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# Identifying Errors In Reasoning

## Overview

Students are given a collection of statements that are incorrect. Their task is to construct arguments about why the statements are flawed and then correct the flawed statements.

# Key Concepts

- Percent change is a rate of change of an original amount.
- In two situations with the same percent change but different original amounts, the percent amount will be different because the percent amount depends directly on the original amount. For example: 50% of 20 is 10. 50% of 10 is 5.
- Similarly, in two situations with the same amount of increase but different original amounts, the percent change of each amount is different. For example: Suppose two amounts increase by $5. If one original amount is $20, the increase is 25%. If the other original amount is $25, the increase is 20%.

# Goals and Learning Objectives

- Identify errors in reasoning in percent situations.
- Use examples to explain why the reasoning is incorrect.

# Maya Buys Two Shirts

# Lesson Guide

- Have students read the situation about Maya and the salesperson.
- Then have students watch the video showing how Maya critiqued the reasoning of the salesperson.
- Point out that the salesperson’s statement—that Maya saved 40%—was a genuine mistake made in a large department store.
- Have pairs of students share ideas about why the salesperson’s statement is incorrect.

Point out to students that it is very easy to combine percents incorrectly when doing calculations in everyday life, which presents opportunities to work on Mathematical Practice 3.

ELL: When showing the video, monitor that the ELLs are following the meaning of what is presented. If necessary, pause the video and allow them to ask clarifying questions. Alternatively, ask questions to ensure students understand what they are watching.

# Mathematics

This lesson gives students opportunities to use Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

## Opening

# Maya Buys Two Shirts

Watch the video about a sales situation.

The salesperson made a mistake. Maya critiqued the reasoning of the salesperson.

- As you complete today’s Work Time problems, think about how you could finish Maya’s statement.
- How could you show the salesperson that his reasoning is incorrect?

VIDEO: Shirt Sale

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will identify and explain mistakes in reasoning about percents.

SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features or concepts and this will help them to pay close attention to salient information.

## Opening

Identify and explain mistakes in reasoning about percents.

# Bags of Apples

# Lesson Guide

Have students work in pairs on all problems and the presentation.

For each problem, have pairs:

- Discuss whether the reasoning is incorrect.
- Write a convincing explanation stating why the statement is incorrect.
- Give specific examples to show what the correct reasoning should be.

ELL: Provide scaffolding so that ELLs develop the vocabulary and English skills needed to provide written comments that describe common errors. Provide models and examples that are comprehensible.

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

Pay attention to the language that students are using with one another and in their written work. Note how precise they are in their explanations, for example, asking questions like “25% of what?”

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Notice if students are constructing arguments and explanations that directly address the incorrect reasoning illustrated in the statements.

# Interventions

**[common error] Student thinks a statement is correct.**

- Try it with a starting amount that makes sense in the situation, and see if it works.

**Student believes a statement is incorrect, but cannot explain why.**

- Try it with a starting amount that makes sense in the situation, and pay attention to the structure of your calculations. Try again with another starting amount. Describe to your partner what is happening before writing anything down.

**[common error] Student is computing with percents incorrectly to test the statements.**

- Remember that a percent is an amount per 100. Convert the percents to decimal values, and do your computations again.
- Check that your computations clearly show what quantity you are taking a percent of.

# Answers

- Answers will vary.
- Mr. Stevens added the percents he saved on each bag: 25% + 25% = 50%. If this reasoning were true, Mr. Stevens could buy four bags and save 100%. He actually saves 25% of his total purchase, no matter how many bags he buys or how much each bag costs. Check students’ examples.

## Work Time

# Bags of Apples

Mr. Stevens bought two bags of apples. Each bag had a label saying “25% off.” Mr. Stevens figured out that altogether he saved 50%.

- What mistake in reasoning about percents did Mr. Stevens make?
- Using examples, explain why Mr. Stevens’s reasoning is incorrect.

## Hint:

How did Mr. Stevens come up with the number 50%?

# Pay Raises

# Lesson Guide

Have students work in pairs on all problems and the presentation.

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

Pay attention to the language that students are using with one another and in their written work. Note how precise they are in their explanations, for example, asking questions like “5% of what?”

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Notice if students are constructing arguments and explanations that directly address the incorrect reasoning illustrated in the statements.

# Interventions

**[common error] Student thinks a statement is correct.**

- Try it with a starting amount that makes sense in the situation and see if it works.

**Student believes a statement is incorrect, but cannot explain why.**

- Try it with a starting amount that makes sense in the situation and pay attention to the structure of your calculations. Try again with another starting amount. Describe what is happening to your partner before writing anything down.

