revised 8may11

Homework Comments 



\begin{document}
\maketitle
\textbf{ $\C$  2010, Prof. George K. Francis, Mathematics Department, University of Illinois} 

\section{Introduction}
There are general instructions and suggestions in the Advice section of these
notes. Periodically, there is an occasion to address issues that come up 
early on in this course. This is the especially the case when the first 
corrections on the first homework assignments roll in. Please note again that
the role of homework in this course is \textbf{pedagogical not judgmental.} 
That means the homework counts against your grade only if you don't even try
to do it, ignore the format requests, or habitually turn it in late. The 
format requests are not dictatorial rules. They reduce the amount of time
your grader spends figuring out what you have submitted. This time can then
be spent on helping you learn. 

\section{Format}
Please make every answer you give \textbf{ self-contained}. I should make sense to 
any reader who knows geometry. It should not require the reader to go looking
stuff up elsewhere. Every problem you solve must also be stated. Copy and paste
the text of the question, if you like. Do not refer to chapter-and-verse when
a simple key-word will identify the reference to something in the textbook
or in the notes. Whenever you can't figure out the correct LaTeX code for a
formula, give it a try. The grader can then correct your syntax along with 
with your geometry.

\section{Story Problems and Logical Formulas}
As with story problems you learned to solve in gradeschool, it is necessary to
translate them into mathematical abstractions to make absolutely sure that
your argument is correct. While something that has to be learned takes time,
you should endeavor to exercise your skill in exposition diligently. Sometimes
it is important to be entirely precise in what you mean. Then it is usefule
to write something in logical notation. The logical notation is not a 
substitute for writing it out in readable English. Nor is it always necessary.
Take your cue from the notes when you should use it, and when not. Note,
you can usually cut-and-paste and then edit such formulas until you learn
to do it faster by just remembering how.

\section{Capitalization Convention for Models}
We adopt the pedagogically helpful convention of capitalizing terms 
when they refer to primitives of a geometry as interpreted in a 
model, and lower case terms when they refer to the ambient geometry
of the model. 
Since this is risky in an increasingly \textbf{case-insensitive world,} 
we do this sparingly, and only when necessary to avoid confusion.
In general, the context suffices to identify the meaning of the word 
without additional decoration. 

Nevertheless, here you need to distinguish between Points and points where
there is such a distinction. In the Ice-Cream Geometry for example, 
if you interpret children as Points 
and flavors as Lines, then the star is a model, with all ten of its points 
also Points in the Axiom system. Note the caps! 
But if you, instead, interpret flavor as Point, and children as Lines 
then your model will be a star inscribed in a pentagon. Note that only
five of the points of intersection can be Points in the model.
 
\section{Caution on Symbol Abuse}
It is a common stylistic error to use technical mathematical symbols in a 
non symbolic sentence. Thus, while it is correct to read 
$(\exists n \in \mathbb{N})(n > 100)$ as ``There exists a natural number
greater than 100", it is not correct to write `` $\exists$ a natural 
number n > 100". In other words, even though $\exists$ is read as ``there
exists" when it occurs in a formula, to use the symbol to abbreviate 
common prose is bad form. You would never write ``Abraham Lincoln was  > 
Ulysses Grant as president", or ``All men are created = ", would you?
(Well, maybe on Twitter, but not in an essay.)
\section{A word about Quotation Marks in LaTeX }
You may have noticed the funny beginning quote in the last sentence. 
In good typography, there is a difference in the slant of beginning 
quotes and end quotes. LaTeX interprets the keypress " as end quotes.
If you don't believe me, try it in texPad or texWins. To get a beginning
quote, you use two back-tick ` key presses. Sadly, this doesn't work in 
MathML, so you don't see it here.