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Identify three-dimensional objects generated by rotations of two-dimensional figures.
Clarifications
Clarification 1: The axis of rotation must be within the same plane but outside of the given two-dimensional figure.General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Circle
- Cone
- Cylinder
- Prism
- Pyramid
- Rectangle
- Square
- Triangle
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 5, students identified three-dimensional figures. In Geometry, students visualize and discuss a three-dimensional figure as the result of a two-dimensional figure being rotated about an axis. In later courses, students will use rotations to find the volume of a three-dimensional figure (solid of revolution) using integrals. (MTR.5.1)- Instruction begins with using a boundary line of a two-dimensional figure, including
irregular figures, as the axis of the rotation, then move to an axis outside the two-dimensional figure.
- For example, when rotating a rectangle about one of its boundary lines, a cylinder will be generated. If the rectangle is rotated about a line that is parallel to a side and not touching the figure, a cylinder with a hole down its center (a washer) will be generated.
- For example, when rotating a right triangle about an axis containing one of the legs, a cone will be generated. When rotating a right triangle about the hypotenuse, a double cone will be generated.
- For enrichment purposes, include cases where an axis of symmetry is the axis of rotation;
this will help make the connection of reflections to rotations about an axis.
- For example, if a non-square rectangle is rotated about a line of symmetry, then the three-dimensional figure that would be generated is a cylinder.
- Instruction includes exploring irregular shapes that can be rotated to generate vases or exploring the rotation of a sphere about an axis that is outside to generate a torus.
- Problem types include two-dimensional figures presented on the coordinate plane with vertical and horizontal axis of rotation. When presented on the coordinate plane, students can identify some attributes of the three-dimensional object generated by the rotation like the height or the radius.
- Instruction includes the use of models, such as straws and cardboard, or animations.
Common Misconceptions or Errors
- Students may oversimplify when they try to visualize these rotations at first. To help address this, have students utilize physical models.
Instructional Tasks
Instructional Task 1 (MTR.4.1, MTR.5.1)- Trapezoid DCFE is shown on the coordinate plane below.
- Part A. If the trapezoid is extended to create right triangle EFB, what are the coordinates of point B?
- Part B. If triangle EFB is rotated about line = 1, what figure will it generate?
- Part C. Determine the volume of the generated from Part B.
- Part D. If trapezoid DCFE is rotated about line = 1, describe the figure that is generated.
- Part E. Determine the volume of the generated from Part D.
- Part F. If trapezoid DCFE is rotated about line = 3, what figure will it generate?
- Part G. Determine the volume of the generated from Part F.
Instructional Task 2 (MTR.3.1)
- Describe the figure that would be generated from the result of rotating a circle about a line outside the circle.
Instructional Items
Instructional Item 1- Which real-world object could be used describe the figure generated by rotating a rectangle
about a line that is parallel to a side but not touching the rectangle?
- a. A doughnut
- b. A piece of plastic tubing
- c. An ice cream cone
- d. A shoebox
- e. An egg
Related Courses
This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
Related Access Points
Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.4.AP.2: Identify a three-dimensional object generated by the rotation of a two-dimensional figure.
Related Resources
Vetted resources educators can use to teach the concepts and skills in this benchmark.
Formative Assessments
MFAS Formative Assessments
2D Rotations of Rectangles:
Students are given the coordinates of the vertices of a rectangle and asked to describe the solid formed by rotating the rectangle about a given axis.
2D Rotations of Triangles:
Students are given the coordinates of the vertices of a right triangle and asked to describe the solid formed by rotating the triangle about a given axis.
Student Resources
Vetted resources students can use to learn the concepts and skills in this benchmark.
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
