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Factoring Quadratics: Find Your Zeros!

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Lesson Plan

Factoring Quadratics: Find Your Zeros!

Students will learn to factor quadratic expressions into binomials and use the Zero Product Property to find the function's zeros, building a foundation for solving quadratic equations and understanding parabolic graphs.

Factoring quadratics is a crucial skill for solving complex equations, analyzing real-world scenarios like projectile motion, and understanding the behavior of curved graphs.

Audience

9th Grade Students (Tier 2 Small Group)

Time

30 minutes

Approach

Direct instruction, guided practice, and independent application.

Materials

Smartboard or Whiteboard, Markers/Chalk, Slide Deck: Factoring Quadratics, Worksheet: Zeroing In!, and Answer Key: Zeroing In!

Prep

Teacher Preparation

10 minutes

Step 1

Introduction & Warm-Up

5 minutes

  • Begin by asking students what they remember about factoring simple expressions (e.g., common factoring).
    - Introduce the concept of quadratic expressions and how finding their 'zeros' relates to where a graph crosses the x-axis. Use Slide 1 and Slide 2 of the slide deck.

Step 2

Understanding Zeros & Factoring

10 minutes

  • Explain that zeros are the x-values where the function equals zero.
    - Demonstrate factoring a quadratic expression (e.g., x^2 + 5x + 6) step-by-step using a method like 'splitting the middle term' or 'guess and check' with Slide 3 and Slide 4.
    - Guide students through an example together, emphasizing the importance of finding two numbers that multiply to 'c' and add to 'b'.
    - Introduce the Zero Product Property: If a*b = 0, then a=0 or b=0. Show how to apply this to factored quadratics to find the zeros on Slide 5.

Step 3

Guided Practice & Worksheet Introduction

10 minutes

  • Work through one or two more examples with the students, prompting them for each step. Use Slide 6.
    - Distribute the Worksheet: Zeroing In!.
    - Have students attempt the first few problems independently, circulating to provide immediate feedback and support.
    - Address common misconceptions as a group.

Step 4

Wrap-Up & Cool Down

5 minutes

  • Review one of the challenging problems from the Worksheet: Zeroing In! as a group.
    - Ask students to summarize in their own words how to find the zeros of a quadratic function once it's factored. Use Slide 7.
    - Collect worksheets for review or assign remaining problems for homework.
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Slide Deck

Factoring Quadratics: Find Your Zeros!

What are we learning today?

  • How to factor quadratic expressions.
  • How to use factoring to find the "zeros" of a quadratic function.

Why is this important?

  • Helps us solve quadratic equations.
  • Understands where a graph crosses the x-axis.
  • Useful in science, engineering, and sports!

Welcome students and briefly review what they remember about simple factoring. Introduce the idea that quadratics have a special shape (parabola) and that 'zeros' are important points on that shape.

What Are 'Zeros'?

Imagine throwing a ball...

  • It goes up, and then it comes down!
  • The path it takes is a parabola.

Zeros are where the parabola crosses the x-axis!

  • These are the points where the function's output (y-value) is ZERO.
  • We want to find the x-values where this happens.

Explain that zeros are the x-intercepts of the parabola. Visually show a simple parabola and point out where it crosses the x-axis. Emphasize that at these points, y = 0.

Let's Factor!

Our Goal: Turn this... (x² + 5x + 6)

Into this! (x + _)(x + _)

How?

  1. Look at the last number (the constant term).
  2. Look at the middle number (the coefficient of x).
  3. Find two numbers that:
    • Multiply to the constant term
    • Add to the middle term

Introduce a basic quadratic expression. Explain that factoring helps us break down the expression into simpler parts (binomials). Remind them of finding two numbers that multiply to 'c' and add to 'b'.

Example: x² + 5x + 6

Step 1: Identify the numbers

  • Constant term: 6
  • Middle term coefficient: 5

Step 2: Find two numbers

  • What two numbers multiply to 6 and add to 5?
    • Think: (1, 6), (2, 3)
    • 2 and 3!

Step 3: Write in factored form

  • (x + 2)(x + 3)

Walk through the example x^2 + 5x + 6 step-by-step. Guide students to identify that 2 and 3 multiply to 6 and add to 5. Show how to write the factored form. Then, introduce the Zero Product Property.

Zero Product Property!

