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\centerline{MONT 104Q -- Mathematical Journeys: From Known to Unknown}
\centerline{Oral Presentations on Euclid's \emph{Elements}, Book I}
\centerline{September 26, 2015}
\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
On October 5 and October 7, as announced in the course syllabus, you will be doing
an oral presentation with a partner (or in one case with two partners).  There will be
three or four of these each day, so aim for 10 - 15 minutes, with each person 
in the team contributing to the development and the delivery in class.  

The goal of your presentation will be to give the class an overview of one or two of the 
propositions from the next segment of Book I of the \emph{Elements} and the proof(s), 
plus the significance of the proposition, how it is similar to or different from 
something you have seen before in high school, etc.  For each assigned proposition 
below, there are some comments to guide you as you think about what you want to say.  

It would be best to do your presentations using either PowerPoint or Prezi since
you will need to keep them to a fairly tight time restriction and you don't want
to have write a lot on the board during the talk.  (If absolutely necessary,
a chalk-talk is OK too if you would prefer that, or if you are unfamiliar with either
software option.) If you do use the software, please keep distracting ``bells and 
whistles'' like slide animations, silly graphics, downloaded images, etc. to an 
\emph{absolute minimum}.  Your presentation should be 
\emph{about the mathematics}, not about how clever you can get with PowerPoint or Prezi. If
you want to include diagrams, but are not sure how to create them, you can consult
with me and we can figure out an appropriate way to do that.   

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\noindent
{\it Teams and Assigned Propositions}
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\noindent
For Monday, October 5:
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\begin{enumerate}
\item[1)]  Proposition 29 -- Isabel B. and Alex C.  This one is \emph{very significant} in the logical 
development of Book I for a reason you should briefly explain.  Make sure you understand 
and point out the precise relation between Proposition 29 and Proposition 27 from before.
\item[2)]  Proposition 32 -- Katie E. and Brendan E.  This one is a very basic and useful
fact.  It should look very familiar and you may have seen this exact proof before.  You 
will need to mention Proposition 31 (the construction of parallel lines) -- but don't try to 
prove that one too.  But pay careful attention to how that is used and how Proposition 29 comes
into it.    
\item[3)]  Proposition 33 -- Lizzie F. and Alex G.  This is an important fact that implies
key facts about parallelograms that will be used repeatedly later on.  Be sure you understand 
how earlier results like Propositions 27 and 29 are being used. 
\item[4)]  Propositions 35 and 36 -- Jack H. and Callan H.  You should state and prove 35
and then mention that the next thing Euclid does is 36 (state it, but it's not necessary to prove that one).
It is important to realize that these statements are really about \emph{the areas of the 
parallelograms}.  We would express this with
an area formula that implies the statement is true.  But Euclid does not have the notion
of a numerical measure for area (like a number of square inches).  So he has to resort
to showing that the parallelograms can be cut up into regions that are congruent.  It's not
necessary to fill in the details about the other cases that Euclid doesn't consider.  But
you should mention that his proof works only in one particular case and ignores the others.  
\end{enumerate}
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\noindent
For Wednesday, October 7:
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\begin{enumerate}
\item[(1)]  Propositions 39 and 40 -- Audrey H. and Michael M.  You should state and prove 39
and then mention that the next thing Euclid does is 40 (it's not necessary to prove that one).
It is important to realize that these statements are really about \emph{the areas of the 
triangles}.  We would express this with a
an area formula that implies the statement is true.  But Euclid does not have the notion
of a numerical measure for area (like a number of square inches).  So he has to resort
to showing that the triangles can be cut up into regions that are congruent.  It's not
necessary to fill in the details about the other cases that Euclid doesn't consider.  But
you should mention that his proof works only in one particular case and ignores the others.  
\item[(2)]  Proposition 41 -- Mary M. and Paolo P.  The special case where the triangle
is formed by a diagonal of the parallelogram is ``obvious.'' But be aware and point out 
that the statement here is much more general.  Your proposition uses the ideas from 35,36,39,40 
so be sure you look at those too and understand how they are being used.
\item[(3)]  Proposition 46 -- Frank S., Francois V. and Zack W.  Since you are the only 3-person
group, I'm going to ask you to do a bit more than the others.
 I'd like you to give the statement and proof of this proposition, then say something
about the following question:  Which regular polygons can be constructed using the methods
Euclid presents in Book I?  We know the equilateral triangle can be constructed from Proposition 1. 
This proposition gives a construction of the square.  There are some comments about further
cases treated later in the \emph{Elements} in the version from David Joyce's web site.  
Look those up and if you're feeling ambitious, see if you can understand how the simplest 
one works.  
\end{enumerate}
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