In this activity students will contribute a set of random points to a graph of a circle of radius 1 inscribed in a square. They will estimate the area of the circle using geometric probability.
Before the Activity
Students should know how to solve a proportion. They should also know the formulas for the area of a square and a circle.
During the Activity
Instruct students to log into the Navigator network and enter the "Activity Center". On the computer go into "Activity Center" and configure the activity in the following way:
1. Contribute: Lists (configured so that students can resubmit lists and students start with empty lists)
2. Click on "Graph-Equation" and graph "y = sqrt(1-x^2)", y = -sqrt(1-x^2)", "x = 1", and "x = -1". Note: To graph an equation of the form x = a, click on "view" and select "Show X-Entry". This will show a circle inscribed in a square.
Then, instruct the students to select several phone numbers, making sure that no two students submit the same number. Instruct students that they will take each phone number and create an ordered pair in the following way:
If the last four digits are 4538, then
1. Write an ordered pair (.45, .38)
2. If the digits form an even number then the number will be positive. If the digits form an odd number then the number will be negative. (-.45, .38)
Start the activity and have the students enter their ordered pairs in lists (where the first coordinate goes in L1 and the second coordinate goes in L2). The points will be automatically graphed on the coordinate plan within the square.
From the results shown on the screen, have students determine the ration of the number of points that fell within the circle to the total number of points. For example, if 25 students submit 4 points each, 100 points will be graphing in total, of which about 75 will fall within the circle. This ratio will be approximately equal to the ratio of the area of the circle to the square. It will be clear to the students that the square has area 4. Let x represent the area of the circle and have students set up and solve a proportion. Use "Quick Poll" to collect their results. Students should realize that the result is an approximation for pi (3.14).
After the Activity
The teacher could place other images in the activity center. This could be done by loading the image as a background image (perhaps a map of the state students reside in). Construct a rectangle around this image whose area is known and use geometric probability to estimate the area of the background image.