What is this How do the state get the votes Its all in the math Apportionment is The process of dividing or assigning the proper proportion to groups by a plan Back to the College ID: 212182
Download Presentation The PPT/PDF document "Apportionment" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
ApportionmentSlide2
What is this????Slide3
How do the state get the votes???Slide4
It’s all in the math!!!!!Slide5
Apportionment is…
The process of dividing (or assigning) the proper proportion to groups by a plan.Slide6
Back to the College
Look at the change between 2008 and 2012.
Why?Slide7
How do they divide the votes???
By MathematicsSlide8
And processes created by these guysSlide9
Terms
States- The name for players that are wanting to parts of the division
Seats- The indivisible identical spots that have to be filled
Population- A positive number that will be used to measure how many seats each state will getSlide10
More Terms
Standard Divisor (SD)- The ration of population to seats
Standard Quota- The fraction that we get when we divide the states population by the standard divisor
Upper and Lower Quota- When rounding the fractions you get the upper and lower quotas. Rounding down gives you the lower quota (L) and rounding up gives you the upper quota (U)
Slide11
Child
Alan
Betty
Connie
Doug
Ellie
TOTAL
Minutes
150
78
173
204
295
900
Mother has 50 pieces of candy and wants to give it to her 5 children. She wants to reward them for the work they have done.
How many pieces should each child get based on the work they have done?
ExampleSlide12
1st
Find the standard divisor.
Slide13
Find each States Standard Quota
Child
Alan
Betty
Connie
Doug
Ellie
TOTAl
Minutes
150
78
173
204
295
900
Standard
Quota
8.33
4.33
9.61
11.33
16.38
50
2
nd
Slide14
Child
Alan
Betty
Connie
Doug
Ellie
TOTAl
Minutes
150
78
173
204
295
900
Standard
Quota
8.33
4.33
9.61
11.33
16.38
50
Lower Quota
8
4
9111648
Upper Quota9510121753
Round all the numbers down for the lower quota
Round all the numbers up for the upper quota
Find Upper and Lower QuotaSlide15
OH NO!!!
As you can see if you take only the lower quota we have 2 seats (or candy) left over.
If we take the upper quota we have 3 too many seats (or candy).
How do we solve this tricky puzzle???Slide16
“By My Way”
-Alexander Hamilton
no a direct quota on this topic
Alexander Hamilton's Method
Used on the Electoral College from 1850 to 1900.
Still in use in Costa Rica, Namibia, and SwedenSlide17
Basic Idea
1. Find each state’s standard quota
2. Everyone gets their lower quota
3. The remaining seats go to the states with the highest
residueSlide18
Term Defined
Residue- in simple terms it is the faction part that is attached to a number
Example: 3.2564488
The residue is 0.2564488Slide19
Child
Alan
Betty
Connie
Doug
Ellie
TOTAL
Minutes
150
78
173
204
295
900
Standard
Quota
8.33
4.33
9.61
11.33
16.38
50
Lower Quota
8
4
9111648
Residue0.330.330.610.330.38Order of surplus
1
st
2
nd
Apportionment
8
4
10
11
17
50
Find each states residue and find the highest, second highest, etc. until all the surplus are gone.
Back to HamiltonSlide20
YEAH IT WORKS!!!
What is working for it:
It satisfies the Quota Rule.
You might ask what the quota rule is?
Well good thing it is on the next slide.Slide21
Quota Rule
No state should be apportioned a number of seats fewer than its lower quota or more than its upper quota
If a state has less than its lower quota it is a
LOWER QUOTA VIOLATION
If a state has more than its upper quota it is an
UPPER QUOTA VIOLATIONSlide22
Alabama Paradox
Oh ALABAMA you have to be
sooooo
difficultSlide23
The Year is 1882
The House of Representatives are increasing from 299 seats up to 300 seats.
Alabama gets upset because the increase will have them lose a seat.
