Contemporary College Algebra:
Data, Functions, Modeling

A reformed college algebra course focused
on meeting the quantitative needs of students
for academic, workplace and society.

The text which follows is the Preface to Contemporary College Algebra: Data, Functions, Modeling, By Don Small, The United States Military Academy, West Point, New York, published by McGraw-Hill Primis Custom Publishing, © 2002 by The McGraw-Hill Companies. Editorial Board members are listed in the left hand sidebar.

Textbook and a companion Interactive CD Rom are available from McGraw-Hill or, for assistance, contact Don Small directly.

"The mind is not a vessel to be filled, but a flame to be kindled."
— Plutarch

Primary Goal

Our philosophy is to educate students for the future rather than train them for the past. We therefore have made a conscious effort to incorporate into our goals the common qualifications for entering the work force as enunciated by social, business, and industrial leaders.

The primary goal of this text is to empower students to become exploratory learners, not to master a list of algebraic rules. Each section contains Queries that engage students in questioning and exploring the material being presented. Exercises that explicitly ask students to explore, ask what-if type questions, make up examples, further investigate worked examples, iterate for the purpose of recognizing a pattern and developing a sense for the behavior of a solution, and graphically fit a curve to a data set are some of the means that are used to establish an exploratory environment for the students.

Other Goals of this Text

1. Improve communication skills - reading, writing, presenting, listening.
      The large majority of the exercises are presented in the story problem format in order to address the reading aspect of this goal. The story problem format also addresses the applicability aspect of college algebra as real-life situations are usually described verbally or in written form rather than in terms of equations.
   
   
2. Small group work - in-class group activities and out-of-class group projects. In-class activities culminate in student presentations to the class and out-of-class projects culminate in both a written report and a student presentation.
   
   
3. Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.
4. Modeling - to empower students to use mathematics to quantify real-life situations.
   
   
5. Confidence - develop personal confidence as a problem solver. Develop confidence in the iterative process: ``try something, note the errors, modify previous attempt to lessen the errors, and try again'' until a satisfactory approximation has been obtained. The initial attempt is usually informed by sketching a picture.
6. Enjoy applying mathematics to meaningful situations.

This text is to be read, studied, and annotated. Students should study the worked examples for the purpose of understanding the concepts and reasoning involved. Students are expected to personalize their text by filling in missing details, making up examples and illustrations, and raising questions. The purpose of the exercises is to help clarify and expand the reasoning process. As such, working exercises is secondary in importance to studying the written material in the sections.

Real-world Contexts

Concepts and techniques are introduced and motivated by real-life situations. Computational techniques are introduced in response to the need to solve real-life situations. For example, the quadratic formula is introduced in Chapter 4 in order to solve motion problems that involve quadratic equations. The ability to understand elementary data analysis, to extract function relations from data, and to mathematically model real-life situations in different disciplines is fundamental to the liberal arts education of every student.

Fun Projects

Fun Projects are small group (three to five students) out-of-class projects. The projects are designed for six to ten hours of work and culminate in a written report. Instructors are encouraged to assign two or three projects during the course. The purpose of the projects are to provide opportunities

1. Mathematically model real-world situations.
   
   
2. Research a topic (usually on the Internet)
   
   
3. Provide a writing assignment
   
   
4. Provide a small group experience
   
   
5. Have fun exploring and creating solutions to meaningful problems
   
   

Cover Page (creative design by students) Title Page (project name, date, instructor name, students' names) Executive Summary (one page abstract of the problem, approach used, results obtained) Supporting Data (computations, labeled drawings, labeled computer plots and/or printouts) Group Log (time, date, location, and brief description of each meeting) Evaluation Summary of the group's learning experience in working on the project List of references consulted

All group members should be involved in answering each of the questions. In addition, each member of the group should be assigned a particular responsibility in connection with the project, such as one of the following:

Leader: Responsible for developing the group. Responsible for seeing that the project is completed in a satisfactory manner and on time.

Recorder: Arranges group meetings and records group activities.

Checker: Checks accuracy of all computations. Checks to see that all questions are answered.

Typist: Types Executive and Evaluation Summaries.

Reader: Responsible for proofreading and final assembly of the report.

A Few Suggestions to Students

1. Be an exploratory learner: sketch pictures, question, create what-if questions, make up examples, question the reasonableness of results, look for applications.
   
