Lesson 18

Inequalities in Context

These materials, when encountered before Algebra 1, Unit 2, Lesson 18 support success in that lesson.

18.1: Inequalities (5 minutes)

Warm-up

The purpose of this warm-up is to elicit students’ current understanding of the inequality symbols, which will be useful when students use inequalities to represent situations later in this lesson. In addition, the activity is designed to elicit students’ understanding and possible misconceptions about inequalities involving negative numbers. 

Launch

Students should complete the task individually and be prepared to explain their reasoning for their answers. 

Student Facing

Place a < or > to correctly complete the inequality.

1. 5 \(\underline{\hspace{.5in}} \)10

2. 5 \(\underline{\hspace{.5in}} \) -10

3. -5 \(\underline{\hspace{.5in}} \) -10

4. \(\frac15 \) \(\underline{\hspace{.5in}} \) \(\frac{1}{10}\)

5.\(\frac{-1}{5} \) \(\underline{\hspace{.5in}} \) \(\frac {-1}{10}\)

Student Response

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

Activity Synthesis

Discuss what students know about inequality symbols. Ask students to read an inequality out loud. If students struggle to read them accurately, demonstrate how to say, “five is less than ten” and “one fifth is greater than one tenth". Ask students how their thinking changes when the comparison involves a negative number. 

18.2: Sweater Weather (20 minutes)

Activity

The mathematical purpose of this activity is to provide a context for students to reason about inequalities and a range of values that can work for a given situation. In addition, this activity provides a context for reasoning about inequalities involving negative numbers.

This activity builds on students' prior work with negative numbers in grade 6, in which contexts and location on the number line were used to support students’ understanding of ordering negative numbers.

This work will be helpful when students solve inequalities and interpret the solutions in their Algebra 1 class, especially when working with inequalities involving negative numbers.

Launch

Ensure that students understand what a sleeping bag and a down jacket are. Ask students if they have any experience camping and whether their sleeping bag was warm enough, and what kinds of winter jackets they know of and which are warmer. Show pictures of different kinds of jackets and sleeping bags if needed.

Student Facing

Elena and Lin are planning some cold-weather camping and are studying jackets and sleeping bags. Here is what they learned about some different equipment.

  • A down jacket is rated as comfortable when the temperature is between \(\text{-} 20^\circ\text{F}\) and \(20^\circ\text{F}\).
  • A down sleeping bag is rated as comfortable when the temperature is \(\text{-} 20^\circ\text{F}\) or above.
  • A synthetic sleeping bag is rated as comfortable when the temperature is \(20^\circ\text{F}\) or above.
a photo of down jackets hanging in a store
  1. What are 2 examples of temperatures that are comfortable for the down jacket?
  2. For which gear would a temperature of \(\text{-}16.5^\circ\text{F}\) be comfortable?
  3. Is it possible to list all the temperatures for which the down jacket is comfortable? Explain your reasoning.
  4. Which jacket’s comfort rating matches each inequality?
    1. \(x \geq 20\)
    2. \(\text-20 \leq x\)
    3. \(\text{-}20 \leq x \) and \(x \leq 20\)
  5. Here are five graphs of inequalities. Match each graph to a situation in this activity.

    A

    Number line with region to the right and including-20 shaded 

    B

    Number line. 20 and all number to the right shaded

    C

    Number line region between and including -20 and 20 is shaded

    D

    Number line, region to the left, not including -20 is shaded 

    E

    Number line, region to the right and not including 20 is shaded 
  6. Are there any temperatures at which the sleeping bags and the jacket would all be comfortable?
  7. Write one or more inequalities representing the range of temperatures in your area in the winter. Which gear, if any, would you recommend based on that?

Student Response

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

Activity Synthesis

The goal of this discussion is to help students make sense of the inequality statements given in the problem. Begin by asking students which gear they matched to each inequality.

Display the three inequalities for all to see, labeled with which piece of gear they go with:

  1. down jacket: \(\text{-}20 \leq x\) and \(x \leq 20\)
  2. down sleeping bag: \(\text-20 \leq x\)
  3. synthetic sleeping bag: \(20 \leq x\)

Call on different students to read each inequality out loud. If students read the inequalities as: “Negative twenty is less than or equal to \(x\) and \(x\) is less than or equal to 20,” ask them to translate that into a statement that is easier for customers to make sense of, such as, “The down jacket is suitable for temperatures between -20 and 20.”

Discuss how students matched the ineualities with graphs. Here are questions for discussion:

  • "How did you determine which graph matches the down jacket?" (It is the only description that has 2 constraints, and its graph is the only graph to show stopping points at 2 different values.)
  • "What is the difference between the meanings of graph B and graph E?"(Graph B has a closed circle, meaning the sleeping bag is comfortable at or above \(20^\circ\text{F} \). Graph E has an open circle, meaning the sleeping bag is comfortable at any temperature that is above \(20^\circ\text{F}\).)
  • "On the graph of an inequality, what is the difference between an open circle and a closed circle?" (An open circle means the value is not included in the solution, and a closed circle means that the number is included in the solution.)

18.3: Representing Inequalities (15 minutes)

Activity

The purpose of this activity is to practice representing situations with inequalities. This will be helpful for students in the associated Algebra 1 lesson when they represent constraints of a more complex situation. Students reason abstractly and quantitatively (MP2) as they attempt to symbolically represent a situation.

Launch

Allow students to work individually. Encourage students to use actual values in their inequalities whenever possible. 

Student Facing

For each statement, write an inequality to represent it. If a variable is used, be prepared to explain what it represents.

  1. Han has 5 pencils, and Andre has 8.

  2. Noah has more books than Kiran.

  3. Clare has more than \$200 in her savings account.

  4. The most the mechanic will charge for an oil change is \$60.

  5. Diego scored 1,200 points in a game, breaking the record for highest score.

  6. Jada is younger than Tyler.

  7. Animal World has at least 400 different species of animals.

  8. Mai’s bowling score is more than Clare’s and Han’s combined.

Student Response

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

Activity Synthesis

Display these inequalities for all to see. Select several students to read each of these inequalities. Encourage them to read what the statement means, not just decode each symbol—for example, reading “\(x \) is between 3 and 7” rather than “3 is less than or equal to \(x \)" and "\(x \) is less than or equal to 7.”

  • \(3 \leq x\) and \(x \leq 7\)
  • \(\text-5 > x\)
  • \(x < \frac32\)

After students read the second and third inequality ask, “What is another way to say the same relationship?” If not brought up in students’ explanations, discuss how “-5 is greater than \(x \)” is equivalent to “\(x \) is less than -5.”

Give students 1 minute of quiet think time to think about answers to these questions:

  • “What do you remember learning about inequalities before today?” (I remembered the meaning of the symbols, and that inequalities can also include \(\le\) and \(\ge\).)

  • “What did you get reminded about as a result of today's lesson?” ( I was reminded that it can be confusing to think about values in relation to a negative number.)