Solving Two-Step Equations

Lesson Context

Grade Level: 9-12
Subject Focus: Mathematics
Curriculum Standards:
Aligned to the US Common Core State Standards for Mathematics:

• CCSS.MATH.CONTENT.HSA.REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
• CCSS.MATH.PRACTICE.MP2: Reason abstractly and quantitatively.

This 45-minute lesson is designed for a small group of 10 high school students, focusing on solving two-step equations involving positive and negative integers. It emphasizes both procedural fluency and conceptual understanding, ensuring that students can analyze and solve these equations correctly and effectively.

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Lesson Objectives

By the end of the lesson, students will:

1. Demonstrate an understanding of the structure and purpose of two-step equations.
2. Utilize inverse operations to isolate the variable and solve two-step equations with positive and negative integers.
3. Verify their solutions for accuracy using substitution.

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Materials Needed

• Whiteboard and markers
• Student whiteboards and markers (one per student)
• Handouts with sample problems
• Highlighters or colored pens (to support visual learning of positive and negative integers)
• A deck of “Equation Challenge” flashcards (created by the teacher beforehand, details provided below)

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Lesson Outline

1. Introduction (5 minutes)

Objective: Activate prior knowledge and introduce key concepts.

• Engagement: Write the equation 3x + 4 = 10 on the board. Ask students, “What do you think we need to do to find out what x is?”

  • Facilitate a quick 1-minute brainstorm where students share ideas.
  • Write down their responses and confirm the focus of today’s lesson: solving two-step equations using inverse operations.

• Provide a clear definition: “A two-step equation is an equation that requires two inverse operations to isolate the variable.”

Real-Life Connection: Example: "Imagine saving money for a concert ticket and knowing you spent a fixed amount before saving the rest. Solving for how much you saved at the beginning requires a two-step equation!"

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2. Direct Instruction (10 minutes)

Objective: Model solving two-step equations step-by-step while emphasizing key strategies.

1. Modeling Example 1: Positive Integers
   Write on the board: 2x + 3 = 11

   • Step 1: Subtract 3 from both sides (inverse of addition).
     2x = 8
   • Step 2: Divide both sides by 2 (inverse of multiplication).
     x = 4
   • Substitute x = 4 back into the original equation to check.

2. Modeling Example 2: Negative Integers
   Write on the board: -5x - 7 = -22

   • Step 1: Add 7 to both sides (inverse of subtraction).
     -5x = -15
   • Step 2: Divide both sides by -5 (inverse of multiplication).
     x = 3
   • Substitute x = 3 back into the original equation for verification.

• Highlight any common misconceptions (e.g., forgetting to adjust the sign of numbers when dividing by a negative).

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3. Guided Practice (10 minutes)

Objective: Students work through solutions with teacher facilitation.

1. Provide a handout with 4 progressively challenging sample equations:

   • 4x + 6 = 18
   • -3x - 5 = -20
   • 5x - 2 = 13
   • -7x + 8 = -6

2. Have all students solve the first equation independently on student whiteboards, while the teacher circulates to observe.

3. Solve the problem as a class, explicitly discussing strategies for error-checking.

4. Proceed to the second and third problems in pairs, encouraging peer discussions.

5. For the fourth problem, ask different students to explain each step aloud while solving on the board.

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4. Independent Practice (10 minutes)

Objective: Build procedural fluency by solving equations independently.

• Hand out an additional worksheet with 6 two-step equations covering a variety of integer combinations (e.g., mixing positive constants with negative coefficients).
• Ask students to solve on their own, showing all work.
• Encourage early finishers to challenge themselves by creating their own equations for peers to solve.

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5. Interactive Game: “Equation Challenge” (5 minutes)

Objective: Reinforce learning in a fun and collaborative way.

• Use a teacher-made deck of flashcards that contain two-step equations (e.g., -4x + 3 = 15, 3x - 8 = 1). Two cards should also have incorrect solutions to discuss potential errors.
• Shuffle the deck, and have students take turns drawing a card and solving the equation in front of the class.
• If they solve correctly, they keep the card. If not, another student is given the chance to solve.

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6. Closure and Review (5 minutes)

Objective: Recap learning and clarify misunderstandings.

• Ask students to articulate the two main steps for solving any two-step equation. (Subtract or add, then divide or multiply.)
• Pose a real-world challenge:
  “If 5 times a number minus 3 equals 17, what is the number? Could you write an equation to solve this?”
• Summarize: “Today, we mastered solving two-step equations. This is a foundational skill as we move toward solving more complex equations with variables on both sides.”

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Assessment

• Formative: Teacher observation during guided practice, pair discussions, and the interactive game.
• Summative: Accuracy of student solutions on the independent practice handout and responses during the closure activity.

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Differentiation Strategies

1. For Struggling Learners:

   • Provide a checklist of steps to solve two-step equations.
   • Use color coding for positive and negative integers to visually distinguish them during modeling.

2. For Advanced Learners:

   • Introduce equations with fractional coefficients (e.g., 1/2x + 5 = 7) as extension problems.
   • Challenge them to write multi-step real-world problems to share with peers.

3. For Visual Learners:

   • Include number-line visuals for equations involving negatives (optional).

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Teacher Reflection

• Did students engage actively during guided practice and the game?
• Were any steps particularly confusing for the group?
• Who might need additional support tomorrow, and how can I provide it?

End of Lesson Plan