
AMATYC Review Spring 2006

The AMATYC Review A refereed publication of the American Mathematical Association of TwoYear Colleges Editor: Barbara S. Rives, Lamar State College Production Manager: John C. Peterson   Vol. 27, No.2, Spring 2006 issue, Abstracts  Authors   James Metz   The Radical Axis: A Motion Study
 Ray McGivney and Jim McKim   Graphical Representation of Complex Solutions of the Quadratic
Equation in the xy Plane
 Todd McDonald   Extending the Rule of 72 Through Linear Approximations (no abstract available)
 Stephen Kcenich and Michael J. Boss´   The Harmonic Series Diverges Again and Again
 Steven J. Kifowit and Terra A. Stamps   Sandra DeLozier Coleman   Extending Rules for Exponents and Roots Utilizing Mathematical
Connections
 Michael J. Boss´e and Stephen Kcenich   James Metz   The Calculus of Elasticity
 Warren B. Gordon   Building Buildings with Triangular Numbers
 David L. Pagni   Edited by Sandra DeLozier Coleman   Mathematics For Learning With Inammatory Notes for the Mortication of Educologists and the Vindication of "Just Plain Folks"
 Alain Schremmer   Edited by Brian E. Smith Reviewed by Marion S. Foster, and Tomball College   The Problems Section (no abstract available)
 Edited by Stephen Plett and Robert Stong 
The AMATYC Review Spring 2006, Vol.27, No.2  
  The Radical Axis: A Motion Study
Ray McGivney and Jim McKim   Ray McGivney is Professor of Mathematics at the University of Hartford. He earned his AB and MA in mathematics at Clark University and his PhD in mathematics at Lehigh University. He has served as mathematics consultant for several school systems in Connecticut and presented at numerous local, regional and national professional meetings. Email: mcgivney@hartford.edu   James McKim, now at Winthrop University, holds a PhD in mathematics from the University of Iowa. He has taught mathematics and computer science for more than 30 years, the last 15 mainly to working professionals. He is the coauthor (with Ray McGivney and Ben Pollina) of two mathematics textbooks and the author of several articles in both computer science and mathematics. Email: mckimj@winthrop.edu  Interesting problems sometimes have surprising sources. In this paper we take an innocent looking problem from a calculus book and rediscover the radical axis of classical geometry. For intersecting circles the radical axis is the line through the two points of intersection. For nonintersecting, nonconcentric circles, the radical axis still exists. We are interested in the relationship of this line to the two circles in this latter case. We take an algebraic approach to its formula so we can see this relationship as we move and scale the defining circles. This approach culminates in the discovery that if the two circles grow so that their areas increase at equal rates then the radical axis remains constant and in fact is the eventual line of intersection of the two circles. 

  Graphical Representation of Complex Solutions of the Quadratic Equation in the xy Plane Todd McDonald   Todd McDonald is an adjunct developmental mathematics instructor at Volunteer State Community College and the quality manager at Crowley Tool Company in Hendersonville, Tennessee. Todd received his BS in mathematics from Middle Tennessee State University and is currently working on a graduate degree. His passions are his family, teaching mathematics, and extreme skateboarding. Email: todd.mcdonald@crowleytool.com  This paper presents a visual representation of complex solutions of quadratic equations in the xy plane. Rather than moving to the complex plane, students are able to experience a geometric interpretation of the solutions in the xy plane. I am also working on these types of representations with higher order polynomials with some success. 

