Lesson 4

How Do We Choose?

4.1: Which Was “Yessier”? (10 minutes)

Optional activity

This activity gives students a chance to recall and use various ratio strategies in the context of a voting problem. Two classes voted on a yes or no question. Both classes voted yes. Students are asked to determine which class was more in favor (“yessier”). Students need to make sense of the invented word “yessier” by thinking about how it might be quantified (MP1, MP2).

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

Launch

Arrange students in groups of 2–4. Ask students to use the mathematical tools they know to answer the question.

Student Facing

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

yes no
class A 24 16
class B 18 9

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

Anticipated Misconceptions

Students may not understand the question. The word “yessy” was invented by other students solving a similar problem. It is not standard English.

If students do not understand what comparing the two classes means, give a more extreme example, such as comparing 20 to 1 with 11 to 10. The 20 to 1 class is much more yessy because almost everyone said yes. The other class had almost equal yesses and nos.

Activity Synthesis

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, at least one who used a ratio of yesses to nos, and another who used a ratio of yesses to all students. Make sure to present a solution using percentages.

If more than a few students did not use multiplicative techniques (for example, if they compared only the yesses, or subtracted to find how many more yesses than nos) remind them of the rate comparisons they did earlier in the year, in which they could check by tasting a drink mixture or looking at the color of a paint mixture.

Speaking, Listening, Representing: MLR2 Collect and Display. Use this routine to display examples of the different methods students used to determine which class was “yessier." Circulate and collect examples of formal and informal language, calculations, tables, or other writting. Ask students to compare and contrast percentage calculations with fractions and equivalent ratios.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

4.2: Which Class Voted Purpler? (10 minutes)

Optional activity

This activity is the same type of situation as the previous one: comparing the voting of two groups on a yes or no issue. However, the numbers make it more difficult to use "part to part" ratios. Again, students need to be thinking about how to make sense of (MP1) and quantify the class voting decisions (MP2).

Student Facing

The school will be painted over the summer. Students get to vote on whether to change the color to purple (a “yes” vote), or keep it a beige color (a “no” vote).

The principal of the school decided to analyze voting results by class. The table shows some results.

In both classes, a majority voted for changing the paint color to purple. Which class was more in favor of changing?

yes no
class A 26 14
class B 31 19

Anticipated Misconceptions

If students do not understand what comparing the two classes means, give a more extreme example, such as comparing 20 to 1 with 11 to 10. The 20 to 1 class is much more purple because almost everyone said yes. The other class had almost equal yesses and nos.

Students may be stuck with the difficult-looking numbers, expecting to be able to do calculations to create equivalent ratios mentally. Suggest that they find the total number of votes in each class.

Students may compute percentages incorrectly, forgetting that percentages are rates out of 100. So it is not correct to say that room A has 54% yesses. However, you can carefully make sense of this percent as a comparison: Class A had 54% as many nos as yesses; Class B had 61% as many nos as yesses. This means that Class B was more no-y, so it was less yessy than Class A.

Activity Synthesis

Invite students to show different ways of solving the problem, including using equivalent ratios and percentages. Ask students to explain their thinking. Correct ideas with incorrect calculations are still worth sharing.

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Once students have determined which class was more in favor of changing, pause the class. Invite students to share their strategies for finding each fraction using percents and equivalent ratios to justify their reasoning. Create a display that includes each strategy labeled with the name and the fraction they represent of a total. Keep this display visible as students move on to the next problems.
Supports accessibility for: Conceptual processing; Memory

4.3: Supermajorities (10 minutes)

Optional activity

This activity introduces the idea of requiring a supermajority. A supermajority is a voting rule that is used for issues where it is important to have more than just barely above half of the voters agreeing. To win, a choice must have more than the given fraction of the votes. In this activity, two supermajority rules are given: one as a fraction, one as a percent. Students find a fraction of the total votes and a percentage of the total votes. They then compare the fraction and the percent.

This activity encourages students to think about votes in ratios and percents. Students make sense of problems and reason quantitatively (MP1 and MP2).

Launch

Arrange students in groups of 2–4. Provide access to a four-function calculator.

Explain the difference between a majority and supermajority: "In many voting situations, a choice that wins a majority of the votes wins. A majority is more than half the votes. So if 1,000 votes were cast, a majority is any number over 500; 501 is the smallest number of votes that can win.

Many groups have special election rules for very important issues. Sometimes they require a supermajority: to win, you need more than a certain fraction that is more than half. For raising taxes, some governments require a $$\frac23$$ supermajority. To change (amend) the U.S. constitution, an amendment must get a $$\frac23$$ supermajority of both the Senate and House of Representatives, and be ratified by $$\frac34$$ of the states. Sometimes supermajorities are described as percents, such as 60%."

