Instructor Tips
Introduction
These teaching tips are written for teaching assistants and instructors who may be teaching from the text COLLEGE ALGEBRA & TRIGONOMETRY through Modeling and Visualization for the first time. This text is designed to be flexible and meet a variety of student and instructor needs. The unifying concept throughout this text is that of a function. The graphing calculator, modeling, and a variety of real-data applications are integrated throughout this unique text. The rule of four (verbal, graphical, numerical, symbolic) is used to represent mathematical concepts. This text is graphing calculator flexible, which allows instructors to use the graphing calculator as little or as much as desired. Instructors have the flexibility to place more emphasis on graphical and numerical techniques, rather than symbolic techniques. However, instructors can also emphasize symbolic techniques over graphical and numerical techniques if desired. Students are often asked to make conjectures and interpret their results, thereby giving meaning to the mathematics that is being learned.
ORGANIZATION
This text has 8 chapters and 42 sections. In addition there is Chapter R, which is a reference chapter. Each chapter contains from 3 to 7 sections. These chapters cover the standard college algebra curriculum by using an innovative approach that breathes life into the course. In this text, sections generally represent one class day. However, it may be necessary to spend an extra day on some of the more difficult topics. One goal of this text is to have students understand how mathematical concepts are related to one another and how these concepts apply to real life. This goal has been well received by the students. Be sure to spend time answering questions on homework assignments. You may want to discuss certain homework problems even if students do not ask questions. Many times students fail to ask questions even when they do not understand the material. Discussing homework is particularly important early in the course.
THE GRAPHING CALCULATOR
Try to obtain a graphing calculator view screen with an overhead projector so that students can see your calculator screen. By seeing how you operate a graphing calculator, students often learn more quickly how to use their graphing calculators. However, it is not necessary to bring a view screen to class every day. Many times after you have demonstrated a technique a few times you do not have to do it again. Be aware that Appendix B in the text gives students "just in time" help with keystrokes on the TI-83 and TI-83 Plus graphing calculators. Appendix B is intended to not only help students learn how to use a graphing calculator, but also to save class time for the instructor. Appendix B relieves the instructor from having to frequently remind students how to use their graphing calculators. There are many resources for students to learn how to use a graphing calculator. Although the text does not usually spend time on specific keystrokes for a particular graphing calculator, numerous graphing calculator screens with detailed explanations are often included throughout the text. There are additional resources on the Web site for this text. For detailed information on the features of your calculator, consult your graphing calculator owner's manual.
You may or may not choose to emphasize square viewing rectangles when using a graphing calculator. Square viewing rectangles results in a square appearing square rather than rectangular and a circle appearing circular rather than elliptical. See page 46 of the text. There are times when it may be simpler for your students not to use square viewing rectangles. For example, a square viewing rectangle may not be practical when solving applications because real data often varies greatly in size. One of the most difficult things for students to do with a graphing calculator is to find an appropriate viewing rectangle or window. Sometimes it is important for students to find their own viewing rectangles; other times it can become a major distraction. This text tends to supply more difficult viewing rectangles to the students. However, it provides ample opportunity for students to find their own viewing rectangles. Setting a viewing rectangle can lead to a worthwhile discussion about the domains and ranges of functions.
APPLICATIONS
Applications are integrated throughout this text in both the discussions and the exercises. Many times applications are used to motivate mathematical concepts and help students learn how mathematics is used in our society. However, it is not necessary to present any particular application. It is important to present some applications in class. Students learn by watching how the instructor solves problems. Afterwards they are more comfortable trying to solve application exercises on their own. It is difficult for students to master solving application and modeling exercises if they are not discussed in class. However, applications do not have to be done at the expense of skill-building exercises. You will find that there is ample time to work examples involving mathematical skills too. Discuss the chapter and section introductions with your students. These introductions often set the tone for the coming chapter or section and often discuss real applications, historical events, and other topics that give life to mathematics.
It is not necessary to assign a large number of applications each day, but it is important to assign at least a couple of applications as part of most assignments. Include one or more applications on every exam. Over time students become more proficient at solving these problems. Select applications that interest you or your students. If you are interested in the application, you will create interest in your students. Strike your own balance between skill-building exercises and exercises that involve applications, modeling, and graphical interpretation. You are an expert and in the best position to judge your students' needs.
COLLABORATIVE LEARNING OPPORTUNITIES
There are many opportunities for collaborative learning in this text. One simple technique is to use Checking Basic Concepts that occur throughout the text. You might want to allow students to work together the last 10-20 minutes of a period on some of these exercise sets. Require students to work in groups of 2-5. Have them turn in one paper with everyone's name on it. During this time, walk around the room and answer their questions. This is not a test. You will be surprised how much you learn about your students. Extended exercises or even-numbered exercises can also be used. It generally works best not to assign exercises for which students are given the answers. Collaborative learning exercises can be beneficial for both the students and the instructor.