**[common error] Student is computing with percents incorrectly to test the statements.**

- Remember that a percent is an amount per 100. Convert the percents to decimal values, and do your computations again.
- Check that your computations clearly show what quantity you are taking a percent of.

# Answers

- Answers will vary.
- This statement would only be true if Lucy’s sister and her brother have the exact same pay before the raise. Check students’ examples.

## Work Time

# Pay Raises

Lucy’s sister got a 15% pay raise. Her brother got a 10% pay raise. Lucy determined that her sister’s pay raise is 5% greater than her brother’s pay raise.

- What mistake in reasoning about percents did Lucy make?
- Using examples, explain why Lucy’s reasoning is incorrect.

## Hint:

How did Lucy come up with the number 5%?

# Cows

# Lesson Guide

Have students work in pairs on all problems and the presentation.

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

Pay attention to the language that students are using with one another and in their written work. Note how precise they are in their explanations, for example, asking questions like “20% of what?”

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Notice if students are constructing arguments and explanations that directly address the incorrect reasoning illustrated in the statements.

# Interventions

**[common error] Student thinks a statement is correct.**

- Try it with a starting amount that makes sense in the situation and see if it works.

**Student believes a statement is incorrect, but cannot explain why.**

- Try it with a starting amount that makes sense in the situation and pay attention to the structure of your calculations. Try again with another starting amount. Describe what is happening to your partner before writing anything down.

**[common error] Student is computing with percents incorrectly to test the statements.**

- Remember that a percent is an amount per 100. Convert the percents to decimal values, and do your computations again.
- Check that your computations clearly show what quantity you are taking a percent of.

# Answers

- Answers will vary.
- If the herd was reduced by 20%, then the remaining 80% would need to produce $\frac{100}{80}$ or l25% to have the same milk production. The increase would need to be 25%. Check students’ examples.

## Work Time

# Cows

A farmer’s herd of cows is 20% smaller than it was the previous year. The farmer figured out that if each cow can increase milk production by 20%, then milk production will be the same as it was last year.

- What mistake in reasoning about percents did the farmer make?
- Using examples, explain why the farmer’s reasoning is incorrect.

## Hint:

- What amount would you take 20% of the first time you see the 20% in the problem?
- What amount would you take 20% of the second time you see the 20%?

# Birds

# Lesson Guide

Have students work in pairs on all problems and the presentation.

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

Pay attention to the language that students are using with one another and in their written work. Note how precise they are in their explanations, for example, asking questions like “20% of what?”

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

# Interventions

**[common error] Student thinks a statement is correct.**

- Try it with a starting amount that makes sense in the situation and see if it works.

**Student believes a statement is incorrect, but cannot explain why.**

- Try it with a starting amount that makes sense in the situation and pay attention to the structure of your calculations. Try again with another starting amount. Describe what is happening to your partner before writing anything down.

**[common error] Student is computing with percents incorrectly to test the statements.**

- Check that your computations clearly show what quantity you are taking a percent of.

# Answers

- Answers will vary.
- You do not add the percents; you multiply. If you start with 100 birds, after 5 years there would be 100 × 1.2 × 1.2 × 1.2 × 1.2 × 1.2 = 248.832, or approximately 250 birds, which is more than double. Actually, the population would double (approximately 207 birds) in 4 years. Check students’ examples.

## Work Time

# Birds

Karen read that the population of birds on an island increases by 20% each year. She figured out that the population would therefore double after five years.

- What mistake in reasoning about percents did Karen make?
- Using examples, explain why Karen’s reasoning is incorrect.

## Hint:

What does it mean for a population to “double” in terms of percent? How do you think Karen came up with this percent based on what she read?

# Prepare a Presentation

# Preparing for Ways of Thinking

Look and listen for students who:

- Show familiarity with the incorrect reasoning, either from their own thinking or from experiences with others.
- Understand the importance of “the whole,” how the whole is represented mathematically, and how it fits within the mathematical structure of the situation.
- Presentations will vary.

# Challenge Problem

## Answers

Jack is right. Possible explanation: The percent change is different because the original amount in each case is different.

## Work Time

# Prepare a Presentation

Prepare a presentation that shows your work for one of the problems. Create an alternate “conclusion” to replace the mistaken conclusion, and explain why your conclusion is correct.

# Challenge Problem

Jack said that the percent change from $20 to $25 is 25% and the percent change from $25 to $30 is 20%. Lucy said that Jack cannot be right: because both the differences are $5, the percent change must be 25% for both sets of numbers.

Who is right? Explain why.

# Make Connections

# Lesson Guide

Ask pairs of students to explain why one of the statements is incorrect. Invite students to present written examples that show their reasoning.

