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\usepackage{precalc}

\begin{document}

\pctitle{Precalculus Syllabus}

\noindent 2008--2009 \\
Instructor: Brent Yorgey, \texttt{byorgey@gmail.com} \\
Website: \url{http://www.seas.upenn.edu/~byorgey/precalc/}

\section{Schedule of topics}
\label{sec:schedule}

Note: this schedule can and probably will change slightly! 

The year will be divided into three main units.  The first unit will
cover some of the foundations of modern mathematics in depth, and
allow for some review of topics from previous years. The remaining two
units will correspond to the two major threads of mathematics over the
last several hundred years---the continuous and the discrete.
Continuous mathematics concerns itself with measurement, motion, and
change; discrete, with counting, pattern, and structure.  \bigskip

\tablehead{Week & Date & Topic \\ \hline \\}

\begin{supertabular}{cll}

    & & \textbf{Foundations} \\ \\

  1 & 9/2--5 & \LaTeX\ introduction, set theory I \\
  2 & 9/8--12 & Functions \\
  3 & 9/15--19 & Numbers \\
  4 & 9/22--26 & Set theory II \\
  5 & 9/29--10/3 & Logic, proof, and problem-solving I \\
  6 & 10/6--10 & Logic, proof, and problem-solving II \\
  7 & 10/13--17 & Interlude: Logic circuits \\
  8 & 10/20--24 & Algebra/geometry review \\

    \\ & & \textbf{Continuous} \\ \\

  9 & 10/27--31 & Foundations of trigonometry I \\
 10 & 11/3--7 & Foundations of trigonometry II \\
 11 & 11/10--14 & Inverse trigonometric functions \\
 12 & 11/17--21 & Trigonometric identities \\

 \\ & 11/24--28 & \emph{Thanksgiving} \\ \\

 13 & 12/1--5 & Triangle laws \\
 14 & 12/8--12 & Interlude: Acoustics \\

 \\ & 12/15--1/4 & \emph{Christmas/New Year} \\ \\

 15 & 1/5--9 & Matrices \\
 16 & 1/12--16 & Rectangular and polar coordinates \\
 17 & 1/19--23 & Vectors and vector functions \\
 18 & 1/26--30 & Complex numbers \\
 19 & 2/2--6 & Interlude: Fractals and computer graphics \\
 
 \\ & & \textbf{Discrete} \\ \\

 20 & 2/9--13 & Combinatorics and probability I \\
 21 & 2/16--20 & Combinatorics and probability II \\
 22 & 2/23--27 & Sequences \\
 23 & 3/2--6 & Series I \\
 24 & 3/9--13 & Series II \\
 25 & 3/16--20 & Induction and recursion \\
 26 & 3/23--27 & Number Theory I \\
 27 & 3/30--4/3 & Number Theory II \\
 28 & 4/6--10 & Interlude: Cryptography \\

 \\ & 4/13--17 & \emph{Easter} \\ \\

 29 & 4/20--24 & Group Theory I \\
 30 & 4/27--5/1 & Group Theory II \\
 31 & 5/4--8 & TBD \\
 32 & 5/11--15 & TBD \\
 33 & 5/18--22 & Overflow \\
 34 & 5/25--29 & Overflow \\
 35 & 6/1--5 & Revise final assignment and wrap-up \\
\end{supertabular}

\section{Format}
\label{sec:format}

You will receive weekly assignments in the form of PDF documents
containing instruction mixed with problems for you to solve.  You will
receive each new assignment at the beginning of the week (exact day
and time to be determined), and your solutions to the problems will be
due exactly one week after receiving the assignment. 

