Lesson 7

Comparing Numbers and Distance from Zero

7.1: Opposites (10 minutes)

Warm-up

The purpose of this warm-up 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 \(\text-a\).

Launch

Arrange students in groups of 2. Give students 5 minutes of quiet think time, then 2 minutes of partner discussion. Follow with whole-class discussion.

Student Facing

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

    Blank number line, 21 evenly spaced tick marks, middle tick mark labeled zero.
    1. Based on where you plotted \(a\), plot \(\text- a\) on the same number line.
    2. What is the value of \(\text- a\) that you plotted?
  2. 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

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Anticipated Misconceptions

For problem 3, students might assume that \(\text-a\) is always a negative number. Ask these students to start with a negative number and find its opposite. For example, starting with \(a = \text-3\), we can find its opposite, \(\text-(\text-3)\), 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 whole-class discussion, ask a student who chose \(a\) to be positive to share their reasoning about how to plot \(\text-a\) 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 \(a=\text-3\), write \(\text-(\text-3) = 3\) 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.

Vertical number line.

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 whole-class discussion.

Representation: Access for Perception. Activate or supply background knowledge. Give students 1–2 minutes to review the first row of the table that shows a possible elevation for Clare. Invite 1–2 students to think aloud and share connections they make between the display with the sticky notes, and the values in the table. Record their thinking on a display of the table and keep the work visible as students work.
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.
  1. Complete the table as follows.

    1. Write a possible elevation for each person.
    2. Use \(<\), \(>\), or \(=\) to compare the elevation of that person to that of the submarine.
    3. Use absolute value to tell how far away the person is from sea level (elevation 0).
    As an example, the first row has been filled with a possible elevation for Clare.
    possible
    elevation
    compare to
    submarine
    distance from
    sea level
    Clare 150 feet \(150 > \text-100\) \(|150|\) or 150 feet
    Andre
    Han
    Lin
  2. 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

For access, consult 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.

Speaking: MLR8 Discussion Supports. To support students’ use of vocabulary related to absolute value and positive and negative numbers, provide sentence frames related to each column heading. Some examples include: “_____ could have an elevation of _____ because _____,” “Comparing _____’s elevation to the submarine’s, I notice _____,” or “_____’s distance from sea level is _____ because _______.”
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:

Two problem cards. Two data cards. 

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.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to determine the location of unknown points on the number line. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
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:

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

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. 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.

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

  4. Read the problem card and solve the problem independently.

  5. Share the data card and discuss your reasoning.

 

Student Response

For access, consult 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 whole-class discussion.

Action and Expression: Provide Access for Physical Action. Create alternatives for physically interacting with materials. Consider creating a set of cards for each of the numbers and inequality symbols that students can select from and sequence to create true comparison statements. Invite students to talk about their statements before writing them down.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: MLR5 Co-Craft Questions. To create space for students to produce the language of mathematical questions themselves, display only the array of numbers that the students will be using in this activity. Ask students to think about the values of the numbers and write a mathematical question using two or more numbers from the array. Students may generate questions such as “How many values are greater than zero?” or “Which numbers are opposites?” Notice students that have questions about comparing and ordering the numbers and ask them to share their questions. This will help students use conversation skills to generate, choose, and improve their questions as well as develop meta-awareness of the language used in mathematical questions.
Design Principle(s): Support sense-making; Maximize meta-awareness

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

\(|\text-8|\)

\(<\)

\(\text{-}\frac {6}{3}\)

-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

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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?

  1. \(x < y\) and \(|x| < y\).
  2. \(a < b\) and \(|a| < |b|\).
  3. \(c < d\) and \(|c| > d\).
  4. \(t < u\) and \(|t| > |u|\).

Student Response

For access, consult 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 \(|\text-8|\) 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 \(|\text-8| < 5\) 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: Cool-down - True or False? (5 minutes)

Cool-Down

For access, consult 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 \(\text-10<12\) 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 \(|\text-10|<|12|\) 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: \(|\text-47.5|\) is 47.5 units away from 0, and \(|5.2|\) is 5.2 units away from 0, so \(|\text-47.5|>|5.2|\).
  • \(|\text-18|>4\) means that the absolute value of -18 is greater than 4. This is true because 18 is greater than 4.