Course 2 Unit 3 - Coordinate Methods

In Course 1 Unit 6, Patterns in Shape, students developed the ability to visualize and describe two- and three-dimensional shapes, to represent them with drawings, to examine shape properties through both experimentation and careful reasoning, and to use those properties to solve problems. Topics studied were the triangle inequality, congruence conditions for triangles, special quadrilaterals and quadrilateral linkages, the Pythagorean Theorem, properties of polygons, tilings of the plane, properties of polyhedra, and the Platonic solids.
In Course 2 Unit 2, Matrix Methods, students developed the ability to use matrices to represent and solve problems. Matrices are used in Lessons 2 and 3 of this unit to represent transformations of geometric figures in the coordinate plane. The sample material below is from Lesson 1. (See the descriptions of Course 2 Units.)

Unit Overview
In this unit, students develop an understanding of coordinate methods for representing and analyzing relations among classes of geometric shapes and proving geometric properties. They use coordinates to represent geometric transformations and to understand their effects and that of their compositions. For more information on this unit, see the sample material from pages T161-T161D below.) Technology, and in particular animation of figures, is the context for this unit. This material makes use of the public-domain CPMP-Tools computer software.

Objectives of the Unit
  • Use coordinates to represent points, lines, and geometric figures in a plane and on a computer or calculator screen
  • Use coordinate representations of figures to analyze and reason about their properties
  • Use coordinate methods and programming techniques as a tool to implement computational algorithms, to model rigid transformations and similarity transformations, and to investigate properties of shapes that are preserved under various transformations
  • Build and use matrix representations of polygons and transformations and use these representations to create computer animations

Sample Overview
The sample student material below is from Lesson 1, Investigation 3. In Investigations 1 and 2, students created shapes using geometry software, developed the distance and midpoint formulas, and considered how the slope of lines can be used to determine whether or not lines are perpendicular. This knowledge was then used to identify special triangles and quadrilaterals, such as isosceles triangles and parallelograms. In the third investigation, students learn how to represent circles in a coordinate plane with equations.

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
You will need the free Adobe Acrobat Reader software to view and print the sample material.

How the Geometry and Trigonometry Strand Continues
In Course 2 Unit 7, Trigonometric Methods, students develop an understanding of trigonometric functions and the ability to use right triangle trigonometry to solve triangulation and indirect measurement problems.
     In Course 3 Units 1 and 3, students extend their ability to reason formally in geometric settings. Deductive reasoning is used to prove theorems concerning parallel lines and transversals, angle sums of polygons, similar and congruent triangles and their application to special quadrilaterals, and necessary and sufficient conditions for parallelograms. Circular functions (sine and cosine) are used to model periodic change in Unit 6, Circles and Circular Functions.
     In Course 4: Preparation for Calculus, geometry and algebra become increasingly intertwined. Students develop understanding of two-dimensional vectors and their application and the use of parametric equations in modeling linear, circular, and other nonlinear motion. In addition, students intending to pursue programs in the mathematical, physical, and biological sciences, or engineering extend their ability to visualize and represent three-dimensional surfaces using contours, cross sections, and reliefs; and to visualize and sketch surfaces and conic sections defined by algebraic equations. They also extend their understanding of, and ability to reason with, trigonometric functions to prove or disprove trigonometric identities and to solve trigonometric equations. They also geometrically represent complex numbers and apply complex number operations to find powers and roots of complex numbers expressed in trigonometric form. (See the CPMP Courses 1-4 descriptions.)

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