TITLE:

·        “Don’t give me that angle” -- Delving into the Pythagorean theorem.

 

GRADE LEVELS:

·        9-12

 

CURRICULUM AREAS:

·        Math

·        History

 

COLORADO CONTENT STANDARDS: 

 

COLORADO MODEL CONTENT STANDARDS
FOR MATHEMATICS

  • Become mathematical problem solvers. To be problem solvers, students need to know how to find ways to reach a goal when no routine path is apparent. To develop the flexibility, perseverance, and wealth of strategies that are characteristic of good problem solvers, students need to be challenged frequently and regularly with non-routine problems, including those they pose themselves.
  • Learn to communicate mathematically. The development of students' power to use mathematics involves learning the signs, symbols, and terms of mathematics. This is best accomplished in problem situations where students have an opportunity to read, write, and discuss ideas in the language of mathematics. As students communicate their ideas, they learn to clarify, refine, and consolidate their thinking.
  • Learn to reason mathematically. Students who reason mathematically gather data, make conjectures*, assemble evidence, and build an argument to support or refute these conjectures. Such processes are fundamental to doing mathematics.

STANDARD 2

Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning used in solving these problems.

In order to meet this standard, a student will

  • identify, describe, analyze, extend, and create a wide variety of patterns in numbers, shapes, and data;
  • describe patterns using mathematical language;
  • describe the connections among representations of patterns and functions, including words, tables, graphs, and symbols.  

GRADES 9-12

As students in grades 9-12 extend their knowledge, what they know and are able to do includes

  • modeling real-world phenomena (for example, distance-versus-time relationships, compound interest, amortization tables, mortality rates) using functions, equations, inequalities, and matrices;
  • representing functional relationships using written explanations, tables, equations, and graphs, and describing the connections among these representations;
  • solving problems involving functional relationships using graphing calculators and/or computers as well as appropriate paper-and-pencil techniques;
  • interpreting algebraic equations and inequalities geometrically and describing geometric relationships algebraically.

STANDARD 4:

Students use geometric concepts, properties, and relationships in problem-solving situations and communicate the reasoning used in solving these problems.

In order to meet this standard, a student will

  • connect various physical objects with their geometric representation;
  • connect mathematical concepts from across the standards with their geometric representations;
  • make, investigate, and test conjectures about geometric ideas;

 

GRADES 9-12

As students in grades 9-12 extend their knowledge, what they know and are able to do includes

  • finding and analyzing relationships among geometric figures using transformations (for example, reflections, translations, rotations, dilations) in coordinate systems;
  • deriving and using methods to measure perimeter, area, and volume of regular and irregular geometric figures;
  • making and testing conjectures about geometric shapes and their properties, incorporating technology where appropriate; and
  • using trigonometric ratios in problem-solving situations (for example, finding the height of a building from a given point, if the distance to the building and the angle of elevation are known).

STANDARD 6:

Students link concepts and procedures as they develop and use computational techniques, including estimation, mental arithmetic, paper-and-pencil, calculators, and computers, in problem-solving situations and communicate the reasoning used in solving these problems.

In order to meet this standard, a student will

  • model, explain, and use the four basic operations - addition, subtraction, multiplication, and division - in problem-solving situations;
  • develop, use, and analyze algorithms; and
  • select and apply appropriate computational techniques to solve a variety of problems and determine whether the results are reasonable.
  • decimals, percents, and integers in problem-solving situations from among mental arithmetic, estimation, paper-and-pencil, calculator, and computer methods, and determining whether the results are reasonable.

GRADES 9-12

As students in grades 9-12 extend their knowledge, what they know and are able to do includes

  • using ratios, proportions, and percents in problem-solving situations;

 

COLORADO MODEL CONTENT STANDARDS
FOR HISTORY

 

STANDARD 2:

Students know how to use the processes and resources of historical inquiry.

2.1 Students know how to formulate questions and hypotheses regarding what happened in the past and to obtain and analyze historical data to answer questions and test hypotheses.

GRADES 9-12

As students In grades 9-12 extend their knowledge, what they know and are able to do includes

 

  • formulating historical hypotheses from multiple, historically objective perspectives, using multiple sources; and
  • gathering, analyzing, and reconciling historical information, including contradictory data, from primary and secondary sources to support or reject hypotheses.

