Section 9.1

The Apportionment ProblemSection 9.1Objectives:Understand and illustrate the Alabama paradox.Understand and illustrate the population paradox.Understand and illustrate the new-states paradox.Understand Balinski and Youngs Impossibility Theorem.Textbook:Page 508 509/Understanding ApportionmentRead through to Table 9.1In your own words, succinctly tell what this passage is about.

Example 1:Identify the integer and the fractional part.

21.075Example 2:Identify the integer and the fractional part.

0.567Using the Hamilton MethodThe Hamilton method uses fractional parts to apportion representatives.Determine the exact number of board members: percent of stockholders X size of boardAssign the integer partIf there are more members to be allocated, then go to step 3.Assign additional members according to the fractional parts The 1st additional member goes to the company having the largest fractional partThe 2nd additional member, if any, goes to the company with the 2nd largest fractional part. Continue in this manner until you have assigned all additional members.

Example 3:Using the Hamilton Apportionment Method12 Member BoardCompanyPopulationDetermine PercentageStep 1:Board Members DeservedStep 2:Assign Integer PartNaxxon47Aroco37Eurobile16Total100Example 4:TB pg. 517/4Jungle World Theme ParkEmployeesPopulationPercent of EmployeesStep 1:Members DeservedStep 2:Assign Integer PartPerformersFood WrkrsMaintenance WrkrsTotalSection 9.1 AssignmentTB pg. 516/1 7 oddRemember to write problems and show ALL work.Section 9.1 Part IIAlabama Paradox and Truncating the Fractional Part of a NumberKey Terms:Alabama Paradox an increase in the total number of items to be apportioned results in the loss of an item for a group.

Apportion to divide according to a plan; to allot.

Truncate to shorten by cutting off.Note: sometimes it is necessary to truncate a number to keep the percentage from adding up to more than 100%.Alabama Paradox:Illustrating the Alabama Paradox:A small country with a population of 10,000 is composed of 3 states. According to the countrys constitution, the congress will have 200 seats, divided among the 3 states according to their respective populations.Illustrating the Alabama Paradox.StatePopulation% of Pop.Step 1:Step 2:Step 3:A5015B4515C470Total10,000Example 5:Using Hamiltons Method determine deserved seats. Illustrating the Alabama Paradox.StatePopulation% of Pop.Step 1:Step 2:Step 3:A5015B4515C470Total10,000Example 6:What happens if the number of seats in congress increases to 201.Illustrating the Alabama Paradox.StatePopulation% of Pop.Step 1:Step 2:Step 3:A5015B4515C470Total10,000Section 9.1 Assignment Part 1TB pg. 517/11 and 12 (worksheet online)Remember to write problems and show ALL work.Example 7: Alabama ParadoxUsing Hamiltons MethodAssume that there are now 10 members on the board. Oil Consortium BoardCompany% Stockholders(in thousands)Step 1:Step 2:Step 3:Naxxon47Aroco37Eurobile16Total100Example 8: 183.6574893Truncate the number to:

Hundredths

Ten thousandths

Example 9: 284.135792753Truncate the number to:

Tenths

Thousandths

Example 10: Presenting Survey ResultsAdjusting a list of numbers.Exact Percentage (column A)Percentage rounded to tenths (column B)Original Date Truncated to TenthsPart That is DiscardedFinal ResultsTaxes34.4235Education13.456Crime14.75Health Care37.3705Total100Example 11: Adjusting a list of numbersA group of consumers was asked how they expected their spending to change in the next six months. Adjust the percentages in the following table so that they are shown to the tenths place and their sum is 100.00%Exact %% Rounded to tenthsOriginal Data TruncatedDiscarded PartFinal ResultsSpending Increase32.029Spending Decrease24.733Spending Stay the Same22.7489Unsure20.4891Total100.00%Historical HighlightTB pg. 513/Apportionment U. S. HistorySection 9.1 Assignment Part 2Class work:TB pg. 517/13 22 Remember you must write problems and show ALL work to receive credit for this assignment.Section 9.1 Part IIIAverage Constituency, Absolute Unfairness, and Relative UnfairnessKey Term:Average Constituency the quotient:

population of state number of representatives from state

NOTE: Comparing the representatives of two states A and B, we saw that state A is more poorly represented than state B, if the average constituency of A is larger than the average constituency of B.Example 12: Average ConstituencyFinding the average constituency

If the 420-member electricians union has three representatives on the United Labor Council, what is the average constituency of this group?Example 13: Average ConstituencyDetermine which group is more poorly represented.If the 420-member electricians union has three representatives on the United Labor Council, what is the average constituency of this group?If the 440-member plumbers union has four representatives on the council, are the electricians or the plumbers more poorly represented?Key Term:Absolute Unfairness (of a state) the difference between the larger average constituency and the smaller one. If State A has the larger average constituency, then the absolute fairness is:

(avg. constituency of A) (avg. constituency of B)

NOTE: if the two states have the same average constituency, then we say that the two states are equally well represented.27Example 14: Find Absolute UnfairnessAssume that state X has a population of 974,116 with four representatives and state Y has a population of 730,779 with three representatives. Compute the absolute unfairness for this apportionment.Example 15: Find Absolute Unfairness Suppose the Weavers Guild, with 1,672 members, has six delegates on the National Art Commission and the Artists Alliance, with 1,535 members, has five delegates. Calculate the absolute unfairness for this assignment of delegates.Key Term:Relative Unfairness (for 2 states) the quotient:

the absolute unfairness of the apportionment the smaller average constituency of the two states

Example 16: Determine Relative UnfairnessIf state A has a population of 11,710 and five representatives and state B has a population of 16,457 and seven representatives, calculate the relative unfairness of the apportionment.Example 17: Determine Relative UnfairnessSuppose state A has a population of 935,000 and five representatives, whereas state B has a population of 2,343,000 and eleven representatives. Determine the relative unfairness of this apportionment.Section 9.1 Assignment Part 3TB pg. 518/23 - 32Remember to write problems and show ALL work.