# Chapter 9 - Apportionment

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```Agenda
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Goals
Info sheet
Overview
Apportionment Activity
Begin Lesson 9.1- Apportionment
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Hamilton Method
Adjusting a list
HW Guide
Discrete vs. Continuous Data (Variables)
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Discrete Math is not easily defined. NCTM
states DM is “the study of mathematical
properties of sets and systems that have only a
finite number of elements.”
Discrete – values that can be counted
Continuous – all values in an interval between
two specific values, such as temperature.
Statisticians gain information about situations by
collecting data for random variables.
Data
Qualitative:
categorical such as gender,
religious preference,
geographic locations
Quantitative:
Numerical and can be ordered or ranked,
such as age, heights, weights,
body temperature
Discrete: (countable) such as
Continuous: can assume
Number of children in a family,
students in a class,
people in a state, or emails in a day
all values in an interval,
Temperature, time,
distance, mass
Mathematics All Around.
Set and
Number
Theory
Ch 1, 4 & 5
Graph Theory
Ch 3
Thomas L. Pirnot
Apportionment
Ch 9
Voting
Methods
Ch 10
Discrete Math
Modeling
Ch 6 & 7
Descriptive
Statistics
Ch 14
Consumer Math
Ch 11
Counting,
Permutations,
Combinations
Ch 12
Portfolio Rubric
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Vocabulary Page
Homework – Assignment # and date in top right
corner. Due in pencil on due date; corrected in pen
for final portfolio.
News article / typed response
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Why did you choose the article?
How does it relate to the unit of study
What did you learn?
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Extra-credit topic (extension - not discussed in
class)
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Portfolios should represent your own work and
understanding; these are individualized for each learner.
This allows us (student, teacher, parent) to see your
strengths/weaknesses for assessment purposes. Late
portfolios will be evaluated for half-credit.
Chapter 9 - Apportionment
• Vocabulary
• Apportionment
Methods
– Hamilton
– Huntington-Hill
– Jefferson
– Adams
– Webster
Hamilton Method
of Apportionment
•Gives each state integer part
•Additional reps allocated by
largest fractional part of exact
representation
Example 1- In a recent census…
1) Alabama – Population 4041
2) Mississiippi – Population 2573
3) Louisiana – Population 4220
Allocate 19 members of the U.S. House of
Representatives to these three states.
• 1st find the percentage for each state’s
population to the total population
State
AL
Miss
LA
Total
% of Delegates
Delegates
deserved
Integer
Part
Fractional Assign
Part
additional
Members
Your Turn
2) The Civics Arts Guild is having a show.
There is room for 31 booths and the guild
has decided that the booths will be
assigned in proportion to the type of
members in the guild. The guild has 87
painters, 46 sculptors, and 53 weavers.
Assign the booths to the three groups.
Group
Painters
Sculptors
Weavers
Total
% of Booths
Booths
deserved
Integer Fractional Assign
Part
Part
additional
Booths
Adjusting a List by Truncating
• Rank your biggest concerns about school.
(1 representing your top concern)
• Choose from the following:
– Peer Pressure
– My friends/social life
– My grades
– My safety
State
Peer
Pressure
My
Friends/
Social Life
My Grades
Safety
Total
# ranked Original Truncated
1st
%
to tenths
Discarded
Portion
Adjusted
List
Your Turn
• #17 on page 518.
Warm-Up
1. p.517 # 11
2. p.518 #17
Fairness of Apportionment
• Average constituency =
Population of state/ number of reps from state
• Absolute unfairness=
l(avg constituency of A) – (avg constituency of B)l
• Relative Unfairness=
absolute unfairness / smaller average constituency
Example
• Find the average constituency to
determine which state is more poorly
represented. State A has a population of
27,600 and 16 representatives and state B
has a population of 23,100 and 14
representatives.
• What is the absolute unfairness of this
apportionment?
• What is the relative unfairness?
27, 600
 1, 725
• Average Constituency of A =
16
23,100
 1, 650
• Average Constituency of B =
14
State A is more poorly represented.
• The absolute unfairness is
1,725 – 1,650 = 75
75
 0.045
• The relative unfairness =
1, 650
Your turn
• # 26 on page 519
Warm-Up
• Do # 29 on p. 519.
Lesson 9.2
The Huntington- Hill Apportionment Principle
– The student will use the Apportionment
Criterion and the Huntington-Hill
Apportionment Principle to determine the
addition of representatives most fairly.
Apportionment Criterion
• When assigning a representative among
several parties, make the assignment so
as to give the smallest relative unfairness.
