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Lesson Plans
The Proof of the Century!
Overview Procedures for Teachers Organizers for Students


Procedures for Teachers is divided into three sections:
Prep -- Preparing for the Lesson.
Steps -- Conducting the Lesson.
Extensions -- Additional Activities.


Prep

Student Prerequisites:
Prior knowledge of the Pythagorean theorem is not required. Students will be asked to use the exponential function keys on their scientific calculators.

Materials:
  • Videotape: THE PROOF. Available from NOVA Videos, c/o WGBH Boston Video, PO Box 2284, South Burlington, VT 05407-2284, (800) 255-9424. Information about the video is available on NOVA Web site
    (http://www.pbs.org/wgbh/nova/proof/).
  • Videotape: THE THEOREM OF PYTHAGORAS. Available from Project MATHEMATICS!, c/o Joyce, California Institute of Technology Bookstore, I-51, Pasadena, CA 9125, (626) 395-6161. Information about the video is available on Project MATHEMATICS!
    (http://www.projmath.caltech.edu).
  • Copies of the article, "Fermat's Last Stand," by Simon Singh and Kenneth A. Ribet, SCIENTIFIC AMERICAN, November, 1997.
  • Scientific calculators.
  • Construction paper for the proof cutout exercise.
  • Scissors.
  • Tape or glue.
Computer Resources:
You will need at least one computer with Internet access to complete this lesson. While many configurations will work, we recommend:

  • Modem: 28.8 Kbps or faster.
  • Browser: Netscape Navigator 3.0 or above or Internet Explorer 3.0 or above.
  • Macintosh computer: System 7.0 or above and at least 16 MB of RAM.
  • IBM-compatible computer: 386 or higher processor with at least 16 MB of RAM, running Windows 3.1. Or, a 486/66 or Pentium with at least 16 MB of RAM, running Windows 95.

For more information, visit What You Need to Get Connected in wNetSchool's Internet Primer.

Bookmarks:
The following sites should be bookmarked:

Information Concerning the Pythagorean Theorem:

  • Pythagorean Theorem Site
    http://www.cut-the-knot.com/pythagoras/

    This site contains 28 geometric proofs of the Pythagorean theorem and links to more.

  • United States Naval Academy Mathematics Department
    http://www.nadn.navy.mil/MathDept/mdm/pyth.html

    The site includes an animated proof of the Pythagorean theorem.

  • Math Forum
    http://mathforum.org/library/problems/sets/middle_pythagorean.html

    Several good animated proofs of the Pythagorean theorem are displayed.

    Information Concerning Pythagorean Triples:

  • Plimpton 322
    http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html

    This site provides information about an ancient Babylonian, baked clay tablet written in cuneiform script called Plimpton 322. The tablet, dating from 1900-1600 BCE, contains a list of 15 Pythagorean triples.

  • Picturing Pythagorean Triples
    http://nrich.maths.org/public/viewer.php?obj_id=1332&part=index&refpage=articles.php

    The site includes examples and diagrams of Pythagorean triples.

    Information Concerning Mathematical Proofs:

  • University of Idaho
    http://www.cs.uidaho.edu/~casey931/mega-math/gloss/math/proof.html
    The site provides explanations of what a mathematical proof entails.

  • Ask Dr. Math--What is Math?
    http://forum.swarthmore.edu/dr.math/faq/faq.why.math.html

    The site provides explanations of practical applications of mathematics.

    Information Concerning Fermat's Last Theorem:

  • NOVA Web site
    http://www.pbs.org/wgbh/nova/proof/

    This site provides information on the NOVA program, THE PROOF, about Fermat's Last Theorem and Wiles' proof.

  • MacTutor History of Mathematics Archive
    http://www-groups.dcs.st-and.ac.uk/~history/
    HistTopics/Fermat's_last_theorem.html

    An exploration of the history of the Fermat problem that includes a brief chronology of the famous mathematicians who tried to prove this theorem. The site includes links to biographies of the mathematicians mentioned.

  • Charles Daney's Fermat Page
    http://www.best.com/~cgd/home/flt/flt01.htm

    This site contains information about Fermat's Last Theorem that is beyond the scope of high school mathematics but is useful for answering students' questions about why the problem was so difficult to solve.

