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. The New-States Paradox • The addition of a new group changes the apportionments of other groups. • When Oklahoma became a state, they decided that they would get 5 representatives, raising the number of seats from 386 to 391. • This, however, changed the apportions for other states.

15. The New-States Paradox • A school district has 2 HS, East High (2574 students) and West High (9426 students). The school district has a counseling staff of 100 counselors. • Apportion the counselors to the two schools.

16. The New-States Paradox • Standard Divisor: • 120

17. The New-States Paradox • Suppose that North High School is added to the district with 750 students. The district hires 6 counselors for this new school. • What is the new apportionment of counselors?

18. So what?

19. Homework Bring your old tests and any clarifying questions that you may have about problems that were difficult for you. Final Class: Friday, May 28th!!! Math 132: Foundations of Mathematics