• Example: Suppose that State A has a
population of 13,680 and 5
representatives, and State B has a
population of 6,180 and 2 representatives.
Use the apportionment criterion to
determine which state is more deserving
of one additional representative.
• If State A gets the rep then:
A) Calculate the average constituencies for
each.
B) Calculate the relative unfairness for
each.
•
If State B gets the rep then:
– Repeat steps A and B
•
The smaller relative unfairness is the
best
Your Turn
• #1 on p.524
Huntington-Hill Apportionment
• Gives new rep preference by using the
Huntington-Hill number.
• H-Hill number (based on Geometric mean)
2
(population of Y )
(population of X )
?
y( y  1)
x( x  1)
2
• The larger number indicates which state should
receive the additional representative.
Example
• Iowa has a population of 2.8 million people and
6 representatives and Nebraska has a
population of 1.6 million people and 3
representatives. Use the Huntington-Hill
Principle to determine which state is more
deserving of an additional rep.
Iowa
(2.8)2 7.84
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 0.186667
67
42
Nebraska
2
(1.6)
2.56
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 0.213333
3 4
12
Nebraska’s H-Hill number is larger so Nebraska
deserves the additional rep.
Your Turn
• The Oil Consortium Board currently has 2
members from Naxxon, 2 from Aroco, and
1 from Eurobile. Use the Hamilton-Hill
Apportionment Principle to decide which
company should receive the next member
on the board if Naxxon has 4,700
stockholders, Aroco has 3,700 and
Eurobile has 1,600.
• Naxxon should get the next rep.
Warm-Up
• Use the Huntington-Hill apportionment
principle to decide which state is more
deserving of an additional rep.
– The population of Alaska, New Hampshire,
and Wyoming are .6 million, 1.1 million and .5
million, respectively. Alaska has 1 rep, New
Hampshire has 2, and Wyoming has 1.
Lesson 9.3
• Applications of the Apportionment
Principle.
– The student will use the Huntington-Hill
principle to apportion representatives or other
objects in order.
• Now we are going to determine how to
apportion a given number of
representatives and in what order they
should be distributed.
• Use the H-Hill method to apportion 10 reps
among Utah, Idaho, and Oregon.The
recent populations are Utah, 1.7 mil;
Idaho, 1.0 mil; and Oregon, 2.8 mil. Begin
by giving one rep to each state. List the
order in which the reps are apportioned.
Calculate the H-Hill #’s
Current Representation Utah
1
2
3
4
5
6
7
Idaho
Oregon
Seat #
1
2
3
4
5
6
7
Goes to
# of add’l reps
Utah
# of add’l
reps Idaho
# of add’l reps
Oregon
Your Turn
• #9 p.533
Warm-Up
We have 15 seats to apportion between the
following groups:
Freshmen with 420 students, Sophomores
with 375 students, Juniors with 350
students and Seniors with 293 students.
A) Use the Hamilton Method of
apportionment to distribute the seats.
B) Determine the relative unfairness
between the freshmen and juniors.
C) Which group is more poorly represented?
Group
% of Seats
Seats
Integer
deserved Part
Freshmen
29.2%
4.38
4
0.38
4
Sophomores
26.1%
3.915
3
0.915
4
Juniors
24.3%
3.645
3
0.645
4
Seniors
20.4%
3.06
3
0.06
3
Total
100%
15
13
Freshmen: AvgC= 105
Fractional
Part
Assign
additional
Seats
15
Juniors: AvgC = 87.5
The relative unfairness is 0.2.
Which group is more poorly
represented?
Average Constituencies:
• Freshmen – 105
• Sophomores – 93.75
• Juniors – 87.5
• Seniors – 97.7
The freshmen are more poorly represented.
Lesson 9.4 Other Paradoxes &
Apportionment Methods
• The student will discuss other paradoxes
and investigate and use other methods of
apportionment to assign additional
representatives/objects.
Standard Divisor / Quota
• Standard Divisor =
Total population / number of reps being allocated
Computed once (a single number) used for the entire
apportionment process. The standard divisor is the # of
constituents that each representative must represent.
Standard Quota =
State’s population / standard divisor
Must be computed individually for each state/client. The
standard quota is the # of representatives that a state
deserves.
Calculate the standard divisor and
each state’s standard quota.
•
•
•
•
We want to apportion 8 representatives among:
State A with a population of 3 million.
State B with a population of 4 million.
State C with a population of 5 million.
