Visualization of Three-Dimensional Figures
Visualization of Three-Dimensional Figures
Objectives
In this unit, students will foster and increase visualization skills. Students will:
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visualize two-dimensional nets as solids and visualize solids as two-dimensional nets.
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classify solids.
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explore relationships between solids.
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calculate surface area and volume.
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use spatial estimation.
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solve real-world problems.
Essential Questions
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What are the properties of solids and how do they relate to one another?
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.
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http://nlvm.usu.edu/en/nav/frames_asid_129_g_4_t_3.html?open=activities&from=category_g_4_t_3.html
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http://nlvm.usu.edu/en/nav/frames_asid_128_g_4_t_3.html?open=instructions&from=category_g_4_t_3.html
- http://nlvm.usu.edu/en/nav/frames_asid_126_g_4_t_3.html?open=instructions&from=category_g_4_t_3.html
Formative Assessment
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View
Multiple Choice Items
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Which of the following has the same number of faces as a square pyramid?
A
rectangular prism
B
triangular prism
C
triangular pyramid
D
hexagonal prism
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Which of the following is not a prism?
A
triangular prism
B
rectangular prism
C
cube
D
square pyramid
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The ratio of the length of the sides of two cubes is 5:7. What is the ratio of their volumes?
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A
25:49
B
35: 245
C
125: 343
D
78125: 16807
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Which is a possible net for a triangular prism?
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A
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B
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C
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D
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Which of the following has the greatest volume?
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A
cone with r = 2 in and h = 5 in
B
rectangular prism with l = 2 in, w = 1 in, and h = 5 in
C
draw Square pyramid with s = 2 in and h = 5 in
D
cylinder with r = 2 in and h = 5 in
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With an increase in the radius, which solid will have a surface area that increases most rapidly?
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A
sphere
B
cone
C
cylinder
D
They will increase at the same rate.
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Which pool has the greatest volume to surface area ratio?
A
Pool A l = 6 ft, w = 7 ft, and h = 8 ft
B
Pool B l = 7 ft, w = 7 ft, and h = 7 ft
C
Pool C l = 8 ft, w = 8 ft, and h = 8 ft
D
Pool D l = 7 ft, w = 8 ft, and h = 9 ft
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What bathroom design would give the least surface area for a bathroom with volume of 720 ft³?
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A
Bathroom A l = 8 ft, w = 11.25 ft, h = 8 ft
B
Bathroom B l = 8 ft, w = 6 ft, h = 15 ft
C
Bathroom C l = 8 ft, w = 7.5 ft, h = 12 ft
D
Bathroom D l = 8 ft, w = 9 ft, h = 10 ft
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Which of the solids has the greatest volume?
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A
cylinder with r = 2 and h = 2
B
cone with r = 2 and h = 2
C
square pyramid with s = 2 and h = 2
D
cube with s = 2
Multiple-Choice Answer Key
1. B
2. D
3. C
4. D
5. D
6. A
7. C
8. D
9. A
Short Answer Items:
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10. Write Euler’s formula and describe its applicability to the study of polyhedra. Give an example.
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11. Choose three solids and draw one net for each.
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12. A right rectangular prism that is not a cube has a surface area of 52 square centimeters. Find the dimensions, length, width, and height of the prism.
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Short-Answer Key and Scoring Rubrics:
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10. Write Euler’s formula and describe its applicability to the study of polyhedra. Give an example.
Points
Description
3
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The student writes the following:
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F + V = E + 2 or a correct variation thereof.
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Euler’s formula is used to represent the relationship that exists between the number of faces, edges, and vertices of a polyhedron. The formula offers the ability to determine the missing properties in any polyhedra. (This is purely a sample response.)
- In examining a triangular pyramid, there are 4 faces, 6 edges, and 4 vertices. 4 + 4 = 6 + 2 because 8 = 8. (This is purely a sample response.)
2
- The student correctly writes two of the three answers.
1
- The student correctly writes one of the three answers.
0
- The student does not provide an answer or incorrectly answers all three parts.
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11. Choose three solids and draw one net for each.
Points
Description
3
- The student correctly draws all three nets.
2
- The student correctly draws two of the three nets.
1
- The student correctly draws one of the three nets.
0
- The student does not provide an answer or incorrectly draws all three nets.
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12. A right rectangular prism that is not a cube has a surface area of 52 square centimeters. Find the dimensions, length, width, and height of the prism.
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Answers will vary; for example, 2 cm by 3 cm by 4 cm:
surface area = 2 ((2 × 3) + (2 × 4) + (3 × 4)) = 52
Points
Description
3
- The response includes the correct calculation of the area of two or three pairs of congruent faces with a total surface area equal to 52.
2
- The response includes the correct calculation of the area of two or three pairs of congruent faces that are not compatible with the dimensions of a rectangular prism, e.g., 2 ((3 × 3) + (1 × 7) + (2 × 5))
1
- The response includes the correct
calculation of the area of two or three pairs of congruent faces
that have total surface area of 48 or 54, e.g.,
2 ((2 × 2) + (2 × 2) + (2 × 8)) or 2 ((1 × 3) + (1 × 6) + (3 × 6))
0
- The response does not include an answer or incorrectly finds a surface area not equal to 52.
Performance Assessment:
You have been asked to write an article titled “Relationships Between Solids” for a national newspaper. You are asked to provide and support five examples throughout the article. Creativity and ingenuity are encouraged.
Performance Assessment Scoring Rubric:
Points
Description
5
- The student writes a thorough and creative article that is mathematically correct, citing and supporting five examples.
4
- The student writes a thorough and creative article that is mathematically correct, citing and supporting four examples.
3
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The student correctly cites and supports only three examples.
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OR
- The student cites four to five examples, but has three to four largely misguided mathematical errors.
2
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The student correctly cites and supports two examples.
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OR
- The student cites three to five examples, but has several mathematical errors (five to seven) throughout.
1
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The student correctly cites and supports one example.
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OR
- The student cites two to five examples, but has many mathematical errors (eight or more) throughout.
0
- The student does not write anything, or does not accurately describe and support any example.
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DRAFT 11/09/2010