2.2 Students know how to interpret and evaluate primary and secondary sources of historical information.

GRADES 9-12

As students In grades 9-12 extend their knowledge, what they know and are able to do includes

  • explaining how historical descriptions, arguments, and judgments can reflect the bias of the author and/or the prevailing ideas of the culture and time period;
  • interpreting oral traditions and legends as "histories";
  • evaluating data within the social, political, and economic context in which it was created, testing its credibility, and evaluating its bias; and
  • comparing and contrasting the reliability of information received from multiple sources.

STANDARD 3:

Students understand that societies are diverse and have changed over time.

3.2 Students understand the history of social organization in various societies.

GRADES 9-12

As students In grades 9-12 extend their knowledge, what they know and are able to do includes

  • explaining how societies are maintained when individuals see benefits and fulfill obligations of membership;
  • analyzing how forces of tradition and change have influenced, altered, and maintained social roles and the social organization of societies throughout history;
  • explaining how, throughout history, social organization has been related to distributions of privilege and power; and
  • describing how societies have become increasingly complex in responding to the fundamental issues of social organization.
  • describing the economic reasons why people move to or from a location (for example, explorers, nomadic people, miners, traders).

STANDARD 6:

Students know that religious and philosophical ideas have been powerful forces throughout history.

6.1 Students know the historical development of religions and philosophies.

GRADES 9-12

As students In grades 9-12 extend their knowledge, what they know and are able to do includes

  • describing basic tenets of world religions that have acted as major forces throughout history including, but not limited to, Buddhism, Christianity, Hinduism, Islam, and Judaism;
  • tracing the history of how principal world religions and belief systems developed and spread;
  • explaining how, throughout history, conflicts among peoples have arisen because of different ways of knowing and believing; and
  • describing basic ideas of various schools of philosophy that have affected societies throughout history (for example, rationalism, idealism, liberalism, conservatism).

6.3 Students know how various forms of expression reflect religious beliefs and philosophical ideas.

GRADES 9-12

As students In grades 9-12 extend their knowledge, what they know and are able to do includes

  • explaining from an historical context why artistic and literary expression have often resulted in controversy; and
  • giving examples of the visual arts, dance, music, theater, and architecture of the major periods of history and explaining what they indicate about the values and beliefs of various societies.

 

TECHNOLOGY CONNECTION:

    • Software, PowerPoint, Microsoft Word
    • Websites for research and demonstration

 

MATERIALS:

Handouts, computer access (use of PowerPoint and/or word preferable), scan converter for group viewing of websites (if available and accessible), library visit (optional), materials for presentations (i.e. PowerPoint, poster boards, cardboard, markers, etc.)

 

 

TEACHER BACKGROUNG INFORMATION:

Generally speaking the Pythagorean theorem is not a difficult a concept for students to handle.  If in no other way the students will usually just choose to memorize this theorem.  The focus, however, of this lesson is then, that students should come to understand and hopefully appreciate mathematics and the beauty in its complexity and simplicity.  With some exploration and discovery students may come to see the elegance of mathematical thought.  Mathematics is circular in nature and becomes easer to learn and appreciate if an overview of the countless ways in which it proves itself is seen.  Not to mention how cool it is!  My point in this lesson is not so much to show students practical applications (as the question “why do I need to learn this?” comes up endlessly) but to explore and discover some fun mathematics, games, links, pure theory represented in simplicity. I would suggest this lesson be taught at the very beginning of the school year to ignite mathematical discovery and thought. *I would further suggest several class periods be allotted.

 

 

PROCEDURES:

 

