Lesson 7
Comparing Numbers and Distance from Zero
7.1: Opposites (10 minutes)
Warmup
The purpose of this warmup is to use opposites to get students to think about distance from 0. Problem 3 also reminds students that the opposite of a negative number is positive.
Notice students who choose 0 or a negative number for \(a\) and how they reason about \(\texta\).
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet think time, then 2 minutes of partner discussion. Follow with wholeclass discussion.
Student Facing

\(a\) is a rational number. Choose a value for \(a\) and plot it on the number line.

 Based on where you plotted \(a\), plot \(\text a\) on the same number line.
 What is the value of \(\text a\) that you plotted?
 Noah said, “If \(a\) is a rational number, \(\text a\) will always be a negative number.” Do you agree with Noah? Explain your reasoning.
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Anticipated Misconceptions
For problem 3, students might assume that \(\texta\) is always a negative number. Ask these students to start with a negative number and find its opposite. For example, starting with \(a = \text3\), we can find its opposite, \(\text(\text3)\), to be equal to 3.
Activity Synthesis
The main idea of discussion is that opposites have the same distance to 0 (i.e., same absolute value) and that the opposite of a negative number is positive. Ask students to discuss their reasoning with a partner. In a wholeclass discussion, ask a student who chose \(a\) to be positive to share their reasoning about how to plot \(\texta\) and whether they agreed with Noah in problem 3. Then, select previously identified students who chose \(a\) to be negative to share their thinking. If not mentioned by students, emphasize both symbolic and geometric statements of the fact that the opposite of a negative number is positive. For example, if \(a=\text3\), write \(\text(\text3) = 3\) and show that 3 is the opposite of 3 on the number line because they are the same distance to 0. If time allows, select a student who chose \(a\) to be 0 and compare to cases where \(a\) is negative or positive. The number 0 is its own opposite because no other number is 0 units away from 0. Sequencing the discussion to look at positive, negative, and 0 values of \(a\) helps students to visualize and generalize the concept of opposites for rational numbers.
7.2: Submarine (15 minutes)
Activity
Students distinguish between absolute value and order in the context of elevation. Students express their ideas carefully using symbols, verbally, and using a vertical number line. Placing possible elevations on the number line serves as a transition to thinking about solutions to inequalities. Look for students who choose positive and negative elevations for Han and Lin to compare in the discussion.
Launch
Arrange students in groups of 4. Distribute one set of sticky notes to each group, where each note contains one name: Clare, Andre, Han, Lin, and Priya. Display the image for all to see throughout the activity.
Ask students to read the instructions for the task and the description of each person's elevation. Give them a few minutes to use their sticky notes, as a group, to decide where each person (except Priya) could be located.
Place Clare’s sticky note on the number line according to the completed first row of the table. Explain the completed first row of the table to students as it pertains to Clare’s description. Use precise language when explaining the symbols in the table:
 One possible elevation for Clare is 150 feet because 150 is greater than 100, and it is also farther from sea level.
 150 is greater than 100.
 The absolute value of 150 is greater than the absolute value of 100.
Ask groups to complete the rest of the table for the other people (except Priya), and then answer the question about Priya. Note that it is possible to come up with different, correct responses that fit the descriptions. Give students 10 minutes to work followed by wholeclass discussion.
Supports accessibility for: Organization; Attention
Student Facing
A submarine is at an elevation of 100 feet (100 feet below sea level). Let’s compare the elevations of these four people to that of the submarine:
 Clare’s elevation is greater than the elevation of the submarine. Clare is farther from sea level than the submarine.
 Andre’s elevation is less than the elevation of the submarine. Andre is farther away from sea level than the submarine.
 Han’s elevation is greater than the elevation of the submarine. Han is closer to sea level than is the submarine.
 Lin’s elevation is the same distance away from sea level as the submarine’s.

Complete the table as follows.
 Write a possible elevation for each person.
 Use \(<\), \(>\), or \(=\) to compare the elevation of that person to that of the submarine.
 Use absolute value to tell how far away the person is from sea level (elevation 0).
possible
elevationcompare to
submarinedistance from
sea levelClare 150 feet \(150 > \text100\) \(150\) or 150 feet Andre Han Lin  Priya says her elevation is less than the submarine’s and she is closer to sea level. Is this possible? Explain your reasoning.
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
The purpose of the discussion is to let students practice using proper vocabulary to express ideas that distinguish order from absolute value with positive and negative numbers. Select previously identified students to share different elevations for Han and for Lin that show both positive and negative possibilities. Encourage students to explain why the elevation they chose satisfies the description in the problem. As students speak, record their statements using \(<,>,=\) and \(\boldcdot \). Allow students to rearrange sticky notes on the vertical number line display. If time allows, use the sticky notes to show the range of possible solutions for each character; this will help to further prepare students for the concept of graphing solutions of an inequality on the number line.
Design Principle(s): Cultivate conversation
7.3: Info Gap: Points on the Number Line (15 minutes)
Optional activity
In this info gap activity, students use comparisons of order and absolute value of rational numbers to determine the location of unknown points on the number line. In doing so students reinforce their understanding that a number and its absolute value are different properties. Students will also begin to understand that the distance between two numbers, while being positive, could be in either direction between the numbers. This concept is expanded on further when students study arithmetic with rational numbers in grade 7.
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
Launch
Arrange students in groups of 2. In each group, distribute the first problem card to one student and a data card to the other student. After debriefing on the first problem, distribute the cards for the second problem, in which students switch roles.
Supports accessibility for: Memory; Organization
Design Principle(s): Cultivate Conversation
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:

Silently read your card and think about what information you need to be able to answer the question.

