Discrete Mathematics Modeling Our World What is it anyway? 1. The focus of the Teacher Leader sessions for the 2002-2003 School Year is Discrete Mathematics. This area of mathematics has become so important in today’s technological society that its concepts must be developed across the grades K-12 mathematics curriculum. We introduce this strand with a short overview of some of its applications.

Graph Theory Euler Paths and Circuits In order to minimize cost to the city, how should weekly garbage collection routes be designed for Detroit’s 350,000 households? 2. Paraphrase the text on the slide.

Graph Theory Traveling Salesman Problem Sears, Roebuck and Company manages a fleet of over 1000 delivery vehicles to bring products they sell to customers’ locations. How should Sears determine an efficient delivery plan for each day? SEARS 3. Paraphrase the text on the slide. Operational research consultants provide Sears with a number of algorithms for scheduling daily deliveries. SEARS

Matrices Computer Representation of Graphs How can problems like the Detroit garbage collection or Sear’s delivery service be modeled in order to utilize technology for the solution? 4. One can model the application using graphs such as shown where the vertices represent certain objects in the situation and the edges represent some relationship between pairs of the objects. To take advantage of the power of computers to investigate a problem, a graph can be represented by a matrix which can then be entered into the computer (Select an example (i, j) position and note how its value describes the number of edges having vertex i and j as endpoints). The given matrix completely represents the graph.

Matrices Solving Systems of Equations “Problems we solve nowadays have thousands of equations, sometimes a million variables.” Professor George Dantzig, Stanford University How do telecommunications companies determine how to route millions of long-distance calls using the existing resources of long-distance land lines, repeater amplifiers, and satellite terminals? 5. Note that matrices are used to enter systems of equations into the computer.

Matrices Geometric Transformations Have you ever wondered how your favorite cartoon characters become animated? 6. You got it! Matrix operations are used to perform the rotations, translations, etc. necessary for the animation. We took these pictures of Chuck Allan on his last day as Math Consultant for the MDE. Note that Chuck retired as Math Consultant last July and is currently working under a separate contract with MDE to complete the revision of the Mathematics Curriculum Framework. We thank Chuck for his outstanding leadership during the past 17 years.

UR4T82 Counting & Arranging How secure are your passwords? If your password consists of 3 letters and 3 numerals, how likely is it that someone could successfully guess the configuration? 7. Here is a topic which we have already explored: Algorithms for solving this type of problem involve the application of the Counting Principle, permutations and combinations. Introducing Bruce Budzynski, Mathematics Education Consultant, Michigan Department of Education. UR4T82

Coding Information Identification Numbers What mathematics is involved in the design of UPC codes? 8. The binary numeration system as well as “remainder arithmetic (mod 10)” are used in the Universal Product Code. Place a face with the voice of M3RP , introducing Sue Simons, Project Secretary.

Coding Information Error-Detecting Codes Did you know that many identification codes contain check digits to help catch errors? 9. The Vehicle Identification Number indicates that the truck pictured above was manufactured in Canada (digit 1), Ford (digit 2), truck (digit 3), describes its restraints (digit 4), F-250 (digits 5-7), Automatic V-8 engine (digit 8), 1994 (digit 10), and the accessories and sequence number on the assembly line during production (digits 11-17). The 4 in the 9th position is a check digit. It is the remainder upon dividing a weighted sum of the other digits by 11 after literal symbols have been replaced by the numerical equivalent. Thus if there is an error in transmitting the VIN or if it has been defaced, the computer will recognize the incorrect VIN. Most codes including the UPC contain a check digit. VIN 2FTHF26H4RCA06058

Social Choice Voting Sydney Wins! News Clip 2000 Summer Olympics Kansas City Star go to Australia September 24, 1993 Sydney, Australia, edged out Beijing Thursday for the right to hold the 2000 Summer Olympic Games. Beijing, which was considered the slight favorite, led in each of the first three rounds of voting but could not gain on overall majority. Here’s how the International Olympic Committee voted. A simple majority was required to win. First Second Third Fourth round round round ** round ** Beijing 32 37 40 43* Sydney 30 30 37 45 Manchester, England 11 13 11* Berlin 9 9* Istanbul, Turkey 7* * Eliminated ** One member did not vote 10. In a democratic society decisions are frequently made by more than one person. When the number of candidates or alternatives is three or more the outcome of the election may depend upon the voting system used rather than the will of the people. Note how Plurality Voting (the candidate with the most first place votes wins) would have selected Beijing as the site on the first vote, however since it did not receive more than 50% of the first place votes (Majority Rule), the vote was repeated after eliminating the one site receiving the fewest first place votes. This voting method is called the Sequential Runoff Method.

Social Choice Apportionment Algorithms U.S. Constitution: Seats in the House of Representatives “shall be apportioned among the several states within this union according to their respective Numbers …” 1792, First Presidential Veto: George Washington vetoes the apportionment bill 11. One of the most politically charged fair distributions problems in this country is the apportionment of the 435 seats (since 1910) in the U.S. House of Representatives. Using the 2000 Census, 288,424,177 ÷ 435 = 646,952 people represented by each seat in the House. The problem arises when a state’s population is divided by this “ideal district size” and the result (called its quota) has a fractional part. Alexander Hamilton’s algorithm for dealing with this problem in 1790 when there were 15 states and 105 Representatives was to assign the “extra” seats to the states having the largest fractional part (Vetoed by Washington but passed by Congress in 1850 and used by Congress until 1900). Methods developed by Thomas Jefferson and Daniel Webster were used until 1940 when the Hill-Huntington algorithm for dealing with fractional parts of seats was adopted. This method uses the geometric mean of the whole number portion, n, of a state’s quota and n+1 in its calculation. Each 10 years some states lose seats prompting challenges in the courts. Following the 1990 Census, the states colored red in the map lost seats and those colored green gained seats. Both Montana and Massachusetts sued to restore lost seats arguing that other algorithms should be applied. 1991 LEGAL CHALLENGES What’s it all about?

Discrete Mathematics Nature of Problems Existence of Solutions Number of Solutions Algorithms for Generating Solutions Optimization 12. The nature of each of the problems associated with the applications of Discrete Mathematics revealed in this short presentation involve questions such as the following: Does this problem have a solution? How many solutions? Is there an algorithm which will provide solutions to this type of problem? Is there an optional solution?

Mathematics Curriculum Framework Probability and Discrete Mathematics “Contemporary uses of mathematics demand that students learn to deal with uncertainty, to make informed decisions based on evidence and expectations, to exercise critical judgment about conclusions drawn from data, and to apply mathematical models to real-world phenomena. The technological world in which we live also depends upon information and communication of information and upon applications of systems with separate (discrete) entities. Topics of discrete mathematics such as counting and permutation problems, matrix operations, vertex-edge networks, and relationships among finite sets have significant real-world applications that students will encounter in diverse fields of work and study.” 13. The importance of the Discrete Mathematics Strand in the grades K-12 curriculum is reflected in the following quote introducing Strand VI. Probability and Discrete Mathematics of the Michigan Mathematics Curriculum Framework. (Read the slide.)