Interactivate

Cross Sections
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Abstract

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:

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

Teacher Preparation

  • Access to a browser
  • A copy of the worksheet for each student

Key Terms

coordinate planeA 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
graphA visual representation of data that displays the relationship among variables, usually cast along x and y axes.
polygonA closed plane figure formed by three or more line segments that do not cross over each other
polyhedraAny solid figure with an outer surface composed of polygon faces
rigid motionA 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.
rotateTo perform a rotation
rotationA 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

  1. 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.

  2. 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.

  3. Teacher Input

    • Introduce cross sections by leading the class in a discussion.
    • If the students have trouble with the concept, demonstrate with styrofoam objects.

  4. Guided Practice

    • Bring up the Cross Section Flyer activity on an overhead projector to demonstrate how the activity works.
    • 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.

  5. Independent Practice

  6. Closure

    As a class, discuss the findings of the lesson:

    • Review the questions on the worksheet and compare results.
    • Ask students if they now know a better way to predict what a particular cross section will look like.

Alternate Outline

If only one computer is available for the classroom, this lesson can be rearranged in the following way:

  • The teacher may do this activity as a demonstration.
  • The class can work together to answer the questions on the worksheet, with one student controlling the applet on a projector for everyone to see.

Suggested Follow-Up

Students who understand the basic concept of cross sections can explore algebraic equations of conic sections using the Conic Section Flyer.

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