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%N:TChapter 4 The Mathematics of Apportionment++&Each state has two senators.
Each state has representatives based on the state s population.
Constitution does NOT state the equation to use for finding the number of representatives.
This is now called The apportionment problem .4.1 Apportionment Problems apportion to divide and assign in due and proper proportion or according to some plan.
In apportionment problems: i) We are dividing and assigning things and ii) we are doing this on a proportional basis and in a planned, organized fashion.
What if a mom had 50 pieces of the same candy and has 5 children many of us would believe this to be an easy fair division problem give each child 10 pieces.
Let s say that mom wants to divide the candy up based on how many hours each child helps with the chores.P50 Pieces vs. 5 ChildrenPreface TerminologyThe states term used to describe all players involved in the apportionment. (if no names are given, we will use A1, A2, & ,AN) (children)
The seats term which describes the set of M (identical, indivisible objects) that are being divided among N states. (candy)
The populations set of N positive numbers which are the basis for the apportionment of the seats. (minutes)
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f0)fe f@Number Terminology numerology? J*!&&&2The standard divisor this is the ratio of total population to seats SD=P/M. (900 minutes/ 50 pieces = 18 minutes per piece)
The standard quotas fractional (2-3 decimal places) number of seats a state would get use q1, q2, & qN.
quota= (state s) population/SD (Alan worked 150 minutes& so, 150/18 = 8 1/3)
Lower quota(L1, L2, & LN) quota rounded down (8)
Upper quota(U1, U2, & UN) quota rounded up (9)
OUR GOAL this chapter to use a procedure that
i) Will always produce a valid apportionment (exactly M seats are apportioned, and ii) Will always produce fair apportionment.
Ex. Turtles, Turtles, Who Gets the Turtles? Page 1
Classwork/Homework Pg. 150: 1-9 odd.ZMZZZYZ/f0fT f-fffYf, *4.2 Hamilton s MethodAlso known as Vinton s method or the method of largest remainders used in the US between 1850 and 1900.
Every state gets at least its lower quota.
Step 1 Calculate each state s standard quota.
Step 2 Give to each state (for the time being) its lower quota.
Step 3 Give the surplus seats (one at a time) to the states with the largest fractional parts until there are no more surplus seats.ff)f.ff~f
4Flaws to Hamilton s MethodzA state with a fractional part of 0.72 may end up with one more seat than a state with a fractional part of 0.70 major flaw in the way it relies entirely on the size of the fractional parts without consideration of what those fractional parts represent as a percent of the state s population creates a bias in favor of larger states. Should be population neutral.
Alabama Paradox occurs when an increase in the total number of seats being apportioned, in and of itself, forces a state to lose one of its seats.
Population Paradox occurs when state A loses a seat to state B even though the population of A grew at a higher rate than the population of B.
The New-States Paradox the addition of a new state with its fair share of seats can, in and of itself, affect the apportionments of other states.
When using Hamilton s method, all three paradoxes can occur definitely not a good thing! (more detail on paradoxes in section 4.3)
Good points of Hamilton s method 1) it is easy to understand, and 2) it satisfies an extremely important requirement for fairness called the quota rule.>Pm3f3fMf3f|f!lfThe Quota Rule\Definition a state should not be apportioned a number of seats smaller than its lower quota (lower-quota violation) or larger than its upper quota (upper-quota violation).
Remember, Step 2 of Hamilton s Method satisfies the quota rule.
Ex. Apportionment Page 1
Classwork/Homework Pg. 152: 11-21 odd
^/Qf f@Bf .4.4 Jefferson s Method ^Jefferson s Method was the first apportionment method used in US House of Reps. (terminated in 1832)
Since in Hamilton s Method there is always a surplus, Jefferson s Method involves tweaking the standard divisor. If you lower the standard divisor (call this the modified divisor D), the quotas increase. Likewise, if you increase the standard divisor, the quotas decrease.
Our Goal to apportion M seats without any surplus!!!
FBf}8Steps for Jefferson s MethodStep 1 find a suitable divisor D a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded down.
Step 2 Each state is apportioned its lower quota (using the suitable divisor D).
Biggest problem with Jefferson s Method It can produce upper-quota violations!!! The upper-quota violations tend to favor the larger states.
Ex. Turtles Page 2
Classwork/Homework Pg. 152: 23, 25
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"4.5 Adam s MethodStep 1 find a suitable divisor D a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded up. (opposite of Jefferson s) This means that quotas have to be made smaller by using a larger divisor (larger than the standard divisor).
Step 2 Each state is apportioned its upper quota (using the suitable divisor D).
Biggest problem with Adam s Method It can produce lower-quota violations!!! Adam s Method was never passed to apportion the House of Reps.
Ex. Turtles Page 3
Classwork/Homework Pg. 153: 33, 35OZff{f
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(4.6 Webster s MethodCompromise between Jefferson s Method (rounding down) and Adam s Method (rounding up) rounding up if 0.5 or higher and rounding down if less than 0.5 BUT, using a modified divisor.
Step 1 find a suitable divisor D a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded the conventional way. (a suitable divisor CAN be the standard divisor& always check the SD first!!!!)
Step 2 Find the apportionment of each state by rounding its quota the conventional way (using the suitable divisor D).
Ex. Apportionment Page 2
Classwork/Homework Pg. 153: 43, 45
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%:TChapter 4 The Mathematics of Apportionment++&Each state has two senators.
