The AMATYC Review
A refereed publication of the
American Mathematical Association of TwoYear Colleges
Table of Contents: Spring 2005 issue



From the Editor’s Keyboard
EMAIL: The editor of The AMATYC Review


Unexpected
Constructible Numbers 
By: Thomas J. Osler 

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.
Email: Osler@rowan.edu 
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Teaching about Inverse Functions 
By: Warren Esty 

Warren Esty is a professor of mathematics in the Department of Mathematical Sciences at Montana State University in Bozeman, Montana. He has published extensively in probability theory, statistics, and mathematics education. He has written two books, The Language of Mathematics and Precalculus. In his spare time he studies ancient Rome and Greece.
Email: westy@math.montana.edu

In their sections on inverses most precalculus texts emphasize an algorithm for finding f ^{1} given f. However, inspection of precalculus and calculus texts shows that students will never again use the algorithm, which suggests the textbook emphasis may be misplaced. Inverses appear primarily when equations need to be solved, which suggests instruction about inverses should emphasize their use in solving the equation "f(x) = c.” Instruction, and the algorithm used, should take advantage of the possibility of perfectly paralleling the process for solving "f(x) = y” for x (not solving "f(y) = x” for y). Switching letters after solving, rather than before solving, preserves the parallel. When f is not onetoone (such as f(x) = x^{2} or f(x) = sin x), students frequently fail to find the second solution. By discussing inverses in terms of solutions to "f(x) = c,” this difficulty is naturally addressed. Furthermore, the terms onetoone and range have natural definitions in this context and the Horizontal Line Test is also natural. The algorithm for finding inverses and the associated terminology can best aid in proper conceptual development if they focus on the primary context —solving "f(x) = c” for x.


Some Unusual Expressions For the Inradius of a Triangle 
by: 
Thomas J. Osler &
Tirupathi R. Chandrupatla


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.
Email: Osler@rowan.edu
Tirupathi R. Chandrupatla is professor of mechanical engineering at Rowan University. His areas of interest include finiteelement analysis, design, optimization, and manufacturing engineering. He is author of the books Introduction to Finite Elements in Engineering and Optimization Concepts and Applications in Engineering, both published by Prentice Hall and used in universities throughout the world. He recently won a prestigious award for his book of Indian poetry.
Email: chandrupatla@rowan.edu

Several formulae for the inradius of various types of triangles are derived. Properties of the inradius and trigonometric functions of the angles of Pythagorean and Heronian triangles are also presented. The entire presentation is elementary and suitable for classes in geometry, precalculus mathematics and number theory.

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Beyond Pascal’s Triangle 
By: Darrell P. Minor 

Darrell Minor is a professor of mathematics at Columbus State Community College. His academic interests include number theory, game theory, the history of mathematics, and helping students discover the beauty of mathematics. Away from work, he enjoys bicycling, reading, playing softball, and spending time with his wife and three sons.
Email: dminor@cscc.edu 
In "Beyond Pascals Triangle” the author demonstrates ways of using "Pascallike” triangles to expand polynomials raised to powers in a fairly quick and easy fashion. The recursive method could easily be implemented within a spreadsheet, or simply by using paper and pencil. An explanation of why the method works follows the several examples that are provided.


The Power of Power Functions 
By: Florence S. Gordon 

Florence S. Gordon is Professor of Mathematics at New York Institute of Technology. She is a coauthor of Functioning in the Real World, coauthor of Contemporary Statistics: A Computer Approach and coeditor of the MAA Notes volumes, Statistics for the Twenty First Century and A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus. She has published extensively in mathematics and statistics education.
Email: fgordon@nyit.edu 
Traditional college algebra courses focus almost exclusively on power functions such as y = x^{2} and y = x^{3} rather than the more general y = x^{p}. However, it is the more general form that is the basis of the mathematical models that arise throughout the natural sciences in a host of unexpected and highly interesting, applications. This article demonstrates a variety of applications drawn from biology and other areas that lead to power functions and some of the kinds of questions that can truly motivate students to find value in the mathematics they are learning. For instance: Why can’t a turkey fly? How much did a pterodactyl weigh? How many different species can the island of Puerto Rico support? How large does an island have to be to have 100 species inhabit it? How does the size of an organism relate to how fast it can run? Or swim? Or fly?


