How Do We Choose?

15 min

Teacher Prep
Setup
Students in groups of 2–4.

Narrative

This activity gives students a chance to recall and use various ways to reason about ratios in the context of a voting problem. Two classes voted on a yes-or-no question. Both classes voted yes. As students think about how to quantify the degree of preference for something, they practice making sense of a problem and persevering in solving it (MP1).

Monitor for students who use different ways to make sense of the problem.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

Launch

Arrange students in groups of 2. Introduce the context of two classes voting on whether to answer math questions in the form of poetry. Use Co-Craft Questions to orient students to the context, and elicit possible mathematical questions.

  • Display only the problem stem and related table, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, and then to compare their questions with a partner.
  • Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as deciding which result is more popular.
  • Reveal the question “Was one class more in favor of math poetry, or were they equally in favor? Find two or more ways to answer the question.” and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
    • “Which of your questions is most similar to or different from the ones given? Why?”

Give groups 2–3 minutes to answer the question, and follow that with a whole-class discussion.

Student Task

Two sixth-grade classes, A and B, voted on whether to give the answers to their math problems in the form of poetry. The “yes” choice was more popular in both classes.

Was one class more in favor of math poetry, or were they equally in favor? Find 2 or more ways to answer the question.

yes no
class A 24 16
class B 18 9

Sample Response

Sample response:

  • Compare equivalent ratios where one quantity is the same for both classes. For example, Class A voted 24 yes to 16 no, which is equivalent to 3 yes to 2 no. Class B voted 18 yes to 9 no, which is equivalent to 3 yes to 1.5 no. For the same number of yeses there are fewer nos in Class B, so Class B is more in favor of answering math problems in poetry.
  • Compare the unit rates. For each no in Class A, there are 1.5 yeses (24÷16=1.524 \div 16 = 1.5). For each no in class B, there are 2 yeses (18÷9=218 \div 9 = 2). This means class B is more in favor.
  • Compare the percentages of yes votes. In Class A, 2440\frac{24}{40} or 610\frac{6}{10} of the class voted yes, so 60% of the class voted yes. In Class B, 1827\frac{18}{27} or 23\frac{2}{3} of the class voted yes, so about 66.7% voted yes. Class B has a higher percentage of yes votes.
     
Activity Synthesis (Teacher Notes)

The goal of the discussion is to connect the idea of voting to rate comparisons. Students should recognize that the situation is mathematically the same as other rate comparison problems, such as comparing the tastes or colors of two mixtures.

Invite several students to present different methods. If no students compare the ratio of yes votes to all votes or use percentages, be sure to present such solutions.

Anticipated Misconceptions

If students compare only the yes votes rather than reasoning about ratios, or if they compare additively (by finding the differences between the yes votes and no votes) rather than comparing multiplicatively, consider using ratios that are more extreme to illustrate the issues. For instance:

  • If students look only at the yes votes, ask: "Suppose Class P voted 10 yes to 10 no, and Class Q voted 15 yes to 20 no, would you say that class Q likes the proposal better because it has more yes votes than class P does? Why or why not?"
  • If students compare additively, ask: "Suppose Class X voted 30 yes to 10 no, and Class Y voted 15 yes to 0 no. Is Class X more in favor of the proposal because it has 20 more yes votes than no votes, while Class Y has 15 more yes votes than no votes? Why or why not?
     
Standards
Addressing
  • 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  • 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

10 min

20 min