Pi machine Week 3: Reversing the process

Make a mathematical prediction

Last week's investigation showed that if you have two concentric circles where the radius of the outer circle is twice the radius of the inner circle, then only 25% of points randomly generated in the circles will fall inside the inner circle. In this investigation, you will try to make 50% of the points fall in the inner circle. You will experiment with the length of the radius of the inner circle so that half of the points fall inside the inner circle, and the other half fall in the outer circle. Which of the following choices do you think best estimates how big the radius of the inner circle would have to be in order to make 50% of the points fall in the inner circle?

1. Inner circle radius would be 0.25 times the size of the outer circle radius.
2. Inner circle radius would be 0.5 times the size of the outer circle radius.
3. Inner circle radius would be 0.7 times the size of the outer circle radius.
4. Inner circle radius would be 0.9 times the size of the outer circle radius.

Prediction:

The inner circle radius needs to be [...insert a decimal fraction here...] of the size of the outer circle radius for 50% of the points to fall in the inner circle, because [...insert your explanation here...]

Activity

Collect data to investigate your prediction

One way to check your predictions is to run an experiment. You can use the computer simulation provided to do that.

In this simulation, the outer radius is fixed to 20 units and you can change the inner radius.

To run your experiments, please follow these steps carefully and in order:

1. If you want to change the inner radius, select the Hand tool in the worksheet and click it on the little plus or minus agents on the right of the circle to increase or decrease the value of the inner radius. The inner radius should take integer values from 5 to 20. Hint: To check the prediction you made in the previous page, you need to calculate how big the inner radius needs to get to be, for example, 1/4 of the outer radius. Remember that the outer radius is 20 units.
2. Change the NumberOfPoints simulation property to reflect the number of points you want the simulation to generate. We suggest that you use 100, 1000, 10000, and 100000 points. Please note, that the more number of points, the more accurate your results. However, you should be aware that experiments with smaller radii and large number of points will take a long time to run.
3. Click the "Run" button.
5. When the simulation is done (the "NumberOfPoints" will count down to 0), record the results of your experiment.
6. Before running the simulation again (for a different radii or number of points), you need to hit the "Reset" button. Wait for the picture in the "Worksheet" area to completely reload and repeat these steps.

When you are done experimenting, please answer the activity questions below.

Complete the following:

Approximately 50% of the points fall in the inner circle when the outer circle has a radius of 20, the inner radius is [...insert a number here...]. This happens because [...insert a mathematical explanation here...].