Part+eight


 * PART EIGHT:** Read the article “Exploring the Birthday Problem with Spreadsheets” in the May 1999 //Mathematics Teacher//, //92//(5), 407-411. You can ignore the calculus section of the article. Focus on discussing the value-added strategy of using spreadsheets to explore the birthday problem and how you would adapt this approach with high school students using the current version of Excel.

The value-added strategy of using spreadsheets to explore the birthday problem is evident when looking at the concepts needed to understand the solution to the problem. For example, the article describes the need for a recursive formula to calculate the number of people who can possibly share the same birthday in the room. This recursive formula can be better explained through simulations and computations made in a spreadsheet. In the same manner, because computations are made in the spreadsheet, it is easier for students to the results for bigger numbers compared to just smaller ones. This is only a matter of dragging the formula to the other cells as compared to computing each recursion with a formula, which would be very time consuming if done by hand as the numbers increases. Another advantage of using a spreadsheet is the ability to capture graphs based on the data computed. This will allow students to make comparisons and see how values change. For example, Fig. 2 in the article demonstrates the graph of the probability depending on the number of people in the room. This graph was obtained from Fig. 1, and students are able to visually analyze how the probability seems to increase faster for the first few people but then increases slower as more people walk in the classroom. This visual aid really adds value to the learning experience, especially for those who are visual learners. When adapting this approach with high school students, I would make some clarifications and observations about the birthday problem to avoid any confusion or misconceptions of the real objective. First, it is important to notice that the problem is asking for how many people you need to have at a party so that there is a better-than-even chance that two of them will share the same birthday. This is not asking, however, the probability of the people to share the birthday with you. This last one brings the number of 183 people for a 50%, which is usually the first answer people think of when dealing with the birthday problem. Unfortunately, this common answer is incorrect because of the lack of distinction between these two ideas. This now leads to the idea of finding the probability of pairs, which will make conceptually more sense to the students when developing the recursive formula. Once these clarifications have been made, I would go through a simple example (or simulation) of what would be the probability for the first few cases, that is, the probability of two people sharing the same birthday (364/365), then three ((364/365) x (363/365)), four ((364/365) x (363/365) x (362/365)), and so on. Once they see the pattern (and done these examples in the spreadsheet), the general recursive formula would be given for the other cases. When the students are finished, they will see that the approximate 50% corresponds to 23. After the answer has been explained and discovered by the students, they will create the graph as in Fig. 2 to observe the pattern behavior (the steps to follow are given in the article’s appendix). The visual representation, as previously mentioned, will help students see the pattern representation of this recursion. One last thing to notice is that the instructor must act as a facilitator throughout the entire lesson. Probability is one subject in which students have a lot of difficulty in, and they should be assisted and guided by a competent instructor to make sure learning is being taken place in the classroom. With this consideration at hand, this problem provides will serve as a great learning experience for both students and teachers.