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All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Concrete Wheel

Stage: 3 Challenge Level: Challenge Level:1

Alan Schoenfeld suggests that this problem tends to provoke immediate and widely divergent intuitive reactions (see some of the possibilities listed in the Hint section).

He writes that he used this problem in a class where the discussions "focused on what it means to have a compelling mathematical argument. The general tenor of these discussions followed the line of argumentation outlined in Mason, Burton and Stacey's (1982) Thinking Mathematically : First, convince yourself; then, convince a friend; finally, convince an enemy. (That is, first make a plausible case and then buttress it against all possible counterarguments.) In short, we focused on what it means to truly understand, justify, and communicate mathematical ideas."

Teachers may want to use this problem in their classrooms to serve the same purposes.