Solving Linear Equations

Objective

By the end of the lesson, 12th-grade students will be able to:

• Solve linear equations in one variable, including equations with variables on both sides and those involving fractions or decimals.
• Apply algebraic techniques to isolate variables accurately.
• Verify solutions by substitution.

This aligns with the following Common Core State Standards (CCSS) for 12th Grade (Algebra):

• 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.D.11: Explain why the solution set of an equation is the set of points that satisfy the equation.
• CCSS.MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them.
• CCSS.MATH.PRACTICE.MP6: Attend to precision.

Duration

60 minutes
Class size: 5 students

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Materials Needed

• Whiteboard and markers
• Student notebooks and graph paper
• Algebra tiles or virtual algebra manipulatives app
• Calculators (optional for checking answers)
• Printed worksheets with tailored equation problems
• Mini dry erase boards for students (optional but beneficial)

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Lesson Breakdown

1. Introduction & Objective Setting (5 minutes)

• Begin with a brief discussion on the importance of solving equations in real-world problem-solving.
• Write a basic example on the board:
  3x + 5 = 20
• Discuss what it means to find the value of x.
• State the day’s learning objective and connect it to CCSS standards.

Teacher tip: Emphasize the precision in isolating variables and identifying equivalent equations.

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2. Review & Prior Knowledge Activation (10 minutes)

• Quick refresher on:
  • Properties of equality (addition, subtraction, multiplication, division)
  • Combining like terms
  • Distributive property
• Students will solve two pre-prepared equations on mini whiteboards and show answers simultaneously.
  Examples:
  1. 2x - 4 = 10
  2. 5(x - 3) = 20
• Discuss common pitfalls - e.g., forgetting to apply an operation to both sides.

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3. Guided Practice: Complex Linear Equations (15 minutes)

• Introduce equations with variables on both sides, fractions, and decimals:
  Examples:
  • 4x - 7 = 2x + 9
  • (3/4)x + 5 = 11
  • 0.5x - 3.2 = 1.8
• Work through the first example as a class:
  • Step 1: Subtract 2x from both sides.
  • Step 2: Add 7 to both sides.
  • Step 3: Solve for x.
• Emphasize maintaining balance and precision (MP6).
• Provide students with two similar problems to solve independently while teacher circulates to assist.

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4. Real-World Application Mini-Project (15 minutes)

• Present a real-world scenario requiring solving an equation:
  “You are creating a budget for a party. You know the total cost is $150, the rental for the venue is $100, and the rest is for buying drinks. If each drink costs $5, how many drinks can you buy?”
• Students write an equation and solve it:
  Let d = number of drinks: 100 + 5d = 150.
• Discuss how to set up equations from word problems.
• Challenge students to create their own real-world problem requiring a linear equation, exchange with peers, and solve each other’s problems.

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5. Formative Assessment & Reflection (10 minutes)

• Distribute a short worksheet with problems of varying difficulty:
  • Solve 7x + 3 = 24
  • Solve (2/3)x - 4 = 8
  • Solve 5(x + 2) = 3x + 16
• Use exit tickets where each student writes one strength and one area for improvement regarding solving equations.
• Collect and review to inform next steps.

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Homework Suggestion

Assign problems that include:

• Linear equations with variables on both sides
• Equations with fractions and decimals
• Word problems requiring equation setup

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Additional Teacher Notes

• Encourage students to verbalize their thought process during problem-solving to deepen understanding (MP1).
• Use manipulatives for students struggling with abstract algebraic operations.
• Reinforce the importance of precision (checking each step and verifying solutions).

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Summary

This lesson plan hones in on mastering linear equations aligned explicitly with CCSS standards, preparing students not only to solve algebraic expressions but to apply these skills in real-world contexts, promoting both procedural fluency and conceptual understanding.