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Math from the Past Introduction | Task | Process | Guidance | Conclusion | Questions Introduction Emmy Noether was a German-born mathematician and professor who taught in Germany and the United States. She made important contributions in both mathematics and physics. The table below shows the titles of some of the workshops for students at a recent Emmy Noether Mathematics Day. Several of the workshops covered topics in discrete mathematics. Title of Workshop Topics Sister Celine and sums of Binomial polynomials, patterns, and binomial Coefficients coefficients Juggling and Algebra number sequences, patterns, counting principles Factoring Integers to Break Codes using computers and factoring to find the meaning of a coded message On Infinity exploring the history of large numbers and the misuse of infinity Source: www.math.ttu.edu In this project, you will research a mathematician of the past and his or her role in the development of discrete mathematics. The Task You are a mathematics professor at a university. One requirement of your job is to make presentations regularly at mathematics meetings or conferences in the U.S. or other countries. You have been selected to make a presentation at a mathematics conference with the theme of mathematicians and mathematics history. You need to prepare a one-hour talk about a mathematician of the past who contributed to the field of discrete mathematics. You will place the materials for your presentation into a binder to use as reference during your presentation. Be sure that your binder contains the following: an outline for your talk, including the name of the mathematician, the time period the person lived, and a history of the person's life and contributions to the area of discrete mathematics; at least three sheets that can be made into transparencies to enhance your presentation; at least one example of a mathematical idea or problem that the person worked on in discrete mathematics. This example will be one of your transparencies. You will get some ideas about mathematical ideas or problems that could be used in a talk from the exercises in Unit 4 in your textbook. The Process To successfully complete this project, you will need to complete the following items. Select a mathematician from the past that interests you. However, you must make sure that the person worked at some time in discrete mathematics. For ideas on discrete mathematics topics, see the lessons in Unit 4 of your textbook. For help, try these Web sites. www-groups.dcs.st-and.ac.uk:80/~history aleph0.clarku.edu/~djoyce/mathhist/mathhist.html www.cc.gatech.edu/classes/cs6751_97_winter/Topics/stat-meas/probHist.html www.mala.bc.ca/~johnstoi/darwin/sect4.htm math.truman.edu/~thammond/history/Probability.html www.britannica.com and search for math history mathforum.org/isaac/mathhist.html www.studyweb.com/links774.html archives.math.utk.edu/topics/history.html www.memphis-schools.k12.tn.us/admin/tlapages/math.htm www.unm.edu/~maryrobn/wbsts.htm Find a problem or mathematical idea that the mathematician of your choice worked on. Use the Web sites above to help you. If necessary, use books listed in bibliographies on the Web sites to find out more about specific ideas or problems. Prepare the outline for your presentation and include the name of the mathematician, the time period, and a history of the person's life and career. Prepare the sheets for your three transparencies. Be creative. Add some additional data, information, or even pictures to your presentation pages. Guidance Here are some additional questions and ideas you may want to consider for your project. 1. Prepare a timeline for the mathematics topic you choose. For example, you might plot the dates for some of the major advancements in probability. 2. Determine what other mathematicians worked closely with the mathematician that you are highlighting. On what projects did they collaborate? 3. When and how were special mathematics symbols developed, for example, summation notation? Conclusion Here are some ideas for concluding your project. Present your project to your class or at a family night. Consider preparing your presentation with presentation software that includes video and audio. Present the information on a Web page. Have other students critique your project and help you to make improvements to your project. Write a research paper on the mathematician you selected. Include at least one mathematics idea or problem the person worked on. Interview a mathematics professor at a university. What training is needed to pursue this career? What expectations are there for maintaining a position as a professor? Questions Lesson 11–7 Tahani is preparing a presentation on Blaise Pascal (1623-1662), who is credited with the discovery of the famous pattern of numbers known today as Pascal's triangle. She plans to show Pascal's triangle on a transparency and highlight the various applications of this triangle to discrete mathematics. In her research she finds this interesting application that is not credited to Pascal. 1. Neatly construct an equilateral triangle, like the one below, in which you can display the first eight rows of Pascal's triangle. Finish filling in the rows. 2. After you have completed Pascal's triangle, shade in all the numbers that are not odd and all of the triangles with no numbers. Describe the pattern. Can you identify this figure? (Hint: See Lesson 11-6B.) 3. What is the ratio of the shaded area to the non-shaded area of the triangle in Exercise 2? Lesson 12–1 Patricia is preparing a presentation on Pierre de Fermat (1601-1665), who was a practicing lawyer, but studied mathematics as a hobby. He corresponded with Pascal over one of the first important probability problems, often referred to as the "problem of the points." According to Howard Eves, author of a mathematics history book, the problem can be stated as follows. Determine the division of the stakes of an interrupted game of chance between two supposedly equally-skilled players, knowing the scores of the players at the time of interruption and number of points needed to win the game. (Eves, p. 288) Fermat provided a solution for one case of this problem. He assumed that player A needed 2 points to win and player B needed three points. (The winner scores 1 point for each win.) You can see that in at most 4 more plays one of the players must win. 1. Let a represent a win by player A and b represent a win by player B. Finish this list of possible outcomes for the next 4 plays. aaaa baaa abaa 2. How many possible outcomes are there? 3. For how many outcomes does player A score at least 2 points? For how many outcomes does player B score at least 3 points? 4. In what ratio should the remaining money be divided to be fair according to the possible outcomes?