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## Circular Reasoning

Measurement concepts are difficult for students of all ages (including adults). Some specific difficulties have been well documented by researchers:

• Distinguishing between linear and area units in given situations
• Realizing the effect of changes in linear dimensions on area
• Using the correct vocabulary--perimeter vs. area vs. circumference
• Understanding the concept of area itself, separate from the use of a magical formula Given these difficulties, math teachers need to use a variety of tools and teaching aides to help students develop skills and meaning relative to measurement taks. Tandi Clausen-May (UK) has recently posted some Powerpoint demonstrations that use dynamic measurement visuals to help students.

The best of her work includes two visual demonstrations, both recently described in her article (see reference below). First, using an innovative approach I have not seen previously, Tandi uses moving points on circles and hexagons to illustrate why the value of pi is greater than 3. Second, Tandi explains why A=(pi)r2 is the appropriate formula for calculating the area of a circle by dynamically unfolding a "beaded" circle into a triangle. These demonstrations are good, innovative, and useful.

Less useful are her demonstrations as to either why A = lw is the formula for calculating the area of a rectangle or how this formula relates to the calculation of the area of a triangle or parallelogram.

And my thanks to Janet Mock (WWU) for connecting me with Tandi's ideas.

Source: T. Clausen-May's "Going Round in Circles," Micromath, Nov. 2006, pp. 42-44