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PrefacePedagogical Decisions

The authors 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 will certainly be 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 written as a review, and is not intended to teach these topics from first principles.

  • 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 round, not truncate. And we use the \(\approx\) symbol. For example \(\pi\approx3.142\) and Portland's population is \(\approx609500\text{.}\)

  • We intend to offer alternative video lessons associated with each section. These are intended to provide readers with an alternative to whatever we have written on a topic. We have produced videos for Chapters 1–4. In later chapters we sometimes use videos from YouTube, but intend to produce videos at some point in the future. The YouTube videos more than likely do not cover 100% of what our written content covers. And such videos may use notation and approaches that differ from ours.

  • Traditionally, a math textbook has “examples” throughout each section. This textbook generally uses two different types of such “examples”.

    Static

    These are labeled “Example.” Static examples may or may not be subdivided into a “statement” followed by a walk-through solution. This is basically what traditional examples from math textbooks do.

    Active

    These are labeled “Exercise,” not to be confused with the exercises that come at the end of a section that might be assigned for homework, etc. In the HTML output, Active examples have WeBWorK answer blanks where a reader could try submitting an answer. In the PDF output, Active examples are almost indistinguishable from Static examples. Generally, a walk-through solution is provided immediately following the answer blank.

    Some HTML readers will skip the opportunity to try an Active example and go straight to its solution. Some readers will try an active example once and then move on to the solution. Some readers will tough it out for a period of time and resist reading the solution.

    For readers of the PDF, it is expected that they would read the example and its solution just as they would read a Static example.

    It is important to understand that a reader is not required to try submitting an answer to an Active example before moving on. It is also important to understand that a reader is expected to read the solution to an Active exercise, even if they succeed on their own at finding an answer.

    Interspersed through a section there are usually several exercises that are intended as active reading exercises. A reader can work these examples and submit answers to WeBWorK to see if they are correct. The important thing is to keep the reader actively engaged instead of providing another static written example. In most cases, it is expected that a reader will read the solutions to these exercises just as they would be expected to read a more traditional static example.

  • We believe in nearly always opening 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. At first this may feel backwards to some instructors, with some “easier” exmaples (with no context) coming later than some of the contextual examples.

  • Linear inequalities are not strictly separated from linear equations. The same section that teaches how to solve \(2x+3=8\) will also teach how to solve \(2x+3\lt8\text{.}\) There will be sufficient subdivisions within sections so that an instructor may focus on equations only or inequalities only if they so choose.

    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.

  • 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 soliving equations (or systems), every example will 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.

  • 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, 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 \(x\neq-1\text{.}\)