Pythagorean Theorem Project
Create a Theodorus' Wheel Lesson
Theodorus of Cyrene Bio
Theodorus of Cyrene was a greek mathematician born in Cyrene Lybia, in the 5th century BC. He was the tutor of the famous greek philosopher Plato. Plato travelled to Egypt many times. During these trips he spent time with Theodorus in Cyrene. Lybia and Egypt are neighboring countries, so that's why they were able to meet each other. Theodorus did not live in Cyrene all of his life. He also lived and taught in Athens. He lived in Athens at the same time that Socrates lived. He was around and conversed with the great Greek philosophers from ancient times.
Theodorus' legacy is his contribution to mathematics in developing the irrational numbers.
Irrational numbers are real numbers that cannot be expressed as a ratio of integers, in other words as a fraction.
He constructed the Theodorus' wheel or the spiral wheel with 16 contiguous right triangles, in other words triangles attached to each other. The base leg of one triangle becomes the hypotenuse for the next triangle - In his original wheel the hypotenuse lenghts varied from √2, √3, √4, …, √17. When the hypotenuse reached the square root of 17 he stopped, he did not build any more triangles, that's why his wheel diagram doesn't overlap. This wheel is very interesting because if you keep building triangles the wheel keeps growing infinitely.
There is not much knowledge of Theodorus' life - the only knowledge of his contribution to mathematics was written in one of Plato's passages. Plato did not say that Theodorus proved that the square root of 2 was an irrational number. Some say that Pythagoras was the one who proved that the square root of 2 was an irrational number.
So, Theodorus contributed to the irrational numbers, but perhaps not the irrational numbers proof according to Plato's passage.
Theodorus' Wheel Student Examples:
Inquiry to derive the hypotenuse:
Predict the Hypotenuse of the 5th Triangle
Predict the hypotenuse of the 5th triangle - the above Theodorus' wheel has 5 triangles. The first triangle has hypotenuse length of √5, the second triangle has hypotenuse length of √6, the third triangle has hypotenuse length of √7. If the pattern continues what is the length of the hypotenuse of the fifth triangle?
Use Trial and Error
Use Trial and error to figure out the numerical value for the hypotenuse - use a combination of the following operations (+, *, √, =) to figure out the length of the hypotenuse for the 5th triangle.
Instructions to create your own Theodorus' Wheel
Things you will need:
11 x 17 paper
Ruler
Protractor
Pencil
First Step:
Draw a point on the center of the paper. First measure 8.5 inches horizontally and make a dot. Then measure 5.5 inches down from the mark of 8.5 inches. Draw a center dot to start the Theodorus' wheel
Second Step:
Now you will draw the first triangle with base 1 unit and height 1 unit. Use the ruler to draw a line segment starting from the center point to the right (fig 1). Place the protractor on the top of the line segment. The end of the line segment is lined with the 90 degrees on the protractor. Make a dot right on the 90 degrees (fig 2). Place the ruler aligned between the end of the line and the dot you just drew (90 degrees dot) (fig. 3) . Draw a 1 unit perpendicular line. The two legs of
the first triangle are ready. Now draw and calculate the hypotenuse (line segment AC) (fig. 4). Label the hypotenuse.
Third Step:
The hypotenuse AC will be the leg for the next triangle. Now place the protractor on the top of the hypotenuse (AC). Measure a right angle and mark a dot on the top to use as reference (fig. 5). Measure and draw 1 unit line (fig. 6) for the second triangle. Draw and calculate the hypotenuse (fig.7). The second triangle is ready. Start on the third triangle.
Fourth Step:
Continue using the hypotenuse as the leg for the next triangle until you have the Theodorus' wheel complete.
Final Step:
Now create a background design of your choice to incorporate the Theodorus' wheel as the sample pictures showed above.
1. How to Construct the Wheel of Theodorus retrieved on October 28, 2015 from
https://msmilesmath.wordpress.com/2012/04/02/how-to-construct-the-wheel-of-theodorus/
2. The Wheel of Theodorus http://questgarden.com/56/36/6/071019070357/process.htm
References:
