Algebra- by stariya


									Math from the Past

Introduction | Task | Process | Guidance | Conclusion | Questions

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
On Infinity                          exploring the history of large
                                     numbers and the misuse of infinity

   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
      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. and search for math history
      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
      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.

Here are some additional questions and ideas you may want to consider for your

   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?

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


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.
   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?

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