Warm Up – 3/3 - Monday Describe in your own words the steps to each of the following fair division processes and when to use each process. Divider-Chooser Lone-Divider Lone-Chooser Method of Sealed Bids

Quiz Review

Test Wednesday Fair Division Divider-Chooser (Pizza, Subs, and Cake) Definition of fairness Individual’s fair shares Divider-Chooser (Pizza, Subs, and Cake) Lone-Divider Lone-Chooser Method of Sealed Bids

Apportionment Based on how much the kids worked, how much candy would you give each kid? (Remember, candy is indivisible).

Terminology

Example #1 Find the standard divisor (give units) Find the standard quota for each child.

Example #1: Solution The standard divisor is the total population divided by the number of seats (in this case pieces of candy) 900 50 =18 min 𝑝𝑒𝑟 𝑝𝑖𝑒𝑐𝑒 𝑜𝑓 𝑐𝑎𝑛𝑑𝑦 I take each person’s minutes and divide them by the standard divisor to see how much candy they should receive.

Apportionment

Example #2: The Tophat republic is a small country consisting of four provinces. 𝑃 1 3,310,000 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑃 2 2,670,000 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑃 3 (1,330,000 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛) 𝑃 4 690,000 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 If there are 160 seats in Congress: A) Find the standard divisor B) Find the standard quota for each Province.

Example 2 Solution Adding up the four populations gives us 8,000,000 people. We divide this by 160 seats to get: 8,000,000 160 =50,000 𝑝𝑒𝑜𝑝𝑙𝑒 𝑝𝑒𝑟 𝑠𝑒𝑎𝑡 To get each states quota, we divide their population by the SD. 𝑃 1 = 3,310,000 50,000 =66.2 𝑠𝑒𝑎𝑡𝑠 𝑃 2 =53.4 𝑠𝑒𝑎𝑡𝑠 𝑃 3 =26.6 𝑠𝑒𝑎𝑡𝑠 𝑃 4 =13.8 𝑠𝑒𝑎𝑡𝑠

Apportionment Homework

Hamilton Method Alexander Hamilton came up with a very simple way of assigning seats and dealing with those extra 1/ 3 ′ s and 2/ 3 ′ 𝑠 of seats (residues).

Hamilton’s Method of Apportionment

Example #1 We found the standard quota’s to be: A:8 1 3 B: 4 1 3 C: 9 11 18 D: 11 1 3 E: 16 7 18 We give each person their lower quota: A: 8 B: 4 C: 9 D: 11 E: 16 This is a total of 48 pieces of candy!

Example #1 We then give the extra pieces to the kids with the highest residues (the biggest fraction or decimal). In this case 11 18 and 7 18 are the biggest residues. Therefore Connie and Ellie will each get an extra piece and our final apportionment is: A: 8 B: 4 C: 10 D: 11 E: 17