Surface Area and Volume
Surface Area and Volume
Objectives
Students will explore and visualize three-dimensional solids to gain a greater understanding of surface area and volume. Students will:
- create and/or match a solid to its net.
- determine various ways to calculate surface area and volume.
- calculate surface area and volume.
- connect surface area and volume to real-world applications.
- describe the relationship of measurements.
- investigate the effects of changes in dimension on surface area and volume.
Essential Questions
- How can recognizing repetition or regularity assist in solving problems more efficiently?
- How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real-world situations or solve problems?
- How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?
- How can geometric properties and theorems be used to describe, model, and analyze situations?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- NCTM’s Geometric Solids http://illuminations.nctm.org/ActivityDetail.aspx?ID=70
- Wolfram Demonstrations http://www.demonstrations.wolfram.com/CubeNet
- Math.com http://www.math.com/tables/geometry/surfareas.htm
- Math.com http://www.math.com/school/subject3/lessons/S3U4L1GL.html
- Math.com http://www.math.com/school/subject3/lessons/S3U4L2DP.html
- Shodor http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
- Interactives from Annenberg Media
http://www.learner.org/interactives/geometry/area_surface.html
http://www.learner.org/interactives/geometry/area_volume.html
- Wolfram Demonstrations http://demonstrations.wolfram.com/RectangularPrisms/
- Shodor’s Surface Area and Volume Interactive Applet http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/
- Wolfram Demonstrations
http://demonstrations.wolfram.com/TheRatioOfSurfaceAreaToVolumeForACubeAndASphere/
Formative Assessment
-
View
Multiple-Choice Items:
1. A student draws a net for a rectangular prism that has dimensions 12 cm by 8 cm by 6 cm. She notices that the areas of the six rectangular faces are in three pairs. Which three pairs of areas describe all six faces of the prism?
A
36, 64, and 144 square cm
B
48, 72, and 144 square cm
C
36, 72, and 64 square cm
D
48, 72, and 96 square cm
2. The surface area of a cube is 600 square inches. What is its volume?
A
1,000 cubic inches
B
600 cubic inches
C
300 cubic inches
D
100 cubic inches
3. A rectangular prism that is not a cube has a volume of 24 cubic meters. Its surface area is 68 square meters. What are its dimensions?
A
2m by 3m by 4m
B
1m by 2m by 12m
C
6m by 2m by 2m
D
1m by 6m by 4m
4. A cube has a volume of 729 cubic units. What is the length of one side?
A
9 units
B
54 units
C
unitsD
243 units
5. A rectangular prism has a volume of 96 cubic feet. The length of the prism is 2 feet and the width is 8 feet. What is the height of the prism?
A
6 feet
B
8 feet
C
80 feet
D
112 feet
6. Which is the best cost estimate for paint to cover the four sides and bottom of a rectangular swimming pool with dimensions of length 40 feet, width 20 feet, and depth 8 feet?
The paint costs $40 per gallon and one gallon covers 220 square feet.
A
$1760
B
$1160
C
$320
D
$240
7. A rectangular shipping box has a volume of 540 cubic inches and its smallest dimension is 3 inches. If that dimension was reduced to 2 inches and the other measurements remained the same, what would be the volume of the new smaller box?
A
180 cubic inches
B
270 cubic inches
C
320 cubic inches
D
360 cubic inches
8. The volume of a cube-shaped box is 1728 cubic inches. If all three of its dimensions were halved, how much smaller would its new surface area be?
A
the original surface areaB
the original surface areaC
the original surface areaD
the original surface area9. What is the largest volume of a cube that can be constructed with surface area of 726 square units?
A
4356 cubic units
B
2178 cubic units
C
1331 cubic units
D
726 cubic units
Multiple-Choice Answer Key:
1. D
2. A
3. D
4. A
5. A
6. C
7. D
8. B
9. C
Short-Answer Items:
10. An appliance company needs to wrap a refrigerator of size 4 ft. by 3 ft. by 3 ft. in a plastic covering. What is the least amount of plastic covering material needed to cover the refrigerator?
11. A materials cost estimate for a rectangular prism-shaped building is $280 per square meter of surface area. The building’s dimensions are 22m by 11m by 15m. What is the estimate for the total materials cost?
12. What is the largest possible volume of a cube that has 150 square feet of surface area?
Short-Answer Keys and Scoring Rubrics:
10. An appliance company needs to wrap a refrigerator of size 4 ft. by 3 ft. by 3 ft. in a plastic covering. What is the least amount of plastic covering material needed to cover the refrigerator?
2(4 × 3) + 2(4 × 3) + 2(3 × 3) =
2(12) + 2(12) + 2(9) =
66 square feet
Points
Description
2
- Student calculates 66.
- Student’s supporting work includes appropriate arithmetic operations.
1
- Student provides correct solution, 66, without appropriate supporting work or calculations, OR
- Student’s supporting work includes finding at least one face equal to 15 sq. ft.
and one face equal to 9 sq. ft.
0
Student provides no appropriate supporting work or calculations.
11. A materials cost estimate for a rectangular prism-shaped building is $280 per square meter of surface area. The building’s dimensions are 22m by 11m by 15m. What is the estimate for the total materials cost?
