Course 3 Unit 6 - Circles and Circular Functions

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
  • State, prove, and apply various properties of a line tangent to a circle, central angles, chords, arcs, and radii of a circle
  • State, prove, and apply the Inscribed Angle Theorem and the property that angles that intercept the same or congruent arcs are congruent
  • Analyze situations involving pulleys or sprockets to determine angular velocity and linear velocity
  • Use sines and cosines to model aspects of circular motion and other periodic phenomena using both degrees and radians

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
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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.)

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