1. Chapter 14: Apportionment Lesson Plan
For All Practical Purposes
The Apportionment Problem
The Hamilton Method
Divisor Methods
The Jefferson Method
The Webster Method
The Hill-Huntington Method
Which Divisor Method Is Best?
Mathematical Literacy in Today’s World, 7th ed. 1
© 2006, W.H. Freeman and Company

2. 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% 2

3. Chapter 14: Apportionment The Apportionment Problem
Example: Apportionment Problem
In the U.S. Constitution, the seats of the House of Representatives shall be apportioned according to the state’s population.
An apportionment problem occurred in 1790:
The House of Representatives was to have 105 members.
The country’s total population of the 15 states was 3,615,920.
The average congressional district should have a population of 3,615,920 ÷ 105 = 34,437 (standard divisor).
To find the fair share of House seats, we can divide each state’s population by this average, or standard divisor.
Virginia = 630,560/34,437 = and was only given 18 seats.
President Washington came from Virginia. Furthermore, Virginia’s apportionment was not rounded up to 19, and he may have been biased. Hence, he did not agree this method was fair to all states.
When Congress passed the apportionment proposed bill and sent it to President Washington, he vetoed it! 3

4. 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 = 4

5. 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. 5

6. 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 6

7. Chapter 14: Apportionment The Hamilton Method
Read Example 6 on p. 512
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. 7

8. 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 8

9. Chapter 14: Apportionment Divisor Methods
Use the Jefferson method for the mathematics teacher example
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. 9

10. 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). 10

11. Chapter 14: Apportionment Divisor Methods
Use the Webster method for the mathematics teacher example
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. 11

12. Chapter 14: Apportionment Divisor Methods
Read Spotlight 14.2 on p. 530
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 12

13. Chapter 14: Apportionment Divisor Methods
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 receive a seat. 13

14. Chapter 14: Apportionment Divisor Methods
An apportionment method is said to satisfy the quota condition if in every situation each state’s apportionment is equal to either its lower or its upper quota.
The Hamilton method satisfies the quota condition since each state starts with it lower quota, and some states are given their upper quotas to fill the house.
The Jefferson method does not satisfy the quota condition. The 1820 apportionment is an example.
Webster's and the Hill-Huntington method violate the quota criterion much less frequently than Jefferson's method.
The Jefferson method, however, is not troubled by the Alabama and population paradoxes.
The Balinsky and Young Theorem (1982) states that no apportionment system is free from the paradoxes troubling Hamilton's method and always satisfies the quota criterion. In other words, it is impossible to construct an apportionment system that is always fair.