In this Mastery Mission students will have the opportunity to explore the historical development of the concept of irrational numbers. They will research the life and times of the Greek mathematician Theodorus and will construct and analyze his famous “Wheel of Theodorus."
Introduction
When early Greek mathematicians attempted to measure the hypotenuse of right angle triangles, they discovered something shocking. They realized that it wasn’t always possible to do so using the numbers that they had available to them at the time.
Pythagoras, one of their most influential leaders, believed that all mathematical puzzles could be solved by using integers and their ratios (called rational numbers). In fact, legend has it that the Pythagoreans, as they were known, were so sure that they were right, that when one of their students told his teachers that he disagreed; he was promptly put to death!
One of the great pioneers in the early work showing the existence of irrational numbers was Theodorus of Cyrene. In this Mastery Mission you will construct his famous “Wheel of Theodorus” and explore the interesting dilemma he faced when trying to measure what he had created. So, choose a partner to work with, and move onto the Task section to find out the details of your mission.
Task
You and your partner will complete the following tasks:
1. Research Theodorus of Crete by using the links that are provided.
2. Use the information that you have gathered to complete the “Who was Theodorus?” worksheet.
3. Construct the Wheel of Theodorus by following these instructions.
4. Use your wheel to explore some interesting mathematical ideas on the number system by completing the “Analyzing the Wheel” worksheet.
Process
1) Research
Explore these sites to research Theodorus of Crete, and learn more about the secret society of Pythagoreans to which he belonged.
Information on Theodorus:
Theodorus of Cyrene
Wikipedia - Theodorus
Information on the Pythagoreans:
History of the Pythagorean Theorem
Pythagoras
Information on irrational numbers:
Definition of Irrational Numbers
Irrational Numbers
Other cool spirals:
Spirals
Animation of the Snail Spiral. Where does this name come from?
2) Summarize
Print the “Who was Theodorus?” worksheet and complete it to summarize your findings.
3) Construct
Construct the Wheel of Theodorus by following these instructions and using the web sites and documents below. You can draw your spiral with a compass and ruler. You can also use a template or something like an index card. Watch the videos and read the instructions first.
Then, create a wheel of Theodorus neatly and in color. Mark the unit measures of all your triangle sides. Feel free to decorate your wheel in a way that demonstrates this spiral in the real world. Attach a sheet of paper with your calculations for the first five triangles.
Making a Right Triangle Spiral
Wheel of Theodorus Projects
Video Instructions (kind of grainy video but good instructions)
4) Analyze
Use your wheel to explore some interesting mathematical ideas and complete the "Analyzing the Wheel" work sheet. Here is the “Analyzing the Wheel” worksheet.
Evaluation
RUBRIC Beginning Developing Accomplished Exemplary
"Who was All the questions Nothing A few questions attempted and Most questions attempted Theodorus?" attempted and answered submitted. answered correctly. and answered correctly. worksheet. correctly.
Construction of Did not attempt Construction completed with Construction completed the Wheel of Construction partially completed. construction. a few errors. with no errors. Theodorus.
"Analysis of the Table partially completed, with Table mostly completed, with Table completed and Nothing Wheel" many errors, and a limited only a few errors, and some accurate with thoughtful submitted. worksheet. attempt at analysis. attempt at analysis. analysis.
Work ethic Was disruptive and disturbed Needed frequent reminders to Needed occasional reminders Worked consistently and others in the stay on task. to stay on task. with focus. computer lab. Conclusion
Congratulations, you and your partner have now completed the mission. Well done!
I hope that you have enjoyed this experience and have learned something interesting about the importance of the “Wheel of Theodorus” in the historical development of the concept of irrational numbers.
Credits
Created by Rosemary Bunker, University Of Pittsburgh. Modified by Kent Roberson, Jeff Roberson, Disney Elementary School.
Artists
Thanks to the artists whose beautiful paintings, quilts, and illustrations have been on display throughout this web mission. To find more of their work follow these links:
Fellowship Paintings
Mathematical Quilts
Indiana University Mathematics Department Gallery
Writers
Kudos to the Connected Math series writers for their production of an interesting and mathematically rich curriculum.
Teachers
Many thanks to Milan Sherman, Dept of Education, University of Pittsburgh, for this introduction to Quest Garden web site.
Mathematicians
And let's not forget to give credit to Theodorus, and the many other innovative mathematicians who have followed in his footsteps.
Permissions We all benefit by being generous with our work. Permission is granted for others to use and modify this WebQuest for educational, non-commercial purposes as long as the original authorship is credited. The modified WebQuest may be shared only under the same conditions. See the Creative Commons Attribution • Non-Commercial • Share-Alike license for details.