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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 figure 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 figure 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 figure 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 to 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
Problem-Solving Task
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.
Problem-Solving Task
Tennis Balls in a Can:
This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder
Type: Problem-Solving Task
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
Problem-Solving Task
Tennis Balls in a Can:
This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder
Type: Problem-Solving Task
