in individually or in pairs
 3 sheets of paper with rectangles A, B, and C
 rulers
 tape
 scissors
1. Cut out the rectangles shown on the attached pages. Tape them together on the sides indicated to form cylinders.
2. Answer the following questions:
a. If the cylinders had tops, what would be the diameter of each top?
b. What would be the radius of each top?
c. What would be the area of each top? Round to the nearest tenth.
d. For the area of the tops, find A:B and A:C.
3. Measure the heights of the three cylinders. For the heights, find:
a. A : B ____________
b. A : C ____________
4. Predict the ratio of the volumes of the following cylinders:
a. A and B __________
b. A and C __________
5. Find the volume of each cylinder. For the volumes, find:
a. A : B ___________
b. A : C ___________
6. Were your predictions in Question 4 correct?
7. Summarize:
a. If the height of two cylinders is the same and the radii are in ratio of __?__, then their volumes are in the ratio of __?__.
b. If the radius of two cylinders is the same and the heights are in ratio of __?__, then their volumes are in the ratio of __?__.
As a result of this activity, students will be able to model how the changes of a figure in such dimensions as length, width, height, or radius affect other measurements such as perimeter, area, surface area, or volume. http://fcit.usf.edu/math/lessons/activities/CylindT.htm
Parents/Admin
Administrative/Peer Classroom Observation
Students are:
 Teachers are: 
making nets of cylinders.  making sure students are correctly viewing the height of the cylinder; some students get confused when the orientation of the cylinder changes (laying on its side vs. standing upright). 
using grid paper to approximate the area of figures.  providing grid paper. 
using protractors or angle rulers to measure and draw different sectors of a circle.  have nets of cylinders available for students to usethey can fill in the dimensions needed. 
labeling dimensions on cylinders drawn (both threedimensionally and on a net).  decomposing a three dimensional cylinder into a two dimensional net; do this in front of the students so they can see the relationship between the faces of the cylinder to its three dimensional state. 
using correct labels on answers and on diagrams. 

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