I will provide feedback on each submitted solution set no more than
two days after it is turned in.  You will then have the remainder of
the week to revise the solution set, which will be due at the same
time as your solutions to the next assignment.  In all, this means
that each Monday (or whatever day we choose), you will:
\begin{itemize}
\item turn in a revised version of assignment $n - 1$
\item turn in assignment $n$
\item receive assignment $n+1$
\end{itemize}

So, the first few weeks might look something like this, assuming that
Monday is the day we choose as the day to receive new assignments:
\bigskip

\begin{tabular}[htp]{lp{3in}}
  Tuesday, 9/2 & Receive assignment 1 (on Tuesday instead of Monday
  due to Labor Day)\\
  Monday, 9/8 & Receive assignment 2 \\
  Tuesday, 9/9 & Assignment 1 due (1 week after assigned)\\
  by Thursday, 9/11 & Receive feedback on assignment 1 \\
  Monday, 9/15 & Assignment 2 and revised assignment 1 due; receive
  assignment 3 \\
  by Wednesday, 9/17 & Receive feedback on assignment 2 \\
  Monday, 9/22 & Assignment 3 and revised assignment 2 due; receive
  assignment 4 \\
  etc. & 
\end{tabular} \bigskip

Our primary method of communication will be by emailing PDF documents
prepared using the free \LaTeX\ typesetting system (which you will
learn about in the first week's assignment). There are also several
other free tools we can use to communicate.  A simple tool for quick
questions and discussions is Gmail Chat.  For discussions that require
writing mathematical formulas, we can use MathIM
(\url{http://www.mathim.com/}) which allows \LaTeX\ formulas to be
embedded in chat.  Finally, for discussions which require drawing
pictures, we can use ScribLink (\url{http://www.scriblink.com/}).

\section{Expectations}
\label{sec:expectations}

Submitted solutions should be:

\begin{itemize}
\item \textbf{complete}: if you are stuck on some of the problems, you
  are expected to ask me for help or hints during the week.  It is not
  acceptable to turn in an assignment and write ``we didn't know how
  to solve this problem'' if you have not asked for help.  (It is,
  however, perfectly fine if you have asked for help and just still
  don't get it; the important point is to ask for help.)  Of course, a
  corollary is that you should \emph{not} wait until just before an
  assignment is due to work on it!  I suggest taking the beginning of
  the week to solve all the problems on paper, and the last few days
  to type up your solutions and work out any last-minute details.

\item \textbf{well-written}: check your grammar and spelling, and make
  sure your solutions are clear.  Math that exists only in someone's
  head is like a tree falling in a forest with no one around to hear
  it---communicating well is a fundamental mathematical skill, not an
  afterthought.

\item \textbf{concise}: your solutions should be as simple and elegant
  as possible.  I \emph{don't} want to see every single step!  If a
  step or a detail is obvious to you, omit it.  For example, it would
  be perfectly acceptable to write, ``this results in the equation
  $2x^2 - 6 = 2x + x^2$, which has solutions $x = 1 \pm \sqrt{7}$'';
  you wouldn't need to show the process of actually solving the
  equation, since it is obvious to anyone who knows some algebra.

\item \textbf{on time}: solution sets should be turned in on time.  If
  there is a good reason you would like some extra time on an
  assignment (a trip, some sort of emergency, an exceptionally large
  workload in other classes), just ask.  However, you must ask at
  least \emph{24 hours in advance}---I will not grant retroactive
  extensions, except in the case of emergencies. See below for
  details on the grading of late assignments.

\item In addition, at the end of each submitted solution set, you
  should include a section with comments on the problem set.  What was
  particularly interesting or uninteresting?  What was too easy, too
  hard, or just right?  Was it too long?  Too short? Were any parts
  confusing?  How long did you spend on it?  What would you like to
  learn more about?  These particular questions are suggestions, not a
  checklist of questions to answer; feel free to write down any
  comments or questions you might have.  The important point is to
  provide me with some ``meta-feedback'' about the assignments so I
  have an idea of what direction to take future assignments.

\item Finally, note that when submitting your solutions to an
  assignment, you should submit \emph{both} the \LaTeX\ source
  (\texttt{.tex}) file \emph{and} the generated PDF.

\end{itemize}

\section{Grading}
\label{sec:grading}

Each submitted assignment will be graded on a scale of five points.
The scale runs approximately as follows:

\begin{itemize}
\item \textbf{5} --- the solutions are complete, correct, clear and
  concise.  There may be a few minor grammatical or spelling errors,
  but no mathematical errors.

