Lesson 19

Queuing on the Number Line

These materials, when encountered before Algebra 1, Unit 2, Lesson 19 support success in that lesson.

19.1: Notice and Wonder: Shaded Number Line (5 minutes)

Warm-up

The purpose of this warm-up is to elicit ideas that show understanding of a solution to an inequality represented on a number line, which will be useful when students interpret and represent solutions to inequalities on the number line in the associated Algebra 1 lesson. While students may notice and wonder many things about these images, the meaning of the shading is the important discussion point.

Launch

Display the image and inequality for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

\(4>x\)

Number line, negative 10 to 10 by ones. Open circle at 4, line continues left.

Student Response

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Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the connections between the image and the given inequality do not come up during the conversation, ask students to discuss them. Point out the open circle if no student mentions it. Invite students to wonder about it.

Mention that this is a graph of all the solutions to the inequality.

19.2: Pick a Number (15 minutes)

Activity

Students sometimes struggle to decide on their own what values to test in a given inequality. In this activity, they are given specific options to consider and get to choose the value they most want to test. This provides an opportunity to talk about why certain values might be chosen. Students are also asked to think about which values will definitely satisfy and not satisfy the inequalities. This encourages them to analyze the structure of the expression and think about how its value will change when testing large numbers, negative numbers, zero, or other types of values. Students are encouraged to solve the inequality without a calculator, so it is acceptable for students to come up with estimates instead of an exact number. This kind of reasoning helps students check whether their solutions are reasonable. This activity prepares student for work in the associated Algebra 1 lesson when they find solutions to inequalities and compare the situations that the inequalities represent. 

Monitor for students who:

  • choose to test zero

  • consider values that will result in whole numbers when divided by a whole number or multiplied by a fraction

  • strategically choose large numbers

  • strategically choose negative numbers

  • always choose the same number

Student Facing

For each expression, pick a number you would like to evaluate, and tell whether it makes the inequality true. Be prepared to explain what made you choose your number.

  1. \(\frac43y+10>19\)
    1. Pick a number you would like to test in place of \(y\): -1, 0, 1, 3, 4, or 5. Explain why you chose your number.

    2. Does your number make the inequality true?

    3. What is a different number that is definitely a solution? How do you know?

    4. What is a different number that is definitely not a solution? How do you know?

  2. \(2.954x-14.287<13.89\)
    1. Pick a number you would like to test in place of \(x\): -1, -0.5, 0, 0.5, 1, 3, 10, or 1,000. Explain why you chose your number.

    2. Does your number make the inequality true?

    3. What is a different number that is definitely a solution? How do you know?

    4. What is a different number that is definitely not a solution? How do you know?

  3. \(10-3y<5\)
    1. Pick a number you would like to test in place of \(y\): -100, -3, -1, 0,\(\frac13\), \(\frac53\), 33, or 100. Explain why you chose your number.

    2. Does your number make the inequality true?

    3. What is a different number that is definitely a solution? How do you know?

    4. What is a different number that is definitely not a solution? How do you know?

  4. \(\frac{10x}{4} > \frac{3x}{5}\)
    1. Pick a number you would like to test in place of \(x\): -10, -5, -4, 0, 4, 5, 10, or 20. Explain why you chose your number.

    2. Does your number make the inequality true?

    3. What is a different number that is definitely a solution? How do you know?

    4. What is a different number that is definitely not a solution? How do you know?

Student Response

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Activity Synthesis

Ask previously identified students to share their strategies for the values they chose to use with different inequalities. If not brought up in students’ explanations, ask students to think about what kinds of numbers they would choose to make each inequality true or false (positive or negative and large or small).

Ask students to share which numbers that they didn’t test that would definitely make a given inequality true or false. This process of reasoning helps students build their intuition about solutions to inequalities. It also helps to check their reasoning when they try to solve inequalities in the associated Algebra 1 lesson.

19.3: Matching Words and Symbols (20 minutes)

Activity

The purpose of this activity is for students to practice applying their understanding of inequalities. They generate values that do and do not satisfy each inequality, and then match inequalities to a verbal description of the inequality. This will be useful in the associated Algebra 1 lesson when students identify possible ranges of values for situations and use various inequalities to represent them.

In this activity, students reason abstractly and quantitatively (MP2) as they match expressions containing inequalities to verbal descriptions.

Student Facing

For each inequality, write 3 values that make the inequality true, write 3 values that make it false, and choose a verbal description that matches the inequality.

  1. \(x > 13.5\)
    1. Three values that make it true:
    2. Three values that make it false:
    3. Which verbal description best matches the inequality?
      1. \(x\) is less than 13.5
      2. \(x\) is greater than 13.5
      3. 13.5 is greater than \(x\)
  2. \(\text- 27 < x\)
    1. Three values that make it true:
    2. Three values that make it false:
    3. Which verbal description best matches the inequality?
      1. \(x\) is less than -27
      2. \(x\) is greater than -27
      3. -27 is greater than \(x\)
  3. \(x \geq \frac12\) and \(x \leq 2.75\)
    1. Three values that make it true:
    2. Three values that make it false:
    3. Which verbal description best matches the inequality?
      1. \(x\) is between \(\frac12\) and 2.75
      2. 2.75 is less than \(x\) is less than \(\frac12\)
      3. \(x\) is greater than \(\frac12\)
  4. \(x\geq \text-\frac{19}{4}\) and \(x \leq \frac12\)
    1. Three values that make it true:
    2. Three values that make it false:
    3. Which verbal description best matches the inequality?
      1. \(x\) is between \(\frac12\) and \(\text{-}\frac{19}{4}\) 
      2. \(x\) less than \(\text-\frac{19}{4}\)
      3. \(x\) is between \(\text-\frac{19}{4}\) and \(\frac12\)

Student Response

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Activity Synthesis

Discuss how to interpret solutions to an inequality. Here are some questions for discussion:

  • "How do you know if a specific value is a solution to an inequality? " (The values that are solutions to an inequality are the ones in which you substitute for \(x \) and the inequality is still true. The values that are not solutions make the inequality false when you substitute them in for \(x \).)

  • "Why is it important to think about values that satisfy the inequality and values that do not?" (Thinking about this helps to make sense of the solution that you create for an inequality. If I know which values make the inequality true, then I also know if certain values are reasonable answers for an inequality and if they are reasonable for a given context.) 

  • "How do you know which verbal descriptions represent the inequality?" (I think about which values are being described in the verbal description, and determined if some sample values that fit the description also satisfy the inequality.)

  • "Which representation (verbal description or symbols) helps you more in understanding the solution?" (The verbal description helps me understand solutions better than symbols because I can think about more values that make the inequality true rather than thinking more about the meaning of the symbol. Or, the symbols are easier to read than a verbal description. )