Exploring Inequalities

Curriculum Context

Grade Level: 10th Grade
Curriculum Area: Algebra (Aligned with Common Core Standards - HSA.REI.B.3)
Learning Objective: By the end of the lesson, students will be able to solve and graph linear inequalities in one variable on a number line, and understand their real-life applications.

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Lesson Overview (42 Minutes)

1. Introduction & Objective (5 minutes)

• Teacher Explanation (2 minutes): Briefly introduce inequalities as a mathematical concept that represents a relationship of comparison (e.g., less than, greater than) instead of equivalence. Link it to real-world examples such as budget constraints, speed limits, and age requirements.
  • Examples: "You must be at least 16 years old to drive."
  • Relate the phrase "at least 16" to the inequality symbol (≥).
• Objective for the Students (1 minute): Write the objective on the board:
  "Today, we will learn to solve and graph inequalities, and connect them to real-world situations."
• Check Prior Knowledge (2 minutes): Ask students:
  1. "What is the difference between the symbols > and ≥?"
  2. "What does it mean to 'flip' an inequality sign?"
     Collect quick verbal responses to engage students.

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2. Main Lesson (25 minutes)

A. Recap and Build Basic Knowledge (5 minutes)

• Review Key Concepts visually:
  • Write examples on the board:
    1. x > 3
    2. x ≤ -2
       Ask students, “How would we graph these on a number line?”
  • Graph them interactively. Use colored chalk or dry-erase markers for distinction.
  • Teachable Moment: Emphasize the importance of an open circle for < and > vs. closed circle for ≤, ≥.
• Ensure students recognize that solving inequalities is similar to solving equations, except for flipping the inequality symbol when multiplying/dividing by a negative number.

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B. Guided Practice (12 minutes)

• Problem Set #1 (5 minutes): Solve Together

  • Solve step-by-step on the board with input from students:
    1. 5x + 3 > 18
    2. -2x ≤ 8
    3. (x - 4)/2 ≥ 3
  • Identify key "flip" moments and strategies to isolate the variable.

• Problem Set #2 (7 minutes): Collaborative Learning Stations

  • Divide students into 6 groups (4 students each). Each group works on a different inequality problem on laminated sheets:
    1. x/3 - 2 < 4
    2. 3(x + 5) ≥ 15
    3. -4x > -12
    • Groups solve and graph their inequality using mini-whiteboards.
    • The teacher circulates to clarify misconceptions and offer hints.
    • Groups rotate to the next problem after 2 minutes.

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C. Real-Life Context (3 minutes)

• Pose this question:
  "Imagine you’re limited to spending no more than $50 on your favorite hobby. If each item costs $8, write and solve an inequality to show how many items you could purchase."
  • Discuss briefly as a class how math helps in decision-making.

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3. Independent Practice (8 Minutes)

Distribute individual worksheets with the following sections:

1. Solve & Graph (Include Negatives):
   a. 7x - 12 ≤ 2
   b. -3(x - 4) > 9
2. Real-Life Application Problem:
   A movie ticket costs $9, and you have at most $40 to spend, including a $4 snack. Write and solve an inequality to determine how many tickets you can afford.

As students work independently, circulate to provide targeted feedback. Encourage those who finish early to help peers or attempt the challenge problem:
Challenge: Find x: 2x/5 + 4 > 7.

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4. Conclusion (4 Minutes)

Exit Ticket:

Write an inequality on the board: 2x - 7 < 15. Students must solve it, graph it, and write a real-world scenario as an explanation. Collect the exit tickets for formative assessment.

Class Summary (2 minutes):

• Recap the key points: Solving inequalities, flipping the sign when dividing/multiplying negatives, and graphing solutions.
• Relate it to broader algebra questions: "How are inequalities used in other mathematical areas like systems of equations or geometry?"

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Instructional Strategies

1. Active Learning: Collaborative group rotations encourage student ownership of problem-solving.
2. Differentiation: Provide visual aids like number lines for less confident students. Use real-world applications to deepen understanding for advanced learners.
3. Checks for Understanding: Regular pauses for questions and active circulation during activities.

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Materials Required:

1. Dry-erase markers & whiteboards (individual and teacher board).
2. Laminated inequality problems for group stations.
3. Independent practice worksheets (printouts).
4. Mini number-line templates for graphing.

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Post-Lesson Reflection (For Teachers):

1. Were students able to accurately graph inequalities?
2. Did the real-world examples increase engagement?
3. Did any groups require additional scaffolding? Use this to adjust future lessons for better differentiation.

Takeaway for Students: Learning inequalities prepares students to navigate situations requiring analytical thinking, from personal budgets to data interpretation in future STEM fields.