Preface Pedagogical Decisions
The authors from PCC and LCC, 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.
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.
Chapter 1 is mostly written as a review, and is not intended to teach all of these topics from first principles.
We often 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. When we round, we often use the \(\approx\) symbol. For example \(\pi\approx3.142\) and Portland's population is \(\approx609500\text{.}\)
We believe in always trying to open a topic with some level of application rather than abstract examples. From applications and practical questions, we move to motivate more abstract definitions and notation. This approach is perhaps absent in the first chapter, which is intended to be a review only.
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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*}And 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 will often have more summary checks where either order of operations steps are skipped in groups, or we promote entering expressions into a calculator. Occasionally, in later sections the checks will still have finer details, especially when there are issues like with negative numbers squared.
Within a section, any first 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.
There is a section on very basic arithmetic (five operations on natural numbers) in an appendix, not in the first chapter. This appendix is only available in the eBook.
With applications of linear equations (as opposed to linear systems), we limit applications to situations where the setup will be in the form \(x + f(x) = C\) and also certain rate problems where the setup will be in the form \(5t + 4t = C\text{.}\) 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.