Chapter 4: Mathematics of ApportionmentESSENTIAL QUESTIONS:Section 4.1: What are the concepts and the goal of an apportionment problem?Section 4.2: How’s does Hamilton’s Method attempt to solve issue of standard divisors?Section 4.3: What are the 3 paradoxes that can occur in Hamilton’s Method?Section 4.4: How does Jefferson’s Method treat quotas for apportionment?Section 4.5: How does Adam’s Method treat quotas for apportionment?Section 4.6: How does Webster’s Method treat quotas for apportionment?

WORD WALL:ADAM’S METHODALABAMA PARADOXAPPORTIONMENTBALINSKI AND YOUNG’S IMPOSSIBILITY THEOREMHAMILTON’S METHODJEFFERSON’S METHODNEW-STATES PARADOXPOPULATION PARADOXQUOTA RULESTANDARD QUOTALOWER QUOTAUPPER QUOTAWEBSTER’S METHODSTANDARD DIVISORMODIFIED DIVISOR

Section 4.1 Apportionment ProblemAPPORTIONMENT 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.

What would one fair way to divide the chocolate be (Chapter 3)?

However, 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).

Child #1 Child #2 Child #3 Child #4 Child #5 TotalMinutes worked 150 78 173 204 295 900Chocolate Pieces

For how many minutes of chores worked would a child earn one piece of candy?

EXAMPLE #2: Dr. Williams has 60 blank DVDs to give to 4 students in a class. What would one fair way to divide the DVDs be (Chapter 3)?

Dr. Williams thinks that the DVDs should be given to the students when you also consider their score on the last test they all took.

Child #1 Child #2 Child #3 Child #4 TotalTEST SCORE 85 88 92 95 360DVDs Received

For how many points earned on a test would a student earn one DVD?

Based on examples 1 and 2 Are there any challenge to trying to share items proportionally to all individuals?

Elements of Apportionment Problem: o “STATES”: players involved in apportionment

NOTATION for N States: A1, A2, …, AN

o “SEATS”: set of M identical, indivisible objects to be divided

o “ POPULATIONS”: set of N positive numbers which are the basis for the apportionment of seats to state (proportionality criterion)

NOTATION for N Populations:

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 apportionmento 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 12,500,000Standard 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 QuotaLower 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

NormalRounding

Upper Quotas

Lower Quotas

“Good” Apportionments:

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 12,500,000Standard Divisor

12,500,000 ÷ 250 = 50,000 people per 1 seat in congressAll 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 19.76 250

2) Give each state its LOWER quota.Step #2: Lower Quota

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

Final Apportionment for HAMILTON: Assign each state it’s lower quota of seats and any surplus seat. State

***Hamilton’s Method is _________ Population Neutral, so it is biased in favor of larger than smaller states***

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 Total

Minutes 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 seatsState A State B State C State D Total

Populations 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

Section 4.3 Alabama and Other ParadoxesParadoxes or _______________________ Outcomes can occur (seat assignment for state’s) when applying Hamilton’s Method and then a change is made to the basic elements of the apportionment problem.

A change to the seats, states, or population of known states can directly cause a change in the standard divisor which will then affect the standard quotas (particularly the decimals) and possibly affect the surplus statement

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 20,000Suppose the country decides to use 200 representatives. Use Hamilton’s Method to apportion the seats. Standard Divisor =

State A B C TOTALStep #1: SQStep #2: LQStep #3: Surplus

Apportionment

Suppose state C makes the request to have 201 seats instead. Where do you think the extra seat will go? Why?

Part B: Use Hamilton’s Method to apportion the 201 seats. New Standard Divisor =

State A B C TOTALStep #1 9.45 90.75 100.80 201

Step #2 9 90 100 199

Step #3 .4 .75+1

.80+1

Apportionment 9 91 101 201

Does this 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 900Step #1

Step #2

Step #3

Apportionment

Part B: Suppose 10 years go by and the country recounts its population to get the following chart:State Alamos Brandura Canton Dexter Elexion TotalPopulation 150 78 181 204 296 909Which state’s had changes to their population? How will the new apportionment will turn out (still using only 50 seats)?

Perform Hamilton’s Method on new populations:

Step #1

Step #2

Step #3

Apportionment

Does this seem fair?

Population Paradox: A state could potentially lose seats because its population grew.

o 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: Suppose we have the following data for the population of two high schools. The school board needs to determine how many counselors to apportion to the two high schools. There are 100 counselors available. Use Hamilton’s Method to apportion the seats.State North HS South HS TotalPopulation 1045 8955 10,000Step #1 SD = 100

Step #2

Step #3

Apportionment

Suppose a new high school comes along and is added to the new district. The new high school, called New High School, has an enrollment of 525 students. The school board decides to hire 5 new counselors and assign them to the New High School. Does 5 make sense? Why or why not?

