1. Chapter 14: Apportionment Lesson Plan
Example
The Apportionment Problem
The Hamilton Method
Divisor Methods
The Jefferson Method
The Webster Method
The Hill-Huntington Method
Which Divisor Method Is Best? 1

2. 2004, 2008 and 2012 Hong Kong Legislative Council Elections

3. Chapter 14: Apportionment The Apportionment Problem
To round a list of fractions to whole numbers in a way that their sum is maintained at its original value.
The rounding procedure must not be an arbitrary one, but one that can be applied consistently.
Apportionment Method – Any such rounding procedure.
Example: Percentage of season’s results of a field hockey team recorded and
rounded to whole numbers.
The sum does not reach 100%, and this becomes an apportionment problem.
The coach may want to give the extra percent to the games won—“to make it look good for the team.” This would be biased.
High School Field Hockey Team: 2004–2005 Season
Percentage
Round
Games won 18
18/23 × 100% = 78.26%
78%
Games lost 4
18/23 × 100% = 17.39%
17%
Games tied 1
18/23 × 100% = % 4%
Games played 23
100.00%
99% 3

4. Chapter 14: Apportionment The High School Teacher’s Dilemma
A school has one mathematics teacher who teaches all geometry, pre-calculus, and calculus classes. She has time to teach a total of five sections. One hundred students enrolled as follows: 52 for geometry, 33 for pre-calculus, and 15 for calculus. How many sections of each course should be scheduled?
Suppose the standard divisor = 20 students per section
As shown in table, six sections are needed.
Apportionment method is to find an equitable way to round a set of numbers like these two examples without increasing or decreasing the original sum. 4

5. Chapter 14: Apportionment The High School Teacher’s Dilemma5

6. Chapter 14: Apportionment The Apportionment Problem
Let n be the number of states, and let their populations be denoted as
P1, P2, …. Pn
The total population will be
P = P1+ P2+ ….+ Pn 6

7. Chapter 14: Apportionment The Apportionment Problem
Standard Divisor
To find the standard divisor, s:
Find the total population, p, and divide it by the house size, h. p h
The standard divisor is calculated first, then you will be able to find the fair amount of shares to give to each class or house.
Quota
To find the state, or class, quota, qi :
Divide its population, pi, by the standard divisor, s pi s
The quota is the exact share that would be allocated if a whole number were not required.
s =
qi = 7

8. Chapter 14: Apportionment The Hamilton Method
An apportionment method developed by Alexander Hamilton, also called the Method of Largest Fractions.
He realized that when rounding each quota to the nearest whole number, sometimes the apportionments do not add up correctly.
Alexander Hamilton
Hamilton’s Apportionment Method:
After the quotas are calculated, round them down to the lower quota qi. Simply truncate (chop off) the decimal portion.
Start with the state with the largest fraction (largest decimal portion), and round up this state’s quota to its upper quota qi. Keep assigning the states their upper quotas in order of largest fractions until the total seats in the house is reached.
Originally, he wrote the congressional apportionment bill that was vetoed by President Washington in 1792, but later adopted from 1850–1900. 8

9. Chapter 14: Apportionment The Hamilton Method
Steps for Hamilton Method:
First add up the total population of each state, p = ∑pi .
Calculate the standard divisor, s = p /h (total population ÷ house).
Calculate each state’s (or class) quota, qi = pi /s.
Tentatively assign to each state its lower quota qi.
Give the upper quota qi to the states whose quota has the largest fractional (decimal) portion until the house is filled.
Apportion 5 Class Sections:
Population ∑pi = 100.
Standard divisor, s s = p/h =100/5 = 20.
Find quotas qi = pi /s
List lower quotas qi .
Give upper quotas qi to Pre-Calc and Calc.
Teach: 2 Geom, 2 Pre-Calc, 1 Calc.
High School Math Teacher Apportions 5 Class Sections
Course
Population pi
Quota qi
Lower Quota
qi
Apportion-ment, ai
Geometry 52
52/20 = 2.60 2
Pre-Calc 33
33/20 = 1.65 1
Calculus 15
15/20 = 0.75
 1
p = 100
5.00 3 5 9

10. Chapter 14: Apportionment The Hamilton Method
Determine the apportionments for the previous example: The field hockey team
Results
Enrolment
Quota
Lower Quota
Quota Fractional Part
Apportionment
with h = 100%
Games won 18
78.26%
78%
0.26%
Games lost 4
17.39%
17%
0.39%
18%
Games tied 1
4.35% 4%
0.35% 23
100%
99%
House size h
Standard divisor s 10

