Lesson 7
More than Two Choices
7.1: Field Day (5 minutes)
Optional activity
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.
Launch
Students work alone and share solutions with whole class.
Student Facing
Students in a sixthgrade class were asked, “What activity would you most like to do for field day?” The results are shown in the table.
activity  number of votes 

softball game  16 
scavenger hunt  10 
dancing talent show  8 
marshmallow throw  4 
no preference  2 

What percentage of the class voted for softball?

What percentage did not vote for softball as their first choice?
Student Response
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Activity Synthesis
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%.
7.2: School Lunches (Part 1) (30 minutes)
Optional activity
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 teacherdirected 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 precut voting slips of paper.
choice  symbol 

A. Meat Lovers  
B. Vegetarian  
C. Something for Everyone  
D. Concession Stand 
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.
lunch plan  number of votes 

A. Meat Lovers  
B. Vegetarian  
C. Something for Everyone  
D. Concession Stand 
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.
what choice did you rank the winner?  number of people  % of people 

top choice (star)  
second choice (smiley)  
third choice (square face)  
last choice (X) 
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.
What choice did you rank the winner?  Number of people  % of people 

top choice (star)  
second choice (smiley)  
third choice (square face)  
last choice (X) 
Students compute percentages in the last column and work on question 7 in groups.
Launch
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.
Supports accessibility for: Visualspatial processing; Organization
Design Principle(s): Cultivate conversation; Support sensemaking
Student Facing
Suppose students at our school are voting for the lunch menu over the course of one week. The following is a list of options provided by the caterer.
Menu 1: Meat Lovers
 Meat loaf
 Hot dogs
 Pork cutlets
 Beef stew
 Liver and onions
Menu 2: Vegetarian
 Vegetable soup and peanut butter sandwich
 Hummus, pita, and veggie sticks
 Veggie burgers and fries
 Chef’s salad
 Cheese pizza every day
 Double desserts every day
Menu 3: Something for Everyone
 Chicken nuggets
 Burgers and fries
 Pizza
 Tacos
 Leftover day (all the week’s leftovers made into a casserole)
 Bonus side dish: pea jello (green gelatin with canned peas)
Menu 4: Concession Stand

Choice of hamburger or hot dog, with fries, every day
To vote, draw one of the following symbols next to each menu option to show your first, second, third, and last choices. If you use the slips of paper from your teacher, use only the column that says “symbol.”
 Meat Lovers __________
 Vegetarian __________
 Something for Everyone __________
 Concession Stand __________
Here are two voting systems that can be used to determine the winner.
 Voting System #1. Plurality: The option with the most firstchoice votes (stars) wins.

Voting System #2. Runoff: If no choice received a majority of the votes, leave out the choice that received the fewest firstchoice votes (stars). Then have another vote.
If your first vote is still a choice, vote for that. If not, vote for your second choice that you wrote down.
If there is still no majority, leave out the choice that got the fewest votes, and then vote again. Vote for your first choice if it’s still in, and if not, vote for your second choice. If your second choice is also out, vote for your third choice.
 How many people in our class are voting? How many votes does it take to win a majority?
 How many votes did the top option receive? Was this a majority of the votes?

People tend to be more satisfied with election results if their top choices win. For how many, and what percentage, of people was the winning option:
 their first choice?
 their second choice?
 their third choice?
 their last choice?
 After the second round of voting, did any choice get a majority? If so, is it the same choice that got a plurality in Voting System #1?
 Which choice won?

How satisfied were the voters by the election results? For how many, and what percentage, of people was the winning option:
 their first choice?
 their second choice?
 their third choice?
 their last choice?
 Compare the satisfaction results for the plurality voting rule and the runoff rule. Did one produce satisfactory results for more people than the other?
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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?
7.3: School Lunch (Part 2) (20 minutes)
Optional activity
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 runoff 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.
Launch
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 wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Let’s analyze a different election.
In another class, there are four clubs. Everyone in each club agrees to vote for the lunch menu exactly the same way, as shown in this table.

Figure out which option won the election by answering these questions.
 On the first vote, when everyone voted for their first choice, how many votes did each option get? Did any choice get a majority?
 Which option is removed from the next vote?
 On the second vote, how many votes did each of the remaining three menu options get? Did any option get a majority?
 Which menu option is removed from the next vote?
 On the third vote, how many votes did each of the remaining two options get? Which option won?

Estimate how satisfied all the voters were.
 For how many people was the winner their first choice?
 For how many people was the winner their second choice?
 For how many people was the winner their third choice?
 For how many people was the winner their last choice?
 Compare the satisfaction results for the plurality voting rule and the runoff rule. Did one produce satisfactory results for more people than the other?
Student Response
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Anticipated Misconceptions
The voting rules are somewhat complicated. Acting out the voting process should make things more clear.
Activity Synthesis
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.)
7.4: Just Vote Once (30 minutes)
Optional activity
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).
Launch
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 handraising 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.
choice  number of votes for top choice (star)  number of votes for second choice (smiley)  number of votes for third choice (square face)  number of votes for last choice (X) 

A  
B  
C  
D 
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.
points  

A. Meat Lovers  3, 0, ... 
B. Vegetarian  2, 3, ... 
C. Something for Everyone  1, 1, ... 
D. Concession stand  0, 2, ... 
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.
Supports accessibility for: Organization; Attention
Student Facing
Your class just voted using the instant runoff system. Use the class data for following questions.
 For our class, which choice received the most points?
 Does this result agree with that from the runoff election in an earlier activity?
 For the other class, which choice received the most points?
 Does this result agree with that from the runoff election in an earlier activity?

The runoff method uses information about people’s first, second, third, and last choices when it is not clear that there is a winner from everyone’s first choices. How does the instant runoff method include the same information?
 After comparing the results for the three voting rules (plurality, runoff, instant runoff) and the satisfaction surveys, which method do you think is fairest? Explain.
Student Response
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Student Facing
Are you ready for more?
Numbering your choices 0 through 3 might not really describe your opinions. For example, what if you really liked A and C a lot, and you really hated B and D? You might want to give A and C both a 3, and B and D both a 0.

Design a numbering system where the size of the number accurately shows how much you like a choice. Some ideas:
 The same 0 to 3 scale, but you can choose more than one of each number, or even decimals between 0 and 3.
 A scale of 1 to 10, with 10 for the best and 1 for the worst.

Try out your system with the people in your group, using the same school lunch options for the election.

Do you think your system gives a more fair way to make choices? Explain your reasoning.
Student Response
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Anticipated Misconceptions
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:
what choice did you rank the winner?  number of people  % of people 

top choice (star)  
second choice (smiley)  
third choice (square face)  
last choice (X) 
Activity Synthesis
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.
Design Principle(s): Cultivate conversation; Maximize metaawareness
7.5: Weekend Choices (10 minutes)
Optional activity
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).
Launch
Arrange students in groups of 2–4.
Student Facing
Clare, Han, Mai, Tyler, and Noah are deciding what to do on the weekend. Their options are cooking, hiking, and bowling. Here are the points for their instant runoff vote. Each first choice gets 2 points, the second choice gets 1 point, and the last choice gets 0 points.
cooking  hiking  bowling  

Clare  2  1  0 
Han  2  1  0 
Mai  2  1  0 
Tyler  0  2  1 
Noah  0  2  1 

Which activity won using the instant runoff method? Show your calculations and use expressions or equations.

Which activity would have won if there was just a vote for their top choice, with a majority or plurality winning?

Which activity would have won if there was a runoff election?

Explain why this happened.
Student Response
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