### Clarifications

*Clarification 1*: The axis of rotation must be within the same plane but outside of the given two-dimensional figure.

**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 figure 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 figure 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 figure generated from Part F.

Instructional Task 2 (MTR.3.1)

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

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

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Problem-Solving Task

## MFAS Formative Assessments

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.

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

## Problem-Solving Task

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

## Problem-Solving Task

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