Volume 47 of *The College Mathematics Journal* includes two articles exploring creative uses of manipulatives. In the lead article, Fred Kuczmarski uses rubber bands to offer a different way of visualizing functions that provides geometric explanations of many calculus results. Viktor Blåsjö considers how and why Leibniz used string to compute logarithms. Other article topics include extending a mathematical joke of Abbott & Costello, presenting a tricky situation for the famous ball-fetching Elvis, and updating model selection in the classroom. If you have figured out the popular 64 = 65 puzzle, then see how can you handle 47 = 48 in one of the five Proofs Without Words. And a villanelle by Paisley Rekdal begins a series of mathematics themed verse from established poets. -*Brian Hopkins*

Vol 47 No 2, pp 80-160

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## ARTICLES

### Rubber Band Calculus

Fred Kuczmarski

We describe a way to visualize functions using rubber bands. Interpreting the derivative as a local stretching factor leads to vivid geometric images for the chain rule, substitution, and curvature, as well as an unusual approach to the fundamental theorem.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.82

### Proof Without Words: A Surprising Integer Result

Roger B. Nelsen

We wordlessly “prove” that 48 equals 47 and mention a ramification..

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.94

### How to Find the Logarithm of Any Number Using Nothing But a Piece of String

Viktor Blåsjö

We present Leibniz's 1691 recipe for determining logarithms using the catenary and discuss why this odd-looking application in fact made good sense in its historical context.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.95

### Proof Without Words: Integer Right Triangle Hypotenuses Without Pythagoras

Colin Foster

Without reference to the Pythagorean theorem, we show that a right triangle with legs 3 and 4 has hypotenuse 5. The figure can be modified for other integer right triangles.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.101

### Integration by the Wrong Parts

William Kronholm

In this article we disregard common sense and good advice and compute antiderivatives for certain functions by stubbornly applying integration by parts infinitely many times.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.102

### A Canine Conundrum, or What Would Elvis Do?

Michael Maltenfort

The dog Elvis became famous by finding optimal paths that solve various calculus problems. Not all problems, however, have solutions. By giving an unsolvable problem very similar to those Elvis solved, we provide a reminder that it is necessary to prove, rather than assume, that optimal paths exist.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.106

### Proof Without Words: The Golden Ratio

Roger B. Nelsen

We employ a square with area 5 to determine the golden ratio (without using the quadratic formula).

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.108

### Empirical Modeling: Choosing Models and Fitting Them to Data

Glenn Ledder

Treatments in mathematics books of how to select models and fit them to data have generally not been updated to account for improvements in knowledge and computing over the last 40 years. Here, we offer a derivation of the linear least squares formula that requires only precalculus and use it to develop a simple numerical method for fitting a broad class of nonlinear functions, with results that are far superior to methods in common practice. We also discuss the use of the Akaike information criterion (AIC) to choose among competing models that have been fit to a common data set.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.109

### Proof Without Words: Powers of Three and Triangular Numbers

C. David Leach

This proof without words presents a series of images that together verify an identity relating powers of three and the triangular numbers.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.120

### Lattice Paths and Harmonic Means

Marc Zucker

Harmonic means are rarely discussed these days, perhaps because there are few interesting examples of their use. Starting with a question about lattice paths, we present a problem that finds its solution incorporating harmonic means. We also outline the situation beyond two dimensions.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.121

### Proof Without Words: Arithmetic Mean of Two Means

Ángel Plaza

We provide a visual proof that the arithmetic mean of two positive numbers is greater or equal than the arithmetic mean of the geometric mean and the root mean square.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.125

### Abbott-and-Costello Numbers

Howard Sporn

We analyze a mathematical routine from the comedy team of Abbott and Costello and determine all possible numbers that could be used in the joke. We determine a recursive formula and a closed-form expression for the resulting integer sequence, both of which use least common multiples.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.126

### Philip Larkin′s Koan

Paisley Rekdal

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.133

## Classroom Capsules

### A Short and Elementary Proof of the Basel Problem

Samuel G. Moreno

By slightly changing a beautiful and little-known argument by E. L. Stark, we give a short and elementary proof of the celebrated Basel problem.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.134

### De Morgan′s Series Test

C. W. Groetsch

Augustus De Morgan, widely known for his eponymous logic laws, also formulated a little-known convergence test for infinite series in his 1842 calculus textbook. After demonstrating a problem with the original version, we formulate it precisely, bringing it up to date for the modern classroom.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.136

## Problems and Solutions

Problems and Solutions: 138-145

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.138

## Book Review

*How Not To Be Wrong: The Power of Mathematical Thinking* By Jordan Ellenberg

Reviewed by: Peter Ross

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.146

## Media Highlights

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.2.153