MA.912.GR.4.2

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 x = 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 x = 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 x = 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
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

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

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.

Type: Formative Assessment

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.

Type: Formative Assessment

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

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Parent Resources

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