Experiencing Introductory and Intermediate Algebra (2nd Edition)

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9780130356826: Experiencing Introductory and Intermediate Algebra (2nd Edition)
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This book bridges the gap between traditional algebra texts and reform texts, written to promote the AMATYC standards published as Crossroads in Mathematics. It provides users with a sound traditional mathematical foundation, fully integrates graphing calculator technology, and encourages computer activities. This book includes key topics in algebra such as linear equations and inequalities with one variable, systems of equations, polynomial functions and equations, quadratic functions and equations, exponential functions and equations, logarithmic functions an equations, and rational and radical expressions. For professionals who wish to brush up on their algebra skills or enhance them with the use of graphing calculators and computers.

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The second edition of Experiencing Introductory and Intermediate Algebra continues to embrace the goal of promoting a new approach to teaching and learning developmental mathematics. This approach combines a traditional model with the reform movements presented in the National Council of Teachers of Mathematics (NCTM) standards and the American Mathematical Association of Two-Year Colleges (AMATYC) standards. The NCTM goals state that in our present technological society, students should learn to value mathematics, reason and communicate mathematically, become confident of their mathematical abilities, and become mathematical problem solvers. The AMATYC standards for intellectual development state that students will model real-world situations, connect mathematics with other disciplines, and use appropriate technology.

In this second edition, we have incorporated recommendations and suggestions from instructors and reviewers of the text. Instructors who currently use the text valued the real-world application feature and encouraged us to expand it. At the same time, they recommended that the text be streamlined to reduce its volume. The contents of this edition are still organized by families of functions, according to the AMATYC standards. Consequently, the first seven chapters of the text focus on linear expressions, equations, and functions. The next four chapters, Chapters 8 through 11, focus on polynomial expressions, equations, and functions. Chapter 12 presents rational expressions, equations, and functions, Chapter 13 examines radical expressions, equations, and functions, and Chapter 14 features exponential and logarithmic expressions, equations, and functions.

We have condensed the discussion of the real-number system into a review of pre-algebra numeric topics that may be taught as a whole or by sections of choice. We have combined Chapters 1 and 2 of the previous edition into one chapter, postponing the discussion of rational exponents to a later chapter. Also, the coverage of radicals in the combined chapter is limited to square roots and cube roots.

In the previous edition, exponents and polynomials were presented together with factoring, all within one chapter. We have now separated this material into two chapters. The first of the two, Chapter 9, focuses on exponents and polynomial operations. We have added a separate section on polynomial division that outlines polynomial long division in greater detail.

Feedback from reviewers indicated a need for expanding and strengthening the discussion of factoring, so we have created a separate chapter on the subject, Chapter 10. In this new chapter, we offer more examples, more exercises, and a different ordering of topics. In the previous edition, there was a separate chapter on complex numbers. In this edition, complex numbers and equations with imaginary solutions have been placed at the end of Chapter 13, following the discussion of radical expressions, equations, and functions. We believe that that is the appropriate place for complex numbers for two major reasons: students will have just learned the algebra of radicals, and the new location provides a perfect opportunity to revisit the quadratic formula, thereby reinforcing their understanding of the concept. However, the section on complex numbers and equations with imaginary solutions has been written as a stand-alone section, and if an instructor so chooses, it could be presented after the discussion of the quadratic formula in Chapter 11.

The first half of the text presents a balanced discussion of algebraic, numerical, and graphical methods for solving linear equations, so that students have a solid understanding of what the concepts represent. In the second half, the discussion of polynomial equations continues to utilize the same algebraic, numerical, and graphical techniques. This approach provides an opportunity for students who enter the sequence at that late point to gain an understanding of those methods. However, as we progress further into the second half of the text, the emphasis increasingly is on algebraic methods. Numerical and graphical methods are used only for checking solutions, rather than obtaining them. This way of teaching the topics will strengthen the students' algebraic skills for their subsequent math courses.

The new feature in the second edition is the inclusion of a project at the end of each chapter. The project enriches the study of the material presented in the chapter and provides connections to other areas of mathematics and other disciplines. Students may be asked to research the history of mathematical topics, collect and interpret data for use in their mathematical modeling activities, and build on the applications they have studied. The companion Web site provides support for those instructors who need to access data.