11. Chapter 14: Apportionment The Hamilton Method
Alabama Paradox
A paradox is a fact that seems obviously false.
Discovered in 1881, a study was done on several apportionments for different house sizes, and the following was discovered:
With a 299-seat house, three state quotas and apportionments were:
Alabama q =  8, Illinois q =  18, Texas q =  9
With a 300-seat house, Alabama lost a seat, but the house added a seat: Alabama q =  7, Illinois q =  19, Texas q =  10
Alabama Paradox – Occurs when a state loses a seat as the result of an increase in the house size.
The Population Paradox – Occurs when one state’s population increases and its apportionment decreases, while simultaneously another state’s population increases proportionally less, or decreases, and its apportionment increases. 11

12. Chapter 14: Apportionment The Alabama Paradox
A mathematics department has 30 teaching assistants to cover 5 subjects, Algebra, Calculus I, Calculus II, Calculus III and Contemporary mathematics.
The house size is 30.
The total population is 750.
The standard devisor is 750/30 = 25. 12

13. Chapter 14: Apportionment The Alabama Paradox
Calculus III loses 1 section. 13

14. Chapter 14: Apportionment The Population Paradox
A country has four states, A, B, C, and D. Its house of representatives has 100 members.
State
Old Census
New Census A
5,525,381
5,657,564 B
3,470,152
3,507,464 C
3,864,226
3,885,693 D
201,203
201,049
13,060,962
13,251,770 14

15. Chapter 14: Apportionment The Population Paradox
State
Old Census
Quota
Lower Quota
Quota Fractional Part
Apportionment with h =100 A
5,525,381
42.30 42
0.305 B
3,470,152
26.57 26
0.569 27 C
3,864,226
29.59 29
0.586 30 D
201,203
1.54 1
0.540
13,060,962
100 98
House Size h
Standard Divisor
130,609.62 15

16. Chapter 14: Apportionment The Population Paradox
State
New Census
Quota
Lower Quota
Quota Fractional Part
Apportionment with h =100 A
5,657,564
42.69 42
0.693 43 B
3,507,464
26.47 26
0.468 C
3,885,693
29.32 29
0.322 D
201,049
1.52 1
0.517 2
13,251,770
100 98
House Size h
Standard Divisor
132,517.70 16

17. Chapter 14: Apportionment Divisor Methods
A divisor method of apportionment determines each state’s apportionment by dividing its population by a common divisor d and rounding the resulting quotient.
Divisor methods differ in the rule used to round the quotient.
Divisor Methods:
The Jefferson Method
The Webster Method
The Hill-Huntington Method
Thomas Jefferson
Edward V. Huntington
Daniel Webster 17

18. Chapter 14: Apportionment The Jefferson Divisor Method
The Jefferson Method
Proposed by Thomas Jefferson to replace the Hamilton Method.
Instead of using the standard divisor, s, average district population, the focus is on the district with the smallest population, d.
Steps for the Jefferson Method:
First add up the total population, p, of all the states, ∑pi .
Calculate the standard divisor, s = p /h (total population ÷ house) .
Calculate each state’s (or class) quota, qi = pi /s.
The tentative apportionment, ni, is the lower quota ni = pi / s = qi . This apportionment will usually not be enough to fill the house.
Calculate the critical divisor for each state, di = pi /(ni + 1). The state with the largest critical divisor will get the next seat.
Re-compute the apportionments ai = pi / d using this largest critical divisor, d.
List the lower apportionments for ai.
Continue this process until the house is filled. 18

19. Chapter 14: Apportionment The Jefferson Divisor Method
The High School Mathematics Teacher
Course
Enrolment
Tentative Apportionment
Critical Divisor d
Apportionment
Geometry 52 2
17.333 3
Pre-calculus 33 1
16.500
Calculus 15
15.000
100 5
House size h
Standard divisor s 20
di = pi /(ni + 1). 19

20. Chapter 14: Apportionment Divisor Methods
The Webster Method
A divisor method invented by Representative Daniel Webster.
The Webster method is the divisor method that rounds the quota (adjusted if necessary) to the nearest whole number.
Round up when the fractional part is greater than or equal to ½ .
Round down when the fractional part is less than ½ .
Based on rounding fraction the usual way, so that the apportionment for state i is  qi , where qi is the adjusted quota for state i.
The Webster method minimizes differences of representative shares between states (it does not favor the large states, as does Jefferson method). 20