  • Introduce to the students the Pythagorean theorem. That is the basic formula of A^2 + B^2=C^2. I have not included instructions for this as most math teachers have their own ways to explain and present this topic, as well as each text.
  • Distribute handout on Pythagoras. Have one student read the handout. Allow the students to break into groups, or as individuals (depending on class size and computer space.) Have each person record ten facts about the life of Pythagoras in ancient Greece. Note: The illustration on the bottom left of the handout is an artist’s representation of fractal number theory. Students most likely will come across this in their research, it is common. Perhaps its identification can be assigned as extra credit!!
  • Reassemble the group and post the significant or interesting facts on a poster board to be hung in the class. Note: Pythagoras was a fairly wild, freethinking “hippie” type of person. There is political unrest, family strife, unrequited love and religious questions involved in his life history.
  • Explain the assignment. This assignment and corresponding presentations are extremely flexible and allow much adaptation and change.
  • Students are to research using the Internet, the text, the library, old relatives or any other resource they choose. Students are to discover an historical application, concept, or game that illustrates and/or explains the Pythagorean theorem. A quick Internet search should assure you that there exists hundreds of three dimension sculptures type proofs, poems, dissertations and games spanning an almost immeasurable time span.
  • Distribute the handout on the Hanoi Tower. Point out the historical interest of the original box top for the game. Also have the students attempt to decipher the seemingly impossible instructions for the game. Than as a group, or in teams have the students visit the website to see how simple the game and its illustrated number theory is. If time allows they can play the game. An alternate would be to have them make the game out of cardboard. It is extremely simple and instructions are on two of the attached websites. A web search will also reveal many, many more playable versions of this game; some with other names, dating back way before 500 B.C.!!
  • The websites referenced at the bottom of this page should lead students to the general idea. 

 

 

ASSESSMENT:

  • Presentation  (this can easily be done as a group project or an individual project; depending on the size and skill level of the class). The presentation will be done by demonstration on the chalkboard, PowerPoint, posters, or any other medium the student chooses. They could make a sculpture is materials and time permits. The purpose of the presentation is to be sure the student understood the concept clearly and also to share the knowledge with the other students. 
  • Report  (this is not meant to be an elaborate full length report, but merely a short summary of what the student presents in class, it should include at least one diagram/visual aid). A Polaroid, digital photo, or sketch with crayons will do!
  • Note: students will be able to choose the weight of each: report and presentation (this is also good mathematically application practice). The presentations will count for 75% of the grade and the report 25% or vise-versa.  This will give the students the opportunity to take control of a portion of their assessment. Good writers can seize the opportunity to weigh the report more heavily and good public speakers can choose to weigh their presentations higher, but all must complete both.

 

WEBSITES TO ENHANCE THE LESSON:

http://www.usna.edu/MathDept/mdm/pyth.html

 

§        This web site is a fantastic example of the sort of exploration that I am aiming for from the students.  There is a small graphic to describe the theorem, and than a paragraph or two explaining the concept. There is also a link to a picture of a huge sculpture that looks like this in the Metropolitan Museum of Fine Art in New York! Would but that Pythagoras had lived to see!

§         

http://www.richlandone.org/webquests2/reed/#INTRODUCTION

 

§        This web site actually contains some lesson plans and short presentations concerning the Pythagorean theorem and its proofs and postulates.  It also has tons of useful links. This will help with background before lesson presentation.

 

http://chemeng.p.lodz.pl/zylla/games/hanoi6e.html

 

§        This web address will bring students directly to the “Towers of Hanoi” puzzle.  That is the one where the object is to move a stack of discs successfully from one peg to another, using a series of three pegs, without ever stacking a larger disc on a smaller one. There is a popular plastic fisher-price baby toy that looks like this. The Tower of Hanoi is frequently used to explain complex number theory. If the students play the game (and you can on this web site) the pattern of number series emerges.  An artist has developed this web site and there are many other geometric puzzles included, as well as many links to other sites, and interesting original artworks.

 

 

 

 

 

THE TOWER OF HANOI

 

 

 

 

 

 

 

 

 

THE TOWER OF HANOÏ

AUTHENTIC BRAIN TEASER OF THE ANAMITES

A GAME BROUGHT BACK FROM TONKIN

BY PROFESSOR N. CLAUS (OF SIAM)
Mandarin of the College of Li-Sou-Stian!

--------------------


This game was found, for the first time, in the writings of the illustrious Mandarin FER-FER-TAM-TAM, which are being published, in the near future, by order of the government of China.

The TOWER OF HANOI is composed of levels, decreasing in size, variable in number, that we have represented by eight disks of wood, pierced at their centers. In Japan, in China, and at Tonkin, they are made of porcelain.

The game consists of demolishing the tower level by level, and reconstructing it in a neighboring place, conforming to the rules given.

