6 th Grade Math Cylinder Task. c) Draw a net (pattern) for the manufacturer to use to make the can.

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1 6 th Grade Math a) Explain what is meant by surface area. What steps would you take to find the surface area of a cylinder? b) One of the major expenses in manufacturing a can is the amount of metal that goes into it. How many square centimeters of metal would be required to manufacture a can that has a diameter of 8 cm and a height of 20 cm? Estimate and then solve. c) Draw a net (pattern) for the manufacturer to use to make the can. d) Use your work in parts a c to write a rule in words for finding the surface area of a cylinder. Now write your rule using letters, numbers, and mathematical symbols (a formula). e) Michael bakes a round two-layer birthday cake that is to be covered with frosting on the top, sides, and in between the layers. Each layer has a height of 4 cm and diameter of 24 cm. The label on the can of frosting he bought claims that the contents will cover the top and sides of a one-layer rectangular sheet cake that is 32 cm by 22 cm by 4 cm. Will Michael have enough frosting? Show how you know. *********************************************************************** Discussion, Suggestions, Possible solutions: This task deals primarily with the surface area of a right rectangular prism and a right circular cylinder. Parts a d of this task may be used appropriately throughout the unit as an instructional task. Part d could be used as a learning task or should the teacher prefer to divide this unit into two smaller units, it may be used as a culminating task over surface area. Georgia student work is available for this task. a) Explain what is meant by surface area. What steps would you take to find the surface area of a cylinder? The surface area of a three dimensional figure is the sum of the areas of its faces. To find the surface area of a cylinder, you would find the sum of the areas of the two circle bases and the area of the rectangle that is wrapped around the circumference of those bases.

2 b) One of the major expenses in manufacturing a can is the amount of metal that goes into it. How many square centimeters of metal would be required to manufacture a can that has a diameter of 8 cm and a height of 20 cm? Estimate and then solve. To estimate the surface area of the can, one might think each circle has an area of a little more than (3)(15) = 45cm 2 and the rectangle would have an area of about (25)(20) = 500cm 2. So a good estimate would be close to 600cm 2. To solve the problem, you would find the areas of the two circle bases which are ( r 2 ) each. This would be ( ) = 50.24cm 2 each. Then to find the area of the rectangle, it would be (circumference height) or ( d h) which is ( ) = Therefore, the surface area of the metal can would be = cm 2. c) Draw a net (pattern) for the manufacturer to use to make the can. d) Use your work in parts a c to write a rule in words for finding the surface area of a cylinder. Now write your rule using letters, numbers, and mathematical symbols (a formula). To find the surface area of a cylinder: find the areas of the two circles by doubling the product of pi and the square of the radius, then and add the area of the rectangle which would have a length of the circumference and a width that is the height of the cylinder. The formula would be SA = 2( r 2 ) + ( d)(h) when SA= Surface Area of the cylinder, r = radius of a circle base, d = diameter of a circle base, and h = height of the cylinder. e) Michael bakes a round two-layer birthday cake that is to be covered with frosting on the top, sides, and in between the layers. Each layer has a height of 4 cm and diameter of 24 cm. The label on the can of frosting he bought claims that the contents will cover the top and sides of a one-layer rectangular sheet cake that is 32 cm by 22 cm by 4 cm. Will Michael have enough frosting? Show how you know. Finding the area needed to ice the round cake: Area of the top and middle = 2( r 2 ) = 2( ) = 2( ) = 2(452.16) = c Area of the side = (C)(h) = ( d)(h) = ( )(2 4) = (75.36)(8) = cm 2 The round cake will need to have enough icing to cover = 1,507.2cm 2. Finding the area needed to ice the rectangular sheet cake:

3 Area of the top = lw = (32)(22) = 704cm 2 Area of the front and back = 2(wh) = 2(22)(4) = 176cm 2 Area of the left and right = 2(lh) = 2(32)(4) = 256cm 2 The rectangular cake will need to have enough icing to cover = 1,136cm 2. Michael will not have enough icing to frost the round cake because 1,507.2 is larger than 1,136. Georgia student work is available for this task. SIXTH GRADE MATHEMATICS GEORGIA STUDENT WORK SAMPLE: f) Explain what is meant by surface area. What steps would you take to find the surface area of a cylinder? g) One of the major expenses in manufacturing a can is the amount of metal that goes into it. How many square centimeters of metal would be required to manufacture a can that has a diameter of 8 cm and a height of 20 cm? Estimate and then solve. h) Draw a net (pattern) for the manufacturer to use to make the can. i) Use your work in parts a c to write a rule in words for finding the surface area of a cylinder. Now write your rule using letters, numbers, and mathematical symbols (a formula). j) Michael bakes a round two-layer birthday cake that is to be covered with frosting on the top, sides, and in between the layers. Each layer has a height

4 of 4 cm and diameter of 24 cm. The label on the can of frosting he bought claims that the contents will cover the top and sides of a one-layer rectangular sheet cake that is 32 cm by 22 cm by 4 cm. Will Michael have enough frosting? Show how you know. M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems. g. Solve problems involving fractions, decimals, and percents. M6M4. Students will determine the surface area of solid figures (right rectangular prisms and cylinders). a. Find the surface area of right rectangular prisms and cylinders using manipulatives and constructing nets. b. Compute the surface area of right rectangular prisms and cylinders using formulae. c. Estimate the surface areas of simple geometric solids. d. Solve application problems involving surface area of right rectangular prisms and cylinders.

5 SIXTH GRADE MATHEMATICS GEORGIA STUDENT WORK SAMPLE: a) a) 1 1) The student shows a clear understanding of the meaning of surface area. 2 2) The student demonstrates that he/she is able to determine the surface area of a cylinder. (M6M4) b) b) 3 3)Although the estimate of the surface area of the cylinder is reasonable, there no explanation of the process. (M6M4c) 4 4) Even with a slight computational error, it is evident that the student does understand the meaning of addition and multiplication as related to positive rational numbers and is able to use these concepts to solve problems. (M6N1)

6 c) c) 5 5) It is apparent that the student is able to construct a net for the cylinder. (M6M4a) d) d) 6 6) The work shown indicates understanding of the development of the formula for surface area of a cylinder. (M6M4) e) e) 7 7) The computational error mentioned in comment 4 is corrected here. 8 8) All work is accurate and indicates mastery of the standard M6M4 and element M6M4d.

7 SIXTH GRADE MATHEMATICS GEORGIA STUDENT WORK SAMPLE:

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