Quadratic Inequalities Mastery

Curriculum Area and Level

Mathematics, Grade 9
Aligned to Common Core State Standards (CCSS):

• CCSS.MATH.CONTENT.HSA.REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
• CCSS.MATH.CONTENT.HSA.REI.B.4.B: Solve quadratic equations in one variable. Analyze and solve quadratic inequalities graphically and algebraically.

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Lesson Objectives

By the end of this 60-minute lesson, students will be able to:

1. Interpret quadratic inequalities and distinguish between solutions that are intervals and points.
2. Solve quadratic inequalities algebraically using factoring and sign testing.
3. Solve quadratic inequalities graphically, recognizing solution sets visually.
4. Apply quadratic inequalities to solve real-world problems.

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Materials Needed

• Whiteboard/Smartboard
• Graph paper and pencils for students
• Pre-made worksheet (teacher-generated with a mix of practice problems on algebraic and graphical solutions)
• Colored markers (for teacher demonstrations and student work)
• A graphing calculator or graphing app accessible to students (optional but ideal)

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Lesson Structure

1. Warm-Up (5 minutes)

• Write this question on the board as students enter:
  "What does it mean when a parabola intersects the x-axis? How might those points relate to solving quadratics?"
• Students will reflect and write thoughts in their notebooks for 2 minutes, then briefly discuss answers with a partner for 3 minutes.

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2. Hook/Introduction to Lesson (5 minutes)

• The teacher announces:
  "Today we are solving quadratic inequalities, a vital skill that will help us make sense of questions like: 'Given a path of a basketball, when is the ball higher than 10 feet?' or 'For what time range does a company's profit decrease below a certain amount?'"
• Visual demonstration on the whiteboard: Draw a parabola representing a basketball, draw a horizontal line at y = 10, and shade the region where the parabola is above the line.

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3. Direct Instruction (15 minutes)

Step 1: Graphical Understanding (8 minutes)

• Begin by graphing the inequality y > x² - 4x - 5.
• Highlight the key components:
  • Finding the quadratic's roots by solving x² - 4x - 5 = 0 using factoring.
    • Roots are x = -1 and x = 5.
  • Use a test point to find when the parabola is above y = 0.
  • Shade the appropriate regions on the graph.
• Reinforce: Solutions are values of x where the quadratic satisfies the inequality. Emphasize open vs. closed circles (≥ vs. >).

Step 2: Algebraic Understanding (7 minutes)

• Solve the same inequality algebraically: x² - 4x - 5 > 0.
  • Factor: (x + 1)(x - 5) > 0.
  • Use a sign chart: Divide the x-axis into intervals based on the roots (-∞ to -1, -1 to 5, and 5 to ∞).
  • Test points in each interval to determine signs of the factors.
  • Conclude: The solution is x < -1 or x > 5.
• Highlight the connection to the graph.

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4. Guided Practice (10 minutes)

• Pose another problem: x² - 3x ≤ 4.
  • Students work step-by-step with the teacher:
    • Rewrite inequality as x² - 3x - 4 ≤ 0.
    • Factor and solve x² - 3x - 4 = 0.
    • Create a sign chart and determine intervals where the inequality holds.
    • Interpret the solution: [-1, 4], including boundary points.
  • Discuss why the solution is a continuous interval (parabola opens up).

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5. Independent Practice (15 minutes)

• Distribute the worksheet with diverse quadratic inequality problems (both graphical and algebraic).
• Problems should include:
  • Simple inequalities (e.g., x² - 16 > 0)
  • Real-world word problems requiring interpretation.
  • Graph and shade regions for inequalities, like x² + 2x > 3.
• Circulate to help individual students.

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6. Application Challenge (8 minutes)

• Group Activity: Students are grouped into teams of 4. Each team solves this problem:
  A company’s profit is modeled as P(t) = -t² + 6t - 5, where t is time in hours. When is the company’s profit above $0?
• Teams will graph the function, solve algebraically, and write their answer in a contextual sentence:
  "The company's profit is above $0 between ___ hours and ___ hours."

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7. Recap and Exit Ticket (5 minutes)

• Recap Key Ideas:
  1. Quadratic inequalities can be solved by factoring, testing intervals, or graphing.
  2. Solutions are regions or intervals that satisfy the inequality.
• Exit Ticket: Students solve this problem individually:
  Solve the inequality x² - 2x - 8 ≥ 0. Show your solution algebraically and graphically.

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Differentiation

• For Advanced Learners: Include a word problem involving quadratic inequalities where the context is abstract or involves maximizing/minimizing.
• For Struggling Learners: Pair these students with a stronger peer during group activities and provide templates (e.g., blanks for step-by-step solving).

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Assessment

• Formative: Student responses during warm-up, independent problem-solving, and group activity.
• Summative: The exit ticket will assess mastery of the day’s objective.

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Homework Task

• Solve these inequalities:
  1. x² + 5x > 14
  2. -x² + 4x - 3 ≤ 0
• Reflect: Write 3 sentences about a real-world situation where quadratic inequalities might apply.

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Teacher Note

Wow your students by emphasizing how mathematical solutions tell stories—like predicting when a ball will be caught or anticipating financial trends. Help them see the beauty of numbers and their power to describe the world!