Total Pop 12, 000, 000
Standard divisor 
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 1,500, 000
# of Reps
8
Standard Quotas:
5,000,000
4,000,000
3, 000, 000
 3.33
 2.67 C 
A
2 B
1,500,000
1,500,000
1,500, 000
•
•
•
•
Using Hamilton’s Method:
State A would receive 2 reps.
State B would receive 3 reps.
State C would receive 3 reps.
• Remember: Hamilton’s method uses the
fractional parts to award additional reps.
• Using the Hamilton method each state
receives either the number immediately
above or below the standard quota.
Definitions
• Lower quota – the standard quota
rounded down.
• Upper quota – the standard quota
rounded up.
• If in making an apportionment, each state
is allocated a # of reps that is between its
lower and upper quota, then we say the
apportionment satisfies the quota rule.
Restating Hamilton’s
Apportionment Method
• Compute the standard divisor.
• Compute the standard quota for each
state and round down to the lower quota.
Assign reps.
• Assign additional reps according to the
fractional parts of the standard quotas.
Other paradoxes
• Population paradox – occurs when state A’s
population is growing faster than state B’s
population, yet A loses a representative to state
B.
• New-States paradox – occurs when a new
state is added, and its share of seats is added to
the legislature causing a change in the allocation
of seats previously given to another state.
Jefferson’s Apportionment Method
• Compute the standard divisor.
• Rather than computing the standard quota,
compute a modified quota by using a divisor
that is smaller than the standard divisor for
the apportionment.
– Modified quota = state’s population /
modified divisor
• Calculate each state’s modified quota and
round DOWN. Assign representatives.
• Keep varying the modified divisor until the
sum of these assignments is equal to the
number being apportioned.
Ex) Southwest Water Authority
• Assume that California, Nevada, and
Arizona are cooperating to build a dam to
provide water to communities currently
lacking adequate water supplies. Seats
on the 11-member Southwest Water
Authority, which governs the project, are
assigned according to the number of
customers in each state who use the water
from the project. There are 56,000
customers in California, 52,000 in Arizona,
and 41,00 in Nevada.
• Compute the standard divisor. 13,545.45
13,000
• Modified Divisor = _________
Population
Standard
Quota
Modified
quota
Round
Modified
Quota
DOWN
___________
California
Arizona
Nevada
56,000
52,000
41,000
4.13
3.84
3.03
Adams’ Apportionment Method
• Compute the standard divisor.
• Rather than computing the standard quota,
compute a modified quota by using a divisor
that is larger than the standard divisor for the
apportionment.
– Modified quota = state’s population /
modified divisor
• Calculate each state’s modified quota and
round UP. Assign representatives.
• Keep varying the modified divisor until the
sum of these assignments is equal to the
number being apportioned.
• Compute the standard divisor. 13,545.45
14,000
• Modified Divisor = _________
Population
Standard
Quota
Modified
quota
Round
Modified
Quota
UP
___________
California
Arizona
Nevada
56,000
52,000
41,000
4.13
3.84
3.03
Webster’s Apportionment Method
• Compute the standard divisor.
• Rather than computing the standard quota,
compute a modified quota by using trial and
error to find a modified divisor.
• Calculate the modified quota for each state
and round it in the usual way.
• Assign that number of representatives to
each state.
• Keep varying the modified divisor until the
sum of these assignments is equal to the
total number being apportioned.
• Compute the standard divisor. 13,545.45
13,545.45
• Modified Divisor = _________
Population
Standard
Quota
Modified
quota
Round
Modified
Quota
___________
NORMALLY
California
Arizona
Nevada
56,000
52,000
41,000
4.13
3.84
3.03
How perfect are they?
Hamilton Jefferson Adams
Webster
Can have Alabama
paradox
Yes
No
No
No
Can have
population paradox
Yes
No
No
No
Can have newstates paradox
Yes
No
No
No
Can violate quota
rule
No
Yes
Yes
Yes
Revisiting the Oil Consortium
Creating a 9 member board
•
•
•
Naxxon = 4700
Aroco = 3700
Eurobile = 1600
1.
2.
3.
4.
5.
Apportion using Hamilton Method
Apportion using Huntington-Hill Method
Apportion using Jefferson’s Method
Apportion using Adams’ Method
Apportion using Webster’s Method
Textbook Problems
as a class
• Page 548: 39-42
• Page 549: 45
• Page 550: 52
• Partner: Finish Handout if needed.
• You may work on your review for 9/23.
• The test will be on Thursday, 9/25 instead of
9/29. The Extra Credit and portfolio will be due
on 9/29.
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