College Algebra - Mathematics
College Algebra
by Avinash Sathaye, Professor of Mathematics 1
Department of Mathematics, University of Kentucky
A? ryabhat.a
This book may be freely downloaded for personal use from the author's web site msc.uky.edu/sohum/ma109 fa08/fa08 edition/ma109fa08.pdf. Any commercial use must be preauthorized by the author. Send an email to sathaye@uky.edu for inquiries.
September 18, 2008
1Partially supported by NSF grant thru AMSP(Appalachian Math Science Partnership)
ii
iii
Introduction.
This book represents a significant departure from the current crop of commercial college algebra textbooks. In our view, the core material for the (non-remedial) courses defined by these tomes is but a shadow of that traditionally covered material in a reasonable high school program. Moreover, much of the material is substantially repeated from earlier study and it proceeds at a slow pace with extensive practice and a large number of routine exercises. As taught, such courses tend to be ill-advised attempts to prepare the student for extensive calculations using calculators, with supposed "real life" examples offered for motivation and practice. Given the limited time and large number of individual topics to study, the average student emerges, perhaps, with the ability to answer isolated questions and the well-founded view that the rewards of the study of algebra (and of mathematics in general) lie solely in the experience of applying opaque formulas and mysterious algorithms in the production of quantitative answers.
As rational, intelligent individuals with many demands on their time, students in such an environment are more than justified when they say to the teacher: "don't tell me too many ways of doing something; don't tell me how the formula is derived; just show me how to do the problems which will appear on the test!. Individuals, who experience only this type of mathematics leave with a static collection of tools and perhaps the ability to apply each to one or two elementary or artificial situations. In our view, a fundamental objective of the students mathematical development should be an understanding of how mathematical tools are made and the experience of working as an apprentice to a teacher, learning to build his or her own basic tools from "the ground up. Students imbued with this philosophy are prepared to profit as much from their incorrect answers, as from their correct ones. They are able to view a small number of expected outcomes of exercises as a validation of their understanding of the underlying concepts. They are further prepared to profit from those "real world applications through an understanding of them as elementary mathematical models and an appreciation of the fact that only through a fundamental understanding of the underlying mathematics can one understand the limits of such models. They understand how to participate in and even assume responsibility for their subsequent mathematical education.
This text is intended to be part of a College Algebra course which exposes students to this philosophy. Such a course will almost certainly be a compromise, particularly if it must be taught in a lecture/recitation format to large numbers of students.
The emphasis in this course is on mastering the Algebraic technique. Algebra is a discipline which studies the results of manipulating expressions (according to a set of rules which may vary with the context) to put them in convenient form, for enhanced understanding. In this view, Algebra consists of looking for ways of finding information about various quantities, even though it is difficult or even impossible to explicitly solve for them. Algebra consists of finding multiple expressions for the same quantities, since the comparison of different expressions often leads to new discoveries.
iv
Here is our sincere request and strong advice to the reader:
? We urge the readers to approach this book with an open mind. If you do, then you will find new perspective on known topics.
? We urge the reader to carefully study and memorize the definitions. A majority of mistakes are caused by forgetting what a certain term means.
? We urge the reader to be bold. Don't be afraid of a long involved calculation. Exercises designed to reach an answer in just a step or two, often hide the true meaning of what is going on.
As far as possible, try to do the derivations yourself. If you get stuck, look up or ask. The derivations are not to be memorized, they should be done as a fresh exercise in Algebra every time you really need them; regular exercise is good for you!
? We urge the reader to be inquisitive. Don't take anything for granted, until you understand it. Don't ever be satisfied by a single way of doing things; look for alternative shortcuts.
? We also urge the reader to be creatively lazy. Look for simpler (yet correct, of course) ways of doing the same calculations. If there is a string of numerical calculations, don't just do them. Try to build a formula of your own; perhaps something that you could then feed into a computer some day.
A warning about graphing. Graphs are a big help in understanding the prob-
lem and they help you set up the right questions. They are also notorious for mis-
leading people into wrong configurations or suggesting possible wrong answers. Never
trust an answer until it is verified by theory or straight calculations.
Calculators are useful for getting answers but in this course most questions are
designed for precise algebraic answers. You can and should use the calculators freely
to do tedious numerical calculations or to verify your work or intuition. But you
should not feel compelled to convert every answer into a decimal number, however
precise.
In
this
course
6 8
and
1+(2)
1- (7)
are
perfectly
acceptable
answers
unless
the
instructions specify a specific form. A computer system which is capable of infinite
precision calculations can be used for study and is recommended. But make sure that
you understand the calculations well.
A suggestion about proofs. We do value the creation and understanding of
a proof, but often it is crucial that you get good at calculations before you know
what they mean and why they are valid. Throughout the book you will find sections
billed as "optional" or "can be omitted in a first reading". We strongly urge that you
master the calculations first and then return to these for further understanding.
In many places, you will find challenges and comments for attentive or alert read-
ers. They can appear obscure if you are new to the material, but will become clear
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as you get the "feel" of it. Some of these are subtle points which may occur to you long after the course is finished! In other words, don't be discouraged if you don't get these right away.
The reader will quickly note that although there are numerous worked examples, there are very few exercises in this text. In particular there are no collections of problems from which the instructor might assign routine homework. Those exist, but they are in electronic form and provided through the Web Homework System (WHS) at (). The system was created by my University of Kentucky (UK) colleague Dr. Ken Kubota. The problems themselves were prepared by myself and UK colleague Dr. Paul Eakin using the Maple problem solving system with the MCtools macro package developed by our UK colleague Dr. Carl Eberhart.
The initial version of this text was used in pilot sections of College Algebra taught at UK in spring 2005 by Paul Eakin, our colleague and department chair, Dr. Rick Carey, and by then pre-service teachers: Amy Heilman and Sarah Stinson. 2
Following that pilot, the text was extensively revised and used for about 1500 students in a large-lecture format at UK in fall 2005 and about 450 students in spring 2006. The results for fall were:31% As, 20% Bs, 14% Cs,6% Ds, and 9% Fs (UK calls them Es), and 20% Ws.
In the spring the outcomes were: 16.5% As, 20.1% Bs, 15% Cs, 13.5% Ds, and 16.2% Fs, and 18.8% Ws. In the spring, six Eastern Kentucky high school students took the course by distance learning: four made As and two dropped because of conflicts with sports practice.
In summer 2006, I worked with a team of 22 mathematicians: eighteen high school teachers, one mathematics education doctoral student, and three UK math faculty went through a week-long (30 hour) seminar which went (line by line, page by page) through the entire text and all of the online homework problems, discussing in detail such course characteristics as the underlying philosophy, the mathematical content, topic instructional strategies, alignment of homework and text material, alignment with high school curricula, etc. The results of that tremendous amount of effort were incorporated into the third major iteration of the text and course.
The third edition was used in Fall 2006 and Spring 2007 for both college and secondary students in a program called "Access to Algebra which is sponsored by the National Science Foundation (NSF) and the University of Kentucky (UK). In that program secondary students, mentored by their school math teachers, take the UK College Algebra course at no cost. The students take the same course in lockstep with a matched cohort of college students, doing the same homework and taking the same uniformly graded (and hand graded) examinations on the same schedule.
2Amy has now completed her masters degree in mathematics and joined Sarah on the mathematics faculty of Paul Laurence Dunbar High School in Lexington, Ky.
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