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# Using Negative and Absolute Numbers

## Overview

Students answer questions about low temperatures recorded in Barrow, Alaska, to understand when to use negative numbers and when to use the absolute values of numbers.

# Key Concepts

- The absolute value of a number is its distance from 0 on a number line.
- The absolute value of a number
*n*is written |*n*| and is read as “the absolute value of*n*.” - A number and the opposite of the number always have the same absolute value. As shown in the diagram,

|3| = 3 and |−3| = 3. - 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*|, that is always positive unless*n*= 0. For example, |3| = 3 and |−3| = 3. - The absolute value of 0 is 0, which is neither positive nor negative: |0| = 0.

# Goals and Learning Objectives

- Understand when to talk about a number as negative and when to talk about the absolute value of a number.
- Locate the absolute value of
*a*and the absolute value of*b*on a number line that shows the location of*a*and*b*in different places in relation to 0.

# Construct and Critique Arguments

# Mathematical Practices in Action

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

Direct students’ attention to the definition of absolute value. Then watch the video. Have students listen to the dialogue between Jan and Carlos. This video shows students engaged in Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

After students have watched the video, discuss Carlos and Jan’s conversation. Point out that their conversation is an example of constructing viable arguments and critiquing the reasoning of others, which is an important mathematical practice. Have students talk about each discussion question. Emphasize that through making arguments and listening to the arguments of others, they can advance their understanding of mathematics.

As students work in math class, they will be expected to explain their thinking and to defend it using objects, drawings, diagrams, or other models. They will learn to listen to and critique the reasoning of others. Tell students that talking about math will help them understand the math better.

SWD: Students with disabilities may have difficulty with cooperative learning tasks. Support students who have trouble resolving conflicts or disagreements by coaching the students to recognize the importance of considering alternate perspectives. Also, provide suggestions as to how students can navigate differences of opinion in learning situations.

ELL: Oral explanations can be hard to follow when they are in a language that the students don’t fully comprehend. Be sure that your pace is adequate, and, if needed at any point, pose questions to make sure students are following what you are trying to convey.

## Opening

# Construct and Critique Arguments

The *absolute value of a number* is the distance a number is from 0 on a number line. The symbol |*a*| is used to indicate the absolute value of *a*.

Watch the video.

- What did Carlos think absolute value was at the beginning of the video?
- What did Jan suggest doing to help them think about absolute value?
- What argument did Carlos use to say that the absolute value of 12 cannot be –12?
- What model did Jan and Carlos use to help them understand absolute value?
- What would Carlos have said the absolute value of 8 is at the beginning of the video? What would Carlos have said the absolute value of 8 is at the end of the video?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will understand when to use the absolute value of a number.

## Opening

Understand when to use the absolute value of a number.

# Temperatures in Barrow, Alaska

# Lesson Guide

Have students work independently on each problem and then discuss their answers with their partners.

ELL: Some of the words in the questions and prompts can be somewhat difficult for ELLs to follow. If necessary, rephrase using words you know students can understand to allow ELLs to fully participate and to have a fair chance to answer the questions.

# Interventions

**Students have difficulty getting started.**

- What information are you given?
- How can you find the coldest temperature?
- Will you represent the coldest temperature using a negative number or an absolute value? Explain your thinking.
- What are you trying to find when you are asked how far below 0°F the coldest temperature is?
- Will you represent the distance below 0 using a negative number or an absolute value? Explain your thinking.

**Student has an incorrect solution.**

**[common error] Student represents distance using a negative number, not the absolute value of a number.**

- Explain what this problem is asking.
- Does your answer to this problem make sense? Explain why or why not.
- Can a distance be a negative number? Explain your thinking.

**Student has a solution.**

- Explain your strategy for solving each problem.
- Why did you use a negative number in your answer to the first problem?
- Why did you use the absolute value to find the answer to the second problem?
- Could you have used a number line to help you solve the problems?

# Mathematical Practices

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

Listen as students explain their answers to their partners. Identify students who begin to use Mathematical Practice 3 so that you can discuss their conversations during Ways of Thinking. Do students’ explanations make sense? Are they defending their answers using a number line? Are they asking questions when their partner’s explanation is unclear?

**Mathematical Practice 4: Model with mathematics.**

Watch for students who make a number line to model the temperatures. Choose one or two students to share their strategy during Ways of Thinking.

# Answers

- The coldest temperature is –23.6°F.
- The coldest temperature is 23.6 degrees below 0°F.
- The difference between the lowest and highest temperatures recorded in the table is 57.2 degrees. Possible explanation: The highest temperature is 33.6°F, so I know it is 33.6 degrees above 0°F. The lowest temperature is –23.6°F, so I know it is 23.6 degrees below 0°F. So, I added together the two amounts: 33.6 + 23.6 = 57.2.
- Possible answer: When measuring a temperature in relation to 0, you use the negative number. When measuring a change or a difference in the number of degrees, you use the absolute value of a number.

