In this assignment we will investigate further the idea in Areas of Leaves for approximating areas, and we will combine this with some ideas of probability.
Suppose you wish to measure the area of a penny. Pretend that you don't know the area of a circle, so that you cannot calculate the area of a penny. What would happen if you were to use the dot paper method for approximating its area, and you were to use inch dot paper? Note that the area of a penny is less than one square inch. Therefore, when you place the penny on the dot paper, either zero dots or one dot will be covered.
Place a sheet of inch dot paper on your table. Take a penny and throw it onto the paper. Do this as randomly as you can. Count the number of dots covered by the penny, if it stops on the paper. Repeat this fifty times and average your results. Is this an accurate approximation of the area of the penny? To compare this approximation with the actual area, measure the diameter of your penny. Then calculate its area, using the formula for the area of a circle.
Finally, write down the total count for all fifty trials on the blackboard. We will find the average for the entire class to see if this average is a better or worse approximation than those of individual students.