 # Algebra: Form and Function ###### William G. McCallum, Deborah Hughes-Hallett, and Eric Connally
Publisher:
John Wiley
Publication Date:
2008
Number of Pages:
402
Format:
Paperback
Price:
57.95
ISBN:
978-0470226667
Category:
Textbook
[Reviewed by
Jane Ries Cushman
, on
03/29/2008
]

As a framework for this review of Algebra: Form and Function, I will use the CRAFTY Guidelines for College Algebra (approved January 2007) (downloaded from http://www.maa.org/CUPM/ on March 16, 2008). Quotations (in bold) from the Guidelines describe the fundamental experiences and competencies which are to be achieved by a College Algebra course.

Fundamental Experience

a) Emphasizes the use of algebra and functions in problem solving and modeling

b) Students address problems presented as real world situations by creating and interpreting mathematical models

c) Solutions to the problems are formulated, validated and analyzed using mental, paper and pencil, algebraic, and technology-based techniques

d) Provides a foundation in quantitative literacy

e) Supplies the algebra and other mathematics needed in partner disciplines

f) Helps meet quantitative needs in, and outside of, academia

Algebra does emphasize the use of algebra and functions in problem solving but it does not emphasize algebra and functions in modeling. In Chapter 2, Functions and Equations, there are only eight problems that ask students to use data to answer questions. Out of the book's nine chapters, one finds one section that asks the student to use data to model a linear function in each of the following: Chapter 3 (Linear Functions and Equations), Chapter 4 (Powers), and Chapter 7 (Exponential Functions).

There are few problems (in relation to the total number of problems) that ask the student to interpret their response. In the review section of Chapter 3 (randomly selected), students are asked to explain, justify, or interpret in only 41 of the 141 exercises and problems. Algebra does address the algebra and mathematics needed in partner disciplines through its problems with context.

Competencies

1. Problem Solving

a) Solving problems in the context of real world situations with emphasis on model creation and interpretation

b) Developing a personal framework of problem solving techniques (e.g. read the problem at least twice; define variables and parameters, use analytical, numerical and graphical solution methods as appropriate; determine plausibility of and interpret solutions

c) Creating, interpreting, and revising models and solutions of problems

Out of 171 questions in the review section of Chapter 3, fifty have a context. Of those fifty problems, thirty-five ask students to develop a model from the given situation; very few ask students to interpret the model.

2. Functions and Equations

a) Understanding the concepts of function and rate of change

b) Effectively using multiple perspectives (symbolic, numeric, graphic and verbal) to explore elementary functions

c) Investigating linear, exponential, power, polynomial, logarithmic and periodic functions as appropriate

d) Recognizing and using standard transformations such as translations and dilations with graphs of elementary functions

e) Using systems of equations to model real world situations

f) Solving systems of equations using a variety of methods

g) Mastering algebraic techniques and manipulations necessary for problem-solving and modeling in this course

Algebra does assist students in understanding the concepts of function and rate of change. It does discuss a family of functions, namely  y = kx. Symbols, a table of values, a graph, and a verbal description of a function are included in at least one example of each section of the Chapter 2.

Periodic functions are the only type of function that is not addressed in Algebra.There is a section that addresses shifting and scaling of functions (quadratic, rational, and square root functions).

In Chapter 3’s section on Systems of Linear Equations, there are 6 problems in context out of 56 total Exercises and Problems. Three methods of solving systems of equations are addressed in Chapter 3: Substitution, Elimination and Using Graphs.

3. Data Analysis

a) Collecting (in scientific discovery or activities, or from the Internet, textbooks or periodicals), displaying, summarizing, and interpreting data in various forms

b) Applying algebraic transformations to linearize data for analysis

c) Determining the appropriateness of a model via scientific reasoning

Algebra does not ask students to collect or display data. It does display data from other sources and then ask students to summarize and interpret the given graph.

Overall, this seems to be a traditional college algebra textbook with readable examples and a multitude of exercises and problems.

