1. Math 132: Foundations of Mathematics
Amy Lewis
Math Specialist
IU1 Center for STEM Education
May 27, 2010
Math 132: Foundations of Mathematics

2. 14.4 Flaws of Apportionment Methods
Understand and illustrate the following:
Alabama paradox
Population paradox
New-states paradox
May 27, 2010
Math 132: Foundations of Mathematics

3. Math 132: Foundations of Mathematics
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?
May 27, 2010
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
May 27, 2010
Math 132: Foundations of Mathematics

5. Math 132: Foundations of Mathematics
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.
May 27, 2010
Math 132: Foundations of Mathematics

6. Math 132: Foundations of Mathematics
Alabama Paradox
An increase in the total number of items to be apportioned results in the loss of an item for a group.
State A B C
Total
Population
5015
4515
470
10,000
What happens when the number of seat in congress is increased from 200 to 201?
Start by finding the standard divisor for 200 seats.
May 27, 2010
Math 132: Foundations of Mathematics

7. Math 132: Foundations of Mathematics
Alabama Paradox
State A B C
Total
Population
5015
4515
470
10,000
Standard Quota
100.3
90.3
9.4
200
Lower Quota
100 90 9
199
Surplus Seats 1 10
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.
May 27, 2010
Math 132: Foundations of Mathematics

8. Math 132: Foundations of Mathematics
Alabama Paradox
State A B C
Total
Population
5015
4515
470
10,000
Standard Quota
100.8
90.75
9.45
201
Lower Quota
100 90 9
199
Surplus Seats 1
101 91
Is this fair?
May 27, 2010
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.
State A B C
Total
Original Population
19,110
39,090
141,800
200,000
New Population
19,302
39,480
200,582
May 27, 2010
Math 132: Foundations of Mathematics

10. The Population Paradox
State A B C
Total
Original Population
19,110
39,090
141,800
200,000
New Population
19,302
39,480
200,582
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.
May 27, 2010
Math 132: Foundations of Mathematics

11. The Population Paradox
State A B C
Total
Original Population
19,110
39,090
141,800
200,000
Standard Quota
9.56
19.55
70.9
100
Lower Quota 9 19 70 98
Surplus Seats 1
Final Apportionment 10 71
May 27, 2010
Math 132: Foundations of Mathematics

12. The Population Paradox
State A B C
Total
Original Population
19,110
39,090
141,800
200,000
New Population
19,302
39,480
200,582
Percent Increase
State A: %
State B: %
Who should benefit from the increased population?
May 27, 2010
Math 132: Foundations of Mathematics

13. The Population Paradox
State A B C
Total
New Population
19,302
39,480
141,800
200,582
Standard Quota
9.62
19.68
70.69
100
Lower Quota 9 19 70 98
Surplus Seats 1
Final Apportionment 20 71
What happened to state A’s apportionment?
May 27, 2010
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
State
East High
West High
Total
Number of Students
2,574
9,426
12,000
Standard Quota
21.45
78.55
100
Lower Quota 21 78 99
Surplus Seats 1
Final Apportionment 79

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?
State
East High
West High
North High
Total
Number of Students
2,574
9,426
750
12,750
Standard Quota
21.40
78.37
6.24
106
Lower Quota 21 78 6
105
Surplus Seats 1
Final Apportionment 22

18. So what?

19. Math 132: Foundations of Mathematics
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!!!
May 24, 2010
Math 132: Foundations of Mathematics