(Above is a detail from the fresco The School of Athens by Raphael - Click any image to begin)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A WebQuest by Jeff Johnson ([email protected])

For

TRED 701

Dr. Libby Hall and

Dr. Karen Kortecamp

November 4, 2000

 

 

 

 

 

 

TABLE OF CONTENTS

INTRODUCTION

TASKS

PROCESS

RESOURCES

EVALUATION

CONCLUSION

 

 

 

 

INTRODUCTION

Our math textbook tells us that the Pythagorean Theorem describes a relationship between the sides of a right triangle. To be more specific, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse (you may know it as a2+b2=c2, where a and b are the leg lengths and c is the hypotenuse length). Are you asleep yet? I’m starting to doze off myself...

The Pythagorean Theorem is much more interesting if you know that it has been around for approximately 3500 years and has been proven in more than 250 different ways. It provides a valuable link between arithmetic, algebra, geometry, trigonometry, and calculus. It is extremely simple, yet very powerful. In fact, Pythagoras, who is credited with proving the theorem about 2500 years ago, was viewed as divine by many of his followers.

Wait a minute! If the theorem is 1000 years older than Pythagoras, why is it called the Pythagorean Theorem? Here is your chance to find out. In this WebQuest, we will look back on the life of Pythagoras, look at some of the proofs of the theorem, and discuss the importance of the theorem throughout history.

TASKS

At the end of the WebQuest, each team will have done four things:

  1. Created a biographical sketch of Pythagoras
  2. Presented a proof of the Pythagorean Theorem on paper
  3. Presented a (different) proof of the Pythagorean Theorem to the class
  4. Participated in a round table discussion about the importance of the Pythagorean Theorem

The "Evaluation" section will provide more details about how each of these things will look.

PROCESS

 

Teams and Roles

Each team will be made up of four students. Since there are four "end products" of the WebQuest (see "Tasks"), it is suggested that each team member have one of the roles listed below:

  1. Biographer: Writes up the biographical sketch created by the team
  2. Technical Writer: Writes up a proof selected by the team
  3. Spokesperson: Presents a proof selected by the team
  4. Ambassador: Participates in the round table discussion on behalf of the team

 

Suggested Schedule of Activities

Two class periods will be devoted to working on the WebQuest ("work sessions"). At least one computer will be provided for each team during these sessions. Additional class time will be used for presentations and the round table discussion. It is assumed that students will also work outside of class, both with and without a computer. Below is a suggested schedule for the team:

 

Work Session 1:

  1. Become familiar with the WebQuest and decide on roles
  2. Make sure all team members have a basic understanding of the Pythagorean Theorem (see the "Resources" section for some background material on the theorem)
  3. Browse through biographical resources
  4. Agree on rough idea of what to include in the biographical sketch
  5. Browse through resources related to proofs
  6. Select some proofs or types of proofs that the team might want to write up and present
  7. Brainstorm some practical applications of the Pythagorean Theorem

 

Between Work Sessions 1 and 2:

Biographer:

  1. Create an outline of the biographical sketch
  2. Decide how you would like to present the biographical sketch (e.g. text, graphics)

Technical Writer:

  1. Select a few (2 or 3) candidates for the proof that you would like to write up
  2. Decide how you would like to present the proof on paper (e.g. text, graphics)

Spokesperson:

  1. Select a few (2 or 3) candidates for the proof that you would like to present to the class
  2. Decide how you would like to present a proof to the class (e.g. class activity, demonstration of a computer animation)

Ambassador:

  1. Prepare a draft of your team’s "prepared statements" for the round table discussion
  2. Select a practical example that you would like to share at the round table discussion

 

Work Session 2:

  1. Have each member report to the team on the work done since last session
  2. Come to a consensus on what to include in each "end product" (see "Tasks")
  3. Continue research as needed to complete the tasks

 

Between Work Session 2 and Presentations:

Biographer:

Complete the biographical sketch

Technical Writer:

Complete the writeup of the proof selected by the team

Spokesperson:

Prepare a presentation of the proof selected by the team

Ambassador:

Complete the team’s prepared statements and prepare for the round table discussion

 

Presentations:

Class time will be set aside for presentations. Each team will be allowed 8 minutes to present their proof.

 

Round Table Discussion:

Approximately 25-30 minutes of class time will be set aside for a round table discussion. One member from each team will sit on a "panel of experts" and participate in a discussion moderated by the teacher. The purpose of the discussion is to help understand the importance of Pythagoras and the Pythagorean Theorem by attempting to answer three central questions:

  1. If the theorem is 1000 years older than Pythagoras, why is it called the Pythagorean Theorem?
  2. How is mathematics different today than it was in the time of Pythagoras?
  3. What are some practical applications of the Pythagorean Theorem?

