- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Interactive, Text/HTML

# Number Line Contraption

# Number Line Contraption

# Opposite of a Number

## Overview

Students watch a dot get tossed from one number on a number line to the opposite of the number. Students predict where the dot will land each time based on its starting location.

# Key Concepts

- The opposite of a number is the same distance from 0 as the number itself, but on the other side of 0 on a number line.
- In the diagram,
*m*is the opposite of*n*, and*n*is the opposite of*m*. The distance from*m*to 0 is*d*, and the distance from*n*to 0 is*d*; this distance to 0 is the same for both*n*and*m*. The absolute value of a number is its distance from 0 on a number line. - Positive numbers are numbers that are greater than 0.
- Negative numbers are numbers that are less than 0.
- The opposite of a positive number is negative, and the opposite of a negative number is positive.
- Since the opposite of 0 is 0 (which is neither positive nor negative), then 0 = 0. The number 0 is the only number that is its own opposite.
- Whole numbers and the opposites of those numbers are all integers.
- Rational numbers are numbers that can be expressed as
*ab*, where*a*and*b*are integers and*b*≠ 0.

# Goals and Learning Objectives

- Identify a number and its opposite
- Locate the opposite of a number on a number line
- Define the
*opposite of a number*,*negative numbers*,*rational numbers*, and*integers*

# Definitions

# Lesson Guide

Make sure students understand the definitions of *negative number*, *integers*, and *rational* *numbers*. Discuss the idea that 0 is neither positive nor negative.

Explore the Number Line Contraption with the class. Ask various students to predict where the dot will land each time. Start by generating the whole-number number line. Then show students how to generate and use the decimal and fraction number lines. Reinforce that for the first number line, students use whole numbers, or integers, to enter their prediction. In the other two, however, students use rational numbers (fractions or decimals) to enter their prediction.

ELL: When introducing new words, remember to allow ELLs to use a dictionary. Repeat the new words at a slower pace, and write them down, asking some of the students to repeat after you. Be sure ELLs feel comfortable with the pronunciation.

Instructor Note:

The interactive instructions refer to a "blue dot" however the dot in the interactive is orange.

## Opening

# Definitions

Read these definitions:

- A
*negative number*is a number that is less than 0. Example: −3 *Integers*are positive and negative whole numbers. Example: −3 and 3*Rational numbers*are numbers that can be expressed as $\frac{a}{b}$, where*a*and*b*are integers and*b*is not equal to 0. Example: fractions like $\frac{1}{4}$ or decimals like 0.25- Explore the Number Line Contraption interactive with your class.

INTERACTIVE: Number Line Contraption

Note: The interactive instructions refer to a "blue dot" however the dot in the interactive is orange.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explore the location of positive and negative numbers on a number line.

## Opening

Explore the location of positive and negative numbers on a number line.

# Explore the Number Line Contraption

# Lesson Guide

Have students predict where the dot will land each time.

ELL: In providing these prompts and questions, be sure that ELLs understand the meaning of the words contained therein. Be sure that if the student involved is an ELL, your pace is adequate and you are providing ample wait-time to allow for a thoughtful response. At any point, present the questions in writing if you think that will support ELLs. Also allow students to use dictionaries before responding if there are words in your questions that they don’t understand.

# Interventions

**Student has an incorrect solution.**

- Describe where the dot starts on the number line. How far is it from 0?
- Name the point where the dot starts.
- Describe where the dot lands on the number line. How far is it from 0?
- Name the point where the dot lands.
- What do you notice about the start number and the stop number?

**Student has a solution.**

- Explain how you made your predictions.
- Were your predictions correct?
- What do you notice about the numbers to the right of 0?
- What do you notice about the numbers to the left of 0?
- What is similar about the start number and the stop number? What is different?
- How do you express mathematically what is happening using numbers and variables?
- How does your diagram represent what is happening?

# Mathematical Practice

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

As students experiment with the Number Line Contraption, watch how quickly students begin to notice that the dot always lands the same distance from 0 as it starts, but on the other side of the number line.

Look for students who notice that all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. Once students see the regularity in the dot’s movement, they will be able to name the point on which the dot will land.

# Possible Answers

- Possible answer: Each time the dot is tossed, it stops the same distance from 0 as it starts, but on the other side of 0 on the number line. If the start number is positive, the dot lands on a negative number. If the start number is negative, the dot lands on a positive number.

## Work Time

# Explore the Number Line Contraption

- Explain mathematically what is happening.

*INTERACTIVE*: Number Line Contraption

- What happens when you set the lever at a positive integer?
- What happens when you set the lever at a negative rational number?
- How far from 0 is the lever to start? How far from 0 does it land?

# Work With the Number Line Contraption

# Lesson Guide

Students work with the Number Line Contraption.

# Mathematical Practices

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

Look for student presentations that mathematically express the regularity of where the dot lands using numbers and variables and a diagram.

# Interventions

**Student does not understand**

- Make a record of the number where you place the dot and then the number where it lands. Do you see a pattern?
- Try starting with positive and negative numbers.

**Student has a hard time writing an expression of the action**

- If that number is
*a*, what is the number where you land? - How could you set those two numbers equal?

# Answer

- You would get the original number. The dot lands at the opposite of the number—if it starts at –8, it will stop at 8; if it starts at 8, it will stop at –8.
- −(−
*a*) =*a*

## Work Time

# Work With the Number Line Contraption

Suppose you put the dot on a number and tossed it over the center; then, you put the dot at the place where it just landed, and tossed it over the center again.

- Where would the dot land?
- Write an expression for this action.

*INTERACTIVE*: Number Line Contraption

- Use the Number Line Contraption interactive to test what happens if you swing the lever twice.
- Use a negative sign to represent each swing of the lever.
- Use
*a*to represent any number.

# Prepare a Presentation

# Lesson Guide

Be sure students’ presentations accurately explain how the Number Line Contraption works using numbers, variables, and diagrams and include the terms *positive*, *negative*, *integer*, and *rational number*.

# Preparing for Ways of Thinking

Find students to present who:

- Take organized notes about their results.
- Are able to predict where the dot will land.
- Understand that if the dot started at a positive number, it will land on a negative number.
- Understand that if the dot started at a negative number, it will land on a positive number.
- Recognize that the positive numbers are on one side of 0 and the negatives are on the other side of 0.

ELL: When asking the questions above, make sure you point to the numbers that you are asking about. Pointing is a simple and effective strategy to use when working with ELL students.

# Challenge Problem

## Possible Answer

- No. If the dot starts on a negative number, it will always land on a positive number. The dot moves the same distance from 0, but on the opposite side of 0. The opposite of a negative number is a positive number.

## Work Time

# Prepare a Presentation

- Explain how the Number Line Contraption works.
- Use diagrams to explain your thinking.
- Represent the action mathematically using numbers and then again using a variable.
- Make sure you include the terms
*positive*,*negative*,*integer*, and*rational number*in your presentation.

# Challenge Problem

Can the dot start on a negative number and then land on another negative number? Explain your thinking.

- Use the Number Line Contraption interactive to test what happens if you swing the lever twice.
- Use a negative sign to represent each swing of the lever.
- Use
*a*to represent any number.

# Make Connections

# Lesson Guide

Gather students to discuss their work with the Number Line Contraption. Have students make their presentations to the class.

**Ask questions such as the following:**

- What is the same about the numbers on either side of 0?
- What is different about the numbers on either side of 0?
- If the dot starts on a positive number, what do you know about the stopping number?
- If the dot starts on a negative number, what do you know about the stopping number?
- What do you notice about the locations of starting and stopping numbers?
- What kinds of numbers did you work with on the Number Line Contraption?

In the discussion, have students use the dot to show you examples of positive numbers, negative numbers, rational numbers, and integers. Informally bring out the following in the discussion:

- Positive numbers are on the opposite side of 0 from negative numbers.
- Negative numbers are on the opposite side of 0 from positive numbers.
- The opposite of a positive number is negative, and the opposite of a negative number is positive.
- Since the opposite of 0 is 0 (which is neither positive nor negative), then −0 = 0.
- The opposite of the opposite of a number is the number itself.

Introduce the concept of the absolute value of a number as its distance from 0. In this simulation, it is the distance between the dot and the number 0. Students will explore absolute value more in the next lesson.

Have students who did the Challenge Problem share their thinking. Ask class members to evaluate whether their thinking makes sense.

