Pick a Winner: Decision Making in a Democracy

Transcription

1 1 UNIT 1 Pick a Winner: ecision Making in a emocracy 2 Video Support LESSON ONE 3 emocratic Elections in the United States LESSON TWO 10 Improving the Election Process TEHER S GUIE 1 18 HNOUTS H1.1 H SUPPLEMENTL TIVITIES S1.1 S SSESSMENT TRNSPRENIES T1.1 T LESSON THREE 13 Making a Point with Point Systems LESSON FOUR 15 Other Ways Unit Summary 17

2 2 VIEO SUPPORT UNIT ONE: PIK WINNER Video Support VIEO SUPPORT With students in groups, have them read the Video Viewing Guide, Handout H1.1, so that they will be ready to answer the questions as they watch the video. Let students work on the questions in their groups during the video. If necessary, stop the video or replay it so that students have adequate time to discuss and answer the questions. (Note: This is the time to begin developing good group habits. e sure to stress the importance of full participation by and understanding of each member of the group.) The questions for the Video Viewing Guide are listed below. Group reporting is an important part of group work, so you may want to have several groups report on their answer to the final question of the viewing guide. Voting examples offered by groups can be saved and examined later in the unit as students learn more about voting methods. 1. How many major candidates were there in the 1992 presidential campaign? 2. What did polls show was the most important issue in the 1992 presidential campaign? 3. What type of poll is used to assess the way people voted? 4. What percentage of votes did President linton receive in 1992? 5. What is the name of the system used to rate television shows? 6. Why are television ratings important? 7. In , what prime-time TV show was ranked #1 among teens? What program ranked second? 8. Name one other situation in which voting occurs, other than presidential elections and television show ratings.

3 UNIT ONE: PIK WINNER LESSON ONE 3 LESSON ONE emocratic Elections in the United States PREPRTION REING Is the System Flawed? See nnotated Teacher s Edition. TIVITY 1 Elections in the United States See nnotated Teacher s Edition. INIVIUL WORK 1 The Plurality Method See nnotated Teacher s Edition. TEHER KGROUN REING 1.1 Preparing Your Students for Group Work In this curriculum, much of the work is done by students in small groups. Much has been written about group work, its forms, and its effectiveness. Without going into all of the benefits of group work, overall, it has been shown to have a positive effect on student achievement. For group work to be most effective, it must be structured, and students need to be prepared. istribute Handout H1.2, Group Rules, to your students at the beginning of the unit. The stages shown below [Farivar & Webb 1994] may help you to prepare of your students.

4 4 LESSON ONE UNIT ONE: PIK WINNER TEHER KGROUN REING 1.1 Stages of Preparation for Effective Group Work STGES N OJETIVES 1. lass-building: ecome acquainted with other students. Learn classmates names and interests. Feel comfortable in class. 2. Learning how to work with others:. asic communication skills with norms for behavior: Listen attentively. Work with classmates without putting them down. Speak politely without yelling. Participate equally with other group members. Understand the difference among cooperation, competition, and working individually.. Team-building: ecome acquainted with teammates. Learn commonalties with teammates. Feel comfortable in the team. evelop a cohesive group.. Small-group social skills: rticulate ideas. Talk about the group s work. Share ideas and information. Encourage teammates to talk and participate. 3. ommunication and cooperation skills: Understand the value of two-way communication. Share the talking and directing. Use teammates as resources. Work as a group without depending on the teacher. 4. Helping skills: heck each other s understanding, not only the accuracy of each other s work. Give specific feedback about teammates work. Give explanations instead of only the answer. Persist in asking for help. Webb, N.M. and S.H. Farivar. re Your Students Prepared for Group Work? Middle School Journal vol. 25, number 3, pp olumbus, OH: National Middle School ssociation. Used by permission of the publisher.

