This website demonstrates a straightforward Monte Carlo method for estimating Pi. The simulation throws a specified number of darts at a circle inscribed in a square. Four times the ratio of the number of darts that land in the circle to the total number of darts thrown is calculated by the applet. This experimental value provides an estimate for Pi.
Type of Material:
Any standard web browser.
Identify Major Learning Goals:
The student should be able to estimate Pi by executing the method and determining the method's strengths and weaknesses. The documentation provides the justification for the method by deriving an algebraic expression for Pi, using area formulas for a circle and square and simple algebraic steps.
Target Student Population:
Courses in elementary probability, students interested in probabilistic methods. Can be used as a supplement or intro when discussing Monte Carlo methods. This experiment would also be appropriate for a developmental geometry course as a supplement to the traditional discussion of Pi. The site could also be used in a developmental algebra class as an application of ratios, provided the students have mastered the algebraic concepts used to develop the relationship for Pi.
Prerequisite Knowledge or Skills:
Elementary Geometry, including area of a square and circle. Basic knowledge of elementary algebra is required to develop the relationship used to estimate Pi.
The problem is discussed in the applet, and references are given. Concepts are presented clearly.
When using this site in a developmental algebra or geometry class, the instructor may need to provide students with more detailed steps in developing the relationship for Pi. This is expected for that level of student and would not cause problems in using the applet.
Potential Effectiveness as a Teaching Tool
The applet will expose the reader to the most basic Monte Carlo estimate of Pi. The applet is engaging, allowing the user to change the number of darts tossed. The specific number of darts landing in the interior of the circle, as well as those exterior to the circle and interior to the square, are provided so that students may calculate the value of Pi, or read the result on the screen. The use of color is visually effective, particularly as the number of darts may be increased significantly. This simulation allows the student to observe results that could not be effectively demonstrated using a chalkboard.
The focus is narrow, but references to additional Monte Carlo methods and a second way of estimating Pi place this site into a larger context.
Ease of Use for Both Students and Faculty
The applet itself is extremely easy to use. The user selects the number of darts to throw, and different colors are used to distinguish darts that land in the circle's interior from those that land inside the square, but exterior to the circle. Feedback is immediate and instructions are simple.
The scaling in the graphic that keeps track of the Monte Carlo estimates overlaps with the graphics at times. This is confusing and cannot be corrected by the user.
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