ELL: Be sure that your pace is adequate, and, if needed at any point, present the questions in writing. Provide sentence frames as appropriate, but consider that ELLs might not need them, especially if the questions are shown in writing (students can “borrow” some of the words from the questions to construct their answers).

Allow ELLs to use a dictionary. Repeat the new words at a slower pace, and write them down, asking some of the students to repeat after you. Be sure ELLs feel comfortable with the pronunciation.

SWD: Make sure students understand these domain-specific terms:

- Absolute value
- Positive
- Negative
- Integer
- Rational number

Pre-teach these terms to students with disabilities. Provide a written definition and a visual representation of each term. Post them on a Word Wall or on anchor charts for students to copy into their notebook.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about how positive and negative numbers are represented on a number line.

As students present, ask questions such as:

- How did you predict where the lever would land?
- How did you know if a number was negative or positive?
- For one swing of the lever, how are the starting number and the ending number alike? How are they different?
- How did you figure out where the lever would land after two swings of the lever?
- What kind of number was the starting number? What kind of number was the ending number? Was it an integer? A rational number? Explain.

# Find the Opposite of a Number

# Lesson Guide

Students read the definition of the opposite of a number, and then they find the opposite of four numbers.

# Interventions

Ask questions such as the following as students are working:

- How did you get that answer?
- Can you sketch a diagram to show your answer?
- Explain what would happen on the Number Line Contraption with this problem.
- What kind of number is –489.257?
- What kind of number is 412?
- What is the opposite of the opposite of any number? How do you know?

ELL: Encourage students to work cooperatively on this task. Group ELLs with students who can help them work through the language to improve understanding.

# Answers

- The opposite of 24 is –24.
- The opposite of –489.257 is 489.257. The opposite of 489.257 is –489.257. So, the opposite of the opposite of –489.257 is –489.257.
- The opposite of $4\frac{1}{2}$ is −$4\frac{1}{2}$. The opposite of −$4\frac{1}{2}$ is $4\frac{1}{2}$. So, the opposite of the opposite of $4\frac{1}{2}$ is $4\frac{1}{2}$.
- The number 0 is its own opposite, so the opposite of the opposite of 0 is 0.

## Work Time

# Apply the Learning: Find the Opposite of a Number

The *opposite *of a number *n* is the number that is the same distance from 0 as *n* but on the other side of 0 on a number line. The number 0 is its own opposite.

- Find each number. Explain your answer.
- The opposite of 24
- The opposite of the opposite of −489.257
- The opposite of the opposite of $4\frac{1}{2}$
- The opposite of the opposite of 0

Ask yourself:

- What is the same distance from 0 as –489.257 but on the other side of 0 on the number line? What is the opposite of that number?
- What is the opposite of 4 1 2 ?
- Remember 0 is its own opposite.

# Opposites and Negatives

# 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 have trouble with the following:

- Thinking that the opposite of a number is always negative
- Thinking that numbers less than 0 can be positive
- Thinking that integers are non-whole numbers and their opposites, such as $3\frac{1}{2}$ and −$3\frac{1}{2}$
- Thinking that decimals are not rational numbers

Work with students individually or in small groups to eliminate any misconceptions. Include additional experiences with the Number Line Contraption.

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

## Formative Assessment

# Summary of the Math: Opposites and Negatives

**Read and Discuss**

- The
*opposite*of a number*n*is 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. - A
*negative number*is a number that is less than 0. *Integers*are whole numbers and their opposites.*Rational numbers*are numbers that can be expressed as $\frac{a}{b}$, where*a*and*b*are integers and*b*is not equal to 0.

Can you:

- Define the opposite of a number?
- Describe the location of positive and negative numbers on a number line?
- Define integers?
- Define rational numbers?

# 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 opposite of a number.

Create a “reflection checklist” to help students to self-assess the quality of their reflection. This will help struggling writers to self-monitor their work.

SWD: Reflecting on learning can be challenging for some students. Support students who may get lost in the meaning of this task by providing specific questions to which they can respond. Some students may require sentence starters or paragraph frames as writing support. Also support students by providing models from earlier lessons that include the elements of a complete reflection.

## 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.

**One thing I know about the opposite of a number is ….**