The College Mathematics Journal
May 2013 Contents
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The May issue of The College Mathematics Journal begins provocatively with the article "Quiz Today: Should I Skip Class?" by Peter Zizler, exploring the impact on course grades of selective assessment (e.g., counting only the best 3/5 quizzes). Weird fractions return in "How Weird Are Weird Fractions?" by Ryan Stuffelbeam. There is also a new solution to the Basel problem by David Benko and John Molokach in "The Basel Problem as a Rearrangement of Series."
As always, Problems and Solutions challenge readers and Media Highlights keep them well-informed.
Quiz Today: Should I Skip Class?
Peter Zizler
How does selective assessment, wherein one counts only the best k out of n quizzes set, impact grade inflation? Based on our analysis, a specific quantitative answer can be given to a student who plans to skip a quiz, depending, of course, on the student's quiz writing consistency—or inconsistency.Proof Without Words: An Alternating Sum of Squares
Joe DeMaio
A visual proof that alternating some of the first n squares counts the number of edges in the complete graph on n + 1 vertices. The Basel Problem as a Rearrangement of Series
David Benko and John Molokach
We give an elementary solution to the famous Basel Problem, originally solved by Euler in 1735. We square the well-known series for arctan(1) due to Leibniz, and use a surprising relation among the re-arranged terms of this squared series.Series of Reciprocal Triangular Numbers
Paul Bruckman, Joseph B. Dence, Thomas P. Dence, and Justin Young
Reciprocal triangular numbers have appeared in series since the very first infinite series were summed. Here we attack a number of subseries of the reciprocal triangular numbers by methodically expressing them as integrals. Archimedes Curves
Gordon A. Swain
We show that inside every triangle the locus of points satisfying a natural proportionality relationship is a parabola and go on to describe how this triangle-parabola relationship was used by Archimedes to find the area between a line and a parabola. Proof without Words: Triangular Sums
Yukio Kobayashi
A visual proof that the triangular numbers are sums of every other odd number. The Number of Group Homomorphisms from Dm into Dn
Jeremiah William Johnson
We count the number of group homomorphisms between any two dihedral groups using elementary group theory only.Circular Inclusion
James Sandefur and John Mason
Given a family of p ≥ 3 points in the plane, some three of them have the property that the smallest circle encompassing them encompasses all p points. Similarly, we show that for p ≥ 3 circles, there are three of them such that the smallest circle encompassing them encompasses all the circles, and that there are three circles for which the smallest circle encompassing and tangent to them encompasses all the circles.How Weird Are Weird Fractions?
Ryan Stuffelbeam
A positive rational is a weird fraction if its value is unchanged by an illegitimate, digit-based reduction. In this article, we prove that each weird fraction is uniquely weird and initiate a discussion of the prevalence of weird fractions.System Lifetimes, the Memoryless Property, Euler's Constant, and Pi
Anurag Agarwal, James E. Marengo, and Likin C. Simon Romero
A k-out-of-n system functions as long as at least k of its n components remain operational. Assuming that component failure times are independent and identically distributed exponential random variables, we find the distribution of system failure time. After some examples, we find the limiting distribution of system life when n approaches infinity, and apply these results to evaluate two challenging integrals.Understanding Singular Vectors
David James and Cynthia Botteron
A certain weighted average of the rows (and columns) of a non-negative matrix yields a surprisingly simple, heuristical approximation to its singular vectors. There are correspondingly good approximations to the singular values. Such rules of thumb provide an intuitive interpretation of the singular vectors that helps explain why the SVD is so effective in analyzing large data sets.CLASSROOM CAPSULES
A Simple Proof of the Right-Hand Rule
Fuchang Gao
A simple proof of the right-hand rule of the cross product is presented.Sharing the Work
Walden Freedman
N people take turns pulling a heavy rope hanging off the edge of a building. How much of the rope should each pull in order to share the work equally? Which person's fair share is closest to 1/nth of the rope's length? We answer these questions using basic calculus and suggest some related problems for student exploration.PROBLEMS AND SOLUTIONS
REVIEWS
Journey through Mathematics: Creative Episodes in Its History
by Enrique A. González-Velasco
reviewed by Robert E. Bradley
The Manga Guide to Linear Algebra
by Shin Takahashi, illustrated by Iroha Inoue
reviewed by Susan Jane Colley
Math Girls
by Hiroshi Yuki (translated by Tony Gonzalez)
reviewed by Susan Jane Colley
MEDIA HIGHLIGHTS