The Importance of Introductory Statistics
Students Understanding Appropriate
Sampling Techniques 
By: Violeta C. Menil 

Violeta C. Menil is an assistant professor at the Mathematics Department of Hostos Community College of the City University of New York, (CUNY). She obtained her PhD in Mathematics and Statistics Education from New York University. Aside from her extensive academic experience, she worked as the audit statistician of the New York City Comptroller’s Office for three years. Among her research interests are sampling, multidimensional scaling and univariate and multivariate data analysis.
Email: menilv2@aol.com

In this paper the author discusses the meaning of sampling, the reasons for sampling, the Central Limit Theorem, and the different techniques of sampling. Practical and relevant examples are given to make the appropriate sampling techniques understandable to students of Introductory Statistics courses. With a thorough knowledge of sampling techniques, students are equipped with the necessary tools to undertake basic research.


Excellence through Mathematics
Communication and Collaboration (E = mc^{2}):
A new approach to Quality in College Algebra 
By: Gerald L. Marshall and Herbert H. J. Riedel 

Gerald Marshall, department head of mathematics, TriCounty Technical College, received his PhD from Illinois State University in 2000. Earlier degrees were from North Carolina State University, Florida State University, and University of Alabama in Huntsville. He has had articles published in the Mathematics Teacher, The Mathematics Educator, and the Journal of Research and Development in Education.
Email: gmarshal@tctc.edu 
Herbert Riedel obtained a PhD in pure mathematics from the University of Waterloo, Canada. In July 2004, he left TriCounty Technical College and became Deputy Director of Nanoscience Technology at the University of Central Florida to assist with the establishment and administration of a new research center in the emerging interdisciplinary field of nanotechnology. He will continue his involvement in undergraduate education and pursue partnerships with twoyear colleges designed to enhance educational opportunities and position students to benefit from careers in the new economy.
Email: hriedel@mail.ucf.edu 
Within the context of traditional quality management principles, a program designed to enhance soft skills of College Algebra students was piloted in fall 2001. Results of the pilot project include increased retention, persistence, notetaking ability, and positive responses concerning course expectations. Quality enhancement efforts in this course have manifested significant improvements in this course and in followup courses, but overall success rates remain low due to a misalignment of process capability with standards. Possible solutions are presented.


Elementary Algebraic Models in Our World:
A General Education Alternative to College
Algebra 
By: Robert Franzosa and Jennifer Tyne 

Robert Franzosa is a professor of mathematics in the Department of Mathematics and Statistics at the University of Maine. He has a PhD in mathematics from the University ofWisconsin. His interests include applied topology and mathematics education.
Email: franzosa@math.umaine.edu
Jennifer Tyne is a lecturer in Department of Mathematics and Statistics at the University of Maine. She has a MS in operations research from the University of North Carolina. Her interests include mathematics education and curriculum development.
Email: tyne@math.umaine.edu 
Elementary Algebraic Models in Our World (MAT 103) is a general education course at the University of Maine that was developed as an alternative to College Algebra. An important goal in the development of MAT 103 was the improvement of the studentsÕ attitudes about and understanding of simple algebraic models. MAT 103 was developed in conjunction with a new Masters in Science Teaching (MST) degree program at the University of Maine to provide a research laboratory for MST graduate students. In this paper we present an overview of the MAT 103 course development project, including a discussion of the background motivation, the course teaching framework, the course content framework, the course materials, and the initial evaluation of the course based on student surveys and evaluations. Appendices are included presenting a sample of class materials and a summary of the survey data.

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Book Reviews 
Reviewed by Roxane Barrows
Edited by Sandra DeLozier Coleman 

LION HUNTING & OTHER MATHEMATICAL PURSUITS, a collection of mathematics, verse, and stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson & Dale H. Mugler, The Mathematical Association of America, Dolciani Mathematical Expositions, No. 15, United States of America, 1995, ISBN 088385323X.


Mathematics For Learning
Chapter 2: Accounting for Money On the Counter (II) 
By: 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; Chapter 1 was in the Fall 2004 journal. This issue contains Chapter 2. There are four sections in this chapter: "(Decimal) Headings,” "Adding Under A Heading,” "Subtracting Under A Heading,” and "(Decimal) NumberPhrase.”

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Software Reviews 
Reviewed by Marion S. Foster, Tomball College
Edited by Brian E. Smith 

MyMathLab/CourseCompass Course Management System
Producer and Distributor: Pearson Education Math and Statistics
Web addresses: www.coursecompass.com and www.mymathlab.com
System Requirements: Specific system requirements vary depending on your course. Most MyMathLab courses require a Windows^{®} operating system and a supported version of Microsoft^{®} Internet Explorer or Netscape^{®}. However, courses for calculus and statistics also run on certain Macintosh^{®} operating systems with supported versions of Netscape.
Internet connection: Cable/DSL, T1, or other highspeed for multimedia content; 56k modem (minimum) for tutorials, homework, and testing. Price: No charge to instructor for course creation. Students have to obtain a student access code. This may be provided with text adoption or purchased online. Contact Pearson Education for details.

