# Designing Learning Environments for Developing Understanding of Geometry and Space

By Richard Lehrer; Daniel Chazan | Go to book overview

Urban Mathematics Collaborative Project (pp. 196-209). New York: Teachers College Press.

Young, J. W. ( 1925). "Lectures on fundamental concepts of algebra and geometry". New York: Macmillan.

APPENDIX

Sample Problem of the Week

Pythagorean Theorem- March 25-29, 1996

We have done a fair bit with the Pythagorean Theorem this year, so lets take a look at the theorem itself.

One very familiar proof of this theorem is a right triangle with squares on each side. Here's my question: Do we have to use squares? What about hexagons, or other shapes? What would we have to do to use other shapes, if that's at all possible? And why do you suppose squares are usually used for this proof?

Extra: Name a U.S. President who discovered a proof of the Pythagorean Theorem.

Sample Facilitator Response

[Annie facilitator]

Maybe I am being too hard with my grading this week, but by my count, only two folks got this one right! I received 48 incorrect solutions, some of which got only the extra part right; this may partly be a case of my having an answer in mind and not really asking the right questions, so I'll try to be more explicit next time.

A number of people said no, you don't have to use squares, and suggested that other shapes such as regular polygons would work. A couple of people showed that you could use hexagons, and proved it, but their answers weren't general enough-the key here is that you can use any shape you want, as long as the shapes on the three sides are similar! Thomas and Brian both got that idea in; Thomas actually said similar shapes, and Brian said shapes with proportional lengths. So you could really say the proof says that, given similar shapes on all the edges, the area of the shape on one leg plus

-484-

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