21. Chapter 14: Apportionment The Webster Divisor Method
Steps for the Webster Method:
First add up the total population, p, of all the states ∑pi .
Calculate the standard divisor, s = p /h (total population ÷ house).
Calculate each state’s (or class) quota, qi = pi /s.
We will call the tentative apportionment ni =  qi  When the fractional part is greater than or equal to ½, round up; when the fraction part is less than ½, round down (the usual way of rounding).
If the sum of the tentative apportionments is enough to fill the house, we are finished, and the tentative become the actual apportionments.
If not…, we adjust the tentative apportionments.
Re-compute the apportionments ai = pi / d using this critical divisor, d.
Continue this process until the house is filled.
Adjusting the divisor:
If the tentative apportionments do not fill the house, find which state has the largest critical divisor where di + = pi /(ni + ½) will receive a seat.
If the tentative apportionments overfill the house, find which state has the smallest critical divisor where di− = pi /(ni − ½) will lose a seat. 21

22. Chapter 14: Apportionment The Webster Divisor Method
The High School Mathematics Teacher
Course
Enrolment
Tentative Apportionment
d- 
Apportionment
Geometry 52 3
20.8 2
Pre-calculus 33
22.0
Calculus 15 1
30.0
100 6 5
House size h
Standard divisor s 20 22

23. Chapter 14: Apportionment Divisor Methods
The Hill-Huntington Method
The Hill-Huntington method is a divisor method related to the geometric mean that has been used to apportion the U.S. House of Representatives since 1940.
Steps for the Hill-Huntington Method:
First, add up the total population, p, of all the states, ∑pi .
Calculate the standard divisor, s = p /h (total population ÷ house).
Next, calculate each state’s (or class’) quota, qi = pi / s.
Find the geometric mean q* =  qi qi .
Round each state’s quota the Hill-Huntington way by comparing the quota to the geometric mean to obtain a first tentative apportionment.
If the sum of the tentative apportionments is equal to the house size, the job is finished.
If not, a list of critical divisors must be constructed.
The Hill-Huntington rounding of q is equal to:
qi if q < q*
qi if q > q*
where geometric mean q* =  qi qi 23

24. Chapter 14: Apportionment The Hill-Huntington Divisor Method
Steps for the Hill-Huntington Method (continued):
Step 7. (continued) Calculating the critical divisors, if needed …
If the tentative apportionments do not fill the house, then the critical divisor for state i with population pi and tentative apportionment ni for each state is: pi
di + = ni (ni + 1)
The state with the largest critical divisor is first in line to receive a seat.
If the tentative apportionments overfill the house, then the critical divisor for state i with population pi and tentative apportionment ni for each state is:
di − = ni (ni − 1)
The state with the smallest critical divisor is first in line to lose a seat. 24

25. Chapter 14: Apportionment The Hill-Huntington Divisor Method
The High School Mathematics Teacher
Course
Enrollment
Quota (q) q*
Tentative Apportionment
Final Apportionment
Geometry 52
2.60 3
21.229 2
Pre-calculus 33
1.65
23.335
Calculus 15
0.75 1
Total
100 5 6
House size h
Standard divisor s 20 25

26. Chapter 14: Apportionment Which Divisor Method Is Best?
Which Divisor Method is Best? (3 comparisons)
To find which of the three divisor methods is the fairest, compare the apportionments produced by the methods.
Remember, pi is the state’s population and ai is its apportionment.
One Measure of Fairness: Representative Share, ai /pi
The quotient of ai ÷ pi (or ai /pi ) is called the representative share. (Represents the share of a congressional seat given to each citizen of the state.)
In an ideal apportionment, every state would have the same representative share. Instead, we look for the least discrepancy.
The Webster method is the fairest when comparing representative shares.
Another Measure of Fairness: District Population, pi /ai
The quotient of pi ÷ ai (or pi /ai ) is the average population of a congressional district in the state.
This method surprisingly does not agree with Webster being the fairest.
Even though for state i: ai /pi = 1 ÷ pi /ai
representative share = 1 / district population
(The first two measures of fairness are reciprocal of each other but do not give the same results.) 26

27. Chapter 14: Apportionment Which Divisor Method Is Best?
Which Divisor Method is Best? (continued)
One Measure of Fairness: Representative Share ai /pi
Another Measure of Fairness: District Population pi /ai
Absolute and Relative Differences: (Last method of fairness)
Given two positive numbers A and B, with A > B, the absolute difference is A − B and the relative difference is the quotient: (A − B)/B × 100%.
For any two states, the relative difference in district population is equal to the relative difference in representative share.
Therefore, an apportionment method that minimizes relative difference in representative shares will also minimize the relative difference in district population.
The Hill-Huntington method gives the apportionment in which the relative differences in representative shares (or district populations) are as small as possible. 27