The discussion will be structured according to the following format:

  1. Moderator asks some of the experts to read prepared statements addressing the first question.
  2. Experts discuss the statements, attempting to come to a consensus regarding the first question.
  3. Moderator asks some of the experts to read prepared statements addressing the second question.
  4. Experts discuss the statements, attempting to come to a consensus regarding the second question.
  5. Moderator asks the experts for examples which address the third question.
  6. Experts discuss the overall importance of the Pythagorean Theorem in light of the results of the prepared statements and discussions.

 

RESOURCES

There are a lot of resources listed below (a total of 11 different websites). The sites are split up into three different categories and short descriptions are given for each site. The idea is not to spend a lot of time on each site. Rather, you should try to efficiently browse through the sites and see which ones you like.

 

Background and Explanation of the Pythagorean Theorem

http://forum.swarthmore.edu/~sarah/hamilton/ham.namesides.html

Describes the names of the sides of a right triangle

http://www.math.com/school/subject3/lessons/S3U3L4DP.html

Right triangle facts and a description of the Pythagorean Theorem

http://www.pbs.org/wgbh/nova/proof/puzzle/

Basic description of Pythagorean Theorem (also has a proof – see below)

 

Biographical Information

http://euler.ciens.ucv.ve/English/mathematics/pitagora.html

A pretty generic look at Pythagoras – a good starting point

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pythagoras.html

This site provides a little more detail

http://www.utm.edu/research/iep/p/pythagor.htm

This site concentrates more on Pythagoras’ work and philosophy

http://plato.evansville.edu/public/burnet/ch2a.htm

This site has a lot of information and it has a helpful Table of Contents. Don’t be scared off by the first section ("Character of the Tradition"). The rest of the information is a lot more straightforward.

Proofs of the Pythagorean Theorem

http://www.math.ubc.ca/~morey/java/pyth/

An award winning interactive geometrical proof

http://www.davis-inc.com/pythagor/

Animated geometric proof

http://www.pbs.org/wgbh/nova/proof/puzzle/

A good "paper cutout" geometrical proof

http://www.emsl.pnl.gov:2080/docs/mathexpl/ptprove.html

This is a good algebraic proof.

http://www.cut-the-knot.com/pythagoras/index.html

This site has 29 proofs! Check out President Garfiled’s proof (#5) and Leonardo da Vinci’s proof (#16).

EVALUATION

The "Tasks" section of the WebQuest lists the things that each team needs to do. Each of these tasks counts equally towards the grade for the team. Below are descriptions of how each task will be evaluated. From these descriptions, you can figure out what should be included in each.

Biographical Sketch (25 points)

This should be a maximum of two pages long. It is up to the team to decide what to include and how to structure the paper, but there are some minimum requirements for content. Each biographical sketch should include information about:

Papers will be assessed for...

Content (15 points)

  • A 15 point paper would include accurate information from all three categories listed above. It would also cite all references (by title and address of website).
    •

    Style (5 points)

  • A 5 point paper would be free of spelling, grammatical, and typographical errors. It would also be clearly and succinctly written. Remember, you are working in a team; team members other than the Biographer should help by reviewing the paper!
    •

    Format (5 points)

  • A 5 point paper would be visually appealing, neat, and clean.
    •

    Written Proof of the Pythagorean Theorem (25 points)

    There are many different proofs to choose from, so the appearance of the writeup will depend on the proof chosen. However, all written proofs should share some things in common. First is the intended audience. The proof should be written such that someone who is NOT a math student can understand it. Each proof should also include:

    Papers will be assessed for...

    Content (15 points)

  • A 15 point paper would include an accurate proof which includes the three elements listed above. It would also cite all references (by title and address of website).
    •

    Style (5 points)

  • A 5 point paper would be free of spelling, grammatical, and typographical errors. It would also be clearly and succinctly written. Remember, you are working in a team; team members other than the Technical Writer should help by reviewing the paper!
    •

    Format (5 points)

  • A 5 point paper would be visually appealing, neat, and clean.
    •

    Presentation of a Proof of the Pythagorean Theorem (25 points)

    There are many different proofs to choose from, so the type of presentation will depend on the proof chosen. For example, some proofs might work well as a computer animation, narrated by the presenter. Others might work well as a class activity where everyone cuts geometrical shapes out of construction paper. Although the type of presentation might vary, all should have some things in common. First is the intended audience. Gear your presentation towards a group of geometry students who already know the Pythagorean Theorem (don’t spend time defining "leg" and "hypotenuse"). Each presentation should also:

    Presentations will be assessed for...