As students’ present, ask questions such as the following:

- The problem says that Lucy’s sister got a 15% pay raise and her brother got a 10% pay raise. Can you think of a situation in which her brother still got more money? (Answer: If Lucy’s brother makes a lot more than his sister, 10% of his pay might be more than 15% of her pay.)
- Why can’t you add percents like regular numbers? What makes them different? (Answer: Any number can stand for 100%. You have to think about what the percent actually means.)

Finish the lesson by asking students for proposals of ways to finish Maya’s statement in the video. Ask:

- How would you finish Maya’s statement?
- Does [student’s] answer make sense? Explain why or why not.
- What would you say differently?
- Can you show calculations to support your answer on the board?
- Do you agree with [student’s] work? Why or why not?
- Is what Maya is saying still true if the shirts are different prices?

Be sure to have students present their work on the Challenge Problem for discussion and feedback from the class.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

- Ask for comments from other students about the reasoning presented, and have students add to the explanations as needed to make the argument stronger.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ explanations of the mistakes in reasoning about percents that people make.

## Hint:

As your classmates present, ask questions such as:

- How did you decide what numbers to use in your example?
- Can you try another example to see if you get the same result?
- Can you think of another situation in which a person might make this type of mistake?
- Do you think your explanation is convincing? How could you make it better?

# Mistakes With Percents

# A Possible Summary

Percent change is a rate of change of an original amount. In two situations with the same percent change but different original amounts, the amount of change will be different because the percent amount depends directly on the original amount. Similarly, in two situations with the same amount of increase but different original amounts, the percent change of each amount is different. For example, even if two amounts increase by $5, if one original amount is $20, the increase is 25%, and if the other original amount is $25, the increase is 20%.

# Additional Discussion Points

Discuss the following:

- The different kinds of mistakes that people make with percents

## Formative Assessment

# Summary of the Math: Mistakes With Percents

Write a summary about the common mistakes in reasoning with percents that people make.

## Hint:

Check your summary.

- Do you give examples of mistakes?
- Do you explain why the reasoning behind these mistakes is incorrect?

# Salary

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

SWD: Some students with disabilities may struggle with self-assessment; use your knowledge of student strengths and vulnerabilities to inform and create interventions you will put into place for the upcoming lesson.

# Interventions

**[common error] Student assumes incorrectly that a percent increase means the calculation must include an addition.**

- Does your answer make sense? Can you check that it is correct?
- Can you express the increase as a single multiplication?

**[common error] Student assumes incorrectly that a percentage decrease means the calculation must include a subtraction.**

- Does your answer make sense? Can you check that it is correct?
- In a sale, an item is marked “50% off.” What does this mean? Describe in words how you calculate the price of an item in the sale. Give an example.
- Can you express the decrease as a single multiplication?

**Student uses an inefficient method to solve the problem.**

- Can you think of a method that reduces the number of calculator keys you press?
- How can you show your calculation with just one step?

**Student misinterprets what needs to be included in the answer.**

- If you just entered these symbols into your calculator, would you get the correct answer?

# Possible Answers

- Let
*s*= new salary.*s*= 40.85 • 1.06, or*s*= (40.85 • 0.06) + 40.85 - Marcus's dad’s new salary is $43.30 per hour.

## Formative Assessment

# Salary

Complete this Self Check by yourself.

Marcus’s dad earns $40.85 per hour. He has just learned that his company is giving him a 6% pay raise. What will his new salary be?

- Write an equation.
- Write the solution as a complete sentence.

# A Dress on Sale

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

**[common error] Student assumes incorrectly that a percent increase means the calculation must include an addition.**

- Does your answer make sense? Can you check that it is correct?
- Can you express the increase as a single multiplication?

**[common error] Student assumes incorrectly that a percentage decrease means the calculation must include a subtraction.**

- Does your answer make sense? Can you check that it is correct?
- In a sale, an item is marked “50% off.” What does this mean? Describe in words how you calculate the price of an item in the sale. Give an example.
- Can you express the decrease as a single multiplication?

**Student uses an inefficient method to solve the problem.**

- Can you think of a method that reduces the number of calculator keys you press?
- How can you show your calculation with just one step?

**Student misinterprets what needs to be included in the answer.**

- If you just entered these symbols into your calculator, would you get the correct answer?

# Answers

- Let
*p*= the sale price of the dress*p*= 56.99 • 0.55, or*p*= 56.99 – (56.99 • 0.45)

- The sale price of the dress is $31.34.

## Formative Assessment

# A Dress on Sale

Karen’s sister finds a dress that she wants to buy. The regular price of the dress is $56.99, but it is on sale for 45% off. What is the sale price of the dress?

- Write an equation.
- Write the solution as a complete sentence.

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out what students identified as common mistakes when working with percents.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**An example of a mistake that people often make with percents is …**