# History of Pi and Buffon's Needle

This material is replicated on a number of sites
as part of the
SERC Pedagogic Service Project

#### Summary

Students are directed to visit the MacTutor History of Mathematics Archive and to read an extensive online article entitled History of Pi; in addition, they make use of an interactive simulation of Buffon's Needle experiment. Afterwards, they answer several questions on how mathematicians calculated approximations for the value of pi and on the formulas that they used.

In a subsequent class session, the instructor demonstrates a Java applet that simulates Buffon's Needle experiment in a cumulative manner. In a think-pair-share activity or write-pair-share activity, students analyze and discuss Buffon's experiment and its relationship to the value of pi.

## Learning Goals

To enable students to:

- gain an understanding of the history and development of the value of pi
- recognize the usefulness of infinite series with regard to defining pi
- gain an understanding of a probability measure for the value of pi
- apply their knowledge and understanding in analyzing Buffon's Needle experiment

## Context for Use

This activity can be used in a Calculus II class as an introduction to the usefulness of infinite sequences, in a Geometry class when covering the topic of circles, or in a class involving Probability and Statistics.

The activity will require approximately 45-60 minutes outside of class depending on how long students spend on the interactive Java applet. The questions are made available via an online quiz or dropbox (Blackboard, Angel or other LMS) and students submit their answers prior to the related class discussion. The in-class follow-up activity takes approximately 15-20 minutes.

The activity will require approximately 45-60 minutes outside of class depending on how long students spend on the interactive Java applet. The questions are made available via an online quiz or dropbox (Blackboard, Angel or other LMS) and students submit their answers prior to the related class discussion. The in-class follow-up activity takes approximately 15-20 minutes.

## Description and Teaching Materials

Activity description:

- Students are directed to read through an online article on the History of Pi outside of class and make use of a Java applet to simulate Buffon's Needle experiment. As they do so, they answer several assignment questions (Rich Text File 34kB Jan22 07) concerning the material and submit their answers online via a quiz or dropbox.
- MacTutor History of Mathematics Archive

Article on the History of Pi

http://www-history.mcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html - MERLOT description of this resource

http://www.merlot.org/merlot/viewMaterial.htm?id=235021

- Buffon's Needle Estimate of Pi

A Java applet that allows the user to drop 10, 100 or 1000 needles at a time; the site also provides some explanatory background and some questions for analysis

http://www.mste.uiuc.edu/reese/buffon/buffon.html - MERLOT description of this resource

http://www.merlot.org/merlot/viewMaterial.htm?id=74555 - During the following class, the instructor presents an interactive Java applet involving the cumulative results of Buffon's Needle experiment and presents a think-pair-share activity or write-pair-share activity (Rich Text File 36kB Jan22 07) in which students are asked to observe the variability in the estimates for pi and to analyze the experiment with regard to reliability.
- Buffon's Needle

A Java applet that allows the user to drop 1, 2, 10 or 20 needles at a time and view the needles dropped in a cumulative manner.

http://www.math.csusb.edu/faculty/stanton/probstat/buffon.html - MERLOT description of this resource

http://www.merlot.org/merlot/viewMaterial.htm?id=233135

## Teaching Notes and Tips

The MacTutor History site is exceptionally well written and researched. The many cross links in the online articles provide a wealth of related information and the Web sites listed at the end of the articles are often of interest. In particular, the link to R. Knott (Pi and Fibonacci Numbers) is an excellent site for a deeper investigation of the approximations for pi.

Students are generally unaware of Buffon's Needle experiment and the variability of its results with respect to the number of needle drops. The class discussion helps them to deepen their understanding of probability measures.

Students are generally unaware of Buffon's Needle experiment and the variability of its results with respect to the number of needle drops. The class discussion helps them to deepen their understanding of probability measures.

## Assessment

I grade the assignment questions on the percentage of correct answers; the activity grade represents approximately 1-2% of a semester grade.

## References and Resources

- A Chronology of Pi

This site contains a chronological listing of the approximations of the value of pi dating from 2000 B.C. onwards, including recent computer-based calculations.

MERLOT description of "Chronology of Pi" site.

Direct link to the Chronology of Pi site.

- Pi and the Fibonacci Numbers

This site contains a wealth of information about pi and its various formulas as well as connections to the Fibonacci numbers. Excellent resource for courses in Trigonometry and Number Theory.

MERLOT description of the Pi and the Fibonacci Numbers site.

Direct link to the Pi and the Fibonacci Numbers site.