Jane Ries Cushman currently works at Buffalo State College in Buffalo, NY as an assistant professor. She received her doctorate at The University of Texas at Austin in August 2006. She is editor of the Association of Mathematics Teachers of New York State Newsletter and she is the chair of the Association of Mathematics Teacher Educators Affiliate’s Connection Committee. Her research interests include Inquiry-Based Learning, Problem-Solving and Functions-Based Approach to Algebra.

0 Review of Basic Skills 1
0.1 EXPRESSIONS AND EQUATIONS 2
0.2 THE DISTRIBUTIVE LAW 7
0.3 SOLVING LINEAR EQUATIONS 14
0.4 POSITIVE INTEGER POWERS AND THE EXPONENT RULES 18
0.5 ZERO, NEGATIVE, AND FRACTIONAL POWERS 23
0.6 FACTORING 29
1 The Basic Ideas of Algebra 37
1.1 EXPRESSIONS 38
1.2 TRANSFORMING EXPRESSIONS 42
1.3 EQUATIONS 48
1.4 SOLVING EQUATIONS 54
REVIEW PROBLEMS 61
2 Functions, Expressions, and Equations 65
2.1 WHAT IS A FUNCTION? 66
2.2 FUNCTIONS AND EXPRESSIONS 72
2.3 FUNCTIONS AND EQUATIONS 76
2.4 FUNCTIONS AND CHANGE 83
2.5 FUNCTIONS AND MODELING 88
REVIEW PROBLEMS 95
3 Linear Functions and Equations 99
3.1 LINEAR FUNCTIONS 100
3.2 EXPRESSIONS FOR LINEAR FUNCTIONS 107
3.3 LINEAR EQUATIONS 115
Contents xiii
3.4 MORE ON EQUATIONS OF LINES 121
3.5 MODELING WITH LINEAR FUNCTIONS 131
3.6 SYSTEMS OF LINEAR EQUATIONS 137
REVIEW PROBLEMS 147
4 Powers 155
4.1 POWER FUNCTIONS 156
4.2 EXPRESSIONS FOR POWER FUNCTIONS 165
4.3 EQUATIONS INVOLVING POWERS 172
4.4 MODELING WITH POWER FUNCTIONS 179
REVIEW PROBLEMS 186
5 More on Functions 189
5.1 DOMAIN AND RANGE 190
5.2 COMPOSING AND DECOMPOSING FUNCTIONS 197
5.3 SHIFTING AND SCALING 202
5.4 INVERSE FUNCTIONS 212
REVIEW PROBLEMS 217
6 Quadratic Functions, Expressions, and Equations 221
6.2 WORKING WITH QUADRATIC FUNCTIONS 228
6.4 SOLVING QUADRATIC EQUATIONS BY FACTORING 244
REVIEW PROBLEMS 252
xiv Contents
7 Exponential Functions 255
7.1 WHAT IS AN EXPONENTIAL FUNCTION? 256
7.2 INTERPRETING THE BASE 262
7.3 INTERPRETING THE EXPONENT 267
7.4 EXPONENTIAL EQUATIONS 274
7.5 MODELING WITH EXPONENTIAL FUNCTIONS 280
7.6 EXPONENTIAL FUNCTIONS AND BASE e 286
REVIEW PROBLEMS 291
8 Logarithms 295
8.1 INTRODUCTION TO LOGARITHMS 296
8.2 SOLVING EQUATIONS USING LOGARITHMS 304
8.3 APPLICATIONS OF LOGARITHMS TO MODELING 310
8.4 NATURAL LOGS AND LOGS TO OTHER BASES 315
REVIEW PROBLEMS 324
9 Polynomials 329
9.1 POLYNOMIALS 330
9.2 THE FORM OF A POLYNOMIAL 333
9.3 POLYNOMIAL EQUATIONS 341
9.4 LONG-RUN BEHAVIOR OF POLYNOMIAL FUNCTIONS 347
REVIEW PROBLEMS 354
10 Rational Functions 357
10.1 RATIONAL FUNCTIONS 358
10.2 LONG-RUN BEHAVIOR OF RATIONAL FUNCTIONS 364
REVIEW PROBLEMS 378
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