If (A) * (B) = 0, then...

A = 0 OR B = 0

Let's find the zeros for (x + 2)(x + 3) = 0

  1. Set each factor to zero:
    • x + 2 = 0
    • x + 3 = 0
  2. Solve for x:
    • x = -2
    • x = -3

Our zeros are -2 and -3!

Explain the Zero Product Property clearly. Emphasize that if the product of two factors is zero, then at least one of the factors must be zero. Apply this to the factored quadratic from the previous slide to find the zeros.

Your Turn! (Guided Practice)

Factor the following quadratic and find its zeros:

x² + 7x + 10 = 0

  1. What two numbers multiply to 10 and add to 7?


  2. Write the factored form.


  3. Apply the Zero Product Property to find the zeros.


Present a new example for guided practice. Encourage students to participate in each step, from identifying the target numbers to setting factors equal to zero. Provide support as needed.

Recap & Practice

What did we learn today?

  • How to factor quadratic expressions.
  • How to use the Zero Product Property to find zeros.

Practice makes perfect!

Briefly recap the steps. Ask students to share one thing they learned or found challenging. Introduce the worksheet for independent practice.

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Worksheet

Worksheet: Zeroing In!

Directions: For each quadratic expression, factor it to reveal its zeros. Show your work!

Section A: Factoring Practice

  1. Factor the following quadratic expression:
    x² + 6x + 8











  2. Factor the following quadratic expression:
    x² + 9x + 14











  3. Factor the following quadratic expression:
    x² - 7x + 12











  4. Factor the following quadratic expression:
    x² - 2x - 15











Section B: Find Your Zeros!

Directions: Factor each quadratic expression and then use the Zero Product Property to find the zeros of the function.

  1. x² + 10x + 21 = 0















  2. x² - 8x + 15 = 0















  3. x² + 2x - 24 = 0















  4. x² - 3x - 10 = 0















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Answer Key

Answer Key: Zeroing In!

Directions: For each quadratic expression, the steps to factor and find its zeros are provided.

Section A: Factoring Practice

  1. x² + 6x + 8
    • Thought Process: We need two numbers that multiply to 8 and add to 6. These numbers are 2 and 4.
    • Factored Form: (x + 2)(x + 4)
  2. x² + 9x + 14
    • Thought Process: We need two numbers that multiply to 14 and add to 9. These numbers are 2 and 7.
    • Factored Form: (x + 2)(x + 7)
  3. x² - 7x + 12
    • Thought Process: We need two numbers that multiply to 12 and add to -7. Since the product is positive and the sum is negative, both numbers must be negative. These numbers are -3 and -4.
    • Factored Form: (x - 3)(x - 4)
  4. x² - 2x - 15
    • Thought Process: We need two numbers that multiply to -15 and add to -2. Since the product is negative, one number is positive and one is negative. The larger absolute value should be negative to get a negative sum. These numbers are 3 and -5.
    • Factored Form: (x + 3)(x - 5)

Section B: Find Your Zeros!

Directions: Factor each quadratic expression and then use the Zero Product Property to find the zeros of the function.

  1. x² + 10x + 21 = 0
    • Factoring: We need two numbers that multiply to 21 and add to 10. These are 3 and 7.
      (x + 3)(x + 7) = 0
    • Finding Zeros (Zero Product Property):
      x + 3 = 0 => x = -3
      x + 7 = 0 => x = -7
  2. x² - 8x + 15 = 0
    • Factoring: We need two numbers that multiply to 15 and add to -8. These are -3 and -5.
      (x - 3)(x - 5) = 0
    • Finding Zeros (Zero Product Property):
      x - 3 = 0 => x = 3
      x - 5 = 0 => x = 5
  3. x² + 2x - 24 = 0
    • Factoring: We need two numbers that multiply to -24 and add to 2. These are 6 and -4.
      (x + 6)(x - 4) = 0
    • Finding Zeros (Zero Product Property):
      x + 6 = 0 => x = -6
      x - 4 = 0 => x = 4
  4. x² - 3x - 10 = 0
    • Factoring: We need two numbers that multiply to -10 and add to -3. These are 2 and -5.
      (x + 2)(x - 5) = 0
    • Finding Zeros (Zero Product Property):
      x + 2 = 0 => x = -2
      x - 5 = 0 => x = 5
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