Precalculus

Precalculus is a technology-rich study of functions, graphs, relations and their application in nature and scientific studies. Topics of study include: trigonometry, conic sections, vectors, matrices, analytic trigonometry, Composing functions, rational functions and an introduction to calculus topics. In Precalculus, students work collaboratively and individually on hands-on inquiry based activities and investigations.

Learning Modules

A balance of algebraic, graphical, numerical, and verbal methods is integrated in the following learning modules in order to reinforce comprehension, problem solving and critical thinking skills.

Unit 1 - Trigonometry

In this unit, you will explore how the trigonometric ratios from high school Geometry relate to circles. First, you will explore the height of a rider on a Ferris Wheel and develop precise definitions of sine and cosine using transformational geometry. This precision will lead us to a discussion of a mathematically natural unit of rotational measure, called the radian. You will also explore the graphs of periodic functions, and use them to model new situations and discover important properties of trigonometric functions.

Unit 2 - Conic Sections

In this unit you will study symbolic and graphic representations of the conic sections and apply them to real world situations. The conic sections have interesting properties that make them useful for many real-world applications. For instance, a reflecting surface with parabolic cross-sections concentrates light at a single point. Many buildings also employ conic sections in their design. Architects have various reasons for using these curves, ranging from structural stability to simple beauty.

Unit 3 - Hogsmeade by Vectors

This unit covers vectors in the plane, including component form, vector operations, unit vectors, direction angles, applications of vectors, the dot product, angles between two vectors, and finding vector components. These ideas are developed through exploration and discovery, combined with technology resources and hands-on activities that allow students to manipulate the vectors and see connections between the mathematics content and real-world situations.

Unit 4 - Welcome to the Matrix

In this unit you will study dimensional matrices and their applications. First, you are introduced to networks and the value of matrices in counting routes. Later, the geometric effect of matrix operations—matrix product, matrix sum, and scalar multiplication—are examined, and students come to see, geometrically, that matrix multiplication for square matrices is not a commutative operation, but that it still satisfies the associative and distributive properties A third context for the use of matrices is the study of systems of linear equations.

Unit 5 - Complex Numbers

In this unit, you will explore how complex numbers can be used to generate fractals. "The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel."

-Benoit Mandelbrot (Polish/French mathematician; 1924 - 2010)

Unit 6 - Analytic Trigonometry

In this unit you will explore more deeply how trigonometry applies to music theory. In order to promote a book, you will build a homemade musical instrument and tune the instrument using trigonometric graphs. You will create a presentation to persuade publishing companies to publish your book. The presentation will include excerpts from your book relating the instrument you build to its mathematical connections.

Unit 7 - Composition of Functions

In this unit, you will explore how to compose and decompose functions. Graphic courtesy of M. David Green from his online Java Script course material at https://www.sitepoint.com/function-composition-building-blocks-for-maintainable-code/

Unit 8 - Rational Functions

In this unit you will explore symbolic and graphic representations of rational functions and apply them to real world situations, including carbon dating and approximating the speed of a fictional zombie apocalypse.

Unit 9 - Intro to Calculus

In this unit you look at the two central problems of motion much as Newton and Leibniz did, connecting them to geometric problems involving tangent lines and areas. You will see how the obvious geometric solutions to both problems led to algebraic dilemmas, and how the algebraic dilemmas led to the discovery of calculus. The language of limits, which is used in this brief introduction to describe asymptotes, end behavior, and continuity, will prepare students for future studies.

Precalculus Resources

Additional resouices for the study of Precalculus include:

Precalculus Cheat Sheet - from Indiana Dept. of Education
Khan Academy - Online Precalculus class

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