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AMATYC Review Spring 2004
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The AMATYC Review


A refereed publication of the American Mathematical Association
of Two-Year Colleges

Editor:Barbara S. Rives, Lamar State College

Production Manager:John C. Peterson


Spring 2004 issue, Vol. 25, No.2

Table of Contents

Sky-High i’s

Andrew J. Berry

Andrew J. Berry received his BS and MS degrees in mathematics at the University of Illinois at Urbana-Champaign, and his PhD at New York University. He is Associate Professor of mathematics at LaGuardia Community College, City University of New York.
When discussing the topic of elementary complex variables, students are often mystified by the fact that ii is real. After seeing a proof of this statement, a standard question is "well, what about iii or iiii etc., are these real or complex?”

In this paper the meaning of the infinite power-tower iii... is considered both from the "bottom-up” and the "top-down.” Some surprising graphical representations are revealed and issues of convergence and evaluation are discussed.

Finally, we introduce the elementary properties of the Lambert’s W-function and it’s relationship to the function f(x) = xxxx... which Euler explored in 1783. (Back to top)

Poiseuille’s Law - Showing that p Is Inversely
Proportional to R4 Using the Shell Method

John P. Drost, and Rachel A. Georges

John P. Drost earned his PhD in Education Administration with emphasis in Mathematics from the University of Utah in 1975. He is currently a Professor of Mathematics at the University of Wisconsin-Eau Claire. His interests are in teaching undergraduate mathematics, M.C. Escher, symmetric groups, and planar and vortex tessellations. In recent years, he has been integrating technology, graphing calculators and Maple into his teaching.

Rachel A. Georges is a junior at the University of Wisconsin-Eau Claire, pursuing a mathematics major in actuarial science. She is interested in applied mathematics and found Poiseuille’s Law of Resistance to be intriguing. This article is a result of her research into why p is inversely proportional to fourth power of the resistance, R4. Her hobbies include horseback riding, spending time with friends and family, and active sports.

Jean Poiseuille, a physician, developed a mercury filled U-tube to measure blood pressure in 1828. He discovered that pressure in veins is significantly lower than pressure in arteries. As a result, he studied liquid flow in small tubes. A few years later he established Poiseuille’s Law, which states the resistance, p, of the flow of blood as
p = K(L/R4) where L and R are the length and radius of the vessel respectively. K is a positive constant determined by the viscosity of the blood.

It is curious that the resistance is inversely proportional to the fourth power of the radius R. At first glance, most individuals may assume the resistance should be inversely proportional to the second power of the radius since the area of a circle is R2. In this article, mathematics, including finding volume using cylindrical shells, is developed to establish that the resistance is indeed inversely proportional to the fourth power of the radius.

Problems on Divisibility of Binomial Coefficients

Thomas J. Osler, and James Smoak

Tom Osler, professor of mathematics at Rowan University, is the author of 58 mathematical papers. In addition to teaching university mathematics for 43 years, Tom has been competing in long distance running races for the past 50 consecutive years. He is the author of two books on running.

Jim Smoak is a retired mathematician with an insatiable interest in number patterns. Jim served as a ballistic meteorologist in Viet Nam from 1968–70, receiving a bronze star for his efforts. He maintains an active correspondence with some of America’s leading mathematicians, including George Andrews, the world famous number theorist from Penn State University.

Twelve unusual problems involving divisibility of the binomial coefficients are represented in this article. The problems are listed in "The Problems” section. All twelve problems have short solutions which are listed in "The Solutions” section. These problems could be assigned to students in any course in which the binomial theorem and Pascal’s triangle are presented. This includes courses in precalculus mathematics, real analysis, and number theory.

The Centers of Similarity of
Two Non-Congruent Squares
by: Ayoub B. Ayoub

Ayoub is a professor of mathematics at Abington College of the Pennsylvania State University. He received his Ph.D. in Mathematics from Temple University, Philadelphia. Ayoub’s areas of interest are number theory, classical mathematics, and undergraduate mathematics education.