Ask your partner for the specific information that you need.

Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.

Share the problem card and solve the problem independently.

Read the data card and discuss your reasoning.
If your teacher gives you the data card:

Silently read your card.

Ask your partner “What specific information do you need?” and wait for them to ask for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

Read the problem card and solve the problem independently.

Share the data card and discuss your reasoning.
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Anticipated Misconceptions
Students may struggle to make sense of the abstract information they are given if they don't choose to draw a number line. Rather than specifically instructing them to use this strategy, consider asking them a question like “How could you keep track of the information you've learned about the points so far?”
Activity Synthesis
Select students with different strategies to share their approaches. Invite them to share which of the clues they thought were more helpful and which were least helpful. Ask students to explain how drawing a number line helped them and how they decided on the appropriate order for the unknown numbers.
7.4: Inequality Mix and Match (15 minutes)
Optional activity
The goal of this activity is for students to practice comparing rational numbers.
Notice students who compare fractions to decimals, fractions to integers, or who compare absolute values to negative numbers.
Launch
Arrange students in groups of 2. Give students 10 minutes to work before wholeclass discussion.
Supports accessibility for: Visualspatial processing; Conceptual processing
Design Principle(s): Support sensemaking; Maximize metaawareness
Student Facing
Here are some numbers and inequality symbols. Work with your partner to write true comparison statements.
0.7
\(\text{}\frac {3}{5}\)
1
4
\(\text8\)
\(<\)
\(\text{}\frac {6}{3}\)
2.5
2.5
8
\(0.7\)
\(=\)
4
0
\(\frac72\)
\(3\)
\(\text{}\frac {5}{2}\)
\(>\)
One partner should select two numbers and one comparison symbol and use them to write a true statement using symbols. The other partner should write a sentence in words with the same meaning, using the following phrases:
 is equal to
 is the absolute value of
 is greater than
 is less than
For example, one partner could write \(4 < 8\) and the other would write, “4 is less than 8.” Switch roles until each partner has three true mathematical statements and three sentences written down.
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Student Facing
Are you ready for more?
For each question, choose a value for each variable to make the whole statement true. (When the word and is used in math, both parts have to be true for the whole statement to be true.) Can you do it if one variable is negative and one is positive? Can you do it if both values are negative?
 \(x < y\) and \(x < y\).
 \(a < b\) and \(a < b\).
 \(c < d\) and \(c > d\).
 \(t < u\) and \(t > u\).
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
The goal of discussion is to allow students to use precise language when comparing rational numbers and absolute values verbally. Select previously identified students to share their responses that compare fractions to decimals, fractions to integers, or absolute values to negative numbers. Display their responses using absolute value and \(>, <, =\) symbols for all to see. Ask students to indicate whether they agree that each response is true, and ask students to share their reasoning about whether they agree or disagree.
Lesson Synthesis
Lesson Synthesis
During this lesson, students have used precise language to distinguish absolute value from order of rational numbers. Display \(\text8\) and 3 questions for all to see:
 “How do you say this?” (The absolute value of 8.)
 “What does it mean in an elevation situation?” (It’s the distance from 8 feet below sea level to sea level.)
 “What does it mean on a number line?” (It’s the distance from 8 to 0 on the number line.)
 “What is its value?” (8.)
Next, display \(\text8 < 5\) and two questions for all to see:
 “How do you say this?” (The absolute value of 8 is less than 5.)
 “What does it mean on a number line?” (8 is less than 5 units away from 0.)
 “Is it true?” (No, 8 is more than 5 units away from 0.)
7.5: Cooldown  True or False? (5 minutes)
CoolDown
Cooldowns for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
Student Facing
We can use elevation to help us compare two rational numbers or two absolute values.
 Suppose an anchor has an elevation of 10 meters and a house has an elevation of 12 meters. To describe the anchor having a lower elevation than the house, we can write \(\text10<12\) and say “10 is less than 12.”
 The anchor is closer to sea level than the house is to sea level (or elevation of 0). To describe this, we can write \(\text10<12\) and say “the distance between 10 and 0 is less than the distance between 12 and 0.”
We can use similar descriptions to compare rational numbers and their absolute values outside of the context of elevation.
 To compare the distance of 47.5 and 5.2 from 0, we can say: \(\text47.5\) is 47.5 units away from 0, and \(5.2\) is 5.2 units away from 0, so \(\text47.5>5.2\).
 \(\text18>4\) means that the absolute value of 18 is greater than 4. This is true because 18 is greater than 4.