Course 4 Unit 6 - Surfaces and Cross Sections

In Course 1 Unit 6, Patterns in Shape, students represented three-dimensional objects in two dimensions using perspective and orthographic (faceview) drawings. This three-dimensional work along with coordinate representations developed in Course 2 Unit 3, Coordinate Methods, and algebraic reasoning and the ability to write algebraic equations developed in many units provides the background knowledge for this unit. (See the CPMP Courses 1-4 descriptions.)

Unit Overview
Unit 6, Surfaces and Cross Sections, focuses on three-dimensional surfaces and how they may be represented so that a good deal of information may be inferred about the shape of the surface and it can be mentally visualized. Surfaces may be represented in a number of ways. Contour diagrams and topographic profiles introduced in this unit are often used when the object or surface is too complex to be easily and effectively represented in those manners.
     The first lesson introduces students to contour diagrams, to a three-dimensional rectangular coordinate system, and to topographic profiles (vertical cross sections of terrain). The second lesson develops some of the basic geometric concepts in a three-dimensional coordinate system—distance, midpoint, symmetry, and graphs of equations. The ideas are developed as analogs of similar ideas in a two-dimensional coordinate system.
     Overall, this unit helps students to visualize and to algebraically and graphically represent surfaces. Its content draws heavily on, and adds to, students' previously developed connections between algebra and geometry.

Objectives of the Unit
  • Represent three-dimensional objects and surfaces with contour lines or horizontal and vertical cross sections
  • Interpret and describe three-dimensional surfaces or objects represented with contour diagrams
  • Use the three-dimensional coordinate system to locate points and represent data, objects, and surfaces in space
  • Identify and sketch graphs of conic sections represented algebraically and write equations matching graphs of conics
  • Use information revealed by the form of an equation of a three-dimensional surface to visualize, characterize, and sketch the surface
  • Identify and sketch surfaces of revolution and cylindrical surfaces

Sample Overview
The sample investigation below is Investigation 3 of Lesson 2. In this investigation, students learn to find intercepts, symmetry, traces, and other cross sections formed by planes parallel to the coordinate planes from equations of surfaces. They then use this information to determine the type of surface corresponding to the equation. Students visualize and in some cases sketch these shapes. In the final investigation of this lesson, students identify and sketch surfaces of revolution and cylindrical surfaces.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
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