Subject:
Numbers and Operations
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Provider:
Pearson
Tags:
6th Grade Mathematics, Absolute Value, Numbers, Statements
License:
Creative Commons Attribution-NonCommercial 4.0
Language:
English
Media Formats:
Downloadable docs, Text/HTML

Education Standards (6)

Possible or Impossible?

Possible or Impossible?

Lesson Overview

Students analyze whether given statements are possible or impossible using their definitions of absolute value and the opposite of a number. If the statements are possible, students give an example of a pair of numbers that fit the statement. If the statements are impossible, students explain why.

Key Concepts

  • A number and the opposite of the number always have the same absolute value.
  • In general, taking the opposite of n changes the sign of n. For example, the opposite of 3 is −3.
  • In general, taking the absolute value of n gives a number |n|, which is always positive. For example, |3| = 3 and |−3| = 3.
  • Since the opposite of 0 is 0 (which is neither positive nor negative), therefore −0 = 0. The number 0 is the only number which is its own opposite.

Goals and Learning Objectives

  • Find pairs of numbers that satisfy different statements about absolute values and/or the opposites of numbers.
  • State when it is impossible to find a pair of numbers that satisfies the statement and explain why.

Task 1: Critique Denzel’s Statement

Lesson Guide

Have students think about Denzel’s statement for a minute or two and then discuss their thoughts with a partner. As partners converse, encourage the use of Mathematical Practice 3: Construct a Viable Argument and Critique the Reasoning of Others.

Have students share their examples. Have the class discuss whether a number and the opposite of the number can both be positive. Students should come to the conclusion that the statement is impossible; a number and the opposite of the number cannot both be positive. A number and the opposite of the number have the same absolute value but have opposite signs.

Opening

Critique Denzel's Statement

Denzel says that he is thinking of a number and its opposite that are both positive.

  • Is Denzel's statement possible?

In your argument, you should either provide an example that justifies Denzel’s statement or explain why his reasoning does not make sense.

Task 2: Math Mission

Lesson Guide

Discuss the Math Mission. Students will justify whether statements are possible or impossible by using the definitions of absolute value and the opposite of a number.

Opening

Justify whether statements are possible or impossible by using the definitions of absolute value and the oppositeof a number.

Task 3: Possible or Impossible?

Lesson Guide

Have students work in pairs. Tell students they are going to look at a series of statements about the opposites of numbers and/or the absolute values of numbers. Their job is to find a pair of numbers that fits each statement and, if that is not possible, to explain why.

SWD: If students seem unsure of the task, model the steps for solving the problem with them before asking them to do the problems themselves.

ELL: In forming pairs, be aware of your ELLs and ensure that they will have a learning environment where they can be productive.

  • Sometimes, pair them up with native speakers so they can learn from native counterparts’ language skills.
  • Other times, pair them up with students who are at the same level of language skills, so they can take a more active role and they can work things out together.
  • Yet other times, pair them up with students whose proficiency level is lower, so they play the role of the supporter.

Interventions

Student has an incorrect solution.

  • Have you checked your work?
  • Does your answer make sense?
  • Can a number and the opposite of a number have the same sign?
  • Can the absolute value of a number be negative?

Student has difficulty getting started.

  • Can you use a number line to help you?
  • Think about the definitions for the terms opposite of a number and absolute value.
  • Try out different pairs of numbers. Can you find a pair that matches the statement?

Student has a solution.

  • What strategy did you use to determine if the statement was true or false?
  • Which statement was the most difficult to evaluate? Why?
  • What is the difference between the opposite of a number and the absolute value of a number?

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Some of these statements may be difficult for students to justify. Look for students who struggle but persevere to find an explanation.

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

Listen to pairs as they discuss the problems. Identify students who articulate their reasoning clearly and who ask good questions about their partner’s explanations. Watch for students who are able to defend their thinking using a drawing or a diagram.

Mathematical Practice 4: Model with mathematics.

Identify students who make a number line to model the math. Choose one or two students to share their work during Ways of Thinking.

Possible Answers

Examples and explanations will vary.

  • Possible Cards: A number and its opposite that are both integers; A number and its opposite that are both not integers; A number and its opposite that have the same absolute values; A number and its opposite that are the same number; A number and its opposite that is greater than the number; A number and its opposite that is less than the number
  • Impossible Cards: A number and its opposite that are both negative; A number and its opposite in which one is an integer and one is not; Two different numbers that have the same absolute values but are not the opposites of each other; A number and its opposite in which the absolute value of one is greater than the absolute value of the other

Work Time

Possible or Impossible?

Sort the statements into two groups: Possible and Impossible.

  • If possible, give an example.
  • If impossible, explain why.

HANDOUT: Possible or Impossible?

Try different numbers to see if they satisfy the statement.

Task 4: Value of Expressions

Lesson Guide

Students might struggle with the sometimes true, always true, or never true problem. Let them struggle and clarify their thinking in Ways of Thinking.

Interventions

Student has an incorrect solution.

  • What is the absolute value of –3?
  • What is the absolute value of 3?

Student has difficulty getting started.

  • Can you use a number line to help you?
  • Think about the definitions for the terms the opposite of a number and absolute value.

Student has a solution.

  • What strategy did you use to determine if the statement was always true, sometimes true or false?
  • What is the difference between the opposite of a number and the absolute value of a number?

