Classroom Resources Index – Odds and Ends

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Approximation Techniques

A Disquisition on the Square Root of Three, by Robert J. Wisner
The author compares Greek ladder, continued fraction, and Newton’s Method approximations, pointing out that the Greek ladder easily produces both of Archimedes’ famous bounds.

Connecting Greek Ladders and Continued Fractions, by Kurt Herzinger and Robert Wisner
An exploration of two historical techniques for estimating irrational numbers and a link between them.

Extracting Square Roots Made Easy: A Little Known Medieval Method, by Friedrich Katscher
A method for extracting (approximate) square roots used in Italy through the 18th century was introduced in a manuscript by the 12th-century mathematician al-Hassar.

The Classic Greek Ladder and Newton’s Method, by Robert J. Wisner
Greek ladders for approximating square roots may be more ancient than the ancient Greeks. Students at any level can appreciate their beauty and simplicity. Those who have studied calculus can compare them with Newton’s Method for approximating roots.

The Root of the Matter: Approximating Roots with the Greeks, by Matthew Haines and Jody Sorensen
An overview of the geometric and matrix interpretations of the ancient method of Theon's Ladder for approximating roots. Includes a class activity that explores Theon's method and convergence via an interactive applet.

Computation

John Napier’s Binary Chessboard Calculator, by Sidney J. Kolpas and Erwin Tomash
Napier’s lesser known invention: a 5-function calculator via binary arithmetic on a chessboard.

Critical Thinking

Euler’s Letters to a German Princess: Translation and Betrayal, by Dominic Klyve
An exploration of how the translations of Euler’s Letters to a German Princess came to differ from the original text, a tale that can be used in the classroom to underscore the value of critical thinking with students bombarded outside the classroom by multimedia of various quality and promoting various agendas.

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