5 UNIT ONE: PIK WINNER LESSON ONE 5 TEHER KGROUN REING 1.2 Several Election Methods Seven voting methods are discussed in this unit. They are plurality, runoff, point methods, sequential runoff, ondorcet, approval voting, and cumulative voting. Most of these methods can be easily applied if voter preferences are known. In practice, this means that none of the methods requires voters to vote more than once if the voters rank the candidates the first time they vote. s an example, consider the set of voter preferences shown in Figure 1. could prevent from winning by withdrawing from the election. In practice, or might demand favors or concessions from other candidates in exchange for withdrawing from or staying in the election. Note: Voter and candidate manipulation are likely to occur because of the plethora of opinion polls that surround modern elections. The Runoff Method This method requires that a second election be held if no candidate gets over half the votes. The runoff is between the top two candidates. In this example,, with 30% of the votes, and, with 28%, are in the runoff. Since the preferences of the voters are known, a second election is not necessary. The votes of those who voted for and are transferred to because those voters rank higher than. wins the runoff by 70% 30%. (See Figure 2.) 21% 28% 9% 20% 22% Figure 1. Note that with four candidates there are 4! = 24 different preferences possible. In this election, the voters expressed only five of 24 preferences. 21% Figure 2. 28% 9% 20% 22% The Plurality Method This method considers first-place votes only. The winner is the candidate with the most votes. Since received 21% + 9% = 30%, which is more than any other candidate, is the winner. IMPORTNT POINTS OUT PLURLITY It works well when there are only two candidates. It also works well when there are more than two candidates and one of them gets a majority (over 50%) of the votes. It can produce a winner ranked last by a majority of voters, as is the case in this example. It can encourage voters to vote insincerely. For example, in this election the supporters of could switch to their second choice,, in order to prevent their lowest-ranked candidate from winning. Of course, the supporters of can choose a similar strategy. It can encourage candidates to manipulate the election. For example, either candidate or candidate The supporters of and ranked higher than, so their votes are transferred to in the runoff. IMPORTNT POINTS OUT RUNOFFS If voters do not rank the candidates, a second election is necessary when no one gets over half the votes. second election is expensive (the taxpayers must pay for it) and it inconveniences voters. It will not produce a winner that is ranked last by a majority of voters, but the winner might be ranked low by a majority. In this example, the runoff winner is ranked third by 63% of the voters. The runoff method can produce a paradox, which is evident in this example. Suppose the 9% who rank first and second change their minds and decide to rank first and second. now has 37% of the votes and only 21%. The candidates in the runoff are and. beats in the runoff. Thus, by getting more votes, went from winner to loser! The key to this paradox is that giving more votes also gives fewer. s opponent in the runoff is no longer,

6 6 LESSON ONE UNIT ONE: PIK WINNER against whom is strong, but, against whom is weak. (See Figure 3.) 21% Figure 3. 28% 9% and have been interchanged on the third preference diagram. (37%) and (22%) are now in the runoff. The votes of the supporters of and are transferred to, and wins the runoff by 63% 37%. The runoff method is also susceptible to manipulation by voters and candidates. In this example, either the voters who support and, or the candidates themselves, can change the election in the same way they can when the plurality method is used. The Point Method When there are four candidates, the point method usually assigns 4, 3, 2, 1 points for first-, second-, third-, and fourth-place rankings, respectively. In this example the point totals for each candidate are: 20% 22% The Sequential Runoff Method The sequential runoff method differs from the runoff method in that it eliminates only one candidate at a time. It can be described in algorithmic fashion: 1. Eliminate all candidates that have no first-place rankings. 2. ount the first-place votes for each candidate. If any candidate has a majority, stop. Otherwise continue. 3. Eliminate the candidate with the fewest firstplace votes. (In the case of a tie, eliminate all those that are tied.) 4. Look at the preferences of the voters who ranked the eliminated candidate first. Transfer their first-place votes to whichever remaining candidate these voters rank highest. 5. Go to 2. Figures 4 6 show the sequential runoff method applied to the example. : 4(21) + 1(28) + 4(9) + 1(20) + 1(22) = 190 : 2(21) + 4(28) + 3(9) + 2(20) + 2(22) = 265 : 3(21) + 3(28) + 2(9) + 3(20) + 4(22) = 313 : 1(21) + 2(28) + 1(9) + 4(20) + 3(22) = % 28% 9% 20% 22% is the winner. IMPORTNT POINTS OUT THE POINT METHO Figure 4. Vote totals: : 21% + 9% = 30% : 28% : 22% : 20% is eliminated. If the points assigned to each place are changed, the winner can change. In this example, if 13, 3, 2, 1 are used instead of 4, 3, 2, 1, wins. In some cases, the point method produces a winner that is ranked first by no voters. The point method is very susceptible to voter and candidate manipulation. In this example, the 28% who rank first can give the election to by ranking last instead of second. Point systems are sometimes called orda counts after Jean-harles de orda, who proposed the system in the eighteenth century. 21% 28% 9% 20% 22% Figure 5. The first-place votes of s supporters are transferred to. The totals are now: : 21% +9% = 30% : 28% : 22% + 20% = 42% is eliminated.

