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
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
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.
of the Unit
three-dimensional objects and surfaces with contour lines
or horizontal and vertical cross sections
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
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
and sketch surfaces of revolution and cylindrical surfaces
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.
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.
Unit Table of Contents and Sample Lesson Material
You will need the
Acrobat Reader software to view and print the sample material.