From: Pat Ballew (for Teacher2Teacher Service)
Date: Nov 29, 2003 at 11:00:27
Subject: Re: The geometry of the complex roots of graphs

TJ,
  If you search old copies of the Mathematics teacher you will find several 
variations of this same idea dating back to the work of the English 
Mathematician John Wallis, who was Newton's mentor for a while.  I wrote a 
couple of short articles on the same idea to share with teachers a short 
while ago.  One explaining how functions of quadratics on the complex plane 
can be used to explain why this "phantom parabola" of yours exists, and 
another suggesting that with the wider range of students in Algebra classes 
today, the ideas behind completing the square could be replaced by a more 
conceptually based graphic approach using these "phantom curves".  I wish to 
heck I had had the creativity you have to come up with that term....


If you wish to see the articles they are at 
http://www.pballew.net/doclist.htm

The articles are "replacing the quadratic" and "comments on quadratics and 
the complex plane.  

I think that in an environment in which too many teachers use rote memory 
ideas in lieu of teaching understanding of mathematical ideas, this topic is 
worth presenting often from a wide range of perspectives, so by all means, 
publish your ideas on the solution method and how it could be implemented in 
classrooms.  I especially think the term "phantom parabola" would catch on 
and be a motivator to Alg I level students.  

In regard to cubics, the relationship between the ideas seems more difficult 
to develop because the roots are not bilateral around the point of 
inflection,  but if you have ideas about an approach, I would love to hear 
your views.  You can find respond here or through my personal email which 
can be found on the webpages mentioned before.

Good luck

 -Pat Ballew, for the T2T service

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