7 UNIT ONE: PIK WINNER LESSON ONE 7 21% 28% 9% 20% 22% 21% 28% 9% 20% 22% Figure 6. The vote s of s supporters are transferred to. The totals are now: : 21% + 9% = 30% : 22% + 20% + 28% = 70% wins. Figure 8. Select another candidate, say,. You have already seen that can beat, so compare to. is ranked higher than by 37%, so cannot beat and cannot be the ondorcet winner. IMPORTNT POINTS OUT THE SEQUENTIL RUNOFF METHO It does not always produce the same winner as the runoff method. However, it does if there are only three candidates. Like the runoff method, it sometimes results in the paradox of turning a winning candidate into a loser when the candidate gets more votes. It is also subject to manipulation by voters and candidates. The ondorcet Method The ondorcet method requires that the winner be able to beat every other candidate in one-on-one races. In other words, the ondorcet winner must be able to beat every other candidate in a runoff. 21% 28% 9% 20% 22% Figure 9. Select another candidate, say,. ompare to. is ranked higher than by 70%, so beats. You have already seen that beats, so go on to compare to. In theory, the ondorcet method requires that every pair of candidates be compared. (With n candidates there are n 2 = n(n 1)/2 comparisons.) In practice, however, quite a few of the comparisons can be skipped. Figures 7 10 show the ondorcet method applied to the example. 21% 28% 9% 20% 22% Figure 10. is ranked higher than by 80%, so beats. Since beats each of the others, is the ondorcet winner. 21% 28% 9% 20% 22% Figure 7. Select a candidate, say,. ompare to another candidate, say,. is ranked higher than by 30% of the voters. cannot beat, so cannot be the ondorcet winner.

