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Farshad Barman received his PhD in electrical engineering from the University of California in Santa Barbara in 1979. He taught and worked in that eld until 1992. He received his master's degree in mathematics from Portland State University in 1995 and has been teaching mathematics at Portland Community College since then. His current interests are the mathematics of astronomy, stargazing, and baseball.
E-mail: fbarman@pcc.edu

The mathematics for finding and plotting the locations of stars and constellations are available in many books on astronomy, but the steps involve mystifying and fragmented equations, calculations, and terminology. This paper will introduce an entirely new unified and cohesive technique that is easy to understand by mathematicians, and simple enough to fit on one line, and easy to program into a graphing calculator. The result will be a 2 x n matrix of star coordinates that will model the positions of naked-eye visible stars and constellations for a given date and time and location of the observer. This technique is based on coordinate transformations in and mapping from to . The precession of the equinoxes will be explained and included in the calculations, and will therefore make the star plots accurate for approximately two thousand years into the future or the past. This paper provides examples for the application of linear transformations and mappings, using one of the most natural physical phenomena, and is written for readers with limited knowledge of astronomy.

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Aleksandr Davydov is an assistant professor of mathematics at the Kingsborough Community College (KCC) of the City University of New York (CUNY). He earned his MS in mathematics from Samarkand State University (Russia) and his PhD in mathematics from Ural State University (Russia). His primary area of interest is differential equations and their applications.
E-mail: ADavydov@kbcc.cuny.edu

Rachel Sturm-Beiss is an associate professor of mathematics in Kingsborough Community College (KCC) of the City University of New York (CUNY). She earned her PhD in pure mathematics from the Courant Institute of New York University. Her primary area of interest is statistical processes and modeling.
E-mail: RSturm@kingsborough.edu

The orders of presentation of pre-calculus and calculus topics, and the notation used, deserve careful study as they affect clarity and ultimately students' level of understanding. We introduce an alternate approach to some of the topics included in this sequence. The suggested alternative is based on years of teaching in colleges within and outside the US, and on our careful review of textbooks currently used in two-year and four-year colleges.

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Harlan Brothers is Director of Technology at The Country School in Madison, CT where he teaches programming, fractal geometry, and guitar. Having worked for six years with Michael Frame and Benoit Mandelbrot at the Yale Fractal Geometry Workshops, he now lectures on the subject of fractal music. Harlan is also an inventor with five US patents.
E-mail: harlan@thecountryschool.org

A simple restatement of its limit definition formula allows one to derive trigonometric approximations for e. These novel closed-form expressions can then be used as functions that will "convert" the digits of into those of e. Maclaurin series expansions are used to assess rates of convergence for these expressions.

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John teaches mathematics and mathematics education at the University of Maine. In addition to developing and discovering practical applications of math, he enjoys flyshing in the Penobscot River from his kayak, spending time with his wife and 4 kids, all things Mac, and long walks listening to novels on his iPod.
E-mail: john.donovan@maine.edu

To achieve the vision of mathematics set forth in Crossroads (AMATYC, 1995), students must experience mathematics as a sensemaking endeavor that informs their world. Embedding the study of mathematics into the real world is a challenge, particularly because it was not the way that many of us learned mathematics in the first place. This article is about one such example, the effects of fishing on fish populations, but the method of analysis used is widely applicable. The fishing model developed is based on intuitions about how populations change over time. Traditionally such examples are reserved for the study of calculus and differential equations, but qualitative methods of analysis make them accessible to students in precalculus. This example, and others like it, should not be considered add-ons to an already over-burdened curriculum. Rather, such problems provide launching points for students to develop deep understandings of mathematics through investigation of things that are real.