\item \textbf{4} --- the solutions are generally correct and clear, but
  may have some minor errors and/or unclear portions.  Some sections
  may need to be rewritten to be made clearer or simpler.

\item \textbf{3} --- most solutions are generally correct, but there
  may be significant portions which are incorrect, unclear, or overly
  complex.  One or more solutions may have major gaps in logic or use
  an incorrect approach.  No more than two solutions are missing
  entirely.

\item \textbf{2} --- many solutions are incorrect, incomplete, or
  missing.  Solutions are unclear and need substantial rewriting.

\item \textbf{1} --- the assignment was attempted, but is largely
  incorrect or incomplete, or it is so poorly written and confusing
  that no one can tell if it is correct or not.

\item \textbf{0} --- the assignment was not turned in at all, had
  no real content, or was turned in more than 48 hours late.
\end{itemize}

Fractional scores may be awarded.  Of course, this scale is quite
subjective, but I will try to be as consistent and fair as possible.
If you ever have questions about scoring you are of course free to
ask.

Keep in mind that I will provide feedback on the first version of an
assignment, with the expectation that you should be able to
incorporate the feedback in order to receive a 5 (or very close to
it) on the revised version of the assignment.  Your final score for a
given assignment will be a weighted average of your scores on the
first draft and the revised version, with the revised version counting
triple.  For example, if you received a 3 on a first draft, and
revised it to receive a 5, your final score would be $(3 + 3*5)/4 = 4.5$.

Late assignments will be penalized by a number of points $p(t)$, where
$t$ is the number of hours by which the assignment is late, and \[
p(t) = \frac{1}{3}(2^{t/12} - 1). \] Thus, an assignment turned in
$24$ hours late will be penalized by $p(24) = 1$ point; an assignment
turned in $48$ hours late will be penalized by $p(48) = 5$ points
(thus receiving a score of zero no matter what score it would have
received otherwise; assignments cannot receive negative scores).  The
purpose of this system is to allow for reasonable flexibility while
still providing firm consequences for lateness.  For example, it is
well worth it to turn in an assignment one hour late (incurring only a
$p(1) \approx 0.02$ point penalty, hardly noticeable) if it gives you
time to improve it a bit.  It is probably worth it to turn in an
assignment twelve hours late (incurring a $p(12) = 1/3$ point penalty)
if it gives you time to improve it substantially or finish the last
few problems.  It probably isn't worth it to turn in an assignment 24
hours late (1 point penalty), or 30 hours late (1.552 points), or 36
hours late (2.333 points)\dots as you can see, the exponential scale
leads to increasingly dire consequences.

\section{Resources}

Here are a few resources which you may find helpful throughout the
year:

\begin{itemize}
  \item Wolfram MathWorld (\url{http://mathworld.wolfram.com}) is a fantastic
    site with tons of reference material; essentially a comprehensive,
    free mathematics encyclopedia.  Because of its aim of
    comprehensiveness and its encyclopedia-like nature, however, the
    articles can sometimes be quite terse and hard to follow.

  \item Wikipedia (\url{http://en.wikipedia.org}) actually has an excellent,
    and growing, set of mathematics articles.  They are often a bit
    more descriptive and easier to follow than the articles on
    MathWorld.

  \item The Online Encyclopedia of Integer Sequences
    (\url{http://www.research.att.com/~njas/sequences/}) is exactly
    what its name says---an encyclopedia containing almost 150,000
    integer sequences, with data, descriptions, references, and
    relationships for each!  It might sound silly, but it's actually
    pretty cool.  

  \item The Math Less Traveled
    (\url{http://www.mathlesstraveled.com/}) is my mathematics
    blog.  I'd encourage you to follow it during the year---it's quite
    likely I'll be writing about some topics inspired by our course.

  \item Ask Dr. Math (\url{http://mathforum.org/dr.math/}) has a large
    collection of math questions and answers.

  \item Check the website for more resources!
\end{itemize}

\end{document}
