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Seán P. Madden and Louis Lim

Students analyze items from the media to answer mathematical questions related to the article. Exponents and working with large numbers are the underlying mathematical ideas this month.

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Margaret R. Meyer

One of my favorite lessons comes from a problem I first heard posed as an open–ended assessment problem by David Clarke at an NCTM conference years ago:

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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.

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Readers comment on published articles or offer their own ideas.

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Rina Zazkis, Ilya Sinitsky, and Roza Leikin

A familiar relationship—the derivative of the area of a circle equals its circumference—is extended to other shapes and solids.

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Joshua A. Urich and Elizabeth A. Sasse

Students peel oranges to explore the surface area and volume of a sphere.

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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.

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Rick Havens

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.

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John M. Livermore

The study of the history of mathematics can be an important approach to engaging students in learning mathematics. At the same time, mathematics history provides a context that will help students understand how and why certain types of mathematics were developed and used. The study of p has as rich a history as any mathematical topic.