Lesson 13

Reintroducing Inequalities

13.1: Greater Than One (5 minutes)

Warm-up

The purpose of this activity is for students to substitute values into an inequality and check to see if each value satisfies the inequality based on a number line representation. This is the first opportunity (of many to come) to practice this type of substitution with inequalities. 

Launch

Arrange students in groups of 2. Since it may have been a while since students encountered this notation, remind students that \(x>1\) is read “\(x\) is greater than 1.” Give students 2 minutes of quiet work time followed by 1 minute to share their responses with a partner. During the partner discussion, tell students to compare their answers for the first question and see if they agree with their partner’s chosen values for the second question. Follow with a whole-class discussion.

Student Facing

The number line shows values of \(x\) that make the inequality \(x>1\) true.

A number line with the numbers negative 5 through 5 indicated. An open circle is indicated at 1 and an arrow is drawn from the open circle extending to the right.
  1. Select all the values of \(x\) from this list that make the inequality \(x>1\) true.

    1. 3

    2. -3

    3. 1

    4. 700

    5. 1.05

  2. Name two more values of \(x\) that are solutions to the inequality.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may think 700 is not a solution to \(x > 1.\) Tell students that since there is an arrow at the end of the dark line, it includes all values that would fall on that line, even the ones not shown. 

Activity Synthesis

Ask a few students to share their responses for the last question. After each student shares, ask the class whether they agree or disagree. If students focus solely on the number line representation, be sure to tell them that to test whether a value makes an inequality true, you can substitute the value for the variable. Highlight the fact that “greater than 1” does not include 1.

Here are some questions for discussion:

  • What does the open circle at 1 mean? (It means 1 is not included.)
  • Is 1 a solution to the inequality \(x>1\)?

13.2: The Roller Coaster (15 minutes)

Activity

The purpose of this activity is to remind students that the symbol < is read “is less than” and the symbol > is read “is greater than.” Also, remind students of the use of an open circle or closed circle to indicate that the boundary point is included. Then, the symbols \(\leq\) and \(\geq\) are introduced.

Monitor for students who express the answer to the last question using words or using symbols. The responses to the last question will be used to introduce the new notation.

Launch

Allow students 5–10 minutes quiet work time and time to share their responses with a partner, followed by a whole-class discussion.

Representation: Access for Perception. Read all problems aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language
Conversing, Writing: MLR5 Co-craft Questions. Use this routine to help students consider the context of the first problem and to increase awareness about language used to describe situations involving inequalities. Begin by displaying only the initial text and photo of the roller coaster, without revealing the follow-up questions. In groups of 2, invite students to write down mathematical questions they have about this situation. Ask pairs to share their questions with the whole class. Amplify questions that highlight the mathematical language of “at least.”
Design Principle(s): Cultivate conversation; Maximize meta-awareness

Student Facing

A sign next to a roller coaster at an amusement park says, “You must be at least 60 inches tall to ride.” Noah is happy to know that he is tall enough to ride.

Photo of a roller-coaster.
  1. Noah is \(x\) inches tall. Which of the following can be true: \(x > 60\), \(x = 60\), or \(x < 60\)? Explain how you know.
  2. Noah’s friend is 2 inches shorter than Noah. Can you tell if Noah’s friend is tall enough to go on the ride? Explain or show your reasoning.
  3. List one possible height for Noah that means that his friend is tall enough to go on the ride, and another that means that his friend is too short for the ride.
  4. On the number line below, show all the possible heights that Noah’s friend could be.

    A number line with the numbers 52 through 68, in increments of 2, indicated.
  5. Noah's friend is \(y\) inches tall. Use \(y\) and any of the symbols \(<\), \(=\), \(>\) to express this height.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are having trouble interpreting the first three questions or articulating their responses, encourage them to make use of the number line that appears in question 4.

Activity Synthesis

Ask selected students to share their response to the last question. They are likely to write something like “\(y=58\) or \(y>58\).” The < and > symbols are not enough to capture what we need here with a single mathematical statement. In grade 6, students saw graphs of inequalities using open and closed circles. Now, we can introduce new symbols \(\leq\) and \(\geq\), that mean “less than or equal to” and “greater than or equal to.”

