1. Apportionment So now you are beginning to see why the method of apportionment was so concerning to the founding fathers……it begs to question, why not just put a formula in place when the Constitution was written and that article was agreed upon????? Article 1, Section 2 of the Constitution says in part…. “shall be apportioned … to their respective numbers”???????

2. The Hamilton Method The Hamilton Method, one of several methods of apportionment we study, is named after Alexander Hamilton. It was first used to decide the initial apportionment of the seats in the House of Representatives in 1790. That apportionment was vetoed by George Washington and the House was reapportioned that year according to the Jefferson Method. The Hamilton Method did come back into use in 1850 and was then used until 1900. To explain this and other methods, we must define certain terms…

3. The Hamilton Method Suppose the total population of all states is p and that are there are h seats in the house (we call h the house size). That is, house size is the total number of seats available. We define the standard divisor, s, as follows: or

4. The Hamilton Method Next, we define the term quota as follows: or Thus, each state could have a different quota. In the case of apportioning seats in the House of Representatives, the quota is the number of seats that a state would get if they could have a fractional part of a seat. That is, it is a fractional part of the whole before being rounded to an integer.

5. Terms and Definitions Lower Quota is the Standard quota rounded down. Upper Quota is the Standard quota rounded up. Quota Rule is “A state’s apportionment should either be it’s upper or lower quota.”

6. The Hamilton Method Summarized The Hamilton Method of apportionment is as follows: 1.Calculate each state’s quota. 2.Temporarily assign each state it’s lower quota. 3.Starting with state’s having the largest fractional part in their original quota, distribute any remaining seats, in order from largest to smallest, until all remaining seats are distributed. The Hamilton Method does not specify what to do in the case of a tie – if two states had the same fractional part in their quota - but this is unlikely to occur.

7. Forming Our Own Country Our new country is only made of eight provinces….. Each province has a certain population. Our new country is going to have a ruling council with 50 representatives What is the total population? What is the standard divisor? What is each province’s standard quota?

8. Population Assignments Province 1--500,000 Province 2--207,500 Province 3--900,000 Province 4--775,000 Province 5--820,000 Province 6--102,000 Province 7--675,000 Province 8--125,000

9. Example The country of Parador has six states: Azucar, Bahia, Café, Diamonte, Esmeralda, Felicidad. Their populations are respectively, 1,646,000; 6,936,000; 154,000; 2,091,000; 685,000; 988,000 Parador has Congress with 250 seats…distributed through apportionment… So Find: Each state’s lower quota and upper quota….

10. Republic of Parador: Population Data by State StateAzucarBahia Café DiamanteEsmeraldaFelicidad Total Population1,646,0006,936,000154,0002,091,000685,000988,00012,500,000 Standard Quota 32.92138.723.0841.8213.7019.76 Lower Quota Upper Quota 32 33 138 139 3434 41 42 13 14 19 20 246 252 Final Apportionment

11. Hamilton’s Method of Apportionment Step 1: Calculate each state’s Standard Quota… Step 2: Give to each state its lower quota…. Step 3: Give the surplus seats (one at a time) to the states with the largest residues (or fractional parts) until there are no more surplus seats.

12. https://www.youtube.com/watch?v=YWfEqWLz9pc Worksheet Tonight on the Website

13. Warm-Up Exercise Southwest Guilford High School has sophomore, junior, and senior classes of 464, 240, and 196 respectively. The 31 seats on the school's student council are divided among the classes according to the population. How many should each class have if the administration uses the Hamilton Method of Apportionment?

14. Hamilton’s Method of Apportionment Approved by the Congress in 1781 Was vetoed by President Washington First presidential veto Was subsequently adopted and used from 1852 through 1911 when it was replaced by the Webster Method.

15. Vocabulary Important to Hamilton’s Apportionment Standard Divisor = Total Population divided by the total number of seats Standard Quota = Individual Population divided by the standard divisor Lower quota, round down….Upper quota, round up Assign the lower quota to each… Extra seats go one by one to the fractional part not represented… So how many seats does each get?

16. Worksheet #1 SamanthaJennieDonnaRogerDanny Total254,300260,100253,050252,700260,000 Fair Share 50,86052,02050,61050,54052,000 ItemsCabinJewelryFigurinesHouse Car Value of items 26,00006,5009,950172,700 Initial Cash 24,86052,02044,11040,590-120,700 Extra Cash Share 1424

17. Worksheet #2 Step 1: Find the Total Total is 6,335,000 Step 2: Find the Standard Divisor Standard Divisor = Total Population # of Seats (or pieces) 6335000/545 = 11,623 (people per seat)

18. Worksheet #2 continued Step 3: Set up a Chart and Find the Standard Quotas by dividing individual populations by the standard divisor: Standard Quotas Lower Quotas Final Apportionment Andromedia68.8268 Pandora84.3184 Cyrious103.23103 Xenatia67.5367 Rabithia25.3825 Finanthia40.8640 Savagia154.85154 TotalXXXXXX541 (+4)

19. A Little History The first Congressional bill ever to be vetoed by the President of the United States was a bill in 1790 containing a new apportionment of the House based on Hamilton’s Method. The reason for the veto may have been related to the following requirement stated in the U.S. Constitution (Article 1, section 2): The number of people per single seat in the House should be at least 30,000. Remember that the standard divisor represents the average number of people per seat in the nation as a whole. In 1790, there were 15 states and 105 seats in the House. According to the 1790 census, the U.S. population was 3,615,920. Thus the standard divisor would have been 3,615,920/105 = 34,437. So why the veto?

