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# Look for and Use Structure

# Number Line Tool

# Ordering Rational Numbers

# Ordering Rational Numbers

# Inequalities

## Overview

Students identify whether an inequality statement is true or false using a number line to support their reasoning.

# Key Concepts

- The meaning of
*m* - The meaning of
*n*>*m*is that*n*is located to the right of*m*on a number line. The inequality statement*n*>*m*is read “*n*is greater than*m*.” - To decide on the order of two numbers
*m*and*n*, locate the numbers on a number line. If*m*is to the left of*n*, then*m*<*n*. If*m*is to the right of*n*, then*m*>*n.*

# Goals and Learning Objectives

- State whether an inequality is true or false.
- Use a number line to prove that an inequality is true or false.

# Look For and Use Structure

# Mathematical Practices in Action

**Mathematical Practice 7: Look for and make use of structure.**

Have students watch the video. The video shows students engaged in Mathematical Practice 7: Look for and make use of structure.

Discuss the conversation that Emma and Jan had about the number line. Discuss how Emma and Jan used the structure of the number system to order the temperatures. Talk about how confusing it would be if number lines did not have a defined pattern and structure. Ask students to think of situations involving numbers where there would be lots of confusion if there was no order. Have students think of examples besides the mixed-up thermometer, showing how pattern and structure is important (e.g., reading yards on a football field, naming elevations above and below sea level).

Point out that Emma and Jan’s conversation highlights the importance of looking for and making use of structure. Explain that mathematics has a very consistent structure that students can look for and use to help them make sense of math. Encourage students to look for and use structure as they work in math class this year.

## Opening

# Look for and Use Structure

Watch the video to see how Emma and Jan talk about the structure of the number line as they look at temperatures on a thermometer.

- Discuss how Emma and Jan used the structure of the number line to order the temperatures.
- What problems would you see in the world if there was no structure or pattern in the number line?

VIDEO: Look for and Use Structure

# Mathematical Statements

# Lesson Guide

Have students write mathematical statements and then share them with their partners.

Then have students share their statements with the whole class. Elicit that −5°F > −24°F is one way to show this relationship. Some students may write −24°F < −5°F, which also shows a correct relationship between the two temperatures.

## Opening

# Mathematical Statements

Write a mathematical statement to express that –5°F is warmer than –24°F.

- The symbol < means “is less than.”
- The symbol > means “is greater than.”

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will understand the meaning of < and > when using positive and negative numbers.

## Opening

Understand the meaning of < and > when using positive and negative numbers.

# Evaluate Inequalities

# Lesson Guide

Have students work in pairs. Tell students to copy the problems in their notebooks, then direct them to make a number line for each problem.

SWD: Students build math understanding by moving from concrete models to visual representations and then to abstract learning. Know where each student is on their landscape of learning to assist you in their next step and help guide them toward mastery.

# Interventions

**Student has an incorrect solution.**

- Have you checked your work?
- What does the > symbol mean?
- What does the < symbol mean?
- How can you use a number line to support your answer?

**Student presents his/her work poorly.**

- Have you checked your work?
- Have you explained how you arrived at your answer?
- Is your explanation complete?

**Student has a solution.**

- Explain your strategy for identifying whether each inequality was true or false.

# Mathematical Practices

**Mathematical Practice 7: Look for and make use of structure.**

Look for students who use the structure of the number line to determine if the inequality is true or false.

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

Listen to pairs as they discuss whether the inequalities are true or false. Note students who provide clear explanations, who defend their thinking using a model, and who query the thinking of their partners.

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

Identify students who use number lines to model the math. Ask a few students to share their work during Ways of Thinking.

# Answers

- Number lines will vary.

## Work Time

# Evaluate Inequalities

Determine if the inequality is true or false.

Justify your answer on a number line.

- −10 < −15
- −12 < −1
- −2 > −2.5
- −$\frac{5}{2}$ ≤ 1
- −$\frac{7}{3}$ > −2
- −2.53 < −2.23

When you locate the numbers on the number line, what do you know about the number to the right?

# Prepare a Presentation

# Preparing for Ways of Thinking

As students are working, observe how they prove to each other whether an inequality is true or false. Make note of particular students that you would like to ask to share during Ways of Thinking.

Have students work in pairs on the presentation.

Presentations will vary. Make sure students include one inequality and a complete and accurate explanation for why the inequality is true or false.

# Challenge Problem

## Possible Answers

- Use a number line to compare any two numbers. Find where each number is on a number line. The number to the left is less than the number to the right. Or, the number to the right is greater than the number to the left.

## Work Time

# Prepare a Presentation

Explain how you ordered positive and negative numbers. Use your work to support your explanation.

# Challenge Problem

- Write a procedure to explain how to compare any two numbers.

# Make Connections

# Lesson Guide

As students share their presentations, ask these questions:

- How do you know if the inequality is true or false?
- How can you use the structure of a number line to decide if the inequality is true or false?
- Does anyone have a different explanation?
- Did you and your partner disagree on any inequalities? If so, how did you settle on an answer?