  • Wiles, Ribet, Shimura-Taniyama-Weil and Fermat's Last Theorem
    gopher://gopher.math.albany.edu:70/h0/.DEPTS/math/fermat/view

    This site provides information about mathematicians' research activities.

  • Earth/matriX -- Science in Ancient Artwork: The Pythagorean Problem
    http://www.earthmatrix.com/Pitagor3.htm

    This site investigates the Pythagorean problem.

  • Earth/matriX -- Science in Ancient Artwork Series
    http://www.earthmatrix.com/series.htm#SS.

    This site investigates extensions of the Pythagorean problem.

    Steps

    Time Allotment:
    This lesson requires approximately 5-6 class periods.




  • Introduce the topic of the Pythagorean theorem and the notion of Pythagorean triples. Have students, either individually or in groups, conduct Web research. Students should look for historical information about Pythagoras as well as explanations of the Pythagorean theorem and Pythagorean triples. Students should start with the following bookmarks:

    United States Naval Academy Mathematics Department (http://www.nadn.navy.mil/MathDept/mdm/pyth.html)

    Math Forum (http://mathforum.org/library/problems/sets/middle_pythagorean.html)

    Pythagorean Theorem Site
    (http://mathforum.org/library/problems/sets/middle_pythagorean.html)

    Plimpton 322
    (http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html)

    Picturing Pythagorean Triples
    (http://www.nrich.maths.org.uk/mathsf/
    journalf/may98/triples.html)

    Students should then conduct their own research to find more sites about Pythagoras and the Pythagorean theorem. Ask students to answer the questions on Student Organizer #1, Investigating the Pythagorean Theorem and Triples, in Organizers for Students.




  • Using the instructions on Student Organizer #2, Pythagorean Proof Activity Sheet, and Student Organizer #3, Pythagorean Theorem Diagram, in Organizers for Students, have students create their own geometric proof of the Pythagorean theorem. It may be helpful to photocopy the diagram onto construction paper so the cut-out pieces are sturdier. When the activity is complete, you can laminate the students' final proofs (optional).




  • View the THEOREM OF PYTHAGORAS video. You can skip the introduction and overview to avoid unnecessary repetition. The rest of the program lasts about 20 minutes.




  • Ask students to continue their Web research about the Pythagorean theorem and Pythagorean triples using Student Organizer #4, More Proofs of the Pythagorean Theorem, in the Organizers for Students, as a guide.




  • Introduce the theorem: an + bn = cn.

    By now, students should have discovered some Pythagorean triples and have proved the theorem.

    Ask students to complete Student Organizer #5, Extending the Idea of Pythagorean Triples, in Organizers for Students.

    This worksheet asks students to change the exponent of the Pythagorean theorem and look for numbers that still satisfy the equation. The task is impossible to accomplish. After students have unsuccessfully tried to solve the equations, state Fermat's Last Theorem: There do not exist positive integers a, b, c, and n such that an + bn = cn when n>2.




  • View THE PROOF video. This program is 60 minutes long. Unless you are on a block schedule, you will probably need two class periods.


    Extensions

    Fermat's Last Theorem is an ideal departure point for history lessons. Some interesting Internet projects could include researching the lives of Pierre de Fermat, Rene Descartes, Blaise Pascal, Evariste Galois, and Sophie Germain. Students should consider the mathematics that they invented as well as the circumstances of their lives and the societies in which they lived.

    These URLs may be useful:

    NOVA Web site
    http://www.pbs.org/wgbh/nova/proof/

    MacTutor History of Mathematics Archive
    http://www-groups.dcs.st-and.ac.uk/~history/
    HistTopics/Fermat's_last_theorem.html

    Charles Daney's Fermat Page
    http://www.best.com/~cgd/home/flt/flt01.htm

    Wiles, Ribet, Shimura-Taniyama-Weil and Fermat's Last Theorem
    gopher://gopher.math.albany.edu:70/h0/.DEPTS/math/fermat/view

    The following sites are useful for investigating related mathematics beyond the Pythagorean theorem and Fermat's Last Theorem:

    Earth/matriX -- Science in Ancient Artwork: The Pythagorean Problem
    http://www.earthmatrix.com/Pitagor3.htm

    Earth/matriX -- Science in Ancient Artwork Series
    http://www.earthmatrix.com/series.htm#SS.



    Submit a Comment: We invite your comments and suggestions based on how you used the lesson in your classroom.



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