8 8 LESSON ONE UNIT ONE: PIK WINNER IMPORTNT POINTS OUT THE ONORET METHO It requires many comparisons, so it is time-consuming to do by hand. It sometimes does not produce a winner at all because there are circumstances in which no candidate can beat every other candidate. It sometimes produces a winner that is not ranked first by any voters. It is less susceptible to manipulation than the plurality, runoff, point, and sequential runoff methods. pproval Voting pproval voting, like plurality, does not require the voters to rank the candidates. The difference is that approval voting allows voters to vote for as many candidates as they choose. Since approval voting does not require ranking, it is necessary to make some assumptions about the choices of voters in order apply approval voting to this example. (See Figure 11.) 21% 28% 9% IMPORTNT POINTS OUT PPROVL VOTING It is a very recently (1970s) proposed alternative. It is less susceptible to manipulation than other methods. 20% 22% Figure 11. ssume that some voters would approve of their first two choices and some would approve of only their first choice. The totals are: : 21% + 9% = 30% : 28% : 28% + 20% + 22% = 70% : 20% + 22% = 42% wins. umulative Voting umulative voting serves a different purpose from the previously discussed methods. It is designed for use in situations in which there is more than one office-holder elected. For example, it has been proposed to elect members of the United States House of Representatives. It has been proposed for this and other purposes as a way of guaranteeing minority representation. One way to guarantee minority representation is to divide a state into districts so that some of the districts are composed primarily of minority members. For example, if a state has five representatives and 40% of the state s population are black people, then the district lines are drawn so that two of the districts have over 50% black people. nother way to guarantee representation to various groups is to give them the same percentage of seats as they receive votes in the election. This practice is used by many European countries to distribute seats in parliaments. Recent court decisions in the United Sates have frowned on the practice of drawing district lines to encourage minority representation if the districts that result have other unusual characteristics. Thus, cumulative voting has been proposed as an alternative to districting. gain, consider the state that has five representatives. If cumulative voting is used, there are no districts. ll five representatives are elected at large. Each voter has five votes and can distribute them in any way the voter chooses. For example, the voter can cast all five votes for one candidate. minority could ensure itself representation by running a relatively small number of candidates and distributing its votes among those candidates only. Mechanically, cumulative voting is similar to plurality and approval in that it does not require that voters rank the candidates. pproval voting lets voters cast as many votes as they like, but cumulative voting limits the number of votes per voter to the number of seats being filled. umulative voting lets voters cast more than one vote for a single candidate, but approval voting does not. Some political scientists believe it would result in better showings by third-party candidates. Some political scientists believe it would result in better voter turnout.

9 UNIT ONE: PIK WINNER LESSON ONE 9 TEHER KGROUN REING 1.3 The History of Election Methods Modern concern with voting methods began with the merican experiment in democracy, which generated considerable interest in eighteenth-century Europe. mong those whose attention it attracted was the Marquis de ondorcet ( ), a French mathematician, philosopher, and revolutionary. In 1785, ondorcet proposed a new voting system in hope of finding one that would employ his sense of democratic ideals. His idea of selecting the candidate that could beat all others in head-to-head contests seemed to be the solution, but it quickly ran into problems. cyclic paradox was evident in close races when none of the candidates was stronger than the others. For example, might beat, beat, and might then beat. The major problem with his method was that it often didn t produce a winner. Jean-harles de orda ( ), a French cavalry officer, naval captain, mathematician, and friend of ondorcet, devised a voting method of his own. s with ondorcet, orda s search for the ideal method was brought about by his dissatisfaction with the plurality method. It became apparent, however, that orda s method was easily manipulated, prompting orda s response, My scheme is intended only for honest men. The search for the ideal election method continued until the 1950s when Kenneth J. rrow, an economist at Stanford University, conjectured that no voting scheme could be perfect. rrow developed a set of five basic criteria for a good voting system and shocked the world by proving that no voting method could always adhere to his five criteria. rrow s criteria: 1. The preferences of no single voter should determine the election. In other words, there should be no dictator. 2. Each voter should be allowed to order the preferences in any way (including ties). 3. If every voter prefers one candidate to another, than the preferred candidate should finish higher than the other. 4. ny method should give the same results each time it is applied to the same set of preferences. The result should be transitive. (If is ranked higher than and higher than, then should be ranked higher than.) 5. The result between any pair of candidates should not depend on the voter preferences for the remaining candidates. Most of rrow s criteria seem like common sense. For example, the fourth criterion says the method should give the same result each time it is applied to the same set of preferences. method isn t likely to violate this condition unless the method is a lottery that puts the voter preferences in a container and draws one. The fifth criterion is the one that requires the most thought. Put slightly differently, to determine whether beats, it should not be necessary to know how the voters feel about. If, for example, the outcome between and depends on whether is in the race, then the fifth criterion is violated. There are several examples in this unit of situations in which the outcome between two candidates changes when a third leaves or enters the race. (For example, see Supplemental ctivity S1.6.) rrow s result, which helped him win the 1972 Nobel Prize in economics, does not mean we should give up the search. To use rrow s own analogy, one does not give up trying to improve the efficiency of the internal combustion engine just because the engine can never achieve perfect efficiency. pproval voting, which was proposed independently by several people in the 1970s, is a promising new method developed since rrow proved his theorem.

