2. 1 Message to the user... The most effective way to use a PowerPoint slide show is to go to “SLIDE SHOW” on the top of the toolbar, and choose “VIEW SHOW” from the pull down menu. OR, using the shortcut toolbar on the bottom left, choose the rightmost icon (“SLIDE SHOW”) Use the spacebar, enter key or mouse to move through the slide show. Use the backspace key to undo the last animation on a slide uUse the “escape” key at anytime to end the show. TEACHERS : If using this show as part of a lecture, it is helpful to go to “PRINT” in the “FILE” menu and use the drop down menu at the bottom left: “PRINT WHAT.” Printing the “OUTLINE VIEW” will be helpful if you intend to view many slides with your class; or you can print a particular slide to use as a handout. (Many shows will include sound… you may want to turn on your speakers!) Apportionment Show #6 of 7

3. 2 Teachers : Because the animation used in these slide shows takes up a large amount of memory (hence downloading time) I will chose NOT to animate much of the text in future shows. Should you decide to use the shows in a classroom, after downloading them, you may wish to animate some of the text, so that it appears on the screen when you need it to. BEFORE running the slide show (staying in “slide view” is most helpful) you can: 1. Go to the slide you wish to animate 2. Choose “SLIDE SHOW” at the top of the toolbar 3. Scroll down to “custom animation” and follow the directions The “Timing” menu will show you what you can choose to animate The “effects” menu will show you the types of animation you can choose Animating the text is useful if your students are taking notes as you lecture

4. 3 More Apportionment Methods... No matter how simple Hamilton’s Method of apportionment is, the paradoxes we noted are often troubling enough to cause the method to be abandoned. Jefferson’s Method of apportionment, although a bit more complicated, does not have any paradoxes associated with it. That’s not to say it’s perfect… We will find that it, also, has a flaw!

5. 4 JEFFERSON’S METHOD of apportionment Introduction Vocabulary & Example

6. 5 What’s the difference??? When using Hamilton’s Method, recall that every state received either its LOWER QUOTA (standard quota rounded down) Or its UPPER QUOTA (standard quota rounded up) for its final apportionment. And recall that we decided which states would receive their UPPER QUOTA by ranking the decimal portion of the standard quota Hence, Hamilton’s Method is referred to as a “method of largest fractions”

7. 6 What’s the difference??? In the examples we did, we never saw a case where the LOWER QUOTAS added up to the correct apportionment right away. There are examples when that does happen… but it doesn’t happen often. Perhaps one of the things people like about Hamilton’s Method is that the STANDARD DIVISOR (total population/ #seats) is so logical to calculate and understand. And this value is used to find the STANDARD QUOTA, which is then used to calculate the correct apportionment.

8. 7 A new DIVISOR??? Wouldn’t it be great if, after using that divisor and finding the LOWER QUOTAS the sum of the LOWER QUOTAS was already the correct apportionment? There is no law that says we have to use that STANDARD DIVISOR. Can’t we find some other divisor that would work so that the sum of the lower quotas for each state adds to the desired value?

9. 8 A new DIVISOR??? Of course, we don’t want this divisor to vary greatly from the Standard Divisor; After all, the sum of the Lower Quotas was never too far off. Let’s just see what happens if we take a look back at a previous example, but use different divisors. (We’ll call them MODIFIED DIVISORS: MD)

10. 9 MATHLAND example Apportionment of the representative body of Mathland

11. 10 A new DIVISOR??? So, with the SD = 100 people/rep. The Lower Quotas add to 175 And we have 176 seats to apportion. What if we try a divisor that is a bit HIGHER than 100? Let’s try 100.5 people/rep. How would that change the quota for each state?

12. 11 Some “modified” vocabulary... Let’s call the new divisors we try: MODIFIED DIVISORS (MD) And the new quotas that occur: MODIFIED QUOTAS (MQ) As before MQ = state population/ MD Create the chart & fill it in... handout

13. 12 Some “modified” vocabulary... MODIFIED QUOTAS (MQ): As before MQ = state population/ MD The sum here is too low

14. 13 Another “modification”... Let’s try a modified divisor LOWER than 100 This time, let’s try 99 Create the chart and do calculations before continuing! HEY! That’s too high!

15. 14 Try again??? Let’s try another modified divisor This time, let’s try 99.2 Create the chart and do calculations before continuing! WOW! That works perfectly!

16. 15 Is “perfection” unique??? Let’s try another modified divisor This time, let’s try 99.18 Create the chart and do calculations before continuing! BINGO! That works, too!

17. 16 What’s the big idea? This idea of finding a “perfect divisor” is the basis of Jefferson’s Method of apportionment. That’s why his method is called a DIVISOR METHOD So we really only need to find ANY MODIFIED DIVISOR that will work the way we need it to. But, hey … There are infinitely many choices...

18. 17 Infinitely many??? That’s a lot! The next few slides will introduce a way to calculate an apportionment using Jefferson’s Method WITHOUT having to GUESS & check your own ideas for MODIFIED DIVISORS. It will always work… so I hope you like it!

19. 18 Jefferson’s Method… handout For this method, you will need to know both the LQ and the UQ The MD#1 will be calculated by dividing the state’s population by the Upper Quota (LQ+1) The MD#2 will be calculated by dividing the state’s population by the LQ+2 Most of the time, MD#2 will not be important

20. 19 Jefferson’s Method… you do the math! The reason for including MD#2 will be evident in the next slide show!

21. 20 Jefferson’s Method… you do the math! Since Jefferson’s Method is a DIVISOR METHOD, you will RANK the states’ entitlement to the “empty” seat(s) using the MODIFIED DIVISORS NOT just the decimal portions this time! The ENTIRE MODIFIED DIVISOR!!!!

22. 21 FINALLY...Jefferson’s Method… Since Algebra has the highest modified divisor, it is awarded the unfilled seat. Just as in Hamilton’s Method, only the “ranked” states receive their UQ One more seat to fill... This is the highest MD

23. 22 And so it goes... Your text does NOT show this method of calculating an apportionment using Jefferson’s Method. It does, however, allude to this method in the reading… Since this is consistent with the chart, calculations and vocabulary used for Hamilton’s Method, I suggest you seriously consider the benefits of using this chart format!

24. 23 Have we found perfection? As mentioned in the beginning, Jefferson’s Method does have a flaw. The problem comes with the educated choice of modified divisors. If a state receives either its UPPER QUOTA or LOWER QUOTA as its final apportionment, it is said to adhere to the QUOTA RULE It seems fair, doesn’t it? There are times, however, when using Jefferson’s Method, that this rule is violated. Another cause for controversy??????

25. 24 End of show #6 Going on?... Apportionment: Show #7: Jefferson’s Method: Quota Rule violation Prepared by Kimberly Conti, SUNY College @ Fredonia Suggestions and comments to: Kimberly.Conti@fredonia.edu