+ Apportionment Ch. 14 Finite Math

+ The Apportionment Problem An apportionment problem is to round a set of fractions so that their sum is maintained at its original value. The rounding procedure must not be an arbitrary one, but one that can be applied constantly. Any such rounding procedure is called an apportionment method.

+ Terminology States: House size: Populations: The numbers in each category The categories In the problem of rounding percentages the house size is 100

+ Standard Divisor The quotient of the total population, p, divided by the house size is called the standard divisor. If h denotes the house size and s is the standard divisor, then:

+ Quota In an apportionment problem, the quota is the exact share that would be allocated if a whole number were not required. To obtain a state’s quota, divide its population by the standard divisor.

+ Example: The High School Mathematics Teacher Courses to be taught correspond to states. Geometry, Pre-calculus. and Calculus The number of students enrolled in each course correspond to populations. 52,33, and 15 respectively The house size is the total number of sections to be scheduled. One teacher has time for 5 sections The average section will have 20 students 100 students/5 sections=20 This is our standard divisor

+ The quota is each section’s population divided by the standard divisor CoursePopulationQuotaRounded Geometry5252/20=2.63 Precalculus3333/20=1.652 Calculus1515/20=0.751 Totals10056 The purpose of an apportionment method is to find an equitable way to round a set of numbers such as these quotas without increasing or decreasing the original sum.

+ Example- California’s Quota The Census Bureau recorded the apportionment population of the US as of April 1 st, 2000 to be 281,424,177. There are 435 seats in the House of Representatives.; therefore, the standard divisor and California’s quota are: California’s population California’s apportionment, which must be a whole number, was set at 53

The Hamilton Method Each state receives either its lower quota, which is its quota rounded down, or its upper quota obtained by rounding the quota up. The states that receive their upper quotas are those whose quotas have the largest fractional parts.

+ 3 steps for the Hamilton Method 3)Allot the remaining seats, one each, to the states whose quotas have the largest fractional parts, until the house is filled. 2)Tentatively assign to each state its lower quota of representatives. Each state whose quota is not a whole number loses a fraction of a seat at this stage, so the total number of seats assigned at this point will be less than the house size. 1)Calculate each state’s quota

+ Example- Math Teacher CoursePopulationQuotaRounded Geometry5252/20=2.63 Precalculus3333/20=1.652 Calculus1515/20=0.751 Totals )Done above 2)Geo-2/Pre-1/Calc-0 3)Geo-2/Pre-2/Calc-1

+ Paradoxes of the Hamilton Method A paradox is a fact that seems obviously false. The Alabama Paradox When a state loses a seat as a result of an increase in the house size StateHouse SizeInc. in Quota Alabama Illinois Texas

+ The Population Paradox When one state’s population increases, and its apportionment decreases, while simultaneously another states population increases proportionally les, or decreases, and its apportionment increases.

+ Apportioning Seats in Parliament PartyFirst ElectionSecond Election Whigs5,525,3815,657,564 Tories3,470,1523,507,464 Liberals3,864,2263,885,693 Centrists201,203201,049 Totals13,060,96213,251,770 Standard Divisor130, , PartyFirst ElectionSecond Election Whigs Tories Liberals Centrists Totals100 The Tories and Liberals get the extra votes: 42,27,30,1 The Whigs and Centrists get the extra votes: 43,26,29,2

+ Practice Problems on page 515 #’s 2 and 4

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.

+ The Jefferson Method The method proposed by Jefferson to replace Hamilton’s method. This method favors the states with large populations. He used an adjusted quota which is always rounded down at first.

+ The Jefferson Method-examples A district needed at least a population of 30,000 according to the constitution. Hamilton’s method did not account for this so Jefferson used the 33,000 as his divisor. This was the smallest population. Virginia’s apportionment: Delaware’s apportionment: Virginia’s apportionment: Delaware’s apportionment:

+ Critical Divisor for the Jefferson method A state’s critical divisor is the number that can be divided into the state’s population to produce a number just on the borderline for changing the state’s apportionment. Let N be the tentative apportionment obtained as the lower quota from the Hamilton method for a state X.

+ Apportionment of the extra seats Since we always round down, some extra seats will need to be allotted. Here, the states with the largest critical divisors get the seats first. Only states with a critical divisor equal to or greater than the divisor will receive additional seats. When even more seats must be allotted, the states who added seats must recalculate their critical divisor.

+ Example: The Field Hockey Team A field hockey team had 18 wins, 4 losses, and 1 tie last season. The coach wants to display this on a poster in terms of win/loss percentages. House size is 100% Total population is 23 games played This makes the Quotas: The lower quotas are: 78,17,4 This does not add to 100% The lower quotas are: 78,17,4 This does not add to 100%

+ Example: The field hockey team To decide who gets the extra percent we must calculate the critical divisors: “Wins” gets the extra percent! To decide who gets the extra percent we must calculate the critical divisors: “Wins” gets the extra percent!

+ The Quota Condition If every state’s apportionment is equal to either is upper or lower quota. The Jefferson method does not satisfy this, but the Hamilton method does. But, the Hamilton method has the Alabama paradox to face and the Jefferson method does not. No known method avoids all issues…

+ The Webster Method The Webster Method is the divisor method that rounds the quota to the nearest whole number, rounding up when the fractional part is greater than or equal to 0.5. This method also avoids the paradoxes and is neutral unlike the Jefferson method.

+ Procedure for the Webster method: 1)Determine the standard divisor and use it to find each state’s quota 2)Round up if the fractional part is 0.5 or above, if not round down to find tentative apportionment. 3)If the rounded quotas equal the house size you are finished. If not, use the same method for allotting additional seats as used in the Jefferson method, but with a different divisor.

+ Critical divisors for the Webster method: If the rounded numbers fall short of the house size we calculate the following to allot additional seats: If the rounded numbers exceed the house size we calculate the following to remove seats: If the rounded numbers fall short of the house size we calculate the following to allot additional seats: If the rounded numbers exceed the house size we calculate the following to remove seats: N is the tentative apportionment V is the state’s population N is the tentative apportionment V is the state’s population

+ Practice Problems pg.516 #’s 6,7,8

Which method is best? Representative Share If A is the apportionment given to a state of population V, then A/V represents the share of a congressional seat given to each citizen. One would hope that this is the same for all. Representative Share If A is the apportionment given to a state of population V, then A/V represents the share of a congressional seat given to each citizen. One would hope that this is the same for all. The Webster method is the method that makes the representative share the most even.

+ District Population The district population of state X is V/A. This is the average population of a congressional district in the state District Population The district population of state X is V/A. This is the average population of a congressional district in the state Absolute and Relative Differences Given two positive numbers A and B, with A>B, the Absolute difference is A-B and the relative difference is the quotient: The relative difference in district population is the same as the relative difference in representative share. Therefore, either can be used to compare methods. Absolute and Relative Differences Given two positive numbers A and B, with A>B, the Absolute difference is A-B and the relative difference is the quotient: The relative difference in district population is the same as the relative difference in representative share. Therefore, either can be used to compare methods.

+ The Hill-Huntington Method Another divisor method with a different way of rounding related to geometric means. Find the Geometric mean of the higher and lower quota and call it q * If q<q * round down, if not round up Geometric Mean The geometric mean of two positive numbers A and B is equal to the square root of their product: Geometric Mean The geometric mean of two positive numbers A and B is equal to the square root of their product:

+ Critical Divisors for the Hill- Huntington Method N is the tentative apportionment, V is the state’s population.

+ Practice Problems pg. 516 #’s 12, 13, 14