Lesson 7

This is the first of five activities about elections where there are more than two choices. This introductory activity gets students thinking about the fairness of a voting rule. If the choice with the most votes wins, it’s possible that the winning choice was preferred by only a small percentage of the voters.

Students work alone and share solutions with whole class.

Poll the class about the answers. (40% of the class voted for softball, so a majority of the class did not vote for softball as their first choice.) Ask if it is possible to determine whether softball was a highly rated choice by those who voted for another field day activity. (Not without holding another vote.) 

In this voting system the plurality wins, the choice with the most votes, even if it is less than 50%.

This activity presents a method for deciding the winner of an election with more than two choices: runoff voting. If no choice has a majority of votes, then one or more choices with the fewest votes are eliminated and another vote is held between the remaining choices. Repeat until one choice gets a majority of the votes.

Students learn the technique of analyzing the results by holding their own vote. A fictitious story (choosing a company to supply school lunches) is provided for students to vote on a situation with four choices, each of which may have some positive and negative aspects. They follow two different systems of voting rules to see how results can differ depending on the rule system used. Students use quantitative reasoning (MP2) to analyze and compare two different voting rules.

Note: This activity includes a lot of teacher-directed voting activity. Students periodically stop to record information and determine the winner, or the need to do another round of voting, or reflect on the results.

Here is the situation to vote on: Imagine the kitchen that usually prepares our school lunches is closed for repairs for a week. We get to choose which of four catering companies to feed everyone for a week. You can choose only one caterer. The school has found four catering companies that will supply a week of lunches for everyone. No changes in the menus are possible.

Make sure students understand the situation. Students vote by drawing symbols next to the four menu choices or on pre-cut voting slips of paper. 

Voting System #1. Plurality: Conduct a vote using the plurality wins voting system:

If there is an even number of students in the class, vote yourself to prevent a tie at the end. Ask students to raise their hand if the lunch plan was their first choice. Record the votes in a table for all to see. 

Students work through questions 1 and 2 in the activity in groups. Then they discuss question 3 as a whole class.

"How could we measure how satisfied people are with the result? For example, people whose top choice was the winner will be very satisfied. People whose last choice was the winner will be very dissatisfied."

Students vote with a show of hands, and record the votes.

Voting system #2: Runoff

Students work through questions 4–6 alternating between conducting the next round of voting as a whole class and analyzing the results in their groups. "Use your same choices that you recorded. We’ll count the votes in a different, more complicated way. If one choice did not get a majority, we hold a runoff vote. Eliminate the choice that got the fewest votes. Then we vote again. If your first choice is out, vote for your second choice."

• Record the votes in a table like the first one, except that one of the choices will be gone.
• "Did the same choice get the most votes both times?" (Sometimes no. Results may vary in your class.)
• "Did one of the choices get a majority?" (If so, that choice wins. If not, eliminate the choice with the fewest votes and vote again. Repeat until one choice gets a majority of the votes.)

Again ask for satisfaction with the results of the voting. Record numbers in column 2.

Students compute percentages in the last column and work on question 7 in groups.

Arrange students in groups of 2–4.

Introduce the situation: "When there are more than two choices, it’s often hard to decide which choice should win. For example, in the field day question, softball got the most votes, but only 40% of the votes were for softball and 60% were not for softball. But were these really votes against softball, or did some of those people like softball, but just liked another choice more?

In this lesson, we’ll try two voting systems. We’ll vote on an imaginary situation: choosing a caterer to supply student lunches."

See the Activity Narrative for instructions for conducting the activity.

Students may not know what some of the foods are. You can either explain, or tell them that it’s up to them to vote for unknown foods or not.

• Liver is an internal organ, not muscle meat. Many people don’t like it.
• Hummus is a bean dip made of chickpeas. Pita is middle eastern flatbread.

The voting rules are somewhat complicated. Acting out the voting process should make things more clear.

