Objectives: 1. Understand and illustrate the Alabama paradox.
2. Understand and illustrate the population paradox. 3. Understand
and illustrate the new-states paradox. 4. Understand Balinski and
Youngs Impossibility Theorem.
Slide 3
Textbook: Page 508 509/Understanding Apportionment Read through
to Table 9.1 In your own words, succinctly tell what this passage
is about.
Slide 4
Example 1: Identify the integer and the fractional part.
21.075
Slide 5
Example 2: Identify the integer and the fractional part.
0.567
Slide 6
Using the Hamilton Method The Hamilton method uses fractional
parts to apportion representatives. 1. Determine the exact number
of board members: i. percent of stockholders X size of board 2.
Assign the integer part i. If there are more members to be
allocated, then go to step 3. 3. Assign additional members
according to the fractional parts i. The 1 st additional member
goes to the company having the largest fractional part ii. The 2 nd
additional member, if any, goes to the company with the 2 nd
largest fractional part. i. Continue in this manner until you have
assigned all additional members.
Slide 7
Example 3: Using the Hamilton Apportionment Method 12 Member
Board CompanyPopulation Determine Percentage Step 1: Board Members
Deserved Step 2: Assign Integer Part Naxxon47 Aroco37 Eurobile16
Total100
Slide 8
Example 4: TB pg. 517/4 Jungle World Theme Park
EmployeesPopulation Percent of Employees Step 1: Members Deserved
Step 2: Assign Integer Part Performers Food Wrkrs Maintenance Wrkrs
Total
Slide 9
Section 9.1 Assignment TB pg. 516/1 7 odd Remember to write
problems and show ALL work.
Slide 10
Section 9.1 Part II Alabama Paradox and Truncating the
Fractional Part of a Number
Slide 11
Key Terms: Alabama Paradox an increase in the total number of
items to be apportioned results in the loss of an item for a group.
Apportion to divide according to a plan; to allot. Truncate to
shorten by cutting off. Note: sometimes it is necessary to truncate
a number to keep the percentage from adding up to more than
100%.
Slide 12
Alabama Paradox: Illustrating the Alabama Paradox: A small
country with a population of 10,000 is composed of 3 states.
According to the countrys constitution, the congress will have 200
seats, divided among the 3 states according to their respective
populations. Illustrating the Alabama Paradox. State Populatio n %
of Pop. Step 1:Step 2:Step 3: A5015 B4515 C470 Total10,000
Slide 13
Example 5: Using Hamiltons Method determine deserved seats.
Illustrating the Alabama Paradox. StatePopulation % of Pop. Step
1:Step 2:Step 3: A5015 B4515 C470 Total10,000
Slide 14
Example 6: What happens if the number of seats in congress
increases to 201. Illustrating the Alabama Paradox. StatePopulation
% of Pop. Step 1:Step 2:Step 3: A5015 B4515 C470 Total10,000
Slide 15
Section 9.1 Assignment Part 1 TB pg. 517/11 and 12 (worksheet
online) Remember to write problems and show ALL work.
Slide 16
Example 7: Alabama Paradox Using Hamiltons Method Assume that
there are now 10 members on the board. Oil Consortium Board Company
% Stockholders (in thousands) Step 1:Step 2:Step 3: Naxxon47
Aroco37 Eurobile16 Total100
Slide 17
Example 8: 183.6574893 Truncate the number to: a. Hundredths b.
Ten thousandths
Slide 18
Example 9: 284.135792753 Truncate the number to: a. Tenths b.
Thousandths
Slide 19
Example 10: Presenting Survey Results Adjusting a list of
numbers. Exact Percentage (column A) Percentage rounded to tenths
(column B) Original Date Truncated to Tenths Part That is Discarded
Final Results Taxes34.4235 Educati on 13.456 Crime14.75 Health Care
37.3705 Total100
Slide 20
Example 11: Adjusting a list of numbers A group of consumers
was asked how they expected their spending to change in the next
six months. Adjust the percentages in the following table so that
they are shown to the tenths place and their sum is 100.00% Exact %
% Rounded to tenths Original Data Truncated Discarded Part Final
Results Spending Increase 32.029 Spending Decrease 24.733 Spending
Stay the Same 22.7489 Unsure20.4891 Total100.00%
Slide 21
Historical Highlight TB pg. 513/Apportionment U. S.
History
Slide 22
Section 9.1 Assignment Part 2 Class work: TB pg. 517/13 22
Remember you must write problems and show ALL work to receive
credit for this assignment.
Slide 23
Section 9.1 Part III Average Constituency, Absolute Unfairness,
and Relative Unfairness
Slide 24
Key Term: Average Constituency the quotient: population of
state number of representatives from state NOTE: Comparing the
representatives of two states A and B, we saw that state A is more
poorly represented than state B, if the average constituency of A
is larger than the average constituency of B.
Slide 25
Example 12: Average Constituency Finding the average
constituency a. If the 420-member electricians union has three
representatives on the United Labor Council, what is the average
constituency of this group?
Slide 26
Example 13: Average Constituency Determine which group is more
poorly represented. a. If the 420-member electricians union has
three representatives on the United Labor Council, what is the
average constituency of this group? b. If the 440-member plumbers
union has four representatives on the council, are the electricians
or the plumbers more poorly represented?
Slide 27
Key Term: Absolute Unfairness (of a state) the difference
between the larger average constituency and the smaller one. If
State A has the larger average constituency, then the absolute
fairness is: (avg. constituency of A) (avg. constituency of B)
NOTE: if the two states have the same average constituency, then we
say that the two states are equally well represented.
Slide 28
Example 14: Find Absolute Unfairness Assume that state X has a
population of 974,116 with four representatives and state Y has a
population of 730,779 with three representatives. Compute the
absolute unfairness for this apportionment.
Slide 29
Example 15: Find Absolute Unfairness Suppose the Weavers Guild,
with 1,672 members, has six delegates on the National Art
Commission and the Artists Alliance, with 1,535 members, has five
delegates. Calculate the absolute unfairness for this assignment of
delegates.
Slide 30
Key Term: Relative Unfairness (for 2 states) the quotient: the
absolute unfairness of the apportionment the smaller average
constituency of the two states
Slide 31
Example 16: Determine Relative Unfairness If state A has a
population of 11,710 and five representatives and state B has a
population of 16,457 and seven representatives, calculate the
relative unfairness of the apportionment.
Slide 32
Example 17: Determine Relative Unfairness Suppose state A has a
population of 935,000 and five representatives, whereas state B has
a population of 2,343,000 and eleven representatives. Determine the
relative unfairness of this apportionment.
Slide 33
Section 9.1 Assignment Part 3 TB pg. 518/23 - 32 Remember to
write problems and show ALL work.