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Here’s the URL for the applet: http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=43

Description:  Manipulate the size of the circle and the rectangle to explore the probability that a point in the rectangle is also in the circle.

These are the explorations we created:

Turn on the "Show areas" and "Show area ratio" options.

1. Experiment with different values of the rectangle's width. Describe how the width affects the rectangle's area. Describe how the width affects the area ratio.

2. Experiment with different values of the rectangle's height. Describe how the height affects the rectangle's area. Describe how the height affects the area ratio.

3. Experiment with different values of the circle's radius. Describe how the radius affects the circle's area. Describe how the radius affects the area ratio.

4. Click inside the circle to move the circle around inside the rectangle.  Predict whether this will affect the circle's area or the area ratio.

5. Click inside the rectangle but outside of the circle to drag the rectangle around the grid. Predict whether this will affect the circle's area or the area ratio.

6. What probability does the area ratio represent?

7. Find the "Throw 100" button.  All 100 darts that will be thrown will land in the rectangle.  Predict how many will land in the circle.  Push the button 10 times and record the results.

8. Repeat Exploration #7 with the "Throw 1000" button.

9. Turn off the "Show areas" and "Show area ratio" options. Set width = 4.7, height = 3.5, and radius = 2.8.  Calculate the area ratio.  Turn the options back on to see if you were right.

10. If we were able to make any part of the circle move outside of the rectangle, what would this do to the area ratio?  Could we still use it to calculate the probability from Exploration #6?  Why or why not?  If we could make the circle large enough so that the rectangle was inside of it, what would this do to the area ratio?  Could we still use it to calculate the probability from Exploration #6?  Why or why not?

These are the ones on the 3MP site:

Turn on the "Show areas" and "Show area ratio" options. Area Ratios

1.Increase the size of the rectangle without changing the size of the circle. How does this affect the area ratio?

2.Increase the size of the circle without changing the size of the rectangle. How does this affect the area ratio?

3.Decrease the size of the rectangle so all its sides touch the circle. Note the area ratio. Predict what would happen to the area ratio if you reduced the size of the circle and then reduced the size of the rectangle so all its sides touched the circle.  Check your prediction.

Area Probability

1.Click on the "Throw 100" button. Compare the area ratio to the percentage of darts that hit the circle. Repeat this a few times.

2.Make a conjecture about the relationship between the area ratio and the percentage of darts that hit the circle. Explain your reasoning.

3.Repeat question 4 using the "Throw 1000" button. How does throwing more darts affect the relationship between the area ratio and the percentage of darts that hit the circle?

4.Resize the circle and/or the rectangle so that the probability that a dart hits the circle is 0.5. Explain how you did this. Throw darts to test your answer.

5.Resize the circle and/or the rectangle so that the probability that a dart hits the circle is 0.25. Explain how you did this. Throw darts to test your answer.