# MA.912.GR.4.1

Identify the shapes of two-dimensional cross-sections of three-dimensional figures.

### 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.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• 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.

• 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.
• 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

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.1: Identify the shape of a two-dimensional cross section of a three-dimensional figure.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Inside the Box:

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

Type: Formative Assessment

Slice of a Cone:

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

Type: Formative Assessment

Slice It:

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

Type: Formative Assessment

Mudslide:

Students are asked to create a model to estimate volume and mass.

Type: Formative Assessment

Square Pyramid Slices:

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

Type: Formative Assessment

Rectangular Prism Slices:

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

Type: Formative Assessment

Cylinder Slices:

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

Type: Formative Assessment

Cone Slices:

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

Type: Formative Assessment

Working Backwards – 2D Rotations:

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

Type: Formative Assessment

## Lesson Plans

Exploring Cavalieri's Principle:

Students will explore Cavalieri's Principle using technology. Students will calculate the volume of oblique solids and determine if Cavalieri's Principle applies.

Students will also perform transformations of a base figure in a 3-dimensional coordinate system to observe the creation of right and oblique solid figures. After these observations, students will create a conjecture about calculating the volume of the oblique solids. Students will use the conjecture to determine situations in which Cavalieri's Principle applies and then calculate the volume of various oblique solids.

Type: Lesson Plan

Plane Slice:

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Students will use modeling clay to explore the cross sections that result from slicing a 3-dimensional figure.

Type: Lesson Plan

Can You Cut It? Slicing Three-Dimensional Figures:

In this lesson, students will sketch, model, and describe cross-sections formed by a plane passing through a three-dimensional figure. Students will create a cube, right rectangular prism, and right rectangular pyramid using modeling clay dough, and then slice the model using parallel, perpendicular, and intersecting lines. Students will describe the two-dimensional figure resulting from slicing the three-dimensional model.

Type: Lesson Plan

## Original Student Tutorial

Ninja Nancy Slices:

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

## Perspectives Video: Professional/Enthusiasts

Reflections, Rotations, and Translations with Additive Printing:

<p>Transform your understanding of 3D modeling when you learn about how shapes are manipulated to arrive at a final 3D printed form!</p>

Type: Perspectives Video: Professional/Enthusiast

3D Modeling with 3D Shapes:

<p>Complex 3D shapes are often created using simple 3D primitives! Tune in and shape up as you learn about this application of geometry!</p>

Type: Perspectives Video: Professional/Enthusiast

Global Positioning System II:

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.

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

## MFAS Formative Assessments

Cone Slices:

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

Cylinder Slices:

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

Inside the Box:

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

Mudslide:

Students are asked to create a model to estimate volume and mass.

Rectangular Prism Slices:

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

Slice It:

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

Slice of a Cone:

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

Square Pyramid Slices:

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

Working Backwards – 2D Rotations:

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

Ninja Nancy Slices:

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

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Ninja Nancy Slices:

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

Global Positioning System II:

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.

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

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.