Algebra Review: Laying the Groundwork for Pre-Calculus

I've been using Michael Sullivan's Algebra and Trigonometry textbook for the last few years to teach College Algebra/Pre-Calculus/Trigonometry on the quarter system, but we're switching to John Coburn's Algebra and Trigonometry next year. They're both pretty good textbooks.

In building Pre-Calculus curriculum I've drawn mainly from four textbooks:

Mary Dolciani - Modern Introductory Analysis (original copyright 1964, my edition is from 1986)

Richard Brown/David Robbins - Advanced Mathematics (this copyright is from 1984, a newer edition of this book is here)

Paul Foerster - Pre-Calculus with Trigonometry (copyright 1987)

Max Sobel/Norbert Lerner - Pre-Calculus Mathematics (copyright 1995)

I had thought about using the Sobel/Lerner book for the Pre-Calculus course I developed for Clatsop Community College, and even e-mailed Max Sobel asking him about a new edition of that text, but he very kindly replied that he had retired (I also have the Harper & Row Algebra I and II textbooks from his series with Evan Maletsky and like these as well).

One of the things I liked about the Sobel/Lerner textbook was the coverage of rational expressions that several of the other books I had considered didn't have. I like rational expressions because this topic requires a firm grasp of many of the most important concepts from algebra – factoring and multiplying of bi- and trinomials, simplifying complex expressions, and combining like terms in the context of manipulating algebraic fractions. If students understand numerical fractions it helps a lot.

When I teach College Algebra courses at Clatsop CC, I often begin with exercises like (x+7)(2x-3) – (x+1)(x+5) to address these topics and prepare the students for when they see this again in working with rational expressions. Another good example is (x+6)(3x+1) – (x+2)^2, to get them used to seeing the squared binomial. ( I make the squared binomial a regular visitor in most of my algebra classes). These expressions often appear as the numerator of a combined fraction in a problem like (x+7)/(x+1) – (x+5)/(2x-3).

Here is a link to a collection of problems I often assign for this topic.

I recently began to reacquaint myself with the College Board Math II Subject Test which I had taken after taking Pre-Calculus in the spring of 1982. The first question on the sample test I looked at was intriguing, and an understanding of rational expressions is really useful in finding a quick solution:

If 3x+6=(k/4)(x+2), then k=

a) ¼

b) 3

c) 4

d) 12

e) 24

In this problem, dividing through by (x+2) so that 3=k/4 (and 12=k) gives the almost instantaneous answer we need for a timed test. This is an interesting problem because it really gets at the concepts involved in working with factors in an equation.

In the Sobel/Lerner textbook, Sections 1.7, 1.8 and 1.9 cover the algebra review (multiplying polynomials, combining like terms, factoring polynomials and rational expressions) necessary to move on. This is where it is important to illustrate these topics with problems, because when I say “combining like terms,” I don't mean 2x+5x. While this type of problem could be appropriate when first teaching the concept, in the context of review for Pre-Calculus, a problem from the Sobel/Lerner text like (x^3-2x+1)(2x)+(x^2-2)(3x^2-2) is better practice for using these skills together.

This is something that I consider extremely important and that textbooks and assessments often don't include enough of – using the skills together. Learning skills in isolation is useful to grasp each skill individually, but to really DO MATH, a student must be able to make decisions about what to do and when. Something that I like about the Pre-Calculus curriculum is that it lends itself well to the type of problem in which the tools of algebra must be applied in a variety of situations.

The Brown/Robbins text covers complex numbers and the quadratic formula in Chapter 1 (1-4, 1-5) and then the solution of equations involving rational expressions in Section 2-2. Chapter 2 goes on to examine the graphing of quadratic and polynomial curves and finishes with material on finding rational roots.

The Sullivan book covers polynomials and algebra in sections R.4, R.5 and R.7, Coburn covers this in sections R.3, R.4 and R.5.

The Foerster and Dolciani texts don't really cover much algebra review at all, but, as a result, they explore a number of topics the other books don't. I'll probably follow a path similar to the Brown/Robbins book and talk about rational roots next.