13.3: Is the Inequality True or False? (15 minutes)

Activity

The purpose of this task is for students to interpret the notation \(<\), \(>\), \(\leq\), and \(\geq\) to evaluate whether different values make an inequality true. It is not expected that students will solve the inequalities generally. The work in this activity sets students up for success in later lessons when they test points to determine the direction of the inequality in the process of solving.

Notice students who use the following approaches:

  • Substituting a value in for \(x\), evaluating the expression, thinking about whether the statement is true. For example, “Is -300 less than 75? Yes, so this is true.”
  • Drawing a number line for each inequality and using it to reason about different values.

Launch

Keep students in the same groups. Give 5–10 minutes quiet and partner work time followed by a whole-class discussion.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. After students have completed the first 1–2 rows of the table, consider pausing for a brief class discussion before moving on.
Supports accessibility for: Organization; Attention

Student Facing

The table shows four inequalities and four possible values for \(x\). Decide whether each value makes each inequality true, and complete the table with “true” or “false.” Discuss your thinking with your partner. If you disagree, work to reach an agreement.

\(x\) 0 100 -100 25
\(x \leq 25\)
\(100 < 4x\)
\(\text-3x > \text-75\)
\(10 \geq 35 - x\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Find an example of in inequality used in the real world and describe it using a number line.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students who try to apply what they know about solving equations to solve the inequalities algebraically may come up with incorrect solutions. For instance, \(100 < 4x\) may at first glance look equivalent to \(x < 25\), since the “less than” sign appears. Students may incorrectly think that \(\text-3x > \text-75\) is equivalent to \(x > 25\). Ask these students, for example, what the solution to \(100 = 4x\) means (25 is the value of \(x\) that makes \(4x\) equal to 100). Then encourage these students to test values like 24 and 26 to see whether they are solutions to \(100 < 4x\). This will be covered in greater detail in a later lesson, so this understanding does not need to be solidified at this time.

Activity Synthesis

The purpose of the discussion is to note the consequences of an inequality using \(\leq\) versus <, and \(\geq\) versus >. Direct students’ attention to \(100 < 4x\) and \(10 \geq 35 - x\). Substituting 25 for \(x\) in each of these inequalities gives \(100 < 100\), which is false, and \(10 \geq 10\), which is true. The key distinction is that \(\geq\) and \(\leq\) inequalities are considered true when both sides are equal, whereas < and > inequalities are considered false when both sides are equal. Ask students to present different strategies for determining whether a value makes an inequality true. Ask whether students were surprised by or initially incorrect about any of the answers. Emphasize that substituting a value in for \(x\), and thinking about whether the resulting inequality is saying something true, is the most direct way to check whether the value is a solution.

Speaking, Conversing: MLR8 Discussion Supports. Provide students with sentence frames that will help them define what each inequality symbol means. For example, “____ is greater than or equal to ____.", "____ is less than or equal to _____.", " _____ is greater than _____." and "_____ is less than _____.” Create a chart with the matching symbols for each frame and encourage students to refer to this chart when they explain whether they think each inequality is “true” or “false.”
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

By the end of this lesson, students should be able to interpret the symbols \(\geq\) and \(\leq\) and be able to test whether a given value makes an inequality true. Ask students to write an equality to which -5 is a solution, then trade with their partner to see if their partner agrees.

13.4: Cool-down - Some Values, All Values (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We use inequalities to describe a range of numbers. In many places, you are allowed to get a driver’s license when you are at least 16 years old. When checking if someone is old enough to get a license, we want to know if their age is at least 16. If \(h\) is the age of a person, then we can check if they are allowed to get a driver’s license by checking if their age makes the inequality \(h>16\) (they are older than 16) or the equation \(h=16\) (they are 16) true. The symbol \(\geq\), pronounced “greater than or equal to,” combines these two cases and we can just check if \(h \geq 16\) (their age is greater than or equal to 16). The inequality \(h \geq 16\) can be represented on a number line:

Number line, 0 to 28 by fours. An arrow begins at 16, closed circle, extends to the right.