Generating Nice Linear Systems
for Matrix Gaussian Elimination
by: L. James Homewood

L. James (Jim) Homewood is a member of the fulltime mathematics faculty at the Downtown Campus of Pima Community College in Tucson, Arizona. He earned a master’s degree in mathematics at Portland State University, with two additional years as a graduate associate in the doctoral program in mathematics at the University of Arizona. His major interests are analysis and functional analysis.

In this article an augmented matrix that represents a system of linear equations is called nice if a sequence of elementary row operations that reduces the matrix to row-echelon form, through matrix Gaussian elimination, does so by restricting all entries to integers in every step.

Many instructors wish to use the example of matrix Gaussian elimination to introduce their students to algorithms that are capable of handling very large linear systems. Instructors should be able to generate, if they desire, a wide variety of modestly sized nice matrices from which they may choose introductory examples and select exam questions. The formulas for generating nice 2 × 3, 3 × 4, and 4 × 5 augmented matrices are shown in this article, with emphasis on the derivation of the matrix. An instructor may use any of these formulas to generate an augmented matrix representing a "one-solution,” a dependent or an inconsistent system. The author has developed TI-83 and TI-86 programs that generate nice augmented matrices.

Finding Equations of Tangents to Conics by: George Baloglou &
Michel Helfgott

George Baloglou is an associate professor of mathematics at SUNY Oswego, where he has been teaching since 1988. He is currently working on a book on planar crystallographic groups (wallpaper patterns), largely influenced by a symmetry course he has been teaching since 1995. His other mathematical interests include elementary inequalities, convexity, and basic number theory.

Michel Helfgott is an assistant professor of mathematics at SUNY Oswego. His main interests are the history of mathematics and its use in teaching, as well as the use of physics and chemistry as pedagogical tools in the mathematics classroom.

A calculus-free approach is offered for determining the equation of lines tangent to conics. Four types of problems are discussed: line tangent to a conic at a given point, line tangent to a conic passing through a given point outside the conic, line of a given slope tangent to a conic, and line tangent to two conics simultaneously; in each case, a comparison to the standard calculus method is made by way of specific examples. Extending an idea of Descartes, this calculus-free approach is based on the fact that a quadratic has a double root if and only if its discriminant is equal to zero. It should be appropriate for both precalculus and calculus students.

Developing Simultaneous
Linear Equations and Rational Equations
by: Michael J. Boss´e &
N. R. Nandakumar

Michael J. Boss´e is an Associate Professor of Mathematics Education
at Morgan State 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.

N.R. Nandakumar is a Professor of Mathematics at Delaware State
University. He received his PhD in Mathematics and a Master’s in
Computer Science from the University of Illinois. His research interests
include functional analysis, numerical analysis, and computer science.

To demonstrate concepts or rapidly create quizzes, teachers commonly encounter the need to quickly create mathematical examples. Unfortunately, by producing undesirable or overly complex solutions, extemporaneously created examples can become problematic, create tense learning environments and become more confusing than they are worth. Experience reminds teachers that a moment of planning may avoid many difficult classroom scenarios. Solid mathematical understanding of a few techniques can greatly assist teachers to quickly develop appropriate examples with desired results. This paper considers techniques which will assist instructors in quickly developing appropriate examples with "nice” solutions when teaching rational equations and simultaneous linear equations.
On the Integration of Technology
into the Elementary Calculus I Curriculum
by: Warren B. Gordon

Warren Gordon is Chair of the Baruch College mathematics department and has been interested, over the last ten years, in the integration of technology into the mathematics curriculum. This paper reflects the approach taken in his basic calculus course.
E-mail: warren

This paper suggests examples that may be used to better integrate modern technology into the calculus I curriculum, and at the same time extend the student’s understanding of the underlying concepts. Examples are chosen from the usual topics considered in most courses and not limited to any specific form of the technology.