   
2. Read with a pencil in hand. Make your text useful to you by using the margins or additional paper to write explanatory notes, questions, fill in missing computations, make up additional worked problems, and so on. When you personalize your text by augmenting it, you transform it into an effective learning tool for you.
   
   
3. Do not get bogged down in computations. The course is about applying mathematics to real-world situations, not about computations.
   
   
4. Be patient and persevere in your studying. Focus on understanding the reasoning in the worked examples.
   
   
5. Answer the Queries.
   
   
6. Make up examples and what-if exercises.
   
   

Chapter Content

The analysis of data is the starting point for most of the topics in this text. The analysis of data motivates the concept of function for the purpose of drawing predictions from data. Just as data is displayed differently for varied purposes, functions are represented differently (graphically, symbolically, numerically, and verbally) to address varied concerns. The ability to graphically approximate a data set is a key skill in applying mathematics. A strong foundation in elementary data analysis and the function concept prepares the student to model real-life situations.

We study how to read and display data: table, pie chart, scatter and line plots, and bar charts. We learn the meanings, use, and methods to compute the three principle summary measures of a set: average (mean), median, and mode. Our understanding that data is information about a variable, introduces us to an understanding of the meaning of variable and its use as a mathematical pronoun. The exploration of relations between variables leads to the study of straight lines, a fundamental concept in the application of mathematics to real-life situations. Applications of linear equations lead naturally to systems of linear equations, linear inequalities, and their applications in linear programming.

The concept of a function is one of the most important concepts in mathematics. The concept is developed informally through discussion of academic grades, modeling water level in a well, and warming a can of soda. Definitions of a function and related terms are clearly presented and illustrated. Graphically extracting functional relations from data introduces the shapes of the basic functions: power, radical, exponential, logarithmic, and periodic (sine, cosine). The skill to graphically fit a curve to a data plot is enhanced by studying the basic graph transformations of shifting and scaling. The algebra of functions (addition, multiplication, composition, and inverses) is developed graphically, symbolically, and numerically. The ability to display data and to graphically approximate numerical solutions of equations and zeros of functions is an important thread throughout the text. The development of symbolic approximation of data (that is, regression analysis) completes the chapter.

College algebra is a college or university program in the sense that it or an equivalent course is required by all disciplines. Therefore an appropriate goal of \textit{Contemporary College Algebra} is to prepare students to mathematically model real-life situations arising in different disciplines. To illustrate the breadth of the applicability of college algebra, the focus of Chapter 4 is on modeling problems in business, physical and life Sciences, and the arts. The primary modeling techniques are graphical approximations and recursive sequences. The recursive sequence model developed on the reasoning

is applicable across the disciplines. In particular, the recursive sequence model of the accumulation of money in a savings account serves as paradigm for most of the discrete models developed in Contemporary College Algebra.

Exercises

There is a rich assortment of exercises at the end of each section to augment the Queries and worked examples in the section. The purpose of the exercises is to support and expand the conceptual understanding of the material. As such, several of the exercises refer to worked examples in the section. Many of the exercises are suitable for small group in-class activities or small group out-of-class projects.

Labels

Queries are numbered consecutively within sections. Examples, figures, and tables have 3-place labels, the first location denotes the chapter, the second location denotes the section, and the third location denotes the example, figure, or table within the section. For example, Figure 3.4.6 is the sixth figure in section 4 of chapter 3 and Table 3.FunProject2.4 denotes the fourth table in Fun Project number 2 in chapter 3.

Computational Skill

Development of computational skill aided by the calculator and/or computer is an expected outcome of studying this text. Computational skill, including approximation of answers and checking the reasonableness of answers, will develop by working on real-life problems. Each section's exercise set begins with three Computational Skill exercises, the third of which is to make up three additional exercises similar to the first two. The purpose is to highlight the important computational techniques used in the section and to encourage students to ask what-if type questions. Appendix A on Computational Skills and Basic Functions is included for reference. Time should not be spent on drilling to master hand computation skills, such as factoring. In educating students to become contributing and productive members of society, we need to engage them in the use of the computational tools used in society - graphing calculators and computers.

The following table indicates where important algebraic or arithmetic techniques are introduced in the text.

Let us begin an exciting journey through Contemporary College Algebra. Fasten your seat belts, you are driving!

Don Small