22 × 11 × 2 = 484
22 × 15 × 2 = 660
15 × 11 × 2 = 330
484 + 660 + 330 = 1474 square meters
1474 square meters × $280 per square meter = $412,720
Points
Description
2
- Student calculates 412,720.
- Student’s supporting work includes appropriate arithmetic operations.
1
- Student provides correct solution, 412,720 without appropriate supporting work or calculations, OR
- Student’s supporting work includes finding correct number of square meters, 1474.
0
Student provides no appropriate supporting work or calculations.
12. What is the largest possible volume of a cube that has 150 square feet of surface area?
150 square feet ÷ 6 faces = 25 square feet per face
= 5 feet per side (square)53 = 125 cubic feet
Points
Description
2
- Student calculates 125.
- Student’s supporting work includes appropriate arithmetic operations.
1
- Student provides correct solution, 125, without appropriate supporting work or calculations, OR
- Student’s supporting work includes finding square faces each of whose area equals 25 square feet.
0
Student provides no appropriate supporting work or calculations.
Performance Assessment:
Pair students and have them choose and complete one of the following:
A. Create a presentation modeling a real-world problem involving surface area and volume. The problem may use building spaces for business or residential use, familiar shapes of objects in common use such as furnishing, transportation, computing, entertainment, industry, or education. You may include questions of cost and how it limits the design or use of any objects. In your presentation, use comparisons of volume and surface area to support the way an object could or should be designed, and how changes in the design affect either volume or surface area or both.
B. Create a resource kit, portfolio, or set of teaching materials for younger children to learn about nets and their solids. Focus on the visualization of nets as two-dimensional objects and how they can be visualized as solids. In the same way, show how you can visually describe solids as nets. Include some step-by-step directions and instructions to teach children how to make a net from a solid and how to make a solid object from a net. You are only required to include cubes and rectangular prisms, but you may address other solids as well, including pyramids, cones, and cylinders.
Performance Assessment Scoring Rubric:
A. Answers will vary.
Points
Description
4
- Student creates a real-world problem that models surface area and volume.
- Student compares volume and surface area to show how the object could be designed.
- Student accurately calculates surface area and volume.
- Student demonstrates how changes in the design affect the volume, surface area, or both.
- Student demonstrates advanced understanding of the mathematical ideas and processes related to surface area and volume.
- Student’s written comparison is thorough, detailed, and insightful.
- Student worked beyond the problem requirements, possibly by checking steps and/or incorporating technology.
3
- Student creates a real-world problem that models surface area and volume.
- Student compares volume and surface area to show how the object could be designed.
- Student accurately calculates surface area and volume.
- Student demonstrates how changes in the design affect the volume, surface area, or both.
- Student demonstrates substantial understanding of the mathematical ideas and processes related to surface area and volume.
- Student’s written comparison is thorough.
- Student meets all of the problem requirements.
2
- Student creates a real-world problem that models surface area or volume.
- Student compares volume and surface area to show how the object could be designed.
- Student accurately calculates the surface area or the volume.
- Student has difficulty demonstrating how changes in the design affect the volume, surface area, or both.
- Student demonstrates some understanding of the mathematical ideas and processes related to surface area and volume.
- Student’s written comparison is brief.
- Student partially meets the problem requirements.
1
- Student creates a real-world problem but it is inaccurate.
- Student’s project is disorganized and incomplete.
- There are major errors in the calculation of surface area and volume.
- Student demonstrates substantial lack of understanding of surface area and volume.
- Student’s written comparison is incomplete or has inaccuracies throughout.
- Student does not meet several of the problem requirements.
0
- Student’s real-world problem is unfinished or missing.
- No volume or surface area is calculated by student.
- Student demonstrates a complete lack of understanding of the problem.
- Student’s written comparison is missing.
- Student does not meet any of the problem requirements.
B. Answers will vary.
Points
Description
4
- Student creates a resource kit, portfolio, or set of teaching materials.
- Student correctly shows how nets can be visualized as solids.
- Student includes accurate step-by-step instructions on how to construct a net for a cube or rectangular prism.
- Student demonstrates advanced understanding of the mathematical ideas and processes related to surface area and volume.
- Student works beyond the problem requirements, possibly by checking steps and/or incorporating technology.
3
- Student creates a resource kit, portfolio, or set of teaching materials.
- Student correctly shows how nets can be visualized as solids.
- Student includes basic step-by-step instructions on how to construct a net for a cube or rectangular prism.
- Student demonstrates substantial understanding of the mathematical ideas and processes related to surface area and volume.
- Student meets all of the problem requirements.
2
- Student creates a resource kit, portfolio, or set of teaching materials.
- Student’s organization of step-by-step instructions is disorganized.
- Student demonstrates some understanding of the mathematical ideas and processes related to surface area and volume.
- Student partially meets the problem requirements.
1
- Student creates a resource kit, portfolio or set of teaching materials but not all materials are included.
- Student’s project is disorganized and incomplete.
- Student has major errors in the step-by-step instructions.
- Student demonstrates substantial lack of understanding of surface area and volume.
- Student does not meet several of the problem requirements.
0
- Student’s resource kit, portfolio, or set of teaching materials is missing vital information.
- No instructions are included.
- Student demonstrates a complete lack of understanding of the problem.
- Student does not meet any of the problem requirements.
Final 07/12/2013
