p1_intro_project.tex

%%%%%%
%
% PROJECT 1
%
% filename: p1_intro_project.tex
% last modified: 2016-8-29
%
%%%%%%%
%
%
%%%%%%%

\documentclass
[justified,nohyper]
{tufte-handout}

\usepackage{amsmath}

\usepackage{booktabs}
\usepackage{graphicx}
\usepackage{kmath,kerkis} % The order of the packages matters; kmath changes the default text font
\usepackage[T1]{fontenc}

\usepackage{enumitem}


% USEFUL SHORTCUTS FOR MATH
\newcommand{\ds}{\displaystyle}

\newcommand{\dt}[1]{\dfrac{d#1}{dt}}

\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}

\newcommand{\evalat}{\biggr\rvert}

\begin{document}
\begin{fullwidth}
    \mbox{\LARGE PreCalculus BC: Project One - \today }\hfill
\end{fullwidth}
\section*{Introduction}

In this first project in PreCalculus BC, we won't be looking at a
particularly difficult math problem. Instead, the focus should be
on getting comfortable with the new tools used to create these
projects.

Here is the problem we will be solving.

\begin{quote}
    A gardener has 140 feet of fencing to fence in a rectangular
    vegetable garden. Find a function that models the area of the
    garden she can fence. For what range of widths is the area
    greater than 825 square feet? Can she fence a garden with area
    1250 square feet? Find the dimensions of the largest area she
    can fence.
\end{quote}

\section*{The Process}

\subsection*{Explore the Problem}
The first step in the process of solving and writing a report about
this problem is to actually solve the problem and
\textbf{explore the problem space}. What does this mean? Here are
a few things for you think about.

\begin{itemize}
    \item What questions have been asked directly in the problem?
    \item What are the answers to those direct questions?
    \item In the process of finding those answers, what other
    questions came into your mind?
    \item In what ways could the problem be generalized or extended
    beyond what was stated? Would those extensions be useful?
\end{itemize}

\subsection*{Example Problem}
Here is a short example to give you an idea about what exploring the
problem means. Suppose that you are in charge of designing a dating website's
algorithm for determining whether two people are compatible. For each person
that is registered on the site, a software engineer has written an
algorithm that provides you with a rectangle representing that person's
character and personality. This rectangle has a length and a width along
with the coordinates of the lower-left corner.

Each person's rectangle is plotted on a normal 2D plane such as the $xy$-axis.
For any two people, there are three possibilities. First, their rectangles could
have no overlap. Second, their rectangles
could overlap a little bit. Third, their rectangles could be identical in which
case they would completely overlap.

Your job is to take the information for each rectangle and write an algorithm
that determines the amount of overlap and interpret that as how compatible
two people are.

Exploring this problem means that you need to think about multiple possible
solutions and consider the advantages and disadvantages of both. There isn't
going to be one and only one correct answer. However, certain answers may
be more easily justified than others and the quality of the justifcation is
what's important.

Here are some of the things that go through my head when I think about
solving this problem. These are certainly not the only things that you can
investigate, but perhaps they give you some idea about what it means to
explore the problem space.

\begin{itemize}
    \item Given two rectangles, how am I going to compute the overlapping
    area?
    \item Once I have that area as a number, how can I interpret that in terms
    of compatibility. For example, does it matter how large or small each
    individual person's area is compared with the amount of overlap?
    \item If there is 0 overlap, but the rectangles are very close, would that
    have any useful meaning? In other words, could I extend the problem a little
    bit and consider rectangle proximity rather than only overlapping area?
    \item Why do we have to use a rectangle? Could other shapes be used and would
    they have any advantages in terms of the goals of the website?
    \item Can I generate some random rectangles and test my algorithm to make
    sure that it works the way I think it does?
    \item What would be the best way to illustrate compatibility in terms of
    area? 
\end{itemize}

\subsection*{Waiting -- time to percolate}
After you spend some time with the problem and exploring the problem space,
the next step involves waiting. You want to give your brain some time
to think about the work you have just completed. This is important
because when we are confronted with a deadline or need to finish
something by a particular time, our brains aren't really in the best
mode to think creatively. We need time to rest and let things
percolate.

During this time, you will want to think about the problem and talk
about your solution with others. Walk through all the steps and
focus on what justification you had for each step. What kinds of
assumptions have you made? Are they reasonable assumptions?

I cannot emphasize this step in the process enough. You simply must
have time to let your brain wander and develop ideas about the problem.
If you find yourself stuck, it helps to ask why you're stuck. Try to write
down exactly why you feel stuck. When I get stuck, the process of thinking
about why I am stuck really helps to get me unstuck.

\subsection*{Outlining}
After a few days have passed, you will want to start and outline
your solution. There are a number of different ways to do this,
and you may already have a good process from a previous class. I
will give you the same outlining strategy that I have used for
writing mathematical papers. The basic idea is to make use of
a number of notecards. Each notecard will have a single idea
written on it that relates to how you solved the problem. These
notecards are a way of documenting your thinking. The reason
that we use notecards is that it is really easy to lay out all
the cards in front of you and visualize your thought process.

\subsection*{Writing}
Once everything seems to be clear, you can begin writing your
report. The notecards will help because you already have all the
information you need. This last part of writing is perhaps the
most difficult because you will need to think about how to
write in a clear and precise way. We will see some examples
of how this looks in class and talk about ways to write
clearly.

\section*{The Report}

After you have written a first draft, you will bring a hardcopy
to class and we will exchange papers for others to read and mark
up. This is an opportunity for you to get feedback on
things like typos or sentences that are not clear.

Once you have fixed any errors and re-written your report, I will show
you how the report will be submitted from within SageMathCloud.

\section*{Basic Expectations}

For this first project, here are the things I consider important.

\begin{itemize}
    \item Your report is written in \LaTeX\, and contains no spelling errors,
    no grammatical errors, and is written in a clear and precise way.

    \item You have presented your solution to the problem so that the reader
    clearly sees that you have considered multiple methods of solving the
    problem and justified your approach.

    \item If there are equations in your report, they are correctly typeset
    in a mathematical font.

    \item If there are graphs in your report, they are correctly titled with
    the axes labeled and referenced from the text of your report. 
\end{itemize}

\section*{The Key}

The key to these projects is not to wait until the last minute to write them.
Start early. You will certainly encounter issues as you work, and if you are
faced with a deadline you won't have the time to solve these issues. This way
of working can be really challenging, depending on your previous experience.



\end{document}