Amusing and instructive, easy to learn and to play in town, in the country, or on a voyage, it has for its aim the popularization of science, like all the other curious and novel games of professor N. CLAUS (OF SIAM).

We can offer a prize of ten thousand francs, of a hundred thousand francs, of a million francs, and more, to anyone who accomplishes, by hand the moving of the Tower of Hanoi with sixty-four levels, following the rules of the game. We will say immediately that it would be necessary to perform successively a number of moves equal to

18 446 744 073 709 551 615

which would require more than five billion centuries!

According to an old Indian legend, the Brahmins have been following each other for a very long time on the steps of the alter in the Temple of Bernares, carrying out the moving of the Sacred Tower of Brahma with sixty-four levels in fine gold, trimmed with diamonds from Golconde. When all is finished, the Tower and the Brahmins will fall, and that will be the end of the world!

--------------------

 

PARIS, PEKING, TOKYO and SAIGON
At bookstores and novelty shops

1883
--------
All rights reserved

 

Rules and practice of the Game of the TOWER OF HANOI

 

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The base is placed horizontally; the pegs stand upright in the holes in the surface. Then, the eight disks are stacked in decreasing order from base to summit; this creates the Tower.

The game consists of moving this, by threading the disks on another peg, and by moving only one disk at a time, obeying the following rules:
I. -- After each move, the disks will all be stacked on one, two, or three pegs, in decreasing order from the base to the top.
II. -- The top disk may be lifted from one of the three stacks of disks, and placed on a peg that is empty.
III. -- The top disk may be lifted from one of the three stacks and placed on top of another stack, provided that the top disk on that stack is larger.

--------------------


The Game easily teaches itself, in solving first the problem for 3, 4, and 5 disks.

--------------------


The Game is always possible and requires double the time each time that an additional disk is placed on the tower. Anyone who knows how to solve the problem for eight disks, for example, moving the tower from peg number 1 to peg number 2, also knows how to solve it for nine disks. One first moves the eight top disks to peg number 3, then moves the ninth disk to peg number 2, and then brings the eight disks from peg number 3 to peg number 2. Now, in augmenting the tower by one disk, the number of moves is doubled, plus one, from the preceding game.

For a tower of two disks, three moves are required;
-three -- seven ---
-four -- fifteen ---
-5 -- 31 ---
-6 -- 63 ---
-7 -- 127 ---
-8 -- 255 ---

and so on.

At one move per second it requires more than four minutes to move a tower of eight disks.

Variations of the Game. -- One varies, to infinity, the conditions of the problem of the tower of Hanoi as follows. At the beginning, one stacks the disks, in any manner, on one, two, or all three pegs. It is necessary to reconstruct the Tower on one of the pegs designated in advance. For 64 disks, the number of initial arrangements is staggering; the number is more than 50 digits long.

For more details, consult the following work in the chapter on the Baguenaudier:

RÉCRÉATIONS MATHÉMATIQUES

by Mr. Édouard Lucas,
professor of higher mathematics at the Lycée Saint-Louis

Two small volumes, in two colors
--------------------
Paris, 1883, by GAUTHER-VILLARS,
printer of the Académie des Sciences and the Ecole Polytechnique
Quai des Augustins, 55

 

 

   

                           

 

 

http://chemeng.p.lodz.pl/zylla/games/hanoi6e.html

 

http://www.richlandone.org/webquests2/reed/#INTRODUCTION

 

http://www.usna.edu/MathDept/mdm/pyth.html

 

 

 

 

“What is it?” “What is it used for?”

 

                               

 

 

Pythagoras: his theory and his commune

 

Pythagoras explored mathematical thought.  He was a mathematician in 530BC in Samos.  Much of modern mathematical theory is based on his original thoughts.

 

So, was he some high and mighty king or genius professor? Did he propose these theories to make the lives of future generations of high school students miserable? Or was he perhaps some ancient “dude” hanging out and doing cool and interesting math stuff?

 

Using the Internet or the library, research the life of Pythagoras and his commune of math doing wine drinking cohorts.

 

You may try this website to start:

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

 

 

 

 

 

 

 

 

 

 

 

Record ten significant facts in the life of Pythagoras:

1.

2.

3.

2^2.

5.

2*3.

49^1/2

8.

9.

10.