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Irene is an assistant professor in the Department of Postsecondary Teaching and Learning with an EdD from Grambling State University, Louisiana. She taught developmental mathematics since 1990 and was an administrator of developmental programs for over 20 years. Irene is the recipient of the 2007 National Association for Developmental Education's (NADE) Outstanding Research Conducted by a Developmental Education Practitioner Award.
E-mail: duran026@umn.edu

This is the third and final article in this series. The first article reviewed the literature for research studies on successful developmental programs and students. The second article reported on the qualitative research methods and results documented from a purposive sample of twenty successful developmental mathematics students 3-5 years after completing their developmental studies. This article presents more detail on what shaped this qualitative study, identifies specific implications for developmental mathematics educators, and makes recommendations for further research on success in developmental mathematics.

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Russell Euler is a professor in the Department of Mathematics and Statistics at Northwest Missouri State University where he has taught since 1982. His mathematical interests include analysis, differential equations, geometry and number theory. Presently he is the Problem Editor for the Fibonacci Quarterly. Russell enjoys construction, volunteer work at his church, and learning from his three daughters.
E-mail: reuler@nwmissouri.edu

Jawad Sadek is a professor in the Department of Mathematics and Statistics at Northwest Missouri State University. His main mathematical interest is complex analysis. Jawad enjoys soccer and traveling around the world.
E-mail: JAWADS@nwmissouri.edu

Finding the longest rectangular couch with a given width that can be maneuvered around a corner is an old and interesting problem. It has been the subject of numerous research articles. In this note, two open questions that were raised in Moretti's article (2002) about the subject are discussed. In addition, the maximum area of a couch rounding a corner is also found.

Reference

Moretti, C. (2002). Moving a couch around a corner. The Coll. Math. Journal, 33(3), 196-200.

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Ray C. Shiflett received his PhD at Oregon State University. He has published in operator, measure, matrix, and number theory, topology, optometry, science fiction, and mathematics education. He served as Chair of Mathematics at Wells College, Dean of the College of Science at Cal Poly Pomona, and Executive Director of the National Research Council's Mathematical Sciences Education Board. He enjoys golf, fly fishing, writing songs, and wood working.
E-mail: rcshiflett@roadrunner.com

Students were asked to find all possible values for A so that the points (1, 2), (5, A), and (A, 7) lie on a straight line. This problem suggests a generalization: Given (x, y), find all values of A so that the points (x, y), (5, A), and (A, 7) lie on a straight line. We find that this question about linear equations must be resolved using the more advanced tools of quadratic equations. The number of possible values of A can be zero, one or two, depending upon the given point (x, y). Moreover, the three cases are partitioned by an oblique parabola having its axis at an angle of 45 degrees to the Cartesian plane coordinate axes.

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Donald Teets has taught at the South Dakota School of Mines and Technology since obtaining his doctorate from Idaho State University in 1988. He received the Allendoerfer award from the Mathematical Association of America in 2000 for an article on the astronomical work of Gauss, and the Distinguished Teaching Award from the Rocky Mountain Section of the MAA in 2004.
E-mail: donald.teets@sdsmt.edu

This article shows how to use six parameters describing the International Space Station's orbit to predict when and in what part of the sky observers can look for the station as it passes over their location. The method requires only a good background in trigonometry and some familiarity with elementary vector and matrix operations. An included set of exercises leads the reader step-by-step through the computations. Specific instructions are included for implementation of the method using a spreadsheet tool such as Excel. This article gives students the rare opportunity to use classroom mathematics to solve a complicated real-world problem, and to observe the results of their solution in real time.

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Travis Thompson received the PhD degree in mathematics from the University of Arkansas in 1977. He is currently the dean of the college of sciences at Harding University in Searcy, Arkansas.
E-mail: thompson@harding.edu

Arithmetic tests for divisibility of an integer by another integer are well known. This article states and proves conditions for divisibility in binary form.

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New Problems

The BA Problem Set consists of four new problems.

Set AY Solutions

Solutions are given to the five problems from the AY Problem Set from the Fall 2007 issue of The AMATYC Review. In addition, addenda were provided for the solvers of the AW Problem Set from the Fall 2006 issue.

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