    Content (15 points)

  • A 15 point presentation would include an accurate proof which meets the three requirements listed above.
    •

    Style (5 points)

  • A 5 point presentation would be delivered clearly and smoothly. For a demonstration, there would be no technical hang-ups. For a class activity, clear instructions would be given. Remember, you are working in a team; the Spokesperson should practice with other team members as the audience!
    •

    Format (5 points)

  • A 5 point presentation would use technology effectively. "Technology" just refers to the use of tools to get your point across. The tools could be a computer and television screen or scissors and construction paper; whichever works best for your proof.
    •

    Participation in a Round Table Discussion (25 points)

    This will be the concluding activity of the WebQuest. One member from each team will sit on a "panel of experts" and participate in a discussion moderated by the teacher. The purpose of the discussion is to help understand the importance of Pythagoras and the Pythagorean Theorem. Each participant in the discussion (each team Ambassador) should bring three things to the discussion:

    1. A written "prepared statement" which attempts to answer the question:

    If the theorem is 1000 years older than Pythagoras, why is it called the Pythagorean Theorem?

    2. A written "prepared statement" which attempts to answer the question:

    How is mathematics different today than it was in the time of Pythagoras?

    3. A written account of a practical application for the Pythagorean Theorem

    The "prepared statements" involve questions which are rather open-ended. Teams are encouraged to be creative with their answers. For example, we all know from the biographical information that Pythagoras is said to have been the first person to prove the theorem. At least 250 others also came up with proofs, though. Why not name it after the first person(s) to use the theorem? Why not the person with the best proof? Whatever your answers are, they should meet the following basic requirements. Each "prepared statement" should...

    The practical example does not have to be earth-shattering. It simply needs to demonstrate a way that the Pythagorean Theorem could be used effectively.

    Participation in the Round Table Discussion will be assessed for...

    Content of Prepared Statements (15 points)

  • A 15 point participation would include accurate prepared statements which fulfill the three requirements listed above and a practical example which accurately uses the Pythagorean Theorem.
    •

    Style of Prepared Statements (5 points)

  • A 5 point participation would include statements which are written and delivered clearly and smoothly. They would be free of spelling, grammatical, and typographical errors and they would be clearly and succinctly written. Remember, you are working in a team; the Ambassador should practice with other team members as the audience!
    •

    Participation in the Discussion (5 points)

  • A 5 point participation would include attentiveness, politeness, and respect for other participants during the discussion. Ambassadors will not be judged on how much they say but rather the by ways in which they conduct themselves during the discussion.
    •

    CONCLUSION

    Hopefully, the Round Table Discussion will be a useful concluding activity for the WebQuest. The idea is to think about math as more than just numbers, symbols, and geometric shapes. Math has practical applications, which are demonstrated during the WebQuest, but it also has connections to other things, such as history and philosophy. Another important thing to think about is the vast number of different proofs of the same theorem. In math, something that we often ignore is that there is usually more than one "correct" way to go about things.

    Think about some other mathematical concepts that you’ve learned about recently. Who was the first to use these concepts? Were they more important at one time in history than another? Is there more than one way to "look at" the concepts? How are they used today? Thinking about these questions can help you understand the concepts as well as understanding why they are important for us to learn. Keep these questions in mind as you learn new concepts in math.

    Ok, now that you’re an expert on the Pythagorean Theorem, how about a joke?

    There were three Medieval kingdoms on the shores of a lake. There was an island in the middle of the lake, which the kingdoms had been fighting over for years. Finally, the three kings decided that they would send their knights out to do battle, and the winner would take the island.

    The night before the battle, the knights and their squires pitched camp and readied themselves for the fight. The first kingdom had 12 knights, and each knight had 5 squires, all of whom were busily polishing armor, brushing horses, and cooking food. The second kingdom had 20 knights, and each knight had 10 squires. Everyone at that camp was also busy preparing for battle. At the camp of the third kingdom, there was only one knight, with his squire.

    This squire took a large pot and hung it from a looped rope in a tall tree. He busied himself preparing the meal, while the knight polished his own armor. When the hour of the battle came, the three kingdoms sent their squires out to fight (this was too trivial a matter for the knights to join in).

    The battle raged, and when the dust cleared, the only person left was the lone squire from the third kingdom, having defeated the squires from the other two kingdoms.

    I guess this just proves that the squire of the high pot and noose is equal to the sum of the squires of the other two sides.

    (Courtesy of Seth Yoshioka-Maxwell’s Pythagoras’ Theorem website:

    http://www.geocities.com/CapeCanaveral/Launchpad/3740/)

    Hosted by www.Geocities.ws

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