Approach

We have carefully written Experiencing Introductory and Intermediate Algebra in a positive manner to help students build confidence in their ability to do algebra. After completing the course, students should be able to do all of the following:

  • Model real-world situations.
  • Reason mathematically and develop convincing mathematical arguments.
  • Use an appropriate method—numeric, graphic, or algebraic—to solve problems.
  • Connect algebra to other disciplines.
  • Communicate mathematically.
  • Use appropriate technology.
  • Work collaboratively in groups.

To teach these skills, we introduce a problem-solving procedure in Chapter 4 and use this approach throughout the text. Numeric, graphic, and algebraic approaches to solving problems are described, and students are encouraged to choose that method which is appropriate to solve their problems. Every section of the text addresses real-world situations, so students can see reasons for learning algebra and can connect up what they learn with other disciplines, both inside and outside of mathematics. Students are asked to discover mathematical ideas on their own, to strengthen their mathematical reasoning skills, and to communicate their results. We then explain these results mathematically to reinforce the concepts the students have found.

Content

The text is written for a two-semester course in beginning and intermediate algebra. However, it is also flexible enough for use in a one-semester course. In both courses, topics are covered with a minimal amount of repetition. In Chapter 1, we introduce the set of real numbers, develop the properties of the real-number system, and present the rules for operations on real numbers. We complete these numeric topics with discussions of integer exponents, scientific notation, and radicals.

After completing the numeric foundation, we introduce variables, algebraic expressions, and equations in Chapter 2. There, we discuss geometric formulas and other formulas used in the first seven chapters of the text. This early introduction to these formulas allows us to integrate geometric and other applications throughout the book. In Chapter 3, we examine additional topics needed for the study of algebra: ordered pairs, relations, functions, and graphs. This early discussion of functions supports the structure of the remainder of the text, which focuses on the study of various families of functions.

Chapters 4, 5, 6, and 7 cover topics related to linear functions. In Chapter 4, we begin the explicit study of algebra by solving linear equations in one variable and absolute-value equations. Here and throughout the text, we teach how to solve equations numerically, graphically, and algebraically. Chapter 5 focuses on linear equations in two variables and on functions. Chapter 6 presents methods for solving systems of linear equations in two variables, emphasizing solutions by graphing, by substitution, and by elimination. Inequalities and solutions of linear inequalities are discussed in Chapter 7. We recommend the first seven chapters as a beginning algebra text.

We designed Chapter 8 as the first chapter in the second semester of study, allowing for a review of graphical methods and the concept of a function. The remainder of the text follows a standard pattern consisting of the introduction of a family of functions, rules for operating with the expressions that define the functions, and methods for solving related equations numerically, graphically, and algebraically. We follow this pattern in discussing polynomial functions in Chapter 8, polynomial expressions in Chapter 9, factoring polynomials in Chapter 10, and polynomial equations and inequalities in Chapter 11.

In Chapter 11, we solve quadratic equations numerically, graphically, and algebraically by factoring, by completing the square, and by using the quadratic formula. This arrangement allows for closure on quadratic equations and enables the student to choose an appropriate method for solving equations by examining the equation given.

In Chapter 12, we describe rational functions, operations with rational expressions, and the solution of rational equations. We complete the coverage of radical functions, expressions, and equations in Chapter 13, along with functions, expressions, and equations having rational exponents. The chapter also includes a section on the complex-number system, describing equations in one variable with complex solutions. This section is designed to stand on its own or to be incorporated into earlier chapters if desired. Finally, Chapter 14 focuses on inverse functions, exponential functions, and logarithmic functions. It also presents a discussion of the properties of exponents and logarithms and the methods for solving exponential equations and logarithmic equations in one variable.

Pedagogy

Use of Technology

Graphing calculators allow students the freedom to experiment with and explore mathematical ideas. Using graphing calculators helps boost confidence and increase motivation. Skills such as estimating, computing, graphing, and analyzi...

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