Regarding the reflection “On the Area of a Circle” by Cheng, Tay, and Lee (MT April 2012, vol. 105, no. 8, pp. 564-65), it is possible to prove that one can arrange infinitely many sectors of a circle into a rectangle to show that a circle's area is π2. However, the authors' derivation is invalid because they assume their conclusion by using the area of the circle within their proof.
Margaret Rathouz, Christopher Novak, and John Clifford
Constructing formulas “from scratch” for calculating geometric measurements of shapes—for example, the area of a triangle—involves reasoning deductively and drawing connections between different methods (Usnick, Lamphere, and Bright 1992). Visual and manipulative models also play a role in helping students understand the underlying mathematics implicit in measurement and make sense of the numbers and operations in formulas.
The Grazing Goat problem, familiar to many teachers and students, has several variations. The version presented here provides a rich opportunity for engaging students in a project spanning several weeks. Three solutions are discussed: one suitable for a calculus class, one suitable for a geometry class, and one suitable for a precalculus class. Although we start with a calculus approach, most of the article uses only algebra and geometry concepts. Also discussed are the didactics of using projects to open ever-larger fields of mathematics to students.
Kathryn G. Shafer, Gina Severt, and Zachary A. Olson
Two preservice teachers describe how using Google SketchUp, Terrapin Logo, and The Geometer's Sketchpad fosters a deeper understanding of measurement concepts.
Diana Cheng and David Thompson
Labyrinths inspire questions about measuring path lengths and representing patterns.
The term Norman architecture is used to categorize styles of Romanesque architecture developed by the Normans in northwestern Europe, particularly England, in the eleventh and twelfth centuries. The Normans introduced castles, fortifications, monasteries, abbeys, churches, and cathedrals, all with characteristic rounded arches, particularly over windows and doorways, and massive proportions.
Readers comment on published articles or offer their own ideas.
A set of problems of many types.
S. Asli Özgün-Koca, Michael Todd Edwards, and Michael Meagher
The Spaghetti Sine Curves activity, which uses GeoGebra applets to enhance student learning, illustrates how technology supports effective use of physical materials.