M=299
M=299
M=300
M=300
State
Quota
Apportionment
Quota
Apportionment
Alabama
7.646
8
7.671
7
Texas
9.64
9
9.672
10
Illinois
18.64
18
18.702
19Slide24
Another Example
M=200 (SD=100)
State
Population
Quota
Lower Quota
Surplus
Apportionment
Alabama
940
9.4
9
First
10
Washington
9030
90.3
90
0
90
Texas
10,030
100.3
100
0
100
Total20,000200.01991200
M=201 (SD=99.5)
State
Population
Quota
Lower Quota
Surplus
Apportionment
Alabama
940
9.45
9
0
9
Washington
9030
90.75
90
Second
91
Texas
10,030
100.8
100
First
101
Total
20,000
201.0
199
2
201Slide25
Population Paradox
This occurs when a state (lets say California) loses a seat to
a state
(lets say New York) even though the population in California grew at a higher rate than New YorkSlide26
Information gathered Yesterday
State
Population
Standard Quota
Lower Quota
Surplus
Apportionment
Maine
150
8.3
8
0
8
New York
78
4.3
4
0
4
Florida
173
9.61
9
1
st
10Alaska204
11.311011California29516.3816
2
nd
17
TOTAL
900
50
48
2
50Slide27
Information Gathered TODAY
State
Population
Standard Quota
Lower Quota
Surplus
Apportionment
Maine
150
8.25
8
0
8
New York
78
4.29
4
2
nd
5
Florida
181
9.96
9
1st10Alaska
20411.2211011California29616.28
16
0
16
TOTAL
909
50
48
2
50Slide28
LOOK
As we see between the data from today and yesterday California grew one while New York grew zero. Florida grew the most at eight. This changed the apportionment of the states by moving one from California to New York.Slide29
Population Paradox
Under the Hamilton method, it is possible for a state to have a positive population growth rate and lose one (or more) of its seats to another state who had a smaller (or even zero) population growth rate.Slide30
New-States paradox
Time for a history lesson… I know History and math together…
zzzzzzzzzzzzzzzzzzzzzz
WAKE UP!!!Slide31
New-States Paradox
The year is 1907… Slide32
Oklahoma has just been granted statehood.
“How many votes should Oklahoma get?”
“Well why don’t we just give them how many they would have gotten the last time we assigned seats… (does a quick mathematical calculation on his paper)
ok they get five.”
“Sweet we are done, lets argue about some bills now.”
(The mathematician in the back row) “You guys are going to regret that”Slide33
Few WEEKS Later
“Ok lets recalculate the seats to see if any changed when Oklahoma was added.”
(Mathematician in the back) “It has.”
“ok we have the new seats. Looks as though Maine you actually deserve one more seat. You should have four seats.”
(Reps.
f
rom Maine) “Awesome!” (giving high fives)
“Now with Maine getting that seat it looks as though New York, you lose a seat. You have to go from 38 down to 37.”
(Reps.
f
rom New York) “What that is bogus!”
“So which one of you wants to move to Maine. HAHAHA”Slide34
NEW-states Paradox
The addition of a new state with its fair share of seats can, in and of itself, affect the apportionment of the other states
Hamilton’s Method is flawed and there probably is better ones so lets see what else we have.Slide35
Mr. President’s Method
Thomas Jefferson’s Method
Used in the United States from 1792 to 1840.
Still used in Austria, Brazil, Finland, Germany, and the NetherlandsSlide36
Jefferson brakes the mold
Why use the Standard Divisor?
Why can’t we use a different divisor?
A MODIFIED DIVISOR!!!Slide37
Process
First you find the Standard Divisor and test it to see if it works out correctly.
If it does you are done. If not we have to use the modified divisor.
You will have all the states use their
LOWER QUOTASlide38
EXAMPLE
State
La
Mesa
El
Centro
Coronado
Lakeside
El Cajon
Fallbrook
TOTAL
Population
1,646,000
6,936,000
154,000
2,091,000
685,000
988,000
12,500,000
Standard Quota (SD=50,000)
32.92
138.72
3.08
41.82
13.70
19.76250.0Lower Quota
321383411319246
Modified
Quota (SD=49,500)
33.25
140.12
3.11
42.24
13.84
19.96
X
Lower Quota
33
140
3
42
13
19
250Slide39
HOW THE Factorial DID YOU GET 49,500?
Well semi-luck, semi-logic.
You see with Jefferson’s Method you are going to change you Standard Divisor to something smaller.
You will use trial and error until you find a number that will work
Look at the flowchart on page 136Slide40
MORE HISTORY
This method was used through 1832 when there was a major discussion about the major flaw in Jefferson’s Method
In 1832 the house was
dicussing
the amount of seats that the state of New York should receive. (New York: always causing problems)
Well the standard quota said that New York should get 38.59 seats. New York was going to receive 40 votes using Jefferson’s Method.
