1. 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

2. Quiz Review

3. 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

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

7. Terminology

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

9. Example #1: Solution
The standard divisor is the total population divided by the number of seats (in this case pieces of candy)
=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.

10. Apportionment

11. 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 ,000 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
If there are 160 seats in Congress:
A) Find the standard divisor
B) Find the standard quota for each Province.

12. Example 2 Solution
Adding up the four populations gives us 8,000,000 people. We divide this by 160 seats to get: 8,000, =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

13. Apportionment Homework

14. 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).

15. Hamilton’s Method of Apportionment

16. Example #1
We found the standard quota’s to be: A:8 1 3 B: C: D: E: 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!

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