10 10 LESSON TWO UNIT ONE: PIK WINNER LESSON TWO Improving the Election Process PREPRTION REING an the System e Improved? See nnotated Teacher s Edition. TIVITY 2 Finding a etter Way See nnotated Teacher s Edition. TIVITY 3 Trying Them Out See nnotated Teacher s Edition. INIVIUL WORK 2 Popular Election Method See nnotated Teacher s Edition. SUPPLEMENTL TIVITY S1.1 The 1912 Presidential Election Use this activity as a quiz or for additional practice. It is best used after students have been introduced to point systems in Lesson 3, but it can be used earlier if you omit the items that refer to point systems or if your students invent point systems in Lesson 2. SUPPLEMENTL TIVITY S1.2 Throw the ums Out Use this activity if students propose to cure the plurality method s flaw by disqualifying any candidate who is ranked last by a majority of voters. It can also be used as a supplemental activity or for assessment after Lesson 2. SSESSMENT PROLEM 1.1 Find the Ranking To be used after Lesson 2. The problem offers a graphical presentation that is slightly different from what students have seen in the unit. nswers representing both higher and lower levels of thinking are provided. SSESSMENT PROLEM 1.2 The Student oard To be used after Lesson 2. The voting system is democratic: if the less powerful students (the lower grades) unite, they are able to win the first round of voting. If they don t succeed in forming a voting block, there will be a second round and the more powerful students (the upper grades) will have more influence. This system could be deemed more fair than it would be to apply the weights in the first round. This two-round system is more fair than a one-round system using weighted voting. The system is manipulative because it is easy to change voters in favor of a proposal into voters against a proposal just by wording the proposal differently. Items 4 and 5 prepare the student for Item 6. The students show good understanding of this problem if they see in Items 4, 5, and 6 that they have to start with counting votes of 12th-grade students. Students using trial and error to find answers are working at a lower level than students using systematic thinking (like in ssessment Problem 1.1).

11 UNIT ONE: PIK WINNER LESSON TWO 11 TEHER KGROUN REING 1.4 Specialty Software for this Unit: Election Machine Election Machine is designed to facilitate classroom elections. The program is available in both Macintosh and OS versions. oth versions use keyboard input (not mouse input). Students can choose an issue, rank the choices individually, and print and analyze the results. menu offers results by five methods: plurality, runoff, a point system, ondorcet, or sequential runoff. The main menu of the program is shown in Figure 12. Options 3 and 4 let you save your election to disk and recover it later. This is useful if voting is interrupted, but you should always save your work regularly. (You may want to examine the results again later or let additional students vote.) Option 6 prints the preferences of all the voters. You can give a copy of the printout to each student and ask students to analyze the results. The printout in Figure 14 shows the preferences of four voters. Figure 12. Figure 13. Option 1 allows the entry of candidate names. Option 2 is used to vote on the entered candidates. Prior to having students vote, you may want to use option 5 to print a ballot. Give a copy to each student and have them prepare their ballot in advance of going to the polls. lternately, you can collect the ballots from students and enter them yourself. voter ranks four soft drinks (Figure 13). (The program does not let the voter use the same rank twice.) The program asks the voter to confirm that the ranking is correct. If the voter says no, the program asks the voter to vote again. Note that the number of the current voter is shown at the lower right of the screen. When the voter finishes, the computer produces a sound to discourage voting twice. To terminate voting, press RETURN after the last voter has finished. Figure 14. Option 7 presents a submenu from which you can determine the winner by any of the five methods used in this program. This option saves you the time needed to determine the winner, and thus serves as a kind of answer key for your class data.

