Solving Equations Both Sides

Title: Solving Equations Both Sides
Current Content:

Overview

In this lesson, Year 8 students will develop skills in solving linear equations with pronumerals (variables) on both sides. Through explicit instruction, guided practice, and independent activities, students will learn to isolate the pronumeral using algebraic techniques in line with the Victorian Curriculum standards for Year 8 Mathematics. Differentiated strategies will support diverse learning needs, including visual aids and scaffolded tasks for students needing support, and extension activities for advanced learners.

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Victorian Curriculum Alignment

Content Descriptor:

• AC9M8A02: Solve linear equations and one-variable inequalities using algebraic techniques; verify solutions by substitution

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Learning Objectives

By the end of the lesson, students will be able to:

1. Identify equations with pronumerals on both sides.
2. Apply inverse operations to collect like terms on one side.
3. Solve linear equations systematically and correctly.
4. Verify solutions by substituting back into the original equation.
5. (Advanced) Solve simple one-variable inequalities with pronumerals on both sides and use graphing technology to verify solutions.

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Lesson Duration: 60 minutes

Class Size: 25 Students

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Materials Needed

• Whiteboard and markers
• Printed worksheets with scaffolded practice questions (varying difficulty levels)
• Algebra tiles or virtual manipulatives (especially for students needing support)
• Calculator (optional for verification)
• Graphing software or apps (optional extension for advanced students)

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Lesson Structure

1. Introduction & Engagement (10 minutes)

• Begin with a brief review of solving simple linear equations with pronumerals on one side only.
• Present real-life situations that could be modelled by equations with variables on both sides (e.g., comparing costs, distances, or ages).
• Write a simple equation on the board, e.g.
  ( 3x + 5 = x + 11 )
• Ask students what initial steps they think are needed to solve it.
• For students needing support, introduce the equation using algebra tiles or visual models to represent terms on both sides.
• Teacher note: Emphasise that the goal is to isolate the variable on one side to solve easily.

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2. Explanation & Modelling (15 minutes)

• Present step-by-step solving of an equation with pronumerals on both sides.

Example 1:
Solve ( 3x + 5 = x + 11 )

Step	Explanation	Equation
1	Subtract (x) from both sides to collect variables together	(3x - x + 5 = 11)
2	Simplify terms	(2x + 5 = 11)
3	Subtract 5 from both sides to isolate term with (x)	(2x = 6)
4	Divide both sides by 2	(x = 3)
5	Verify by substitution	Check (3(3) + 5 = 3 + 11) → (9 + 5 = 14) = (3 + 11 = 14) ✓

• For students needing support, provide a simplified, scaffolded version of this example with additional visual aids and slower pacing.
• Discuss each step clearly, highlighting the use of inverse operations and maintaining balance of the equation.
• Present a second example with negative terms or more complex coefficients to reinforce understanding.

Example 2:
Solve ( 5x - 7 = 2x + 8 )

• Guide students through similar steps, pausing for questions.
• Use algebra tiles or virtual manipulatives to demonstrate combining like terms for students needing support.

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3. Guided Practice (15 minutes)

• Distribute scaffolded worksheets consisting of 5-6 linear equations with pronumerals on both sides, varying in difficulty:
  • Level 1 (Support): Simple equations with small coefficients and no negatives, e.g. (2x + 3 = x + 7)
  • Level 2 (Core): Moderate difficulty, e.g. (4x + 3 = 2x + 11), (7 + 3x = 5x - 1)
  • Level 3 (Extension): More complex equations, e.g. (6x - 4 = 8 + 2x)
• Work through the first two equations as a class with teacher facilitation.
• Encourage students to verbalise their thought processes and explain the algebraic steps.
• Organise peer tutoring or small group work to support students needing additional help, pairing them with more confident peers.

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4. Independent Practice (15 minutes)

• Students work individually on solving 3 more equations from their scaffolded worksheets, applying what they have learned.
• Teacher circulates to provide support and feedback.
• Optional: For advanced students, pose challenge problems such as:
  • Solve equations with variables on both sides and nested brackets, e.g. ( 3(2x - 1) = 4x + 5 )
  • Solve simple one-variable inequalities with variables on both sides, e.g. ( 2x + 3 > x + 7 )
  • Use graphing software or apps to visually verify solutions of equations and inequalities.

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5. Assessment and Reflection (5 minutes)

• Quick quiz-style oral questioning or exit slip: Write and solve one linear equation with pronumerals on both sides.
• Collect students’ independent practice worksheets as informal assessment.
• Ask reflective questions:
  • What was the most important step in solving these equations?
  • How do you know your solution is correct?
  • (For advanced students) How can graphing help verify your solution?

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Differentiation and Extensions

• For students needing support:
  • Use algebra tiles or physical models to visualise combining like terms and balancing equations.
  • Provide step-by-step guided examples with simpler equations and scaffolded worksheets.
  • Encourage peer tutoring and small group work for collaborative learning.
• For advanced students:
  • Extend to solving simple one-variable inequalities with pronumerals on both sides.
  • Incorporate graphing technology to visually verify solutions of equations and inequalities.
  • Challenge with nested brackets and more complex expressions.
• Consider incorporating digital tools for graphing or equation solving as suggested by the curriculum elaborations.

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Cross-Curricular Connections

• Explore how equations are used in real-world contexts, such as financial planning or science experiments.
• Discuss First Nations perspectives on balancing relationships and apply analogies to algebraic balance.

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Summary

• Review the importance of maintaining equality in an equation while performing inverse operations.
• Reinforce checking solutions by substitution.
• Highlight that practicing these skills builds a strong algebraic foundation for Year 8 and beyond.
• Emphasise the value of using visual aids and technology to support understanding and verification.

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This detailed, structured approach follows the Victorian Curriculum for Year 8 Mathematics, specifically targeting the achievement standard of solving linear equations with pronumerals on both sides using algebraic techniques and verification by substitution, while addressing diverse learner needs through differentiation and extension.