Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
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
- 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 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
- 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)
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
- Describe the figure that would be generated from the result of rotating a circle about a line
outside the circle.
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