# 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-

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Notes

#### Cited page

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

#### Cited page

Designing Learning Environments for Developing Understanding of Geometry and Space

Settings

#### Settings

Typeface
Text size Reset View mode
Search within

Look up

#### Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Help
Full screen
Items saved from this book
• Bookmarks
• Highlights & Notes
• Citations
/ 506

### How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

## Cited passage

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

## Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.