BUYING TIMECindy Boyd Anthony Ice Donna Osborn Margaret SandersClass: Pre-Algebra, Algebra, Statistics, Discrete MathMaterials: Poll sheets, tally sheets, calculator, background information, practice problem, student project sheet.Goals: After gathering information on preferences, examine the different methods for determining a winner -Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet. To enable students to become familiar with oral and written presentations in Discrete Math.Time Required: Approximately 2-4 days. One day to teach methods, if needed. One day to tally the data, then time either in or out of class to apply the decision making methods, make the decisions, and write the report. One day for oral presentations.Background: Knowledge of methods of voting, frequency tables and survey techniques. Following this page are a series of pages illustrating these different methods with different soft drinks.Setting: A client has come into your advertising agency with a new theme park for fun-loving young adults.They want to advertise during one prime time situation comedy.Problem: Take a preferential survey of the top three situation comedies. Use Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet methods to determine winners. Then make a final choice. Each advertising agency should submit a data sheet and a written recommendation which defends and explains their method for determining when to advertise the new product. Also, each agency will orally present their report to the client.Evaluation: One grade will be given to the entire group. Point values will be assigned for the written report and presentation. Points for the written report should be divided among the description of the problem, the list of assumptions, the data sheet with the results of the survey, the results of the decision-making processes, the recommendation for the client, and the explanation of the decision.Extension: 1. For a more challenging project, increase the survey to include four TV shows.2. Call local TV stations to find the cost for advertising during the shows surveyed.3. Choose a specific night or time slot.4. Change the product and/or target audience.Teacher Notes:1. To choose the shows for the survey: Have each student put his/her favorite prime time situation comedy on a slip of paper. Place slips in a container or a box. Draw out three slips (with different shows); these will be the three shows used in the survey. If two classes are doing the project simultaneously, use the same shows for both classes.2. Survey at least 2 classes (50 students).3. Depending on the level of the class, the background concept sheets could be run for all of the students. The background concept sheets are explanations of the different decision making methods.Funded in part by the National Science Foundation and Indiana University 19954. The project can be scaled up or down to match the level of the class by increasing/decreasing the number of methods used to choose a winner.5. The oral presentation should present the written report to the class. It should not be read.6. Enclosed is a student tally sheet with six places for tallies; these are for the six possible rank orders. If four choices are given, then there are 24 possible rank orders.Funded in part by the National Science Foundation and Indiana University 1995PREFERENTIAL VOTINGConsider this set of preferences for soft drinks. Each column represents the rank order, while the number below indicates the number of people voting that preferential order.Pepsi Diet Coke Dr. Pepper Coke Sprite SpriteCoke Sprite Diet Coke Dr. Pepper Diet Coke Dr. PepperSprite Coke Sprite Sprite Coke CokeDr. Pepper Dr. Pepper Coke Diet Coke Dr. Pepper Diet CokeDiet Coke Pepsi Pepsi Pepsi Pepsi Pepsi 18 12 10 9 4 2Some methods of determining the winner:1. Plurality2. Borda Count3. Run-Off4. Sequential Run-Off5. CondorcetWhen a tie occurs for any of the methods, use the other methods to make a decision.The Plurality MethodThe winner is the choice that receives the most first place votes.Pepsi Diet Coke Dr. Pepper Coke Sprite SpriteCoke Sprite Diet Coke Dr. Pepper Diet Coke Dr. PepperSprite Coke Sprite Sprite Coke CokeDr. Pepper Dr. Pepper Coke Diet Coke Dr. Pepper Diet CokeDiet Coke Pepsi Pepsi Pepsi Pepsi Pepsi 18 12 10 9 4 2Diet Coke = 12 Pepsi = 18 Coke = 9 Dr. Pepper = 10 Sprite = 4 + 2 = 6The plurality winner is Pepsi. The plurality winner is not always the majority winner. Note that the pluralitywinner is ranked last by a majority of the voters.Funded in part by the National Science Foundation and Indiana University 1995The Borda MethodIf there are N choices on the ballot, the Borda method awards N points for a first-place ranking, N-1 points for a second-place ranking, etc.Pepsi Diet Coke Dr. Pepper Coke Sprite SpriteCoke Sprite Diet Coke Dr. Pepper Diet Coke Dr. PepperSprite Coke Sprite Sprite Coke CokeDr. Pepper Dr. Pepper Coke Diet Coke Dr. Pepper Diet CokeDiet Coke Pepsi Pepsi Pepsi Pepsi Pepsi 18 12 10 9 4 2In order, the totals for Diet Coke, Pepsi, Coke, Dr. Pepper, and Sprite:Diet Coke = 18(1) + 12(5) + 10(4) + 9(2) + 4(4) +2(2) = 156Pepsi = 18(5) + 12(1) + 10(1) + 9(1) + 4(1) +2(1) = 127Coke = 18(4) + 12(3) + 10(2) + 9(5) + 4(3) +2(3) = 191Dr. Pepper = 18(2) + 12(2) + 10(5) + 9(4) + 4(2) +2(4) = 162Sprite = 18(3) + 12(4) + 10(3) + 9(3) + 4(5) +2(5) = 189Coke wins.Funded in part by the National Science Foundation and Indiana University 1995The Run-Off MethodThe Run-Off method conducts a new election between the two choices with the most first-place rankings.Pepsi Diet Coke Dr. Pepper Coke Sprite SpriteCoke Sprite Diet Coke Dr. Pepper Diet Coke Dr. PepperSprite Coke Sprite Sprite Coke CokeDr. Pepper Dr. Pepper Coke Diet Coke Dr. Pepper Diet CokeDiet Coke Pepsi Pepsi Pepsi Pepsi Pepsi 18 12 10 9 4 2Diet Coke = 12; Pepsi = 18; Coke = 9; Dr. Pepper = 10; Sprite = 4 + 2 = 6The Run-Off is between Diet Coke and Pepsi. Eliminate all but Pepsi and Diet Coke. Next consider each ofthe preference schedules which were eliminated. Dr. Pepper (10) should be awarded to Diet Coke because Diet Coke was rated above Pepsi in that preferential list. Coke (9) will be awarded to Diet Coke. Sprite (4) will be awarded to Diet Coke.