# The College Mathematics Journal - May 2001

### Articles

Teaching Linear Algebra: Issues and Resources
Dan Kalman and Jane M. Day

Here are some section headings: Curriculum issues, Connections with secondary mathematics, Research on how students learn, Ideas from physics education, Technology.

The Average Speed on the Highway
Larry Clevenson, Mark Schilling, Ann Watkins, and William Watkins

"You are driving on the highway and adjust your speed until the number of cars that you pass is equal to the number of cars that pass you." So your speed is the median speed, right? Wrong!

Good Rational Approximations to Logarithms
Tom Apostol and Mamikon Mnatsakanian

How to arrive at 21306/70777 as an approximation to the square root of 2 (and other rational approximations to other square roots, of course) from first principles, using nothing fancy like continued fractions.

Differentiation with Respect to a Parameter
Joseph Wiener

This method tends to get short shrift these days, which is a shame. How would you get (p ln 2)/2 for the integral of x cot x from 0 to p/2 ? ("Looking it up" is an insufficient answer.) If you know how to differentiate with respect to a parameter, you put p = 1.

The Lord Over Better and Worse Births
John Fossa and Glenn W. Erickson

Plato's notorious Nuptial Number is, he says, a "geometrical number", lord over births, and if you reproduce in ignorance of it, "the children will not be well-endowed, either in their constitution or in their fates." What was Plato thinking of? It must have been something. Here is a possible explanation.

On the Tangent Lines of a Parabola
Mikko Stenlund

If you draw two tangents to a parabola they intersect halfway between the points of tangency (in the x-direction). This property could be used as a definition of parabola.

### Fallacies, Flaws, and Flimflam

Edited by Ed Barbeau

A convincing argument that the medians of a triangle intersect at the same point as the angle bisectors, with its refutation.

### Classroom Capsules

What's Harmonic about the Harmonic Series?
David Kullman

It looks good and it sounds good too.

Stephen Kackowski

How to find quadratics so that x2 + ax + b and x2 + ax - b both factor nicely.

Linear Transformations of the Unit Circle in R2
Pratibha Ghatage and Sally Shao

Everybody looks at what happens to the unit square to see the effect of a linear transformation. They should look at what happens to the unit circle too.

Convergence-Divergence of p-Series
Rasul Khan

Everybody uses the integral test to see when the p-series converges and diverges. This is extreme. How much better not to use calculus! It can be done quickly.

Linear Algebra in the Financial World
Barbara Swart

Financial derivatives are quite different from mathematical derivatives, but mathematics, in particular linear algebra, can give information about them.

### Problems and Solutions

Edited by Irl Bivens and Ben Klein

### Software Review

Review of LiveMath Maker 3.0, by Marcia Tharp

### Media Highlights

Edited by Warren Page

### Book Review

Review of two biographies of Paul Erdös by Steven Krantz