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December 2008, Volume 102, Issue 5

FEATURES

Illuminating the Mathematics of Lamp Shades
Michael Matthews; Greg Gross
The problem of creating lamp shades to specific design parameters allows rich and interesting explorations in the mathematics of circles and triangles. This interactive project helps students build their spatial reasoning and is especially appropriate during a unit on either the Pythagorean theorem or similar triangles.

The Histogram-Area Connection
William Gratzer; James Carpenter
An alternative approach to the construction of histograms is demonstrated—one based on the notion of using area to represent relative density in intervals of unequal length. The resulting histograms illustrate the connection between the area of the rectangles associated with particular outcomes and the relative frequency (probability) of those outcomes.

Developing Reasoning through Proof in High School Calculus
John Perrin
Developing students’ ability to reason has long been a fundamental goal of mathematics education. A primary way in which mathematics students develop reasoning skills is by constructing mathematical proofs. This article presents a number of nontypical results, along with their proofs, that can be explored with students in any calculus classroom. Carefully investigating these results with calculus students is meant to strengthen students’ reasoning skills as well as their understanding of calculus.

Area by Dissection
Ioana Mihaila; Ellen Barger
Building sets of problems rooted in the same concept allows students to develop understanding and take on increasingly complex problems. By beginning with “foundation stories” and using strategies similar to those for literary constructs,  students’ mathematical comprehension and fluency have dramatically increased. A problem set focussing on areas, specifically areas of circles, triangles, and rectangles is highlighted.

Thoughts on Why (–1)(–1) = +1
Tina Rapke
Consider why (–1)(–1) = +1 and how and why a teacher might go about explaining this concept to high school students without using pseudoreasoning. Also a precise explanation, through the use of the distributive property, as to why (–1)(–1) = +1.