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Some of the most lasting impressions that we retain from the study of circles come from our informal exposure to them in daily life. It may have been years since we learned, and perhaps forgot, the equation x sup 2 + y sup 2 = r sup 2 , but we can still summon our hands-on experiences to remember the essential features of a circle. The notion of a circle's having a center that is equidistant from all its points gets reinforced whenever we use a compass or twirl a rope or cord with our fingers. The symmetry of a circle becomes apparent each time we turn a knob or struggle to loosen a lid on a jar.
This hands-on philosophy underlies the activities in the book Exploring Conic Sections with The Geometer's Sketchpad (Scher 1995). Rather than move directly into the algebra of conics, the text presents construction techniques that generate them. By using straws, rulers, dental floss, paper circles, and other household items, the class develops a repertoire of construction methods for the ellipse, hyperbola, and parabola. Ultimately, students' hands-on work with the conics gives them an experiential base to rival their informal knowledge of a circle.
The curriculum material that follows originates from a trio of paper-folding activities in the conics book. These paper-folding techniques are not new--reference to them can be found in Gardner (1983), among other sources. However, as the title of the conics book suggests, each activity uses the software program The Geometer's Sketchpad (Jackiw 1991) to complement the folding process. The ability of the software to produce dynamic, moveable figures allows students to model their folded-paper creations on screen. The subsequent activity descriptions show how this marriage of physical constructions and Sketchpad modeling offers benefits that neither instructional medium alone can supply.
FOLDING AN ELLIPSE
Students begin the ellipse activity by drawing and cutting out a paper circle with a radius of at least 16 centimeters. Waxed paper and patty paper make ideal choices for folding, as they retain sharp, visible creases. Students mark the center of the circle C and choose an arbitrary point within the circle (point F in fig, 1). (fig. 1 omitted) They then fold the circle so that any point on its...