Section 4.1 Apportionment Problem
“Representatives and direct taxes shall be apportioned among the several states, which may be included within this union, according to their respective numbers, which shall be determined by adding to the whole number of free persons…”
APPORTIONMENT Problem:
EXAMPLE #1: Mrs. Phillips has 50 identical pieces of chocolate which she is planning to divide among her 5 children (division part). She wants to do this fairly.
A. What would one fair way to divide the chocolate be (Chapter 3)?
B. To teach her children a lesson about hard work she decides to give her children candy based on how much time they work doing chores in the next week (proportionality criterion).
Minutes worked / 150 / 78 / 173 / 204 / 295
Chocolate Pieces
C. How many minutes of chores would a child need to work to earn one piece of candy?
EXAMPLE #2: Dr. Williams has 60 blank DVDs to give to 4 students in a class.
A. What would one fair way to divide the DVDs be (Chapter 3)?
B. Dr. Williams thinks the DVDs should be given to the students using their last test scores.
TEST SCORE / 85 / 88 / 92 / 95
DVDs Received
For how many points earned on a test would a student earn one DVD?
Based on examples: Are there any challenges to share items proportionally to all individuals?
· THREE ELEMENTS of Apportionment Problem:
“STATES”: players involved in apportionment
NOTATION for N States: A1, A2, …, AN
“SEATS”: set of M identical, indivisible objects to be divided
“POPULATIONS”: set of N positive numbers which are the basis for the apportionment of seats to state (proportionality criterion)
NOTATION for N Populations: p1, p2, …, pN
For #1 – 4: IDENTIFY who or what are the seats, populations, and states.
Exp #1: Mr. Gates plan to split $3500 allowances between his 4 children at the end of each quarter. The children will receive their allowances in proportion to their GPA.
Exp #2: There are 40 teachers in an elementary school and the principal plans to apportion teachers to the 5 grade levels (1st through 5th) in the school based on the current enrollment of each grade level.
Exp#3: There are 75 administrative assistants for the entire college which has 10 academic departments. Each department will receive assistants based on the number of students that have declared that department as a major.
Exp #4: The city is planning to reorganize their 5 major bus routes around the city. The average number of passengers on each route will determine how the city’s 36 buses will be apportioned.
Ratios (fractions) are the important measurements for apportionment
o STANDARD DIVISOR, SD: ratio of total population to seats
Standard divisors represents the ______(population unit) per ______
o STANDARD QUOTAS, q: ratio of the state population, p, to the standard divisor
Standard Quota represents the ______number (including decimal) of seats that each state should get (if ONE seat could be divided into smaller parts)
EXAMPLE #3: Consider a nation of 6 states with only 250 seats in their congress with different populations.
State / A / B / C / D / E / F / TotalPopulation / 1,646,000 / 6,936,000 / 154,000 / 2,091,000 / 685,000 / 988,000 / 12,500,000
Standard Divisor
Standard Quota
Normal Rounding
(0.5 Rule)
SPECIAL ROUNDING OF QUOTAS (0.5 rule doesn’t apply)
UPPER Quota, ↑: rounding up the standard quota to nearest integer
a. 45.6↑ = b. 9.2↑ = c. 17.5↑ = d. 108↑ =
LOWER Quota, ↓: rounding down the standard quota to nearest integer
a. 35.6↓ = b. 99.2↓ = c. 16.5↓ = d. 108↓ =
EXAMPLE #3b: find the upper and lower quotas for the 6 nations from Example 3.
State / A / B / C / D / E / F / TotalUpper Quota
Lower Quota
Will the lower and upper quotas be equal?
EXAMPLE #4: The school board wants to assign 30 new teaching assistants among 5 elementary schools based on the current number of students in the schools. Complete the calculations for Standard Divisor and Standard Quota for the table given.
North / South / East / West / Central# of Students / 375 / 297 / 408 / 340 / 380
EXAMPLE #5: For 4 players and 200 seats. Complete the table by finding the standard divisor, standard quotas and appropriately round the standard quotas.