Rotation of Axes and the Mean Value Theorem by: David Price

David Price earned his BS and MS degrees from Southwestern University and the University of North Texas respectively. He teaches mathematics at Tarrant County College in Arlington, Texas.

This article provides a proof of the Mean Value Theorem by rotating a coordinate system through a specified angle. The use of this approach makes it easy to visualize why the Mean Value Theorem is true. An instructor can use the proof as another illustration of the rotation of axis technique in addition to the standard one of simplifying equations of conic sections.

Examining Prospective Teachers’
Understanding of Proportional Reasoning
Richard Kitchen &
Julie DePree

Richard Kitchen is an assistant professor in the College of Education at the University of New Mexico in Albuquerque, New Mexico. His primary interests are problem solving and alternative assessment formats.

Julie DePree is an associate professor in the Department of Mathematics and Statistics at the University of New Mexico- Valencia Campus in Los Lunas, NM. She teaches many of the mathematics courses for teachers and also teaches statistics and algebra courses.

In this article, the authors describe their efforts to assess prospective K-8 teachers’ knowledge of proportional reasoning. Based upon their analysis of prospective K-8 teachers’ work on a mathematics performance task, they discuss the implications for preparing prospective teachers to teach proportional reasoning to their students.

In general, the prospective teachers used good estimation strategies and were capable of engaging in proportional reasoning, but many had misconceptions about fractions, decimals, and proportions. In particular, the prospective teachers had difficulties converting between decimals and fractions. Because of errors made when working with fractions and decimals, most failed to correctly solve the task. The prospective teachers’ errors and misconceptions highlighted the difficulties they had making sense of fractions and decimals.

The lack of facility when converting between a decimal and a fraction illustrated that the prospective teachers need multiple experiences making sense of decimals and fractions, and converting between them. To address prospective teachers’ deficiencies in making sense of fractions and decimals, converting between them, and working with proportions, the authors are placing more emphasis on these concepts in their mathematics content and methods courses for prospective K–8 teachers.

The Open Box Problem by: William B. Gearhart &
Harris S. Shultz

William B. Gearhart received his BS degree in engineering physics
and his PhD in applied mathematics from Cornell University. He
is currently a professor of mathematics at California State University, Fullerton. His research interests include approximation theory, numerical analysis, optimization theory, and mathematical modeling.

Harris S. Shultz, professor of mathematics at California State
University, Fullerton, received his BA degree in mathematics from
Cornell University and his PhD in mathematics from Purdue University. He has directed numerous institutes for secondary mathematics teachers and been a frequent contributer to The AMATYC Review.

In a well-known calculus problem, an open top box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. The task is to find the dimensions of the box of maximum volume. Typically, the length of the sides of the corners that produces the largest volume turns out to be an irrational number. However, there are examples of non-square rectangles for which this length is a rational number. In this article we show how to generate all cases in which integer values for the dimensions of the rectangle produce rational answers. This provides calculus instructors with several rectangles for which the optimal box has "nice” dimensions.


How Mathematics Could Make Sense
to Lots Of People And Why It Does Not:
The Case Against Educology
by: Alain Schremmer

In the Fall 2003 issue of this Review, Schremmer brought the "Notes of The Mathematical Underground” to an end because. He is now writing a proto-textbook along the lines that he has urged all these years. This issue contains the rationale, preface, and contents for the book. Subsequent issues will contain the chapters.

Book Reviews Edited by: Sandra DeLozier Coleman

Spectrum Series, The Mathematical Association of America, Inc.,
United States of America, 2002, ISBN 0-88385-540-2.

Software Reviews Edited by: Brian E.
The Problems Section Edited by: Stephen Plett &
Robert Strong

New Problems
The AR Problem Set consists of four new problems.

Set AP Solutions
Solutions are given to the four problems from the AP Problem Set that were in the Spring 2003 issue of The AMATYC Review.


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