Each state has representatives based on the state s population.
Constitution does NOT state the equation to use for finding the number of representatives.
This is now called The apportionment problem .4.1 Apportionment Problems apportion to divide and assign in due and proper proportion or according to some plan.
In apportionment problems: i) We are dividing and assigning things and ii) we are doing this on a proportional basis and in a planned, organized fashion.
What if a mom had 50 pieces of the same candy and has 5 children many of us would believe this to be an easy fair division problem give each child 10 pieces.
Let s say that mom wants to divide the candy up based on how many hours each child helps with the chores.PDocumentSummaryInformation8h50 Pieces vs. 5 ChildrenPreface TerminologyThe states term used to describe all players involved in the apportionment. (if no names are given, we will use A1, A2, & ,AN) (children)
The seats term which describes the set of M (identical, indivisible objects) that are being divided among N states. (candy)
The populations set of N positive numbers which are the basis for the apportionment of the seats. (minutes)
bxZ
f0)fe f@Number Terminology numerology? J*!&&&2The standard divisor this is the ratio of total population to seats SD=P/M. (900 minutes/ 50 pieces = 18 minutes per piece)
The standard quotas fractional (2-3 decimal places) number of seats a state would get use q1, q2, & qN.
quota= (state s) population/SD (Alan worked 150 minutes& so, 150/18 = 8 1/3)
Lower quota(L1, L2, & LN) quota rounded down (8)
Upper quota(U1, U2, & UN) quota rounded up (9)
OUR GOAL this chapter to use a procedure that
i) Will always produce a valid apportionment (exactly M seats are apportioned, and ii) Will always produce fair apportionment.
Ex. Turtles, Turtles, Who Gets the Turtles? Page 1
Classwork/Homework Pg. 150: 1-9 odd.ZMZZZYZ/f0fT f-fffYf, *4.2 Hamilton s MethodAlso known as Vinton s method or the method of largest remainders used in the US between 1850 and 1900.
Every state gets at least its lower quota.
Step 1 Calculate each state s standard quota.
Step 2 Give to each state (for the time being) its lower quota.
Step 3 Give the surplus seats (one at a time) to the states with the largest fractional parts until there are no more surplus seats.ff)f.ff~f
4Flaws to Hamilton s MethodzA state with a fractional part of 0.72 may end up with one more seat than a state with a fractional part of 0.70 major flaw in the way it relies entirely on the size of the fractional parts without consideration of what those fractional parts represent as a percent of the state s population creates a bias in favor of larger states. Should be population neutral.
Alabama Paradox occurs when an increase in the total number of seats being apportioned, in and of itself, forces a state to lose one of its seats.
Population Paradox occurs when state A loses a seat to state B even though the population of A grew at a higher rate than the population of B.
The New-States Paradox the addition of a new state with its fair share of seats can, in and of itself, affect the apportionments of other states.
When using Hamilton s method, all three paradoxes can occur definitely not a good thing! (more detail on paradoxes in section 4.3)
Good points of Hamilton s method 1) it is easy to understand, and 2) it satisfies an extremely important requirement for fairness called the quota rule.>Pm3f3fMf3f|f!lfThe Quota Rule\Definition a state should not be apportioned a number of seats smaller than its lower quota (lower-quota violation) or larger than its upper quota (upper-quota violation).
Remember, Step 2 of Hamilton s Method satisfies the quota rule.
Ex. Apportionment Page 1
Classwork/Homework Pg. 152: 11-21 odd
^/Qf f@Bf .4.4 Jefferson s Method ^Jefferson s Method was the first apportionment method used in US House of Reps. (terminated in 1832)
Since in Hamilton s Method there is always a surplus, Jefferson s Method involves tweaking the standard divisor. If you lower the standard divisor (call this the modified divisor D), the quotas increase. Likewise, if you increase the standard divisor, the quotas decrease.
Our Goal to apportion M seats without any surplus!!!
FBf}8Steps for Jefferson s MethodStep 1 find a suitable divisor D a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded down.
Step 2 Each state is apportioned its lower quota (using the suitable divisor D).
Biggest problem with Jefferson s Method It can produce upper-quota violations!!! The upper-quota violations tend to favor the larger states.
Ex. Turtles Page 2
Classwork/Homework Pg. 152: 23, 25
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"4.5 Adam s MethodStep 1 find a suitable divisor D a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded up. (opposite of Jefferson s) This means that quotas have to be made smaller by using a larger divisor (larger than the standard divisor).
Step 2 Each state is apportioned its upper quota (using the suitable divisor D).
Biggest problem with Adam s Method It can produce lower-quota violations!!! Adam s Method was never passed to apportion the House of Reps.
Ex. Turtles Page 3
Classwork/Homework Pg. 153: 33, 35OZff{f
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(4.6 Webster s MethodXCompromise between Jefferson s Method (rounding down) and Adam s Method (rounding up) rounding up if 0.5 or higher and rounding down if less than 0.5 BUT, using a modified divisor.
Step 1 find a suitable divisor D a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded the conventional way. (a suitable divisor CAN be the standard divisor& always check the SD first!!!!)
Step 2 Find the apportionment of each state by rounding its quota the conventional way (using the suitable divisor D).
Ex. Apportionment Page 2
Classwork/Homework Pg. 153: 43, 45 (important chart next slide)
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