Have students who did the Challenge Problem share their work and have the class try the procedures to make sure they work.

SWD: Revisit the vocabulary introduced in this lesson. As students present their solutions to the Work Time problems, make note of key words and write them on a chart. Provide plenty of repetition and review of new terminology. Make sure all students have these terms in their notebook: *inequality*, *greater than*, *less than*

ELL: As in other presentations, be sure that all students (including ELLs) participate in the presentation exercise, and monitor that ELLs do not shy away from this activity. Encourage other students to be patient if the pace of ELLs is slower than native speakers, and explain that listening attentively is one way in which we show we care for others.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about how to compare any two rational numbers.

As students present, ask questions such as:

- How do you know if the inequality is true or false?
- How does your number line model the math?
- How did you resolve any disagreements with your partner?
- What advice would you give someone who was trying to determine whether an inequality was true or false?
- What do the inequality symbols < and > mean?

# Order Rational Numbers

# Interventions

**Ask questions such as the following as students are working:**

- How do you know this number is the least (greatest)?
- How do you know this number goes next?
- How do you know the numbers go in this order?
- Is $-4\frac{1}{2}$ less than or greater than −4.6? How do you know?

SWD: Students with disabilities may need re-teaching and review of the skills necessary to successfully convert mixed numbers to decimal form.

# Answer

- Numbers ordered from least to greatest: −4.6, $-4\frac{1}{2}$, −4, 0, 4, $4\frac{1}{2}$

## Work Time

# Order Rational Numbers

Order these numbers from least to greatest:

$4,4\frac{1}{2},-4,-4\frac{1}{2},-4.6,0$

Use the Number Line Tool if you find it to be helpful.

HANDOUT: Ordering Rational Numbers

Can you locate the numbers on a number line?

# Order Decimals

# Interventions

**Ask questions such as the following as students are working:**

- Is−3.256 less than or greater than −0.3456? How do you know?
- How can the structure of a number line help you order the numbers?

# Answer

- Numbers ordered from least to greatest:

−5, −4.2, −3.256, −2.53, −1.25, −0.3456

## Work Time

# Order Decimals

Order these numbers from least to greatest:

−2.53 , −1.25 , −3.256 , −4.2 , −5 , −0.3455

Use the Number Line Tool if you find it to be helpful.

HANDOUT: Ordering Rational Numbers

Can you locate the numbers on a number line?

# Positive and Negative Numbers

# A Possible Summary

You can use a number line to demonstrate which of two numbers is greater or less. You can write an inequality to state which of two numbers is greater or less. The inequality symbol < means “is less than.” The inequality symbol > means “is greater than.”

# Additional Discussion Points

- When comparing two numbers, the greater number is to the right of the lesser number on a number line.
- When comparing two numbers, the lesser number is to the left of the greater number on a number line.
- The inequality statement
*n*>*m*is read “*n*is greater than*m*.”

## Formative Assessment

# Summary of the Math: Positive and Negative Numbers

Write a summary of what you learned about ordering and comparing positive and negative numbers.

Check your summary:

- Do you use the term inequality?
- Do you describe how to use a number line to compare two numbers?
- Do you use the inequality symbols < and >?
- Do you explian how to order numbers?

# Check Your Understanding

# 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. Research shows that scoring is counterproductive, as it will encourage students to compare their scores and will distract them from finding out what they can do to improve their understanding of the mathematics.

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

**Student has difficulty getting started.**

- What do you need to do?
- What are you trying to find?
- Can you use the number line on the handout to help you?

**Student does not seem to understand the opposite of a number.**

- What is the definition for the
*opposite of a number*? - Where can you go to review the opposite of a number?

**Student does not seem to understand absolute value.**

- Where can you go to review absolute value?
- What do these | | symbols mean?

**Student produces a correct solution but does not give an explanation of why it is correct.**

- How do you know your answer is correct?

**Student presents the work poorly.**

- Is your work shown clearly?
- Have you given enough explanation and is it clear?

**Student provides an adequate solution to the problem with a good explanation.**

- Explain your strategy for solving this problem.

SWD: Post the Interventions in the classroom for students to use as a resource as they work. Create and provide an enhanced version of the interventions with embedded text structures (labels, highlights, words in bold) to cue students to pay closer attention to particular terms.

# Answers

- Explanations will vary.
- –6.6
- –2.7
- Yes, −|−7| = −|7| because −|−7| = −7 and −|7| = −7; −7 = −7.
- –6.6 is farther from 1.

## Formative Assessment

# Check Your Understanding

Complete this Self Check by yourself.

Answer the questions. Explain your thinking.

Use the Number Line Tool if you find it to be helpful.

- Which of these numbers is least?

−0.6, −6, −6.06, −6.6, 6.6, or 0.6 - What is the opposite of the opposite of −2.7 ?
- Do the two expressions below have the same value?

−|−7| and −|7| - Which of these two numbers is farther from 1?

−6.6 or 6.

HANDOUT: Integers and Rational Numbers Self Check

# 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 strategies students use to tell which of two numbers is greater.

## 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.

**A strategy I use to tell which of two numbers is greater is…**