Preface Pedagogical Decisions
The authors and the greater PCC faculty have taken various stances on certain pedagogical and notational questions that arise in basic algebra instruction. We attempt to catalog these decisions here, although this list is certainly incomplete. If you find something in the book that runs contrary to these decisions, please let us know.
Basic math is addressed in an appendix. For the course sequence taught at PCC, this content is prerequisite and not within the scope of this book. However it is quite common for students in the basic algebra sequence to have skills deficiencies in these areas, so we include the basic math appendix. It should be understood that the content there does not attempt to teach basic math from first principles. It is itended to be more of a review.
Interleaving is our preferred approach, compared to a proficiency-based approach. To us, this means that once the book covers a topic, that topic will be appear in subsequent sections and chapters in indirect ways.
We round decimal results to four significant digits, or possibly fewer leaving out trailing zeros. We do this to maintain consistency with the most common level of precision that WeBWorK uses to assess decimal answers. We generally round, not truncate, and we use the \(\approx\) symbol. For example, \(\pi\approx3.142\) and Portland's population is \(\approx609500\text{.}\) On rare occasions where it is the better option, we truncate and use an ellipsis. For example, \(\pi=3.141\ldots\text{.}\)
We offer alternative video lessons associated with each section, found at the top of most sections in the HTML eBook. We hope these videos provide readers with an alternative to whatever is in the reading, but there may be discrepancies here and there between the video content and reading content.
We believe in opening a topic with some level of application rather than abstract examples, whenever that is possible. From applications and practical questions, we move to motivate more abstract definitions and notation. At first this may feel backwards to some instructors, with some easier examples following more difficult contextual examples.
Linear inequalities are not strictly separated from linear equations. The section that teaches how to solve \(2x+3=8\) is immediately followed by the section teaching how to solve \(2x+3\lt8\text{.}\) Our aim is to not treat inequalities as an add-on optional topic, but rather to show how intimately related they are to corresponding equations.
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When issues of “proper formatting” of student work arise, we value that the reader understand why such things help the reader to communicate outwardly. We believe that mathematics is about more than understanding a topic, but also about understanding it well enough to communicate results to others. For example we promote progression of equations like
\begin{align*} 1+1+1\amp=2+1\\ \amp=3 \end{align*}instead of
\begin{equation*} 1+1+1=2+1=3\text{.} \end{equation*}We want students to understand that the former method makes their work easier for a reader to read. It is not simply a matter of “this is the standard and this is how it's done.”
When solving equations (or systems of linear equations), most examples should come with a check, intended to communicate to students that checking is part of the process. In Chapters 1–4, these checks will be complete simplifications using order of operations one step at a time. The later sections may have more summary checks where steps are skipped or carried out together, or we promote entering expressions into a calculator to check.
Within a section, any first context-free example of solving some equation (or system) should summarize with some variant of both “the solution is…” and “the solution set is….” Later examples can mix it up, but always offer at least one of these.
With applications of linear equations (not including linear systems), we limit applications to situations where the setup will be in the form \(x + \text{expression-in-}x = C\) and also to certain rate problems where the setup will be in the form \(at + bt = C\text{.}\) There are other classes of application problem (mixing problems, interest problems, …) which can be handled with a system of two equations, and we reserve these until linear systems are covered.
With simplifications of rational expressions in one variable, we always include domain restrictions that are lost in the simplification. For example, we would write \(\frac{x(x+1)}{x+1}=x\text{,}\) for \(x\neq-1\text{.}\) With multivariable rational expressions, we are content to ignore domain restrictions lost during simplification.