This is above its UPPER QUOTA, therefore violating the Quota Rule.Slide41
HISTORY HISTORY
HISTORY
Believing that this was unfair, two men purposed their own methods. Which they are named after.
It did not matter in the end as the House accepted the new apportionment.
This was the last time Jefferson’s method was used in the U.S.Slide42
Mr. President’s Method Part II
Created by Thomas Jefferson
Modified by John Quincy Adams
Never used in the United StatesSlide43
Adam’s Method
You will use the same process as Jefferson, with one exception.
Instead of always taking the Lower
Q
uota of the Modified Quota, you will take the Upper Quota for the apportionment.
This means that when you are looking for your new Divisor,
y
ou will want to pick a number that is bigger than that of the Standard Divisor.Slide44
Example
State
La
Mesa
El
Centro
Coronado
Lakeside
El Cajon
Fallbrook
TOTAL
Population
1,646,000
6,936,000
154,000
2,091,000
685,000
988,000
12,500,000
Modified Quota 1 (D=50,500)
32.59
137.35
3.05
41.41
13.56
19.56Upper Quota
331384421420251
Modified Quota 2 (D=50,700)
32.47
136.80
3.04
41.24
13.51
19.49
Upper Quota
33
137
4
42
14
20
250Slide45
Webster’s Method
Created by Thomas Jefferson
Modified by Daniel Webster
Used in the United States in 1842, 1901, 1911, and 1931Slide46
Why Not Do Normal rounding
Webster uses the same process as Jefferson and Adams with one exception.
If we have a residue that is more than 0.5, why not round up. If it is less than 0.5 we round down.
If the Residue is < 0.5 we use the Lower Quota
If the Residue is ≥ 0.5 we use the Upper QuotaSlide47
Example
State
La
Mesa
El
Centro
Coronado
Lakeside
El Cajon
Fallbrook
TOTAL
Population
1,646,000
6,936,000
154,000
2,091,000
685,000
988,000
12,500,000
Standard Quota (SD=50,000)
32.92
138.72
3.08
41.82
13.70
19.76250.0Nearest
Integer331393421420251
Modified Quota (D=
50,100)
32.85
138.44
3.07
41.74
13.67
19.72
Nearest
Integer
33
138
3
42
14
20
250Slide48
Conclusion: A look at all methods
State
La
Mesa
El
Centro
Coronado
Lakeside
El Cajon
Fallbrook
TOTAL
Population
1,646,000
6,936,000
154,000
2,091,000
685,000
988,000
12,500,000
Standard Quota (SD=50,000)
32.92
138.72
3.08
41.82
13.70
19.76250.0
Hamilton331393421320250
Jefferson
33
140
3
42
13
19
250
Adams
33
137
4
42
14
20
250
Webster
33
138
3
42
14
20
250Slide49
Conclusion: A look at all methods
As we see in the table before we see that El Centro is the one state that has a major swing in votes.
Some states do not have a change (La Mesa) between methods.
No method is perfect and they are all flawedSlide50
Conclusion: A look at all methods
Method
Violates Quota Rule?
Paradoxes?
Favors
Hamilton
NO
YES
Large States
Jefferson
YES (UPPER)
NO
Large
States
Adams
YES
(LOWER)
NO
Small
States
Webster
YES
(BOTH)
NO
NeutralSlide51
LAST ONESlide52
Huntington-Hill Method
Been in use since the passing of the 1941 Apportionment ActSlide53
Huntington-hill
The 1941 Apportionment Act did three things
1. Set the Huntington-Hill method as the
permenant
method that is used
2. This method is self-executing (the Congress does not have to approve the results)
3. The seats are now set in stone at 435 (unless another state joins the union)Slide54
Lets talk means
How do you find the mean of two numbers?
You add and divide by twoSlide55
Lets Talk Mean
That is correct if we are talking Arithmetic
M
ean
If we were talking Geometric Mean
Slide56
a
b
10
10
10
10
10
20
14.14
15
10
40
20
25
10
100
31.62
55
10
1000
100
505
10
9000
300
4505
a
b
10
10
10
10
10
20
14.14
15
10
40
20
25
10
100
31.62
55
10
1000
100
505
10
9000
300
4505
We have the following Inequality
Lets dig deeperSlide57
Difference
The Webster Method uses rounding of the basic
A
rithmetic Mean (0.5)
Huntington-Hill Method takes into account the Geometric Mean
Instead of always rounding at 0.5 the cut off can
be between 0.414 to 0.5