 The Harmonic Series Diverges Again and Again Steven J. Kifowit and Terra A. Stamps   Steve Kifowit is an associate professor of mathematics at Prairie State College. He has a BS degree in physics and applied mathematics and an MS degree in computational mathematics, both from Northern Illinois University. Email: skifowit@prairiestate.edu Terra Stamps is an associate professor of mathematics and the mathematics department chair at Prairie State College. She holds a bachelor’s degree in mathematics from the University of Montevallo and a master’s degree in pure mathematics from the University of Alabama. Email: tstamps@prairiestate.edu  The harmonic series is one of the most celebrated infinite series of mathematics. A quick glance at a variety of modern calculus textbooks reveals that there are two very popular proofs of the divergence of the harmonic series. In this article, the authors survey these popular proofs along with many other proofs that are equally simple and insightful. A common thread connecting the proofs is their accessibility to firstyear calculus students. 
 Universal Paradox Sandra DeLozier Coleman One gigantic set made of all that there is Boggles the mind with paradoxes. For it is greater than all, but smaller than this— The set which consists of the subsets of it. June 1986 
  Extending Rules for Exponents and Roots Utilizing Mathematical Connections
Michael J. Bossé and Stephen Kcenich   Michael J. Boss´e is an associate professor of Mathematics Education at East Carolina University. He received his PhD from the University of Connecticut. His professional interests within the field of mathematics education include elementary and secondary mathematics education, pedagogy, epistemology, learning styles, and the use of technology in the classroom. Email: bossem@ecu.edu   Stephen Kcenich is currently a lecturer at the University of Maryland in College Park. His main interest is actuarial mathematics and its application to undergraduate curriculum. Email: stephenkcenich@yahoo.com  This paper considers rules for multiplying exponential and radical expressions of different bases and exponents and/or roots. This paper demonstrates the development of mathematical concepts through the application of connections to other mathematical ideas. The developed rules and most of the employed connections are within the realm of secondary and elementary college mathematics. 

 
 The Calculus of Elasticity
Warren B. Gordon   Warren B. Gordon is professor and chair of the Department of Mathematics at Baruch College, and has recently been interested in integration of technology and applications to the calculus curriculum. He has just completed a text which will appear in the fall 2006, providing an integrated approach to precalculus and applied calculus, including the use of technology. Email: wgordon@baruch.cuny.edu  This paper examines the elasticity of demand, and shows that geometrically, it may be interpreted as the ratio of two simple distances along the tangent line: the distance from the point on the curve to the xintercept to the distance from the point on the curve to the yintercept. It also shows that total revenue is maximized at the transition point from elastic to inelastic demand; when elasticity is unitary. 
  Building Buildings with Triangular Numbers David L. Pagni   David L. Pagni is a mathematics professor at California State University, Fullerton. His interests range across the spectrum of mathematics education from grades K16. His teaching fields include mathematics, mathematics learning, teaching, and technology. He is currently principal investigator of a National Science Foundation Mathematics and Science Partnership called Teachers Assisting Students to Excel in Learning Mathematics (TASELM). Email: dpagni@Exchange.fullerton.edu  Triangular numbers are used to unravel a new sequence of natural numbers heretofore not appearing on the Encyclopedia of Integer Sequences website. Insight is provided on the construction of the sequence using "buildings” as a viewable model of the sequence entries. A stepbystep analysis of the sequence pattern reveals a method for generating the function. Graphing calculator programs are provided for generating the sequence both recursively and explicitly for different initial "building” sizes. Finally, an explicit formula for the sequence that makes use of the "ceiling” function generalizes the results. 

 Book Reviews Edited by Sandra DeLozier Coleman
 FRACTALS, GRAPHICS, AND MATHEMATICS EDUCATION, Michael Frame and Benoit B. Mandelbrot, editors, The Mathematical Association of America., USA, 2002, ISBN 0883851695. 
 Mathematics For Learning With Inflammatory Notes for the Mortification of Educologists and the Vindication of "Just Plain Folks” Alain Schremmer
 In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an opensource serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring 2004 issue with a new chapter in each subsequent issue of The AMATYC Review. This issue contains a reorganization of the first three chapters and includes sections on Comparing Collections: Equalities and Inequalities and Specifying Collections: Equations and "Inequations.” 
 Software Reviews Reviewed by Tristan Denley, University of Mississippi Edited by Brian E. Smith
 Hawkes Learning Systems Courseware Producer and Distributor: Hawkes Learning Systems Address: 1023 Wappoo Road Suite 6A, Charleston, SC 29407 Web addresses: www.hawkeslearning.com Pricing Information:  Basic Mathematics Textbook and Software Bundle $72.00
 Prealgebra Textbook and Software Bundle $72.00
 Introductory Algebra Textbook and Software Bundle $65.00
 Intermediate Algebra Textbook and Software Bundle $65.00
 College Algebra Textbook and Software Bundle $72.00
 Statistics Textbook and Software Bundle $70.00



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