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Q&A #12605 |
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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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