Representation: Internalize Comprehension. Activate or supply background knowledge about reasoning quantitatively about operations involving decimals and percentages. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

1. Another school is also voting on whether to change their school’s color to purple. Their rules require a $$\frac23$$ supermajority to change the colors. A total of 240 people voted, and 153 voted to change to purple. Were there enough votes to make the change?

2. This school also is thinking of changing their mascot to an armadillo. To change mascots, a 55% supermajority is needed. How many of the 240 students need to vote “yes” for the mascot to change?

3. At this school, which requires more votes to pass: a change of mascot or a change of color?

Activity Synthesis

Choose one or more students’ methods to present.

Conversing, Speaking, Listening: MLR7 Compare and Connect. Before the whole-class discussion, invite students to share their work with a partner or small group. Display the following questions to support conversation: “What was your strategy?”, “Does anyone want to add on to _____’s strategy?”, “Did anyone solve the problem in a similar way, but would explain it differently?”, and “Does anyone have a completely different method for solving?”  Listen for and amplify observations that include mathematical language and reasoning.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

4.4: Best Restaurant (20 minutes)

Optional activity

This activity shows how a few people can make a decision if many people don’t vote. The mathematics involves repeated “percent of” operations, each percent giving a smaller amount than the previous step. The main issue in this problem is to identify “percent of what?” for each percentage. The first percentage is 25% of the people in town subscribe to the newspaper. The second percentage is 20% of the result of the previous number, and the third is 80% of the second result.

Students need to give a written explanation, clearly show their calculations by writing expressions and equations, and make a diagram that accurately shows the sizes of all the groups in the problem (MP3). The diagram might be on a $$10 \times 10$$ grid, or a tape diagram. Graph paper is a good way to make sure the sizes are right. Tape diagrams can also be made with a folded strip of paper, if students are accustomed to folding fractions.

Launch

Arrange students in groups of 2–4. Tell students that sometimes local newspapers or magazines ask their readers to vote for their favorite businesses. In this activity, they think about whether this is a good way to decide which businesses are the best or are most popular. (In other words, is this a scientific survey?) Make graph paper, colored pencils or markers, and scissors available.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to blank square grid and tape diagrams to support information processing. Encourage students to annotate diagrams with details to show how each value is represented. For example, number of all people in town, number of newspaper subscribers in town, number of newspaper subscribers who voted.
Supports accessibility for: Visual-spatial processing; Organization
Design Principle(s): Support sense-making; Cultivate conversation; Maximize meta-awareness

Student Facing

A town’s newspaper held a contest to decide the best restaurant in town. Only people who subscribe to the newspaper can vote. 25% of the people in town subscribe to the newspaper. 20% of the subscribers voted. 80% of the people who voted liked Darnell’s BBQ Pit best.

Darnell put a big sign in his restaurant’s window that said, “80% say Darnell’s is the best!”

Do you think Darnell’s sign is making an accurate statement? Support your answer with:

• Some calculations
• An explanation in words
• A diagram that accurately represents the people in town, the newspaper subscribers, the voters, and the people who liked Darnell’s best

Anticipated Misconceptions

Students may wonder how they can answer the question without knowing how many people are in the town. Encourage them to invent a total number of people (such as 100 or 1,000) or to show the percents as parts of a 10-by-10 square. Remind them that the answer is a percentage, not a number of people. Make sure to discuss the fact that, no matter what the number of people in the town is, the percentage at the end is still the same.

Diagrams may still be too abstract for some. Demonstrate with a large 10-by-10 square: Cut a 5-by-5 square out; this is 25% of the total, representing the subscribers. Put the other 75% aside. Now find 20% of the 5-by-5 square. This is $$\frac15$$ of the square, so is 5 squares. Cut off a strip of 5 to represent the subscribers who voted. Put the rest aside. Now find 80% of 5 squares, which is 4 squares. Cut off and put aside one square. What is left represents 80% of the subscribers who voted. It’s only 4% of the people in the town!

Activity Synthesis

This problem requires students to revise their idea of what is “the whole” three times: initially, it’s the number of people in the town. Then it’s the number of subscribers, then the number of voters. “Percent of what?” is a useful question to ask.

Ask students who chose a specific number of people in the town what percentage they got. Try to find students who chose different numbers of people. All should get 4%.

Choose several diagrams to display and discuss. Ask what part of the diagram represents each quantity:

• all the people in town
• 1% of the people in town
• the people who subscribe to the newspaper (25% of the people in town)
• the people who voted (20% of the subscribers, which is 5% of the people in town)
• the people who voted for Darnell’s (80% of those who voted, 4% of the people in town)

Discuss the fact that Darnell’s sign is misleading, asking students whom you noticed had interesting or well-stated answers. Students may want to discuss the unfairness of the results when only a few people vote. You might want to consult with a civics/social studies teacher about actual numbers in a recent election, and whether this is likely to be too controversial to discuss in your class.