Education Technology

# Activities

• ##### Subject Area

• Maths: Geometry: Analytic Geometry
• Maths: Statistics: Probability and Random Variables

9-12

90 Minutes

• ##### Device
• TI-Nspire™
• TI-Nspire™ CAS
• ##### Software

TI-Nspire™
TI-Nspire™ CAS

• ##### Other Materials
• Each team needs ten toothpicks
• lined paper/poster board on which to drop the toothpicks (the distance between the lines needs to be equal to the length of the toothpicks you are using)
• the aPieceofPi.tns file
• PiecofPiwsimulator.tns file
• PieceofPiTeacherVersion.tns file
• teacher/student activity sheets

## A Piece of Pi: An Investigation into Geometric Probability and Pi

#### Activity Overview

Have you ever noticed in how many places π turns up? Students will take part in a Buffon's Needle experiment and see how close they can get to 3.14.

To set up the lesson, there will be a brief historical tour of Pi as well as a clip from NUMB3RS, SEASON 2, Judgment Call. Next, the students will do an experiment and then transfer the data collected to the TI-nSpire where it will be analyzed and interpreted. Last, they will get to explore the problem using a simulated version.

#### Before the Activity

You will need to load the student .tns file on the class handhelds. Be sure to note into what folder the file saved. Connect to Class can be used to accomplish this quickly.

This investigation is set up to be a cooperative learning activity. If possible, pre-select the groups to save time.

It would also be a good idea to familiarize yourself with Buffon's Needle experiment. There are several sites that have historical information as well as a discussion of the actual investigation. See the Extensions Section of the Student Activity Sheet.

You can do the activity without showing the NUMB3RS clip; however, it is an excellent anticipatory set for your instruction.

#### During the Activity

Begin the classroom discussion by asking students what they know about the number pi. The first part of this document gives a brief historical overview of pi. Students can find out more by going to the Mac Tutor History of Mathematics Archives website: http://www-history.mcs.st-and.ac.uk/ . Discussing the history of the development of mathematics increases the student's interest in and appreciation of mathematics.

Most high school students should be familiar with the number pi. The history of pi is addressed in questions one through seven on the student activity sheet and Problem 1 of the document. This is a good opportunity to review the real number system and the classifications of different types of numbers. Students may need a refresher on this for question 7.

The students then view the NUMB3RS clip and perform the experiment. They analyze their data. There are questions about the experiment to discuss. Then they may use the simulator.

#### After the Activity

1. There are several Buffon's Needle simulations available on the Internet. The investigation you did assumed that the length of the "needle" was equal to the distance between the parallel lines on the paper. Use the following websites to investigate what would happen if the length of the needle was altered.

2. http://www.ms.uky.edu/~mai/java/stat/buff.html
http://www.mste.uiuc.edu/reese/buffon/buffon.html
3. Investigate Buffon's Coin Problem at the following website.

4. http://lhome.wlu.edu/~mcraea/GeometricProbabilityFolder/Introduction/Problem1/Problem1.htm
5. Investigate how trigonometry and calculus could be used in deriving an approximation for the value of pi in the Buffon's Needle Experiment.