## PreCalculus

Precalculus concepts become clear using multiple representations. Instruction is therefore aimed to show the subject from many angles; illustrations include algebraic, graph-based, and real-world examples and feature the use of the graphing calculator as a calculation and visualization tool on advanced forms of algebraic math.

Students will use polynomial, rational, and algebraic functions to write functions (in standard forms) and draw graphs to solve world problems, to find inverse and composite functions, and to analyze functions and graphs. Course topics start with sets, real numbers, complex numbers, and functions. Then, students will explore properties and graphs of polynomial functions and equations, rational functions, exponential, and logarithmic functions. Additional topics include sequences and series, modeling data with linear and non-linear functions.

Review of Algebra: Review slope, lines, distance-time graphs, closed intervals, and inequalities with an explicit goal of creating a foundation for precalculus and calculus.
Functions and Graphing Functions: Use of graphing calculator; Find domain, range intercepts, zeros, asymptotes; predict shape of graphs; describe transformations; describe relationships; represent functional relationships in formulas, tables, and graphs.
Rational and Polynomial Functions: Given an equation, determine the number and nature of roots, find roots, graph them; describe, generalize, and use basic types of functions; work with properties and mechanics of functions.
Exponential and Logarithmic Functions: Use the laws of exponents and logarithms; define, find, and check inverse functions of logarithmic and exponential functions.
Sequences and Series: Understand and use summation notation; apply the sum formulas for arithmetic series and for finite and infinite geometric series.
Data Analysis: Find linear models using linear regression; decide which model gives a better fit; find a non-linear function to model a data set and explain the parameter of the model.
Media Used:
Incorporating new technology (HP 39/40 mg and TI-Nspire)which would allow for a wider variety of approaches (numeric, graphic, symbolic, and verbal) that appeals to all learners.
Sample Websites:
Jefferson Math Project website: jmap.org
Kuta Software: http://www.kutasoftware.com/free.html
Brunnermath: an immense directory of worksheets, tutorials and activities for precalculus
National Council of Teachers of Mathematics (NCTM) Illuminations: http://illuminations.nctm.org
M&M's web site: http://www.mms.com/
Significant Assignments:

Mathematical Lens: Using photos of amusement park rides, water fountains, landscape, house, etc., students model the functions that the shapes of the objects represent

Families of Graphs: Students identify the domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions. (Use paper and pencil methods and graphing calculators.)

Solve word problems involving applications of logarithmic and exponential functions.

Significant Activities or Projects:

M&M Activity: Using M&Ms, students will generate data to explore exponential growth and decay.

Helicopter Rotors Activity: Using cut-out paper models of short and long rotors, students will compare drop times and model functions.

“Shockwave” activities that illustrate real-time correlations between equations and graphs designed to help students visualize and experiment with the graphs of functions, systems, and complex numbers.

Writing: Explain the use the properties of the number systems and order of operations to justify steps of simplifying functions and solving equations.

Research:

Examples of exponential growth and decay occur often in the real world. In the world of finance, for example, there are savings accounts, mortgages, automobile loans. Population growth and the half-lives of radioactive material also provide numerous real world examples of exponential growth and decay. Students will research on selected real-life situation that models exponential growth or exponential decay. Students will produce data through sampling and experiment, organize the data obtained, and explain what information is revealed by their data (i.e., look for patterns and deviations from patterns.

Sample PBATs:
Modeling Movie Gross Income: Is it Exponential Decay? (Use data on top 10 movies (gross incomes), selected years)
Population Growth Model: U.S. Population from 1790-2000. (Students compare with others to see whose prediction model was most accurate.)
Growth of Investments in Stock Market