## Work Time

# Temperatures in Barrow, Alaska

The table shows some low temperatures recorded in Barrow, Alaska, during a 12-month period. Use the table to answer these questions.

- What is the coldest temperature?
- How far below 0°F is the coldest temperature?
- What is the difference between the lowest and highest temperatures, in degrees? Explain how you know.
- When did you use a negative number and when did you use the absolute value of a number?

Ask yourself:

- Represent the temperatures on a number line. Where is the coldest temperature?
- Can you represent distance using a negative number or using an absolute value?
- How many degrees above 0°F is the highest temperature?
- How many degrees below 0°F is the lowest temperature?
- How many degrees are between the highest and lowest temperatures?

# Prepare a Presentation

# Lesson Guide

Check that students understand when to use a negative number and when to use the absolute value.

# Preparing for Ways of Thinking

As students are working, note if there are students who cannot distinguish between a negative number and an absolute value. Note any misconceptions and clarify them during Ways of Thinking.

# Challenge Problem

## Possible Answers

- Problem: The temperature dropped by 12 degrees on Saturday. If the original temperature was 4°F, what was the new temperature? Answer: The new temperature was –8°F.
- Problem: On the number line, how many units are between the points –3 and –7? Answer: There are 4 units between –3 and –7.

## Work Time

# Prepare a Presentation

- Explain when you use a negative number and when you use the absolute value.
- Demonstrate how to take the absolute value of a number.

# Challenge Problem

- Create a story problem that has a negative answer.
- Create a story problem that includes a negative value but has a positive answer.

# Make Connections

# Lesson Guide

Have students share their work on the problems and then give their presentations.

Ask questions such as the following when reviewing the temperature problems:

- How did you solve the problems?
- What do you think of [Name]’s and [Name]’s methods?
- How are they similar?
- How are they different?
- Whose method makes more sense to you? Why?

- Did anyone use a number line to help them? Show us what you did.
- What elements of Mathematical Practice 3 did you use when discussing the problems with your partner?

**[common error] Student shows an answer of –23.6 degrees for the second problem. Help students understand that this problem is asking for a distance—how far below 0°F—and that a distance is always positive.**

In the discussion of the last problem, elicit that in the first problem, students use a negative sign to give the temperature. In the second problem, students look at the distance from 0°F, so the answer is 23.6 degrees. A negative sign is not used for distance. In the third problem, students find the distance from –23.6°F to 33.6°F. Since students are finding the distance, the answer is positive.

Have students who did the Challenge Problems share their work. Have the class solve some of the problems to see if the answers are the kind of answers the writers wanted.

ELL: Be sure that all students (including ELLs) participate in the presentation exercise, and monitor that ELLs do not shy away from this activity, as it is important that they share aloud so that they can hear their own voice and get used to talking in front of large groups.

When critiquing students whose proficiency in English is low, focus on what the student is trying to convey and not the grammar mistakes. If you are not sure you understand, ask the student to repeat in different ways. If other students in the class understand (and you don’t), allow them to help you.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about how to determine the absolute value of a number and when to use the absolute value of a number.

## Hint:

As students present, ask questions such as:

- How can a number line be used to solve the problem?
- In what types of situations is it best to use the absolute value of a number?
- In what types of situations is it best to use a negative number?

# Absolute Value

# A Possible Summary

In situations involving distance, you can give the distance as the absolute value of some number. The absolute value of a number is its distance from 0 on a number line.

# Additional Discussion Points

- The absolute value of a number
*n*is written |*n*| and is read as “the absolute value of*n*.” - 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, except when*n*= 0, since 0 is neither positive nor negative. For example, |3| = 3 and |−3| = 3. - In some situations, it is best to talk about a negative number as a negative number, and in other situations, it is best to talk about its absolute value.

## Formative Assessment

# Summary of the Math: Absolute Value

Write a summary of what you learned about absolute value.

Check your summary.

- Do you include a definition of absolute value?
- Do you explain how to find the absolute value of a number on a number line?
- Do you talk about when to use the absolute value?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what students know about the absolute value of a number and the opposite of a number.

ELL: Before students write their reflection, allow some additional time for ELLs to discuss their ideas with a partner to help them organize their thoughts. Allow ELLs who share the same language of origin to discuss in this language if they wish and to use a dictionary (or dictionaries).

## 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.

**When I look at the absolute value of a number and the opposite of a number, I see these connections …**