This lesson utilizes the concepts of cross-sections of three-dimensional models to demonstrate the
derivation of two-dimensional shapes.

Objectives

Upon completion of this lesson, students will:

understand the concept of cross-sections

gain experience manipulating polygons, ellipses, parabolas, and hyperbolas

learn the difference between ellipses, parabolas, hyperbolas, and circles as they relate to conic sections

gain an intuitive understanding of the relationship between cross sections of three-dimensional objects and two-dimensional figures

discover the relationships between the number of faces of a three-dimensional figure and its two-dimensional cross-sections

Standards Addressed:

Grade 10

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Geometry

The student demonstrates an understanding of geometric relationships.

The student solves problems (including real-world situations).

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 9

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Geometry

The student demonstrates an understanding of geometric relationships.

The student solves problems (including real-world situations).

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Geometry

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section

Geometric Measurement and Dimension

Visualize relationships between two-dimensional and three- dimensional objects

Modeling with Geometry

Apply geometric concepts in modeling situations

Grades 9-12

Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Use visualization, spatial reasoning, and geometric modeling to solve problems

Geometry

Geometry and Measurement

Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Geometry

Algebra

Students will recognize, extend, create, and analyze a variety of geometric, spatial, and numerical patterns; solve real-world problems related to algebra and geometry; and use properties of various geometric figures to analyze and solve problems.

Geometry

Students will investigate, model, and apply geometric properties and relationships and use indirect reasoning to make conjectures; deductive reasoning to draw conclusions; and both inductive and deductive reasoning to establish the truth of statements.

Geometry

Dimensionality and the Geometry of Location

6. The student analyzes the relationship between three-
dimensional geometric figures and related two-dimensional representations and uses these
representations to solve problems.

7. The student understands that coordinate systems
provide convenient and efficient ways of representing geometric figures and uses them
accordingly.

Geometric Patterns

5. The student uses a variety of representations to describe geometric
relationships and solve problems.

Geometric Structure

4. The student uses a variety of representations to describe geometric
relationships and solve problems. The student is expected to select an appropriate representation
(concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

Secondary

Geometry

G.12 The student will make a model of a three-dimensional figure from a two-dimensional drawing and make a two-dimensional representation of a three-dimensional object. Models and representations will include scale drawings, perspective drawings, blueprints, or computer simulations.

Student Prerequisites

Geometry: Students must be able to:

identify and describe two-dimensional figures

identify and describe three-dimensional objects

Algebra: Students must be able to:

work with two-dimensional graphs

Technology: Students must be able to:

perform basic mouse manipulations such as point, click, and drag

use a browser for experimenting with the activities

A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

polygon

A closed plane figure formed by three or more line segments that do not cross over each other

polyhedra

Any solid figure with an outer surface composed of polygon faces

rigid motion

A rigid motion, of the plane or of space, is one that keeps the distances between all pairs of points unchanged. Rotations, reflections and translations are examples of rigid motions.

rotate

To perform a rotation

rotation

A rotation in the plane is a rigid motion keeping exactly one point fixed, called the "center" of the rotation. Since distances are unchanged, all the other points can be thought of as having moved on circles whose center is the center of the rotation. The "angle" of the rotation is how far around the circles the points travel. A rotation in three-dimensional space is a rigid motion that keeps the points on one line fixed, called the "axis" of the rotation, with the rest of the points moving some constant angle around circles centered on and perpendicular to the axis.

Lesson Outline

Focus and Review

Ask the following opening questions:

If you place a cone on the table and cut a slice that is parallel to the table, what will that
face look like?

Use a styrofoam cone, if helpful for the students

Ask the students to sketch what it might look like

After students sketch the resulting cross section, slice the cone to see if they are
correct.

As a class, discuss how you can predict what a particular cross section will look like.

Have students explain their method and why it should work.

Ask students if their methods would work for non-cone objects like prisms or pyramids.

Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

Today, we are going to learn how to find the two-dimensional cross section of a
three-dimensional object.

We are going to use computers to visualize these cross sections, but please do not turn your
computers on or go to this website until I ask you to. I want to show you a little about this
first.

Teacher Input

Introduce cross sections by leading the class in a
discussion.

If the students have trouble with the concept, demonstrate with styrofoam objects.

Explain that the diagram on the left side shows a three-dimensional object that is being
"sliced" by a two-dimensional plane to form a cross-section.

Explain how the cross section is then shown in two dimensions in the graph on the right side.

Walk the students through the applet, showing how to move the slicing plane and how changes in
the slicing plane affect the cross section shown on both the three-dimensional object and the
two-dimensional graph.