12 12 LESSON TWO UNIT ONE: PIK WINNER TEHER KGROUN REING 1.5 Specialty Software for this Unit: Preference iagram Preference iagram is designed to allow rapid manipulation of voter-preference data. It can be used by students to study the effects of changes on election results, to enhance classroom demonstrations, or to develop new sets of data to use in the classroom. The program is available in both Macintosh and OS versions. oth versions use keyboard input (not mouse input). menu offers results by five methods: plurality, runoff, a point system, ondorcet, or sequential runoff. nother menu displays the preference diagrams and lets you manipulate the election by changing one of the diagrams, eliminating one of the diagrams, or adding a new diagram. The program is designed for speed. It requires that the candidates be named with upper-case letters starting with. (Set your keyboard s aps Lock before starting the program.) Whenever possible, the program does not require that RETURN be pressed after entry. n exception to this rule is the vote total attached to a diagram because the totals are often more than a single digit. The program requires that vote totals be whole numbers, so percentages should be rounded to the nearest whole number. Figure 15 shows a set of three preferences with three candidates. Figure 16. fter all preferences have been entered, you can go to a menu that includes options for finding the winner by five different methods, ending the program, starting over, or redisplaying the diagrams. You can also choose to make changes in the diagrams. If you choose the menu option and decide to determine the ondorcet winner, you can choose to display a table that shows how each candidate did against each of the others. (See Figure 16.) Read the candidate s row in the table to see how the candidate did against each of the others. For example, candidate loses to, but beats. runoff diagram is easy to construct from this table. For example, since beats, an arrow would point from to. If you choose the change option (for the order of the candidates and/or the vote total), you can change one of the diagrams, add a diagram, delete a diagram, or jump to the menu to determine winners. (See Figure 17.) Figure 15. Figure 17.

13 UNIT ONE: PIK WINNER LESSON THREE 13 LESSON THREE Making a Point with Point Systems PREPRTION REING Proving a Point See nnotated Teacher s Edition. TIVITY 4 re Runoffs the nswer? See nnotated Teacher s Edition. INIVIUL WORK 3 Point Systems See nnotated Teacher s Edition. SUPPLEMENTL TIVITY S1.3 Point ounts with Replay Use this activity with calculators with a replay feature. It outlines the steps for doing point counts with such calculators. SUPPLEMENTL TIVITY S1.4 Point ounts with Matrices Use this activity with calculators with matrix features. It outlines the steps for applying matrix multiplication to point counts. SUPPLEMENTL TIVITY S1.5 Point ounts with Spreadsheets Use this activity with computers with spreadsheets. It outlines the steps for doing point counts with spreadsheets. SSESSMENT PROLEM 1.3 The est Temperature Use after Lesson 3. The student is confronted with a completely new representation of a preference diagram. Rather than grading this activity, you may wish to let the students get used to playing with what they have learned. POSSILE EXTENSION FOR SMLL-GROUP INVESTIGTION Let the students add preference profiles of two more voters to the five that are given in Figure 2 on their page. What they should find out is that, no matter what the shape of these profiles, there is always a well-defined winner in the pairwise voting system. Looking at the optimal temperatures of all voters, the median one is the winner. fter investigating this phenomena for a number of different profiles, the students can be asked to find arguments for why this is true. This is not easy because they have to understand that the optimal temperature comes after a line segment that is going up, while the line segment is always going down after reaching the optimum. This fact can be used to find out why the median optimum always is the winner, given an odd number of voters. This law is very important in voting theory, because this type of preference profile (single-peaked preferences and an ordinal scale for the different candidates) doesn t suffer from the voting paradox.

