Preface

"Precalculus is not a spectator sport, so GET INVOLVE!" Pin D. Ling, SUNY, New Paltz, 1998.

This online Precalculus textbook is intended as a supplementary text to Functions Modeling Change - A Preparation for Calculus.

Precalculus: Contemporary Models is designed to provide additional learning tools for college math students that is not available from the typical mathematics hard copy textbook such as:

1. Immediate updates to relevant information

2. Interaction with Instructor out side of the classroom

3. Customized text appropriate to students learning styles / backgrounds

3. Math Resource for students to learn at their own pace

4. Powerful Multimedia learning tools

5. Dynamic updates to homework and exams problems

6. Nonlinear way of learning (an explorative, discovery approach to learning)

7. The power and resource of the entire Internet

Technology and mathematics have become essential tools for most of today's professional jobs and careers. Mathematics in particular helps develop critical skills in Problem Solving, Critical Thinking, Analytical Thinking and Decision Making especially when the problems question is not clear or comes with limited information.

All problems started out as word problems, often phrased in the form of a question or a desire to find a practical solution to a puzzling problem. Problems, therefore are opportunities for progress:

Bridges, college education, airplanes, written textbooks, computers, telecommunication, heart transplants, hot-dogs and apple pies are solutions to word problems. Woman: "What I am going to do with all these apples?" Child: "Bake me a pie!" Therefore, almost all professions need thinkers, people who can find applicable solutions to demanding questions.

Precalculus has a long history that includes mathematical models that were solutions to analytical problems. Most of what we do in science and technology works from principles that can be formulated as mathematical solutions. Precaluculs can be considered practical mathematics modeling relationships between two or more attributes or variables: relationships between temperature and weather forecasting, health and weight, seasonal price changes with time of year, distances and angles, investment and money matters, motion and time, social factors and controlling variables are few of the relationships that are dealt with through Precalculus.

Chapters Links:

This online textbook has the following resources that are linked to each chapter:

1. Precalculus Learning Module : A brief theory page that covers the essentials of each math model / chapter and showed worked out examples for each math model. Students may read ahead and examine worked out examples of various types of problems associated with each math concepts.

2. Video Simulation :These are interactive graphical math game modules that cover mathematics concepts through challenging problems. Students can test their mastery of key concepts my progressively working on chapter problems with hints or solutions provided. This graphics learning tool is a model of a later interactive, animated math learning resource and like a video game, students may play it over and over until mastery is second nature. Students may use this tool to strengthen areas or concepts that they are weak in and always may interact on-line with instructor as student worked through each challenged.

3. Workshop : Students may read ahead and should bring to class a copy of workshop - these are in class cooperative learning exercises; Problems / concepts that the entire class or groups work on together to gain extra prospective on how different students learn and teach the same concepts. Students who work together in groups of no more than 4 or 5 tends to improve their understanding of difficult math concepts. You must complete each workshop and return to instructor for evaluation the Monday after its is introduced in class.

4. Calculator help : Throughout the course we use the graphing calculator to explore mathematical concepts through visualization; therefore extensive resource is provided to individual calculator practice and in class demo and exercise.

5. Formula Summary : A summary of relevant formulas are provided for each mathematics concepts / chapters with graphical illustrations (students may click on images to see larger versions).

6. Worked Out Solutions : Problems are prepared especially for the class to match students background and levels of expectations; therefore at least 20% of assigned problems may have some challenge associated with them - i.e. Students may have to draw from their past knowledge in Algebra and Geometry general math usage to solve problems. All homework, quiz, exam problems are worked out completely and provided as reference and additional learning tools to students immediately after homework is dued and exams are taken (except the final exam). Students who reflect upon or compare instructor's solutions to their have a better chance in mastering the material than students who do not. (I anticpate that there will be 2 out of 100 problem solution posted that will contain some error - typo or otherwise - so I welcome your timely feedback when reviewing solutions to problems)

7. Mathnopoly - A video multimedia Precalculus comprehensive review is provided to students shortly before the final exam to review course material. Students are encouraged to review new materials within 1 to 2 hours after each lecture and several hours each day since this course covers at least 34 math concepts at a maddening pace.

Mathnopoly is copyright, 2001, 2002, 2003 Pindling.org

Problems Types:

Precalculus: Contempary Models explore mathematical models in 5 dimentions:

1. Word problem (80+% of the course): Problems stated in word form; students must learn to discovery from problem phrase what mathematical models are best solutions to the problems. A suggested approach is to rephrase or paraphrase each problem asking: What is given to define applicable mathematics models and act as inputs to problem formulation? and what solutionis been sought? 

2. Graphs: A graph is a visual or picture that often identify specific mathematical models or combinations of such. Students are asked to learn the unique or particular visual graphs of appropriate functions or mathematical models. It is suggested that students illustrate all problems when appropriate graphically at the start of each problem solution.

3. Tables: A table is often a good way of showing relationships between attributes or variables by listing pairs or values that are related to each other. Often it is best to plot of graph the related values in the table to get a help identify visually what is been modeled.

4. Formula: Most students coming into college mathematics often learn mathematics by first learning the formula or equations of a math concept and work with manipulations of equations to find unknow(s). But this course at this level will take a more complex approach to mathematical modeling - Students must be able to identify a math problems / solution and at least the first 4 models above: Word Problem, Graphical Problems, Pair-wise Tables of valuse, and Formulas.

5. Multiple Intellengence Mapping: Ways of using innovative musical, visual, multiconecptual methodolgies to learning through one's special learning styles (This area is a new approach and will be developed through interactions with students as they learn mathematics).

Table of Contents 

Introduction to Functions 
Linear Models 
Domain and Ranges 
Exponential Models 
Logarithms 
Piecewise Functions 
Quadratic Functions 
Periodic Functions 
Trigonometry 
Transformation / Composition 
Inverse of Functions 
Polynomials 
Rational Functions 

Teaching Strategies: 

The following techniques used in teaching this course will help your understanding or organization of the topics or concepts covered: the expanded syllabus, Textbooks and its aids, web resource, handouts, student-teacher relationship, office hours, multi-sensory approach (lecture, visual, workshops), Reviews of Material before and after lectures, repetition, attention getting techniques, Questions, projects, opportunity for previewing, Personalize information ( examples used), quizzes, homework, mnemonics, lecture outlines, organization of lectures, group discussions, concepts presentation, use of chalkboard, building lectures on prior knowledge, use of analogies to clarify new information, time limits for tests, presentation language level, definitions of key terms, group collaborations, worked out solutions on web, technology based lectures / calculator demonstrations, Precalculus by examples on web and any other techniques used to facilitate learning. 

Personal Challenge

For some of you this textbook and the course it covers will be a personal challenge as you develop your problem solving skills. Many non mathematicians have been great problem solvers and decision makers whether they took a similar math course or not one thing is certain, they have learned to recognize their special talents and have hone in on personal strategies that have worked for them under tried and proven circumstances.

How can we in a short semester make you a better problem solver? We must provide you with challenging problems to solve, lots of them! You will get out of this text what you put into it; there is no short cut to becoming the best or developing your problem solving skills to its optimal potentials.

                               "Heights that great men 
                                    reached and kept. 
                                 Were not attained by 
                                      sudden flight. 
                                  But they while their 
                                   companions slept. 
                                 Were toiling upwards 
                                  through the night." 

Edited: 1/24/2003