20. A Little History It is likely that at least one reason George Washington vetoed the 1790 House apportionment as calculated by Alexander Hamilton is that under Hamilton’s apportionment, two seats were assigned to Delaware while the 1790 census indicated a population of 55,540 for Delaware. Therefore, there were actually 55,540/2 = 27,770 people per seat for Delaware, a violation of the Constitution. How did this happen? The answer comes from the fact that Hamilton’s method had awarded Delaware the extra seat by rounding up the quota for Delaware. That is, in 1790, Delaware’s quota was q = (state population)/(standard divisor) = (55,540)/(34,437) = 1.613. Following Hamilton’s Method and rounding quota’s, it so happened that Delaware was rounded up to get 2 seats.

21. A Little History Congress was unable to override the Presidential veto, and to avoid a stalemate, they turned to Thomas Jefferson, who had devised another means of apportionment … Jefferson’s Method was the method actually used for the first apportionment of the House, which was finally done in 1792. This method was then used until 1840 when Hamilton’s Method made a return. To use Jefferson’s Method we must find a modified divisor such that when dividing each state’s population by this divisor, and rounding down, the sum of the adjusted quotas (seats) is equal to the total number of seats. One justification for this method is that it may seem more fair in the sense that every state’s quota will be rounded down – based on division by the same modified divisor.

22. Jefferson’s Method – Example #1 Let’s consider a simple example. Suppose there are only three states: Texas, Alabama and Illinois and 100 seats in the House. Suppose the populations are as given in the table. Population Texas10,030 Illinois9,030 Alabama940 Total20,000 We begin by calculating the standard divisor, which is 20,000/100 = 200. Then we determine the lower quota for each state using the standard divisor…

23. Jefferson’s Method – Example #1 Now we compute the initial apportionment, as defined previously, using the standard divisor of 20,000/100 = 200. Population Lower quotas, using standard divisor of 200 Texas10,030 50 Illinois9,030 45 Alabama940 4 Total20,00099 (1 seat still available) Note that not all of the 100 seats have been apportioned

24. Jefferson’s Method – Example #1 We haven’t apportioned all of the seats using the standard divisor of 200, so we will choose another modified divisor… We choose a modified divisor so that when we round all of the quotas down, they add to the required total. PopulationLower quotas (with standard divisor) Texas10,03050 Illinois9,03045 Alabam a 9404 Total20,00099 (so 1 seat still available)

25. Jefferson’s Method – Example #1 We find the sum of the lower quotas is less than the required total of 100 seats. Therefore, we seek a modified divisor to replace the standard divisor. By trial and error (or as described below) we find that a modified divisor of d = 196.6 will work. Population Lower quotas (with standard divisor) Texas10,030 50 Illinois9,030 45 Alabama940 4 Total20,00099 (so 1 seat still available) Is this ok? Is it ok to change the divisor? The answer is yes – there is no constitutional (or mathematical) requirement for using the standard divisor. In fact, we are determining a value representing the lowest ratio of people per seat for any of the states. In this case, it can be found by adding one to each state’s apportionment and dividing into the population and then comparing each of these resulting values and taking the largest, which is 196.67

26. Jefferson’s Method – Example #1 We find the sum of the lower quotas is less than the required total of 100 seats. Therefore, we seek a modified divisor to replace the standard divisor. By trial and error (or as described below) we find that a modified divisor of d = 196.6 will work. Population Lower quotas (with standard divisor) Texas10,030 50 Illinois9,030 45 Alabama940 4 Total20,00099 (so 1 seat still available) By finding the value of 196.6 we are essentially solving a problem of optimization – to find the maximize value of the minimum number of people per seat in any state. This was Jefferson’s problem – he wanted to make sure to satisfy the Constitutional requirement that there were at least 30,000 people per seat in the apportionment of the House. So he was looking at the resulting ratios of people per seat in every state and sought to maximize this value – to make sure it was always more than 30,000.

27. Jefferson’s Method – Example #1 Now, using the modified divisor, we calculator lower modified quotas for each state… PopulationInitial lower quotas Lower quotas (with modified divisor of 196.6) Texas10,030 50 Illinois9,030 45 Alabama940 4 Total20,00099 (so 1 seat still available) Total apportionment = 100

28. Jefferson’s Method – Example #1 Now, using the modified divisor, we calculator lower modified quotas for each state… PopulationInitial lower quotas Lower quota using modified divisor of 196.6 Final apportionment Texas10,030 5051 Illinois9,030 45 Alabama940 44 Total20,00099 (1 seat still available) 100

29. Practice Problem The first grade reading teacher is using stickers to give her students incentive to write more words in their writing journals. She only has 50 stickers and the number of words each of her students wrote is in the table below. Use this information to find the number of stickers each child should receive using the Jefferson Method. ChildJamieJodyDannyJimboSammySharon Words7589651109854

30. Literacy Activity Research the following and choose one to create a “foldable” learning guide much as you did with the different methods in election theory. Your foldable should include: the origins of the method Information about when (the years) the method was used by the United States to apportion seats in the House of Reps The method of calculation with an example, particularly with how the method deals with the fractional pieces left over Choices: Hamilton Method; Jefferson Method; Webster Method; Hill Method