Math 132: Foundations of Mathematics

1. Math 132:Foundations of Mathematics Amy Lewis Math Specialist IU1 Center for STEM Education Math 132: Foundations of Mathematics

2. 14.4 Flaws of Apportionment Methods • Understand and illustrate the following: • Alabama paradox • Population paradox • New-states paradox Math 132: Foundations of Mathematics

3. Apportionment • “The very mention of Florida outraged the Democrats. Florida’s contested electoral votes helped elect a Republican president who had lost the popular vote.” • What election is this quote referring to? • 1976: Rutherford B. Hays v. Samuel J. Tilden • What happened? Math 132: Foundations of Mathematics

4. Fair Apportionment Method • Although Hamilton’s method may appear to be a fair and reasonable apportionment method, it also creates some serious problems: • Alabama paradox • Population paradox • New-states paradox Math 132: Foundations of Mathematics

5. Hamilton’s Method • Calculate each group’s standard quota. • Round each standard quota down to the nearest whole number (the lower quota). Initially, give each group its lower quota. • Give the surplus items, one at a time, to the groups with the largest decimal parts until there are no more surplus items. Math 132: Foundations of Mathematics

6. Alabama Paradox • An increase in the total number of items to be apportioned results in the loss of an item for a group. • What happens when the number of seat in congress is increased from 200 to 201? • Start by finding the standard divisor for 200 seats. Math 132: Foundations of Mathematics

7. Alabama Paradox • Now let’s see happens when the number of seat in congress is increased from 200 to 201? • Calculate the new standard divisor and allocate seats. Math 132: Foundations of Mathematics

8. Alabama Paradox • Is this fair? Math 132: Foundations of Mathematics

9. The Population Paradox • Group A loses items to group B, even though the population of group A grew at a faster rate than group B. • A small country has 100 seats in the congress, divided among the three states according to their respective populations. The table below shows their population before and after the country’s population increase. Math 132: Foundations of Mathematics

10. The Population Paradox • Use Hamilton’s method to apportion the 100 congressional seats using the original problem. • Find the percentage increase in the population of states A and B. • Use Hamilton’s method to apportion the 100 congressional seats using the new population. Math 132: Foundations of Mathematics

11. The Population Paradox Math 132: Foundations of Mathematics

12. The Population Paradox • Percent Increase • State A: 1.004% • State B: .9977% • Who should benefit from the increased population? Math 132: Foundations of Mathematics

13. The Population Paradox • What happened to state A’s apportionment? Math 132: Foundations of Mathematics

14. Adam’s Method • Find a modified divisor, d, such that when each group’s modified quota is rounded up to the nearest whole number, the sum of the whole numbers for all the groups is the number of items to be apportioned. The modified quotients that are rounded up are called modified lower quotas. • Apportion to each group its modified upper quota. Math 132: Foundations of Mathematics

15. Webster’s Method • Find a modified divisor, d, such that when each group’s modified quota is rounded to the nearest whole number, the sum of the whole numbers for all the groups is the number of items to be apportioned. The modified quotients that are rounded are called modified rounded quotas. • Apportion to each group its modified rounded quota. Math 132: Foundations of Mathematics

16. No Homework! Next Session: Thursday, May 27 Last Class: Friday, May 28th!!! Math 132: Foundations of Mathematics