Getting Started |
| Misconception/Error The student is unable to envision the three-dimensional shape that results and only describes a two-dimensional shape. |
| Examples of Student Work at this Level The student writes the coordinates of points of the image reflected about the given axis.
The student writes a statement such as:
- The solid is a rectangle.
- The solid is a square.
- It would form a circle.
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| Questions Eliciting Thinking What is a three-dimensional shape or solid?
How is a three-dimensional solid formed from the rotation of a two-dimensional figure? |
| Instructional Implications Be sure the student understands the distinction between two- and three-dimensional figures. Discuss a variety of real world examples of rotational motion to help the student visualize three-dimensional solids, e.g., a spinning quarter, a hand mixer, or an airplane propeller.
Help the student visualize the solids formed by attaching the edge of a rectangle to the side of a thin straw. Have the student hold the straw horizontally while slowly, and then more quickly, rolling the straw between his or her palms, modeling the desired rotation. The student can repeat this exercise with a variety of two-dimensional shapes. Have the student identify the solid formed by the rotation of each of these shapes.
Demonstrate the solids formed by the rotation of two-dimensional shapes by using interactive websites such as “3D Transmographer” (shodor.org). |
Moving Forward |
| Misconception/Error The student is able to visualize one or both of the solids of rotation as three-dimensional, but identifies and/or describes one or both of the solids incorrectly. |
| Examples of Student Work at this Level The student identifies the first shape as a cylinder but gives the wrong dimensions.
The student writes a statement such as:
- The solid is shaped like a cone or 3D circle.
- The solid is a 3D shape.
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| Questions Eliciting Thinking Describe to me what this solid looks like.
Can you describe the base of the solid?
What are the dimensions of the solid?
How do the dimensions of the solid differ when rotated around the y-axis versus the x-axis?
Describe the base and height of a cylinder.
To what point on the x-axis does the base of the solid extend? |
| Instructional Implications Help the student visualize the solids formed by attaching the edge of a rectangle to the side of a thin straw. Have the student hold the straw horizontally while slowly, and then more quickly, rolling the straw between his or her palms, modeling the desired rotation. Have the student repeat this exercise with a variety of two-dimensional shapes whose dimensions are given. Have the student identify the solid formed by the rotation of each of these shapes, also describing their dimensions. |
Almost There |
| Misconception/Error The student struggles to identify the solid formed by rotating the rectangle around a line that does not contain one of its sides or describes in incorrectly. |
| Examples of Student Work at this Level The student writes a statement such as, “The solid is a cylinder.”
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| Questions Eliciting Thinking Can you describe the solid formed by this rectangle rotating about the y-axis?
What are the dimensions of the cylinder that you are visualizing?
How would the solid formed by the rotation of the rectangle differ if the rectangle were shifted two units to the left before it was rotated?
Can you describe the shape of the “hole” in the cylinder? |
| Instructional Implications Demonstrate the solids formed by the rotation of two-dimensional shapes by using interactive websites such as “3D Transmographer” (shodor.org). Then provide the student with additional problems similar to those in this task in which the student must visualize and describe the result of the rotation.
Consider implementing MFAS task 2D Rotations of Triangles (G-GMD.2.4). |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student writes a statement such as, “The solid in question 1 is a cylinder with a height of 2 units and a base with a diameter of 6 units. The solid in question 2 is a cylinder with another smaller cylinder removed from its interior. The larger cylinder has a height of 3 units and a circular base with a diameter of 8 units. The smaller cylinder has a height of 3 units and a base with a diameter of 4 units.
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| Questions Eliciting Thinking Can you describe the solid formed by rotating the same rectangle about the line x = 1, y = -1, or y = 1? |
| Instructional Implications Review the formulas for volume of a cylinder. Have the student compute the volumes of both solids in this task.
Challenge the student to describe and draw various horizontal and vertical cross-sections of each of the solids. |