Ask students which system seems more fair, plurality or runoff. (The plurality system doesn’t take second, third, etc., choices into account, while the runoff system does.)

In an election in Oakland, California, a candidate won by campaigning to ask voters to vote for her for first choice, and if she was not their first choice, then put her as their second choice. Is this like what happened with one of the votes we analyzed?

In this activity students revisit the situation from the previous activity but they analyze the votes of a different class. In this case the members of different student clubs all voted for the same lunch option. Students repeat the process of the run-off election on the provided data and compare it to a plurality vote. They use quantitative reasoning (MP2) to analyze and compare the two different voting rules.

Students must think through the voting process and determine which choice is eliminated at each round, and what votes the club presidents will turn in at every round of voting.

Arrange students in groups of 4. Tell students that they analyze the results of the vote from a different class for the same lunch caterer situation from the previous activity. Tell them, "There are four clubs in this other class, and everyone in each club agrees to vote exactly the same way, as shown in the table."

Have each group of four act out the voting for this class: each person is the president of a club, and delivers the votes for all the club members. Demonstrate with a group, "This person is the president of the barbecue club. Tell us how many votes you are turning in, and for which choice."

Give students 10 minutes to work through the questions with their group, followed up with whole-class discussion.

The voting rules are somewhat complicated. Acting out the voting process should make things more clear.

Ask for results from the fictitious class.

Notice that after the first round, Meat seemed to be winning. After the second round, Vegetarian seemed to be winning. But the actual winner was Something for Everyone.

Ask students: 

• How did the results of this class compare to our own class?
• What are some advantages and disadvantages of plurality and runoff voting? (Plurality takes less effort but could be less fair.)
• Which system seems more fair, plurality or runoff? (The plurality system doesn’t take second, third, etc., choices into account, while the runoff system does.)

This activity presents another method for choosing among three or more choices when none wins a majority, instant runoff voting. Voters again rank their choices. Each choice is given points, with 0 for the last choice, 1 for the next to last, and so on. The choice with the most total points wins, and no runoff elections are needed. Students use quantitative reasoning (MP2) to compare two models of fairness in voting (MP4).

Arrange students in groups of 2–4.

Introduce another method of voting: "The runoff system sometimes needs more than one election if no choice got a majority. If we are all here together, that’s not a big problem. But if it were a vote with everyone in a city, or county, or state, it would be too complicated and expensive to have more than one vote. The instant runoff system gives each vote points 0 for the last choice, 1 for the next to last, and so on. The choice with the most total points wins, and no runoff elections are needed. Let’s redo our election using this system, and then see what the other class’s votes would choose. Remember what your choices were for the first time we voted for school lunch providers. Write down the points for each of the choices."

Then either ask for votes by hand-raising or ask students to come to the board to record their choices.

Raising hands: "Raise your hand if you gave Meat 3 points (count and record in table). Raise your hand if you gave Meat 2 points, etc." There will be 16 categories. Everyone should raise their hand 4 times.

Come to the board: Have students come up and record their numbers. Each student should have a 0, a 1, a 2, and a 3, one in each category. The table below shows the points for two students’ choices.

After the results are recorded for all to see and students understand the presented information, students work in groups and answer the questions in the activity statement.

Students may not remember how to do the satisfaction survey. Remind them of the work done in the previous activity. Ask them to fill out a similar table:

Poll students about the results of the instant runoff vote. Ask:

• How do the three voting methods we have seen compare?
• Which method should we use the next time our class has to make a decision? Why?

We have seen several methods for fairly deciding between more than two choices. There is no single fairest method. Some methods give one winner, others a different winner with the same vote.

This voting activity helps students summarize the voting systems for more than two choices that were discussed in the previous lessons. Five students vote on three choices of weekend activities. In this activity, students engage in quantitative reasoning (MP2) to compare two mathematical models for fairness in voting (MP4).

Arrange students in groups of 2–4.