GETTING STARTED
It is essential to get students off to a good start in Chapter 1. The first chapter offers a unique approach compared to most mathematics courses. This text does not start by reviewing intermediate algebra. (If you would like to start by reviewing prerequisite material, refer to Chapter R.) Chapter 1 is intended to generate student interest and give students a different look at mathematics from the start. Instead of initially concentrating on symbolic manipulation skills from intermediate algebra that students may be weak on, students are introduced to some essential mathematical concepts such as functions and graphs. The motivation behind mathematics is discussed. Skill building is emphasized more after Chapter 1. If you have any time to spare in your syllabus, spend an extra day in Chapter 1. Students typically experience success in this chapter and attitudes toward mathematics often begin to change.
Students should become familiar with their graphing calculators in Chapter 1. If possible, take time to answer questions about graphing calculators. When the questions become repetitive, refer them to Appendix B. You may not know all the answers-_that is normal. With the rapid growth in technology, no one can answer every question related to graphing calculators. If you are learning the graphing calculator for the first time, concentrate only on the essential features. Don’t be afraid to tell students to learn how to use their own graphing calculators, if they have a model that you are familiar not with. For some questions you might suggest that students consult their owner's manual. Generally, students adapt well to graphing calculators. Many students have already been exposed to graphing calculators in high school. If a student is having genuine difficulty, you may want to suggest that they talk to you during office hours. Collaborative learning is an excellent opportunity for you to help students and for students to help each other with the graphing calculator. The number of calculator skills required in this text is minimal. At the end of the first chapter, students should be able to set a viewing rectangle, graph a function, evaluate mathematical expressions, and make a table, scatterplot or line graph. You may want to have one or more brief quizzes to make sure students are learning the important concepts in Chapter 1.
THE RULE OF FOUR
The rule of four is used throughout this text. It is important in Chapter 1 that students learn what is meant by verbal, graphical, numerical, and symbolic representations of a function. You are probably most familiar with graphical and symbolic representations (graphs and formulas) of functions. A verbal representation of a function simply states what the function computes in words. A numerical representation of a function is a table of values, which you have probably made use of when you plot a graph of a function by hand. Since a function can have an infinite number of ordered pairs in the form (x, y), most numerical representations are incomplete and therefore, called partial numerical representations.
Graphical, numerical, and symbolic methods can be used to solve equations and inequalities. (Of these three methods, numerical representations are emphasized the least in this text.) The solution of equations begins in Chapter 2 and continues throughout the text. A basic graphical method for solving equations is the intersection-of-graphs method. This graphical method is explained in Section 2.2 and used throughout the text. The intersection-of- graphs method is quite intuitive for students when solving equations and inequalities involving real applications. There are many examples and exercises that require students to use symbolic techniques to solve equations.
OTHER IMPORTANT FEATURES
There are several important supplements for this text that can help you and your students, such as sample tests, videotapes, and a Web site. See the preface of this text for details or talk to your Addison-Wesley sales representative. The following are some features that are especially helpful.
- Be sure that students are aware that each section ends with Putting It All Together, which summarizes the important concepts presented in the section.
- Class Discussion exercises occur in most sections and can be used for class discussions, group activities, or extra credit problems. Many times the Class Discussion exercises ask students to take a concept one step further than has been presented.
- The exercise sets are designed to be flexible. There are numerous exercises that test a variety of mathematical concepts and skills. This feature allows an instructor to design assignments that meet their students' needs. Exercises are organized by topic and graded to make it easier to choose an assignment. For example, if you are planning to take two days to cover a section, it is easy to give students a partial assignment on the material covered on the first day. If you choose to skip over a topic, the corresponding topics in the exercise set can be easily omitted. If you choose to not emphasize a component of the rule of four, it is easy to adjust the assignment accordingly. For example, if most students have older models of calculators and cannot create tables, you may decide not to assign exercises that use numerical tables.
- There is an Instructor's Solutions Manual that contains the solutions to the exercises in the text. In contrast, the Student’s Solutions Manual does not have solutions for Checking Basic Concepts, even-numbered exercises, or Extended and Discovery exercises.
- The Extended and Discovery Exercises at the end of certain exercise sets and at the end of each chapter can be used for collaborative learning or extra credit problems. One student, after being assigned an extra credit extended exercise, returned to class the next day and said enthusiastically, "It took me about an hour, but I got it!"