Part B: The school board decides to add the New High School and 5 new counselors. Find the apportionment using Hamilton’s Method for the 3 high schools assuming North and South maintain the same number of students.State North HS South HS New HS TotalPopulation 1045 8955 525 10,525Step #1: SD =

Step #2

Step #3

Apportionment

Does this 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

Section 4.4 Jefferson’s Method (Divisor Method) DIVISOR METHOD: manipulate the divisor (modified) in the apportionment.

GENERAL DIVISION RULE for modified divisor:

;

To create BIGGER quotas, you need to use a ____________________________ divisor.

To create SMALLER quotas, you need to use _____________________________ divisor

Examples: 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 x will result in a smaller

answer. (A) y = 120 (B) y = 123 (C) y = 125 (D) y = 127

4) Consider the expression, where z = 0.35, which new value of x will result in a larger answer.

(A) z = 0.15 (B) x = .275 (C) x = 3.5 (D) x = 0.09

Goal of Jefferson’s Method: make quotas bigger so that LOWER QUOTAS will be an exact apportionment.

JEFFERSON’S METHOD:1) Calculate Standard Divisor, SD

Standard quotas will work in Jefferson’s Method ONLY when they are INTEGERS

2) Find a “suitable” modified divisor, MD such that the sum of LOWER quotas is the number of seats. (exact apportionment with no surplus)

Finding MD: Originally guess should be reasonably < Standard Divisor Current Total LESS than exact number of seats, then a SMALLER modified divisor is needed Current Total GREATER than exact number of seats, then a LARGER modified divisor is needed

GUESSING DIVISORS: Use previous divisor statements to assist

SMALLER MD

T > M

Modified Divisor

1) Calculate the lower quotas with Modified Divisor.

2) Compare total (T) v. # of seats (M)

T < M

You’ve found a SUITABLE MD!!!

T = M

LARGER MD

1) If divisor of 100 created OVER apportionment and divisor of 105 created a BELOW apportionment. Next guess should be:

2) If divisor of 64 created OVER apportionment and divisor of 66.5 created an OVER apportionment. Next guess should be:

3) If divisor of 135 created BELOW apportionment and divisor of 117 created a BELOW apportionment. Next guess should be:

EXAMPLE #1: There are 15 scholarships to be apportioned among 231 English majors, 502 History majors, and 355 Psychology majors. Majors English History Psychology TotalPopulation 231 502 355 1088Standard DivisorStandard QuotasLower QuotasModified Divisor

Modified QuotasLower QuotasModified Divisor

Modified QuotasLower Quotas

EXAMPLE #2: Try 40 scholarships on different subjectsMajors Math Biology Political Science TotalPopulation 245 481 654 1380Standard DivisorStandard QuotasLower QuotasBE SELECTIVE TO GUESS: Take all populations and divide by the next whole number you want

Math Biology Political Science245 481 654

Modified Divisor

Modified QuotasLower Quotas

EXAMPLE #3: Previous example of the 6 nations with 250 seats for congress.State A B C D E F TotalPopulation 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000SD 12,500,000/ 250 = 50,000Q 32.92 138.72 3.08 41.82 13.70 19.76 250LQs

Should your next divisor be bigger or smaller than 50,000?

SHORT HAND NOTATION: Saves space and you round immediately after calculating quota1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000

Modified Divisor

A’s Lower Quota

B’s Lower Quota

C’s Lower Quota

D’s Lower Quota

E’s Lower Quota

F’s Lower Quota

TOTAL

What do you notice about the apportionment and the standard quotas?

Does there have to be exactly one MD for every apportionment problem?

HOMEWORK: p. 152 #24, 25, 30, 31

Section 4.4 Jefferson’s Method Part 2 Use the calculator to assist in the repetitive divisor of the state populations to find quotas to round

Calculator [Y =] method: You will need to use the correct quotas depending on the method and check the total of them to determine your next divisor guess .

1) and X = divisor

2) [2nd] [WINDOW: TBLSET] TblStart = Divisor you want (Next Guess)

3) [2nd] [GRAPH: TABLE] … shows all of the modified quotas for that divisor.

EXAMPLE #4: Banana Republic has states Apure, Barinas, Carabobo, and Dolores with populations given in millions and 160 seats in the legislature.State A B C D TotalPopulation 3.31 2.67 1.33 .69 8

Standard Divisor

Standard Quotas

Lower Quota

SHORT HAND NOTATION: You may need more than the ones provided Modified Divisor

Lower Quotaof State A

Lower Quotaof State B

Lower Quotaof State C

Lower Quotaof State D

TOTAL

EXAMPLE #5: Ms. Gambill has 150 colored pencils to give to 6 students in her class. She plans to give students the pencils based on their exam scores. Find the apportionment of Ms. Gambill’s 6 students under Jefferson’s method.

Zac Kate Eileen AJ Tommy Megan TotalExams 89 75 82 97 78 93 514SD

SQ

LQs

SHORT HAND NOTATION: Saves space and you round immediately after calculating quotaModified Divisor

Zac Kate Eileen AJ Tommy Megan Total

Pros and Cons of Jefferson’s Method:

Section 4.5 Adam’s Method (Divisor Method)Goal of Adam’s Method: make quotas smaller so that when rounding up (UPPER QUOTAS) there is NO SURPLUS.