State / A / B / C / D / TotalPopulation / 125 / 150 / 350 / 275 / 900
Standard Divisor
Standard Quota
Normal
Rounding
Upper Quotas
Lower Quotas
“Good” Apportionments: Produce a valid (exactly M seats given) and a “fair” apportionment. HOMEWORK: p.150 # 2, 3, 5
Section 4.2: Hamilton’s Method and Quota Rule
· Every state will get ______its lower quota.
Example #1 Hamilton’s Method: 6 nations with 250 seats for congress. (Calculations from 4.1 notes)
State / A / B / C / D / E / F / TotalPopulation / 1,646,000 / 6,936,000 / 154,000 / 2,091,000 / 685,000 / 988,000 / 12,500,000
Standard Divisor / 12,500,000 ÷ 250 = 50,000 people per 1 seat in congress
All methods will begin with standard divisor calculation
1) Calculate each state’s STANDARD QUOTAS, q
Step #1:Standard Quota / 32.92 / 138.72 / 3.08 / 41.82 / 13.70 / 19.76 / Total = 250
2) Give each state its LOWER quota.
Step #2:Lower Quota / Total:
3) SURPLUS: Find the decimal for each state’s standard quota. Give one seat to each state from the largest to smallest decimal until out of surplus seats.
Step #3:Standard Quota Decimal / Total:
Final Apportionment for HAMILTON: Assign each state’s lower quota of seats and any surplus seat.
Seats Given Away
/ Total:EXAMPLE 2: Mrs. Phillips has 50 pieces of chocolate to divide among her 5 children. (4.1 Notes)
Child #1 / Child #2 / Child #3 / Child #4 / Child #5 / TotalMinutes worked / 150 / 78 / 173 / 204 / 295 / 900
S1: Standard Quota / 8.333 / 4.333 / 9.611 / 11.333 / 16.388 / 50
S2: Lower Quota
S3: Standard Quota Decimal
Apportionment
EXAMPLE 3: Perform Hamilton’s Method for 4 players and 150 seats
State A / State B / State C / State D / TotalPopulations / 300 / 450 / 800 / 650
Standard Divisor
S1: Standard Quota
S2: Lower Quota
S3: Standard Quota Decimal
Apportionment
EXAMPLE #4: The school board wants to assign 40 new teaching assistants among 5 elementary schools based on the current number of students in the schools. Complete performing all steps of Hamilton’s Method to determine the apportionment of all teaching assistants.
North / South / East / West / Central# of Students / 274 / 372 / 331 / 304 / 259
FAIRNESS CRITERION – “QUOTA RULE”: a state shouldn’t be apportioned a number of seats smaller than its lower quota or larger than its upper quota.
The Hamilton Method ______the quota rule
Lower-quota Violation: a state is apportioned number of seats ______THAN lower quota
Upper-quota Violation: a state is apportioned number of seats ______THAN upper quota
Example: Standard Quota = 14.32 à Lower Quota is 14 and Upper Quota is 15
Lower Quota Violation = Upper Quota Violation =
Section 4.3 Alabama and Other Paradoxes
Paradoxes or Illogical Outcomes can occur (decrease in a state’s seat assignment) when applying Hamilton’s Method once and then recalculating after a change is made to the basic elements of the apportionment problem.
· Basic Idea: A change to the seats, states, or population of the given states can directly cause a change in the standard divisor that then affects all future calculations in Hamilton’s Method.
EXAMPLE #1: 3 states, 20,000 total people and the following data for population per state:
State / A / B / C / TOTALPopulation / 940 / 9030 / 10,030 / 20,000
Suppose the country decides to use 200 representatives. Use Hamilton’s Method to apportion the seats.
Standard Divisor =
State
/ A / B / C / TOTALStep #1:
SQ
Step #2:
LQ
Step #3: Surplus
Apportionment
Suppose state C makes the request to have 201 seats instead. Where do you think the extra seat will go?
Part B: Use Hamilton’s Method to apportion the 201 seats.