Possible Answers

  • −|−3| = −3 and −(−3) = 3. The expressions are not equal since −3 ≠ 3.
  • The opposite of a number is sometimes equal to the opposite of the absolute value of the number. When the number is positive or 0, the statement is true. When the number is negative, the statement is false. For example, the opposite of 5 is −5 and −|5| = −5, and −5 = −5. But the opposite of –7 is 7 and −|−7| = −7, and 7 ≠ −7 so the statement is false in this case.

Work Time

Value of Expressions

  • Do the two expressions below have the same value? Explain your thinking.
    −|−3| and −(−3)
  • Is the following statement sometimes true, always true, or never true? Give an example to support your answer.

The opposite of a number is equal to the opposite
of the absolute value of the number.

  • Sort the cards to show whether the statements on the cards are possible or impossible. Provide an explanation.

INTERACTIVE: Card Sort: Opposite and Absolute Values

  • For the first question, do each step—start inside the absolute value symbols or inside the parentheses.
  • For the second question, test different numbers. Try positive and negative numbers and 0.

Task 5: Prepare a Presentation

Lesson Guide

Make sure students present one possible statement with more than one example to support their reasoning and present one impossible statement with a complete explanation.

Preparing for Ways of Thinking

As students are working, listen to the conversations they are having so that you can identify good conversations to share in Ways of Thinking. Identify students who articulate their reasoning clearly and who ask good questions about their partner’s explanations. Find an example of a student who defends their thinking using a drawing or a diagram.

Challenge Problem

Possible Answers

  • Answers will vary. Possible answer: An integer that is the same as its absolute value
  • Answers will vary. Possible answer: 3 because 3 = |3|

Work Time

Prepare a Presentation

  • Present one statement you said was possible. Provide more than one example using integers, fractions, and decimals to support your reasoning.
  • Present one statement you said was impossible. Provide a complete explanation that the class will understand.

Challenge Problem

  • Write a statement like the ones in the card sort.
  • Give your statement to your partner. Have your partner either find a pair of numbers that satisfies the statement or explain why it is impossible to do so.

Task 6: Make Connections

Lesson Guide

Ask questions such as the following:

  • Are some statements harder to justify than others? Which ones did you work really hard on?
  • Did you use any sketches or diagrams to help you justify an answer? Show us what you did.
  • Did you or your partner ever disagree? How did you resolve your disagreement?
  • For which statements can you find examples? For which statements can you not find an example?
  • What is the difference between the absolute value of a number and the opposite of a number?
  • Whose presentation was complete and easy to understand? Explain why.
  • Did you disagree with anyone’s presentation? Can you construct a viable argument for why you disagree?

Have students who did the Challenge Problem share the statements they wrote with the class. Have students work together to decide if the statement is possible or impossible.

SWD: If you know that some students may need additional time and/or prompting to participate in this discussion, provide them several of the questions ahead of time (printed out or digitally).

ELL: Since the academic vocabulary in this section is substantial, consider going through it (explaining and defining as needed) prior to the discussion and also during the discussion. Encourage students to use the academic vocabulary in their own sentences as a way to apply the academic vocabulary as well as responding to the questions. If ELLs need it, consider providing some sentence frames to help scaffold the activity.

Performance Task

Ways of Thinking: Make Connections

Take notes on which statements belong in the Possible group and which statements belong in the Impossible group. Include examples and explanations in your notes.

As classmates present, ask questions such as:

  • What strategy did you use to decide if the statement was possible or impossible?
  • How do you know this statement is impossible?
  • How do you know your example fits the statement?
  • Can you sketch a diagram to help you justify the statement?
  • Explain each term in the statement and what it told you about the numbers.

Task 7: Absolute Value and the Opposites of Numbers

Lesson Guide

  • Have pairs quietly discuss the definitions and give examples of each. Then discuss the definitions as a class.
  • As student pairs work together, listen for students who may still think that the absolute value of a number can be negative. Work with these students individually or in pairs and use a number line to review the idea that absolute value represents a distance, so it can never be negative.
  • Discuss as a class the following points:
    • A number and the opposite of the number always have the same absolute value.
    • Taking the opposite of n changes the sign of n. For example, the opposite of 3 is –3. Taking the absolute value of n makes n positive. For example, |3| = 3 and
      |−3| = 3.
    • Since the opposite of 0 is 0 (which is neither positive nor negative), –0 = 0. The number 0 is the only number that is its own opposite.

ELL: When discussing absolute value and the opposites of numbers, make a point of writing the questions on the board along with students’ responses. This will assist ELLs by giving them written and oral access to the questions. Have students write all important information in their notebook.

Formative Assessment

Summary of the Math: Absolute Value and the Opposites of Numbers

Read and Discuss

  • Absolute value: The distance a number is from 0 on a number line. The symbol |a| is used to indicate the absolute value of a. |a| is a, if a is positive. |−a| is a, if a is negative.
  • Opposite of a number n: The number that is the same distance from 0 as n but on the other side of 0 on a number line; 0 is its own opposite. The opposite of a is −a.

Can you:

  • Write a definition for the terms absolute value and opposite of a number?
  • Explain what signs a number and its opposite have?
  • Explain what sign the absolute value of a number has?

Task 8: Reflect On Your Work

Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out which statements students had difficulty proving.

ELL: Since reflecting on and summarizing one’s learning entails such a high level of understanding of the topic and command of the language, allow ELLs additional time to come up with the reflection, and encourage them to work in pairs if necessary.

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.

It was difficult to decide whether the statement about _____ was possible or impossible because...