The content includes (but is not limited to) the usual precalculus material (functions, powers, polynomials, logs, exponentials, trig, etc.) in the usual order.

However, that typical material is preceded (in Chapter 1) by
material unlike
the review material of other precalculus texts. Chapter 1 is *not*
just
"review." Research with this audience shows that they have some
mathematical strengths and some glaring weaknesses. Most textbooks
begin with review material
that students do not want or need to review. This text begins with
material
designed to promote the development of essential mathematical concepts
that
the students need but often fail to have.

Chapter 1
shows students how algebra and
algebraic notation work, in general. It emphasizes **how to read
symbolic algebra**. The essential concepts (operations and order) and
their role in word problems, identities, theorems, and definitions are
considered. Placeholders
(dummy variables) are distinguished from unknowns -- the idea that a
theorem
or definition stated using "x" also applies to "x+h" and "z" is
fundamental
to being able to read mathematics. Entirely original examples and
problems
help teach students how to read math -- and learn math by reading math.

Chapter 1 goes on to emphasize that the sequence of operations
expressed in an algebraic expression is the key to solving equations
with that expression. It shows that the processes of *evaluating
expressions* and *solving equations* are precisely inverse
processes. This also explains how equations are solved with graphics
calculators. Sections 1.5 and 2.1 explain more about
the use of graphics calculators than other texts.

For research on the value of this approach, see my article (joint
with Anne Teppo) "Algebraic Thinking, Language, and Word Problems," in
the 1996 NCTM Yearbook, *Communication in Mathematics*, pp.45-53.
Other articles appear in the recent issues (XVI and XVII) of *PME-NA*
proceedings and
the 1992 and 1994 volumes of *FOCUS -- on Learning Problems in
Mathematics*.

The table of contents simply listed.

Chapter 2 concentrates on functions. Functional thought is essential for algebra and calculus. Nevertheless, most other precalculus texts avoid key algebraic topics that turn out to be critical in calculus. This text makes sure that students can distinguish the function, "f ", from its image, "f(x)" so that calculus-style expressions such as "f(x + h)" and "f(g(x))" are well-understood. The usual topics of graphs, windows, composition and graphical changes, and inverses are treated more thoroughly than in most texts.

In Chapters 1 and 2 students see a lot of old algebra in a new way. They can concentrate on the process represented by an algebraic expression rather than on the symbols used to express the process. This is a large conceptual step up from concentrating on examples of the process. Without this conceptual change there is little chance of success at calculus. Chapters 1 and 2 distinguish this text from other precalculus texts.

Chapter 3 becomes more traditional. Section 3.1 has real
applications of
lines the way they are used in calculus. Section 3.2 gives important
properties of quadratics that make connections across topics. Section
3.4 shows how
to factor expressions graphically. Sections 3.5 and 3.6 address the
difficulties students often have with **word problems**. This
emphasis on how to do word problems is not found in other texts. Here
the organization is important. The concepts required to do word
problems are emphasized in all the previous sections.

Chapter 4 discusses powers, polynomials, and rational functions. (I wait till Chapter 4 to do fractional powers since experience shows that students do not master them when they are presented earlier as in some precalculus texts.) Chapter 4 emphasizes modern numerical and graphical methods of solving polynomial and rational equations.

Chapter 5 discusses exponential and logarithmic functions. It teaches that the properties of exponential functions are exactly the properties of powers, with the letters changed. The preparation with symbolism in Chapter 1 pays off. The connections make the material easy to grasp. Other texts do not correlate power functions and exponential functions so closely.

Chapters 6 and 7 are trigonometry for solving triangles and for calculus. The unit-circle interpretation of trig functions is emphasized alongside the usual right-triangle interpretation.

Chapters 1 - 7 constitute the entire course at MSU.

Information about ordering a copy.

Chapters 1 through 7 (through exponentials, logs, and trigonometry) have been used twelve semesters (and five summer sessions) at Montana State University. Chapters 8 and 9 are not bound in and not used at MSU because we don't have time to cover them in a one-semester four-credit course.

The text has now been revised nine times, after having been used by
over 5000 students and 55 different instructors at MSU and elsewhere

There is a complete solutions manual for instructors and an edited
manual for students. There is an instructor's manual.

Return to *Precalculus*

Warren Esty, Department of Mathematical Sciences, Montana State
University