14 14 LESSON THREE UNIT ONE: PIK WINNER SSESSMENT PROLEM 1.4 Tournament Winners and Losers Use after Lesson 3. In this problem, the matrix representation is used in a slightly different way than the students use it in the unit. They have to read carefully to understand what is happening. n important point in answering Item 2 is that the student be consistent in the method he uses: if the student argues that beat so that will be ranked first, then consequently would be higher in ranking than E, because beat E. The student deserves full credit only if she is consistent, no matter what method she uses. SSESSMENT PROLEM 1.5 The TWINKLE Taste Test Use after Lesson 3. Part 1 brings up a mathematical topic that is in the unit, but is not directly addressed: how many different rankings are possible with two, three, or four choices. In this problem, the students play with this topic a bit more. It might be better not to grade this; as with Problem 1.3, you may prefer to use it for additional practice. Part 2 makes a good group activity.

15 UNIT ONE: PIK WINNER LESSON FOUR 15 LESSON FOUR Other Ways PREPRTION REING onsider the lternatives See nnotated Teacher s Edition. TIVITY 5 re Point Systems the nswer? See nnotated Teacher s Edition. TIVITY 6 How to hoose an Olympic ity: Sequential Runoffs See nnotated Teacher s Edition. SUPPLEMENTL TIVITY S1.6 Enter Zalinski This activity considers the effect of the entrance of a new candidate on an election. It demonstrates one of the flaws of point systems. SUPPLEMENTL TIVITY S1.7 Is There Hope in Hybrids? This activity considers the possibility of combining the runoff and point systems. It shows that the new method suffers from, rather than eliminates, the flaws of both methods. TIVITY 7 Round and Round We Go: The ondorcet Method and Pairwise Voting See nnotated Teacher s Edition. TIVITY 8 No More Ranking: pproval Voting See nnotated Teacher s Edition. TIVITY 9 Put More Power in Your Votes: umulative Voting See nnotated Teacher s Edition.

16 16 LESSON FOUR UNIT ONE: PIK WINNER SSESSMENT PROLEM 1.6 Where to Go? Use after Lesson 4, if all of the parts are used. This problem covers all of the voting methods the students learned in the unit and tests the level of the students critical thinking (Items 6, 13, 15, and 19). Justification is always a very important part of the answer to each question. The first two questions are do questions, to give students a comfortable start. In these and several other items, students apply what they learned in the unit. Items 6, 15, and 19 are on a somewhat higher level: the student has to give arguments, so he must really think of what he is doing. Item 13(b) is on an even higher level. Items in which students are asked to prove something can be challenging, so don t expect every student to answer these. SSESSMENT PROLEM 1.7 Two Rounds of Voting Use after Lesson 4. This problem introduces a new way of voting: choosing between one option and all the alternatives together. epending on the outcome, there may be a next step. tree diagram is introduced to support the written description of the two-step process. The items are increasingly challenging. In Item 3 the students have to explain why a given group will be unhappy; in Item 5 they have to think of such a group themselves. In Item 7 the students have to explain why they d do better to vote in the way given in the text; in Item 8 they have to think of an alternative themselves. For Item 11, if a student comes up with the argument that the ondorcet winner will always be found using the last type of agenda, she deserves full credit.

17 UNIT ONE: PIK WINNER UNIT SUMMRY 17 UNIT SUMMRY Pick a Winner: ecision Making in a emocracy Wrapping up Unit One Handout H1.3 is a list of potential student projects. It can be duplicated and given to students if you are assigning individual or group projects in the unit. Encourage students who have Internet access to use the Internet for research. Internet searches often turn up hundreds or even thousands of documents related to a topic. site that has information on a topic often has links to related sites. For an example, visit the enter for Voting and emocracy at Glossary See nnotated Teacher s Edition. SUPPLEMENTL TIVITY S1.8 Final Project: How o You Vote? This is a summary activity. It asks students to list criteria for a good election method and describe a method they think best fits those criteria. Mathematical Summary See nnotated Teacher s Edition.

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