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\section{Math 316 Project Rubric}

\subsection{Main Purpose}

The purpose of this document is two-fold:
\begin{itemize}
\item To provide information on what is expected for your Final Project in Math 316, and how you will be evaluated;
\item To give this latex file as a source for the written component of your Final Project, along with some techniques in latex.
\end{itemize}

In the other sections of this document, I just give a few methods in latex, some of which are also given in this section.  For example, above I used the {\tt itemize} environment, and below I will use the {\tt enumerate} environment.

\subsection{Basic Scoring of Project}
Since your Final Project will count as 30\% of your final grade, I will score it out of 300 points.  If you are doing a group project, then there must be an individual written component done on your own, which I give details about below.  In the case of a group project, 100 out of 300 points will be dedicated to your individual contribution, and 200 points will be dedicated to the group project as a whole.

Your project is required to have a primary topic in geometry, and a secondary topic.  This secondary topic can be related to the history of the geometry topic, or the teaching philosophy of that topic if the format of your project is a lecture and a lesson plan.  Other secondary topics which have been brought up are philosophy, art, and architecture.  As long as there is a very natural connection between the geometric topic and the secondary topic, many things can work.

The geometric topic of your project is the most important, and so 70\% of your score is dedicated to the correctness and depth of that topic, while 30\% of your score is dedicated to your secondary topic.  This is true for either an individual project or a group project.  That is:
\begin{enumerate}
\item If you are doing an individual project, 210/300 points are dedicated to the correctness and depth of the geometric topic of your project, and 90/300 points are dedicated to the correctness and depth of the secondary topic of your project.
\item If you are doing a group project, then for the correctness and depth of the geometric topic of the project, 140/200 points are dedicated to this for the group project as a whole, and 70/100 points are dedicated to this for the individual contribution.  For the correctness and depth of the secondary topic, 60/200 points are dedicated to this for the group project as a whole, and 30/100 points are dedicated to this for the individual contribution.
\end{enumerate}

\subsection{Specific Expectations}  First, every project must have {\it some} written component.  Each written component should be written in Latex, and you should specifically use this latex file as a base for your written component.  That is, use the exact same preamble as in this document so that you have the same margins, font size, spacing, etc.  I'll first address specific expectations for an individual project, and then for a group project.\\
\\
\begin{itemize}
\item For an individual project, it may be that you are writing an expository paper.  In this case, there is not too much to say, except that your paper should be 8-12 pages in length using this specific latex format (unless this paper is also your Major writing requirement, in which case refer to the expectations I sent to you about that).  You should have an introduction giving an overview of your topic, although it is often the case that it is easiest to write the introduction last.  Your paper should have at least 5 references, but there is no upper bound.  If your individual project which is not a paper, then there should still be a written component, although this written component may vary depending on the format of that project.  Please speak to me individually to clarify what should be in your written component if this is the case.\\
\\
\item For a group project, your final project can still be an expository paper, and the expectations are the same in this case as they are for an individual project.  Each individual author must also complete their own written component individually as follows.  {\bf First, you must complete this part completely individually, and not in collaboration with those in your group}.  Use this same Latex format to write a 2 page summary of your contribution to the project, a brief explanation of your references, and a brief explanation of what in your view is the main geometric topic of the paper, and what the secondary topic of the paper is.  If your group project is not an expository paper, you should discuss with me what the group written component of your project should include, but if your project is a lesson plan specifically, then the group written portion should be an analysis of your teaching methods, including any theory behind the methods, your target audience, and the expected mathematical background of your audience.  The individual portion of the written component should be the same as just described, that is, should be 2 pages in this Latex format, with a summary of your contribution, and a brief explanation of the references, the main geometric topic, and the secondary topic.
\end{itemize}

In all cases, all parts of the Math 316 Final Project are due (in my inbox or to me physically) by {\bf 12 Noon on Monday, December 16}.

\section{Another Section}

Say some more things.  We have the complex numbers $\CC$.  The following word is in italics:  {\it boom!}
\\
\\
To define things, you can put the word you are defining in italics, and declare the Definition in bold face, as follows:\\
\\
\noindent
{\bf Definition:  } A {\it triangle} is a polygon with three sides.\\
\\
If you state and prove a theorem it is important to use the Theorem/Proof environment for it to look nice.  In the preamble, I declared Proposition, Theorem, and Lemma as separate names for Theorem environments.  I give examples of each below.


\begin{theorem}  Here is a theorem about $X$.
\end{theorem}
\begin{proof} I just proved this. 
\end{proof}

Here is a fraction within a fraction:

$$ \frac{\frac{1}{t^2} + k}{\frac{1}{t} +k}.$$

To have brackets or parentheses auto-sized, use the syntax {\tt \textbackslash left} and {\tt \textbackslash right} before those brackets.  This will automatically make those brackets the right size to match what they are around, as below:

$$D(x,y) \leq \mathrm{max} \left\{ \frac{\bar{d}(x_1, y_1)}{1}, \frac{\bar{d}(x_2, y_2)}{2}, \ldots, \frac{\bar{d}(x_N, y_N)}{N} \right\}.$$

Also note that you have to be careful if you want text to appear in Latex which is actually a symbol used in Latex commands.  When I wanted to give the syntax for auto-sized brackets above, in order for it to appear as {\tt \textbackslash left} I had to type out a command just to make the backslash since this symbol is used so often.  Another example is the ampersand, which is typed by putting a backslash in front of the ampersand, as in \&.

