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Course 3 Unit 6 - Circles and Circular Functions
©2009
This unit focuses on consequences of the complete symmetry of circles. The progression of the treatment of circles began in the coordinate plane in Course 2 Unit 3, Coordinate Methods. Then in Course 2 Unit 7, Trigonometric Methods, the sine, cosine, and tangent functions were defined based on a point on the terminal side of an angle in standard position. This unit draws on and extends the idea of circles to circular motion.
Unit Overview
Overall, this unit
helps students extend their understanding of proof in geometric settings
and broaden their understanding and application of important geometric
and trigonometric concepts. Again, students are frequently asked to explore
relationships, make conjectures, and reason deductively to verify these
conjectures. Synthetic proofs, mainly based on congruent triangles, and
coordinate proofs are constructed and compared. Topics developed include:
properties of chords, tangent lines, and central and inscribed angles
of circles; linear and angular velocity; and functions modeling periodic
change.
Since the rotation of circles is key to
the functioning of wheels, pulleys, and sprockets, applications involving
these devices are drawn upon in the unit. Such circular motion is a special
case of periodic change. The mathematical description of periodic change
uses the sine and cosine functions of trigonometry and a new unit of
angle measure called radians.
Objectives
of the Unit
|
Sample
Overview
The sample material
provided here is the "Looking Back" Lesson for this unit. (Alignment
of Core-Plus Mathematics with the Common Core State Standards)
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 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.)