# Dueling Candidates

Alignments to Content Standards: 7.RP.A

Joel and Marisa are running for president at their middle school (grades 6-8). After the votes are in, Joel and Marisa are each convinced that they have won the election:

• Joel argues that he has won a larger percentage of the overall vote than Marisa so he should be the new president.
• Marisa argues that she has won a larger percentage than Joel of the 6th grade vote and the 7th grade vote. Since the majority of the grades voted for her, she should be the new president.

Is it possible that both Joel and Marisa are making accurate claims? Explain.

## IM Commentary

The goal of this task is to have students examine some properties of ratios (and fractions) in an important real world context. Students will gain practice working with ratios while investigating some of the complexities of voting theory. This task can be made less open-ended by supplying, for example, the total number of students at the school or even the number of students in each class: the teacher may also wish to discuss the analysis in the first paragraph of the first solution if students are stuck. As written, it is intended to be an engaging, open-ended question and will require ample time. It is an ideal task for group work.

If an extension of the problem is desired, the teacher might ask if it is possible for Joel to win a larger percentage of the overall vote than Marisa while Marisa wins a larger percentage of the vote within every grade level. This scenario is impossible because it would mean that Marisa wins more votes than Joel in each grade but fewer votes when the grades are added up.

It is important for students to understand that both Joel and Marisa have legitimate arguments and so it is essential that the rules governing the election be specified in advance. For a further discussion, the teacher may wish to show students numbers for the 2000 presidential election:

Al Gore received more overall votes than George W. Bush (Joel's argument) but Bush won the election because presidential elections are determined on a state by state basis (Marisa's argument). The rules governing presidential elections follow Marisa's line of reasoning (although there are additonal weights involved as each state has a given number of delegates), where the states play the role of the different grade levels at the school. So George W. Bush was elected president even though he received fewer votes than Al Gore.

Students working on this task will engage in MP2, Reason Abstractly and Quantitatively, as the main work of the task involves constructing and reasoning with numbers which satisfy constraints (which also must be reasoned out from the context). The task also provides an opportunity to work on MP3, Construct Viable Arguments and Critique the Reasoning of Others. This may happen at two levels: first students will critique the supplied reasoning of Joel and Marisa and, secondly, they may well disagree about which line of reasoning is more convincing and then they will examine and critique the reasoning of one another.

This task was designed for an NSF supported summer program for teachers and undergraduate students held at the University of New Mexico from July 29 through August 2, 2013 (http://www.math.unm.edu/mctp/).

## Solutions

Solution: 1 Working with Fractions and Percents (6.RP.3)

We are given that Joel has won the majority of the total number of votes at the school. On the other hand, when the vote is divided up by grade level, Marisa has a higher percentage of the 6th grade and 7th grade votes. Before choosing numbers, note that if Joel wins a higher percentage of the 8th grade votes than Marisa then he has a chance to make up for the ground he lost in the 6th and 7th grade. To see if the two scenarios are consistent we have to choose numbers so that Joel makes up more ground on the 8th grade votes than he lost in the combined 6th and 7th grade votes.

We will assume for simplicity that there are 600 students at the middle school. We will also assume that all 600 students vote for either Joel or Marisa. We need to divide the 600 students between the three grades and begin with the assumption that there are 200 in each grade. As observed above, we need to make sure that Joel wins the 8th grade vote by more than Marisa's combined 6th and 7th grade wins. Suppose the votes go as in the table below:

Joel 80 (40%) 90 (45%) 140 (70%)
Marisa 120 (60%) 110 (55%) 60 (30%)

We can see that Marisa won more of the 6th and 7th grade votes than Joel and so she won a larger percentage of the 6th grade votes and the 7th grade votes. For the overall vote, however, Joel received $80 + 90 + 140 = 310$ votes while Marisa received $120 + 110 + 60 = 290$ votes. Converting to percentages, we find that Joel has won $$100 \times \left(\frac{310}{600}\right) \approx 52\%$$ of the vote while Marisa has won $$100 \times \left(\frac{290}{600}\right) \approx 48\%$$ of the vote. So Joel has won a larger percentage of the overall vote than Marisa.

Solution: 2 Working with ratios (7.RP.3)

We begin with a table representing the scenario described in the problem with question marks where we need to provide information:

6th grade vote 7th grade vote 8th grade vote total school vote
Marisa:Joel ? ? ? ?

For the 6th and 7th grade columns, we know that Marisa has won a larger percentage of those votes than Joel. For the 8th grade and total school columns, Joel has won a larger percentage of those votes than Marisa. We need to see if it is possible to assign numbers so that all 4 of these conditions hold. We begin by setting the size of each class: for simplicity, we will assume that there are 100 students in each of the 6th, 7th, and 8th grade classes and so there are 300 total students.

For the first two columns we start by putting in some numbers where the vote favors Marisa and then see if we can complete the table:

6th grade vote 7th grade vote 8th grade vote total school vote
Marisa:Joel 55:45 60:40 ? ?

Here Marisa has won a larger percentage of the 6th grade vote (55 percent to 45 percent) and a larger percentage of the 7th grade vote (60 percent to 40 percent). Note that if Marisa had won the 6th and 7th grade votes by a margin of more than 100 votes, then it is impossible for Joel to make up the difference with the 100 votes in the 8th grade. With the values we have entered for the 6th and 7th grade, Marisa has won these two by a combined margin of 30 votes. So for Joel to win the 8th grade vote and the overall vote, he needs a margin of more than 30 votes in the 8th grade. Here is what we get if we assume a 70 to 30 split of the 8th grade vote in favor of Joel:

6th grade vote 7th grade vote 8th grade vote total school vote
Marisa:Joel 55:45 60:40 30:70 145:155

The table above gives values which match the situation described: Marisa has won a larger percentage of the 6th and 7th grade votes. The 8th grade vote splits as 70% for Joel and 30% for Marisa. There are 300 votes total so to find the percentage of the overall vote won by each candidate we can divide each number in the total vote ratio by 3: $$\frac{145}{3}:\frac{155}{3} :: 48.\overline{3}:51.\overline{6}.$$ Here we see that for the total vote, Marisa won a little over 48 percent and Joel won a little less than 52 percent.