Here is an aligned set of equations, which is useful for a string of equalities:

\begin{align*}
\sum_{j=1}^n j & = 1 + 2 + \cdots + n \\
             & = \frac{n(n+1)}{2}.
\end{align*}

\begin{prop} The above thing is a fraction
\end{prop}

\section{Yet Another Section}

Yet some more things!  Here is how you cite a result from a source:  The following appears in \cite[Lemma 4.17]{Au01}.  I have also given it a label (with the syntax {\tt \textbackslash label\{CplxLemma\}}), so that we can refer to it later.

\begin{lemma}\label{CplxLemma} [I. M. Author] Suppose that $L$ is a Lemma, and $\CC$ is the field of complex numbers.  Then $L$ is true.
\end{lemma}
\begin{proof} The proof is by contradiction.  Suppose $L$ is false.  That contradicts what I say is true.  Thus $L$ is true. 
\end{proof}

Later, we can say by Lemma \ref{CplxLemma}, other stuff follows.  This way we will be able to refer to it, and Latex will automatically refer to it by the right number without having to keep track or change it later.

Notice that when writing syntax for latex, I use the following {\tt font}.  This is also useful when referencing websites.  I will give an example of a website reference in the bibliography below. 

\section{Your Bibliography}
In a mathematics paper, the standard is that references are listed in alphabetical order by the last name of the first author.  If there is no specific author, like for a blog post with no listed author, you should go by the name of the blog instead of an author.  A very useful website for mathematical references is the following, which you can access if you are on the W\&M system:\\
\\
\noindent {\tt https:\slash\slash matscinet.ams.org\slash mathscinet\slash search.html}\\
\\
If you are not on the W\&M system, you can access this database through the following steps:
\begin{enumerate}
\item[(i)] Go to the main Swem webpage:  {\tt https:\slash\slash libraries.wm.edu}.
\item[(ii)] Click on the Databases icon.
\item[(iii)] Click on Mathematics \& Computer Science.
\item[(iv)] Under Starting Points, click on MathSciNet (via AMS).
\item[(v)] You will be prompted to enter your W\&M login info.  After you do this, you have access to MathSciNet.
\end{enumerate}

This database can help you with precise citations.  To list articles from journals, follow the format in the references \cite{Au01} and \cite{Sw19} below.  The reference \cite{Au01} is of course fictional, and is just there to demonstrate the basic format for this type of reference.  The article \cite{Sw19} is a real one, and you should look it up on MathSciNet to make sure you see where the citation comes from.  To reference books, the basic format is given in \cite{Gr93} below, which is our course textbook.  For book references, you can end the entry with the year of publication.  There is no need to list page length or ISBN codes, which are sometimes given on MathSciNet.  For another example of a book, see \cite{Eu} below.  This is a different kind of example, since it is a translation of Euclid's {\it Elements}, of which there are many.  Finally, to reference a website, see \cite{Geom} for an example.   The most important part here is that the last two parts of the entry should be the full web address, along with the date accessed.  Depending on whether you are citing a blog post with no specific author (like the one below), or with one author, or whether you cite a comment in some forum, the first part of the entry here could vary quite a bit.

\begin{thebibliography}{10}

\bibitem{Au01}
I. M. Author, How I wrote a Math 300 paper, \emph{Journal of Papers} \textbf{10} (2001), no. 2, 110--137.

\bibitem{Eu}
Euclid, Euclid's {\it Elements}.  All thirteen books complete in one volume.  The Thomas L. Heath translation.  Edited by Dana Densmore.  \emph{Green Lion Press, Santa Fe, NM}, 2002.

\bibitem{Geom}
GeometryCoach.com, {\it How to Teach Proofs with Uno}, {\tt http:\slash\slash geometrycoach.com\slash introducing-proofs-with-uno-cards}, accessed October 24, 2019.

\bibitem{Gr93}
Marvin Jay Greenberg, Euclidean and non-Euclidean geometries.  Development and history.  Third edition.  \emph{W. H. Freeman and Company, New York}, 1993.


\bibitem{Sw19}
Eric Swartz, On generalized quadrangles with a point regular group of automorphisms, \emph{European J. Combin.} \textbf{79} (2019), 60--74.

\end{thebibliography}




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