### Clarifications

*Clarification 1*: Instruction includes the use of manipulatives and models to visualize cross-sections.

*Clarification 2*: Instruction focuses on cross-sections of right cylinders, right prisms, right pyramids and right cones that are parallel or perpendicular to the base.

**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 the elementary grades, students classified two- and three-dimensional figures. In middle grades, students worked with nets of three-dimensional figures when determining surface area. In Geometry, students begin to understand the concept of the two-dimensional cross-sections of familiar three-dimensional figures in preparation to discuss the formulas for the volume of these figures. In later courses, students will relate cross-sections to conic sections and will use cross-sections as a basis for finding the volumes of other three-dimensional figures.- For the purposes of this benchmark, there is no expectation for students to master cross-sections of non-right, or oblique, three-dimensional figures. Instruction can include these as an enrichment or as comparison to cross-sections of right figures. As an additional enrichment, composite figures can also be utilized within instruction.
- Instruction includes the student understanding that among the shapes of the two-dimensional cross-sections can be circles, rectangles, squares, triangles, trapezoids and all other polygons that can be used as the base of right prisms and right pyramids (when the plane is parallel or perpendicular to the bases).
- When the vertical cross-section of a right cone does not include the apex, it is one piece of a hyperbola. Since many students at this level are not familiar with hyperbolas, the expectation is not to name the hyperbola, but be able to draw or visualize this cross-section. For enrichment, instruction may include showing that it is not a parabola, since it has diagonal asymptotes.
- Instruction focuses on pyramids with bases that are either equilateral triangles or squares, with the vertical cross-sections being parallel to a side of the base so that the vertical cross-sections are isosceles triangles and isosceles trapezoids. Since vertical cross-sections of pyramids include triangles that are not isosceles, and a variety of irregular polygons it may be difficult for students to visualize or name.
- Instruction includes utilizing objects, such as soda cans, cereal boxes or party hats, as
models to explore their cross-sections. Additionally, students can explore other cross-sections using manipulatives such as clay and string to cut through the three-dimensional
figure.
*(MTR.7.1)*

### Common Misconceptions or Errors

- Students may oversimplify when they try to visualize cross-sections. To help address this misconception, use real-world three-dimensional figures to explore their cross-sections, as well as animations.
- Students may have difficulty with vertical cross-sections of pyramids and cones. To help address this, utilize manipulatives and physical models within instruction.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1, MTR.4.1)*

- Part A. Draw and name right three-dimensional figures that could have a triangular cross-section.
- Part B. Draw and name right three-dimensional figures that could have a circular cross-section.
- Part C. Compare your answers from Parts A and B with a partner.

*Instructional Task 2 (*

*MTR.3.1*)- Part A. Fill in the blank below. Both a right cylinder and a right prism have _______ cross-sections when cut perpendicular to the base.
- Part B. Draw some cross-sections that are perpendicular to the base for each figure below.

### Instructional Items

*Instructional Item 1*

- Which of the following polygons are cross-sections that are parallel or perpendicular to the
base of a regular pentagonal pyramid? Select all that apply.
- a. Triangle
- b. Parallelogram
- c. Trapezoid
- d. Pentagon
- e. Hexagon
- f. Octagon

**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

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Professional/Enthusiasts

## Problem-Solving Tasks

## MFAS Formative Assessments

Students are asked to sketch and describe the two-dimensional figures that result from slicing a cone.

Students are asked to sketch and describe the two-dimensional figures that result from slicing a cylinder.

Students are asked to identify and draw cross sections of a rectangular prism and to describe their dimensions.

Students are asked to sketch and describe two-dimensional figures that result from slicing a rectangular prism.

Students are asked to identify and describe two-dimensional cross sections of three-dimensional solids.

Students are asked to sketch, describe, and compare three horizontal cross sections of a cone.

Students are asked to sketch and describe the two-dimensional figures that result from slicing a square pyramid.

Students are given a solid and asked to determine the two-dimensional shape that will create the solid when rotated about the *y*-axis.

## Original Student Tutorials Mathematics - Grades 9-12

Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial.

## Student Resources

## Original Student Tutorial

Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Tasks

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Type: 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 Tasks

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Type: 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