New Standard Divisor =
State / A / B / C / TOTALStep #1
Step #2
Step #3
Apportionment
COMPARE the results of your apportionment answers to your second answer: Do these results seem fair?
· ALABAMA Paradox: an increase in the total number of seats being apportioned, in and of itself, forces a state to lose one of its seat.
EXAMPLE #2: We have the following data for a continent with 5 countries with a population of 900 (in millions). And there are 50 seats to be apportioned.
SD =
State / Alamos / Brandura / Canton / Dexter / Elexion / TotalPopulation / 150 / 78 / 173 / 204 / 295 / 900
Step #1
Step #2
Step #3
Apportionment
Part B: Suppose 10 years go by and the country recounts its population in the new chart:
State / Alamos / Brandura / Canton / Dexter / Elexion / TotalPopulation / 150 / 78 / 181 / 204 / 296 / 909
Which states had changes to their population? What do you expect will happen in the apportionment based on these changes?
Perform Hamilton’s Method on new populations: NEW STANDARD DIVISOR =
Step #1Step #2
Step #3
Apportionment
COMPARE the results of your apportionment answers to your second answer: Do these results seem fair?
POPULATION Paradox: A state could potentially lose seats because its population grew with no other changes .(State A loses a seat to state B even through the population of A grew at a higher rate than the population of state B.)
EXAMPLE #3: The school board decides to apportion 100 COUNSELORS to the two high schools in the county based on their current enrollment of students using HAMILTON’S METHOD.
State / North HS / South HS / TotalPopulation / 1045 / 8955 / 10,000
Step #1
Step #2Step #3
Apportionment
Suppose New High School was added to the school district with an enrollment of 525 students. The school board decides to hire 5 NEW COUNSELORS.
Why does hiring 5 new counselors make sense based on the Standard Divisor?
Part B: The school board decides repeats Hamilton’s Method with all 3 high schools and the additional 5 new counselors. Find the apportionment assuming North and South maintain the same number of students.
State / North HS / South HS / New HS / TotalPopulation / 1045 / 8955 / 525 / 10,525
Step #1: SD =
Step #2
Step #3Apportionment
COMPARE the results of your apportionment answers to your second answer: Do these results seem fair?
· NEW - STATES Paradox: the addition of a new state with its FAIR SHARE of seats an, in and of itself, affect the apportionment of other states.
o Fair share is based on original SD and new state’s population.
o Expect states to maintain same apportionment if fair share added
HOMEWORK: p.152 #12, 17, 19 - 22
BASIC REVIEW OF DIVISION AND FRACTION MATH:
For each comparison (Y > 0), Circle the Fraction that is LARGER
#1) v. #2) v.
#3) v. #4) v.
For each comparison (Y > 0), Circle the Fraction that is SMALLER
#1) v. #2) v.
#3) v. #4) v.
DIVISOR METHOD:
Manipulate (Modify) the divisor to create quotas for the apportionment.
GENERAL DIVISION RULE for modified divisor:
the population is constant or fixed for each state when creating new quotas
To create BIGGER quotas, you need to use a SMALLER divisor.
To create SMALLER quotas, you need to use BIGGER divisor
Examples of choosing possible modified divisor: Circle all that apply
1) Consider the expression, where x = 6, which new value of x will result in a LARGER answer.
(A) x = 5 (B) x = 6 (C) x = 7 (D) x = 8
2) Consider the expression, where x = 9, which new value of x will result in a SMALLER answer.
(A) x = 6 (B) x = 10 (C) x = 8 (D) x =12
3) Consider the expression, where y = 125, which new value of y will result in a SMALLER answer.
(A) y = 120 (B) y = 123 (C) y = 125.01 (D) y = 127
4) Consider the expression, where z = 0.35, which new value of z will result in a LARGER answer.
(A) z = 0.15 (B) z = 0.275 (C) z = 3.5 (D) z = 0.09
5) Consider the expression, where x = 123.5, which new value of x will result in a LARGER answer.
(A) x = 123.7 (B) x = 122.7 (C) x =124.5 (D) x = 123.49