Apportionment Practice Worksheet Math Topics
Name _____________________
The first ever census of the population of the United States took place in 1790. Although it probably missed some people, the census result counted 3,615,920 people in the United States at that time.
Congress met in 1792 to decide the first House of Representatives.
Based on the U.S. Constitution, the first House of Representatives was supposed to allow for each representative to represent 30,000 people but not less than that.
1.
Based on that number, how many representatives would be in the first House of
Representatives?
*Calculate here:
Congress used only the whole number part of this number for the number of seats. How many seats did they decide on? ______________________
2.
The U.S. Constitution states that the number of representatives each state gets should be based on the percent that state’s population is of the total U.S. population.
Hamilton’s method as shown below was devised to do that. This Hamilton’s method is slightly improved from one we’ve worked with before – rather than using each state’s percentage of the total population of the United States it uses a number called a state quota. This is good to use when the total number of seats to be allocated is more than 100. If we didn’t use a state quota, we could have more extra seats to allocate in the second step of Hamilton’s method than we have states to give seats to.

*What is the value of D for 1792?
Calculate the state quota for all 15 states:
State
State Population
VA 630,560
MA
PA
NC
NY
475,327
432,879
353,523
331,589
MD
CT
SC
NJ
NH
VT
GA
KY
RI
DE
278,514
236,841
206,236
179,570
141,822
85,533
70,835
68,705
68,446
55,540

State Quota unrounded State quota rounded down
VA
MA
PA
NC
NY
MD
CT
SC
NJ
NH
VT
GA
KY
RI
DE
3.
The state quota rounded down represents the first step of allocation.
*How many total seats are allocated so far? _______________
*How many seats are still left to allocate? __________________
4.
The second step of Hamilton’s method looks at only the decimal part of the unrounded state quota. The state with the largest decimal part gets the first ‘extra’ seat, followed by the state with the next largest decimal, and so on, until all remaining seats are allocated.

Copy the decimal parts of the unrounded state quotas here:
NH
VT
GA
KY
RI
DE
State
VA
MA
PA
NC
NY
MD
CT
SC
NJ
*Which state has the largest decimal part? ______________
*Which state has the second largest decimal part? _____________
*List the remaining states in order of largest decimal part to smallest decimal part:
*Based on your answers above, which state(s) should be allocated another seat in the House of
Representatives to bring the total number of representatives up to 120?

5.
If Hamilton’s method had been used, how many representatives would each state have had in the first House of Representatives?
SC
NJ
NH
VT
GA
KY
RI
DE
State
VA
MA
PA
NC
NY
MD
CT
(Quick check: make sure the total is 120)

6.
Remember that the Constitution had required that each member of the House represent
30,000 people. Let’s find out how Hamilton’s method did at meeting that requirement. To do that, we must calculate the Average Constituency for each state:
Average Constituency = population of state / number of representatives for that state
State
VA
MA
VT
GA
KY
RI
DE
PA
NC
NY
MD
CT
SC
NJ
NH
7.
If any of the states ended up with better representation than 1 representative per 30,000 people, this allocation would violate Clause 3 of Article 1 of the Constitution which was intended to keep the House of Representatives from getting too large. If the clause were violated,
George Washington would execute the first ever Presidential Veto, preventing this method of allocation from being used.
*What does “better representation” mean in this case?
*Did George Washington issue his first veto in this case? Why or why not?

8.
Let’s examine some issues of unfairness that resulted from Hamilton’s allocation.
*Using his method, which state had the best representation?
About how many people would a representative from that state represent? (Average
Constituency)
*Which state had the worst representation in Hamilton’s allocation?
About how many people would a representative from that state represent? (Average
Constituency)
Absolute Unfairness between two states is the difference between the average number of people per representative in each state, i.e. the difference in average constituencies.
*What is the Absolute Unfairness between the best and worst represented states?
Relative Unfairness is a better measure of unfairness because it takes into account the size of the populations of the states. It is calculated as follows:
Relative Unfairness = Absolute Unfairness / smaller Average Constituency of the two states
*What is the relative unfairness between the best and worst represented states? _________

9. (Extension Question #1)
Ten days after Hamilton’s method was rejected, Congress passed a new method of apportionment, now known as Jefferson’s Method in honor of its creator, Thomas Jefferson.
The “d” used in 1792 was 2363.
The numbers found using this formula would replace those found in problem #2 above, and then would be rounded down to the nearest whole number to get the final allocation.
This method was approved by the President and was used to apportion the U.S. House from 1792 through 1842.
Use Jefferson’s method to find the allocation that was actually used to assign seats to the first House of Representatives.
State State Quota Round Down = Final Allocation
VA
MA
PA
NC
NY
MD
CT
SC
NJ
NH
VT
GA
KY
RI
DE
How many total seats are given out in the final allocation? ____________
What is the relative unfairness between the best and worst represented state? ___________
Did Jefferson’s method reduce unfairness or not? ___________________________________

10.
(Extension question #2)
The 1920s saw Congress move toward the apportionment method it still uses today. Joseph
Hill of the Census Bureau proposed a new method. His goal was to keep one state’s “people per Representative” (i.e. Average Constituency) as close to another state’s as possible. The method is now commonly known as Hill’s Method.
The value of D in 1792 was 120 seats. What would the allocation have been if Hill’s method had been used in 1792?
State State Quota n = Rounded down State Quota Geometric Mean Final #
VA
MA
PA
NC
NY
MD
CT
SC
NJ
NH
VT
GA
KY
RI
DE

*Would Hill’s method have resulted in all 120 seats being filled without revising D? (Check your total for final allotment.)
*Calculate the relative unfairness between the best and worst represented states in Hill’s method.
*How does the unfairness of Hill’s method compare to Hamilton’s and Jefferson’s?
11.
(Extension question #3) Reducing unfairness seems like a good thing in general, but when you are representing a particular state, you might try to vote according to what you think is best for your state even if your choice might not be good for all states.
*What is the average constituency for the best represented state in Hamilton’s method?
*What is that state’s average constituency in the Jefferson method?
*What is that state’s average constituency in the Hill method?
*Which of the three methods provides the best representation for that state?
*Did that method have the smallest relative unfairness out of the three we examined?
*If you were representing that state in Congress, which method would you want and why?