MR- Using the Power of Mathematical Reasoning to Make Decisions Overview

Using the Power of Mathematical Reasoning to Make Decisions Overview  

Would you like to settle the dispute over an age-old question? Who is really the best basketball player in the NBA? Do you believe that there is anyone in the league whose statistics who can compare with Michael Jordan's? Using proportional reasoning, ratios, rates and percents you can figure out whose statistics come close to Jordan's stellar performance. Since we have bought up the topic of Jordan we can also talk about his shoes. These products fly off of the shelves when released but there is something very important that each box of shoes has Universal Product code, better known as a UPC! In this unit, you will gain an understanding of product identification numbers and how they work. You will also solve unknown answers by using estimation techniques that allow you to represent the data mathematically.

Additionally, we are going to look at various voting methods and the advantages and disadvantages of each.  As citizens of a free society, we have both a right and duty to vote. The political, social, financial, and environmental choices we make in elections can affect every aspect of our lives. Voting is only part of the story. The second part, the methods for counting our votes, lies at the heart of the democratic process. In this module, you will learn the role that mathematics plays in finding our collective voice when faced with more than two options in an election. You will see that determining a winning option may depend on the method used for counting votes as well as issues and the preferences of the voters. Is there a method for counting votes that is always fair? In exploring election theory, you will begin to understand some of the mathematical paradoxes of our democracy.

Essential Questions

1. 1. How can we estimate large number quantities? Are we limited to just one estimate?
   2. Do you think we will ever run out of telephone numbers?
   3. How are today's televisions different from older models?
   4. If we change the size of the tires on a car, how are the odometer and speedometer readings affected?
   5. How are numbers used to identify everything from people to things? How can we recognize an error in a Universal Product Code (UPC)?
   6. How can we recognize an invalid credit card number?
   7. How do you calculate a student's final grade average?
   8. How do we compare statistics from different sport players? What costs are associated with sporting events?
   9. What makes a passage of text easier or harder to read?
   10. How do people analyze information to make a good and fair decision?
   11. How are the wishes of many individuals combined to yield a single result? Do the methods for doing so always treat each choice fairly?
   12. If not, is it possible to improve on these methods?
   13. How can a portion of food be divided fairly among two or more children?
   14. What are the advantages and disadvantages of the different elections processes?
   15. Is the meaning of fairness when food is divided among children differently from the meaning of fairness when an estate is divided among heirs or when seats in Congress are divided among states?
   16. Are the methods that are commonly used to divide food, estates, and legislatures necessarily the fairest methods?

Key Terms

• aspect ratio - ratio between the width and the height of the element
• letter box - a film format with black or gray bars above and below the image to preserve images
• pillar box - a film format with black or gray bars on the sides of the screen to preserve images
• proportion - a portion or part of a whole
• weighted average - an arithmetic mean with an assigned weight value on each quantity. The weighted value sets more "significance" on elements.
• Fan Cost Index - the fan cost index determines the cost to take a family to an event
• Apportionment - to round a set of fractions so that their sum is a fixed number
• Borda Count - points are assigned for place on the ballot, one point for last place, two for second to last, etc. The number of points given to candidates for each ranking is determined by the number of candidates standing in the election. Thus, under the simplest form of the Borda count, if there are five candidates in an election then a candidate will receive five points each time they are ranked first, four for being ranked second, and so on, with a candidate receiving 1 point for being ranked last (or left unranked). In other words, where there are n candidates a candidate will receive n points for a first preference, n – 1 points for a second preference, n – 2 for a third, and so on.
• Bottom Up Strategy - a fair division scheme used for optimal strategy for rational players, each knowing the other's preferences.
• Cake-division procedure - a scheme that n players can use to divide a cake among themselves in a way that satisfies each player.
• Divide-and-choose - a fair division method for two people One of the people divides the object into two pieces, and the second person chooses either of the two pieces.
• Knaster inheritance procedure – a fair division method for three or more people
• Last-diminisher method - A cake-division scheme that guarantees proportional shares with any number of players. Like the lone-divider method, it is proportional but not envy-free. A cake-division scheme is said to be proportional if each player's strategy guarantees him a piece of size at least 1/n in his own estimation
• Lone-divider method - A cake-division scheme is said to be proportional if each player's strategy guarantees him a piece of size at least 1/n in his own estimation. It is envy-free if each player feels that no other player's piece is bigger than the one he has received.
• Lower quota for a state is obtained by rounding its quota down to the nearest integer
• Majority - receives more than half of the first place votes
• Pairwise Comparison - an agenda is created to pit the first candidate against a second candidate in an one on one contest. The winner moves on to confront the third candidate and so on until a winner emerges.
• Plurality - winner is the candidate with the most first place votes
• Run Off – election is held between the top two candidates to determine the winner
• Sequential Run Off Method - a variation of the run off method in which you repeatedly eliminate the candidate with fewest 1st place votes until a winner emerges.
• Standard Divisor -Total Population / Number of Seats
• Standard Quota - State population / Standard Divisor
• Taking turns – simplest and most natural of fair-division schemes: Two people often split a collection of assets between them
• Upper quota is obtained by rounding its quota up to the nearest integer

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