ADAM’S METHOD: SAME RULES FOR GUESSING GAME FOR MODIFIED DIVISOR1) Calculate Standard Divisor, SD

Standard quotas will work in Adam’s Method ONLY when they are INTEGERS

2) Find a “suitable” modified divisor, MD such that the sum of UPPER quotas is the number of seats. (exact apportionment with no surplus)Finding MD: Originally guess should be reasonably < Standard Divisor Current Total LESS than exact number of seats, then a SMALLER modified divisor is needed Current Total GREATER than exact number of seats, then a LARGER modified divisor is needed

EXAMPLE #1: There are 15 scholarships to be apportioned among 231 English majors, 502 History majors, and 355 Psychology majors. Majors English History Psychology TotalPopulation 231 502 355 1088

Standard Divisor 1088/15 = 72.53

Standard Quotas

Upper Quotas

Modified Divisor UPPER Quota English

UPPER Quota History

UPPER Quota Psychology

TOTAL

EXAMPLE #2: Try 20 scholarships on four different sciencesMajors Biology Chemistry Physics Astronomy Total

Population 165 208 182 75 630Standard Divisor

630/20 = 31.5

Standard Quotas

5.238 6.603 5.778 2.381 20

Upper Quotas 6 7 6 3 22 Above

Modified Divisor

UPPER Quota Biology

UPPER Quota Chemistry

UPPER Quota Physics

UPPER Quota Astronomy

TOTAL

EXAMPLE #3: Previous example problem of the 6 nations with 250 seats for congress.

State A B C D E F TotalPopulation 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000Standard Divisor

12,500,000/ 250 = 50,000

Standard Quotas

32.92 138.72 3.08 41.82 13.70 19.76 250

Upper Quota

Modified Divisor

UPPER Quota of A

UPPER Quota of B

UPPER Quota of C

UPPER Quota of D

UPPER Quota of E

UPPER Quota of F

TOTAL

What do you notice about the apportionment and the standard quotas?

EXAMPLE #4 Banana Republic has states Apure, Barinas, Carabobo, and Dolores with populations given in millions and 160 seats in the legislature.

State A B C D TotalPopulation 3.31 2.67 1.33 .69 8Standard Divisor

.05

Standard Quotas

66.2 53.4 26.6 13.8 160

Upper QuotaModified Divisor

Upper Quotaof State A

Upper Quotaof State B

Upper Quotaof State C

Upper Quotaof State D

TOTAL

Pros and Cons of Adam’s Method

HOMEWORK: p. 153 #33, 36, 37, 40, 41

Section 4.6 Webster’s Method (Divisor Method)Goal of Websters Method: is to make quotas so that when rounding up and down (CONVENTIONAL/NORMAL Rounding) there is NO SURPLUS.

WEBSTER’S Method: (SAME MD GUESSING GAME and USE CALCULATOR TO HELP)1) Calculate Standard Divisor, SD this is the starting point

2) Find a “suitable” modified divisor, MD such that the sum of conventionally rounded quotas is the number of seats. (exact apportionment with no surplus)

Finding MD: Originally guess should be reasonably < Standard Divisor Current Total LESS than exact number of seats, then a SMALLER modified divisor is needed Current Total GREATER than exact number of seats, then a LARGER modified divisor is needed

EXAMPLE #1: There are 15 scholarships to be apportioned among 231 English majors, 502 History majors, and 355 Psychology majors. Majors English History Psychology Total

Population 231 502 355 1088Standard Divisor 1088/15 = 72.53Standard Quotas 3.185 6.921 4.895 15Normal Rounding

EXAMPLE #2: Previous example of the 6 nations with 250 seats for congress.State A B C D E F TotalPopulation 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000Standard Divisor

12,500,000/ 250 = 50,000

Standard Quotas

32.92 138.72 3.08 41.82 13.70 19.76 250

Normal Rounding

Modified Divisor

Normal Rounding

A

Normal Rounding

B

Normal Rounding

C

Normal Rounding

D

Normal Rounding

E

Normal Rounding

FTOTAL

EXAMPLE #3: Banana Republic has states Apure, Barinas, Carabobo, and Dolores with populations given in millions and 180 seats in the legislature.State A B C D TotalPopulation 3.31 2.67 1.33 0.69 8Standard DivisorStandard QuotasNormal Rounding

EXAMPLE #3: The police department was able to hire 20 new officers and plans to deploy them in 4 precincts based on the previous year’s crimes.

Precinct #1 Precinct #2 Precinct #3 Precinct #4 TotalCrimes 78 92 63 37 270

Standard DivisorStandard QuotasUpper Quotas

Modified Divisor Precinct #1 Precinct #2 Precinct #3 Precinct #4 TOTAL

Pros and Cons of WEBSTER’S METHOD:

BALINSKI AND YOUNG’S IMPOSSIBILITY THEOREM : An apportionment method that does not violate the quota rule and does not produce any paradoxes is a mathematical impossibility.

HOMEWORK: p.153 44 - 47