Tortoise and Hare Race

What is Tortoise and Hare Race?

This activity allows the user to step through the tortoise and hare race, which is based upon Zeno's Paradox about Achilles and the tortoise. In the paradox, Achilles gives the tortoise a head start of half the distance to the finish line. Then Achilles gets to the halfway mark as the tortoise crosses the 3/4 mark, Achilles gets to the 3/4 mark as the tortoise crosses the 7/8 mark, and so on. Who wins??

In this race the participants are a hare and a tortoise -- as in Aesop's fable -- with the tortoise getting a half way (50 mile) head start on the hare.

Zeno of Elea was a Greek philosopher and mathematician from the 5th century BC who used paradoxes like this one to point out problems with the notion of infinity. The notion of infinity and problems with looking at a continuous world by thinking in terms of discrete steps were not resolved until the nineteenth century. What is hidden in this paradox is the fact that while Achilles is always running faster than the tortoise, they are not moving at constant speeds like one would imagine.

Related Resources

• Discussion about Infinity and Iteration
• Discussion about Recursion
• Worksheet Tortoise and Hare Race Exploration Questions
• Worksheet Tortoise and Hare Race Exploration Questions (doc)

How Do I Use This Activity?

This activity allows the user to step through the tortoise and hare race, which is based upon Zeno's Paradox about Achilles and the tortoise.

Controls and Output

• Before the race begins, the user can determine how much of a head start to give the tortoise by entering a percentage in the tortoise head start box and clicking set.
• The tortoise and hare move ahead according to the following algorithm:
  length of track = 100 units
  initial rabbit position = 0
  initial tortoise position = percent * 100
  new position of both tortoise and hare = (100-old position) * percent + old position
  Or they move forward the percent of the track they have left to run.
• The Next Stage and Previous Stage buttons at the top of the applet control which stage of the race is being viewed. The Zoom In and Zoom Out buttons at the top of the applet will allow the user to get either closer or farther from the tortoise and hare, allowing them to see up close who is winning.
• To restart any given race, the user can click the Reset button at the top of the applet.
• As the user advances through the different stages they can view how far the tortoise and hare have moved by looking at the race line.

Description

This activity allows the user to step through the tortoise and hare race, which is based upon Zeno's Paradox about Achilles and the tortoise. This activity would work well in groups of one or two for about ten to fifteen minutes if you use the exploration questions and five minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

• demonstrate non-terminating sequences of fractions and decimals
• motivate the notion of infinity
• motivate the notion of self-similarity

Be Prepared to

• discuss infinity
• explain average speed

Related Resources

• Discussion about Infinity and Iteration
• Discussion about Recursion
• Lesson on Introduction to Fractals: Infinity, Self-Similarity and Recursion
• Worksheet Tortoise and Hare Race Exploration Questions
• Worksheet Tortoise and Hare Race Exploration Questions (doc)