Mathematics · Grade 8

Active learning ideas

Solving Equations with Variables on Both Sides

Solving equations with variables on both sides demands precision in algebraic steps, and active learning turns abstract ideas into tangible understanding. By physically manipulating terms and discussing steps in real time, students build fluency and confidence in balancing equations. This approach addresses common errors before they become habits.

• Ontario Curriculum Expectations
• 8.EE.C.7.B

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Problem-Based Learning
• 35 min · Pairs

Balance Scale Model: Equation Tiles

Provide algebra tiles or cutouts representing variables and constants. Students build both sides of given equations on physical or digital balances, then move tiles equally to isolate the variable. Pairs discuss each step and verify balance before recording the solution.

Explain the strategic steps for isolating the variable when it appears on both sides of an equation.

Facilitation Tip

During the Balance Scale Model, circulate to ensure students are physically removing the same number of tiles from both sides, reinforcing the concept of balance.

What to look for

Present students with the equation 5x + 2 = 3x + 10. Ask them to write down the first step they would take to isolate the variable and explain why they chose that step.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 02

• Stations Rotation
• 45 min · Small Groups

Stations Rotation: Equation Types

Set up stations with cards: simplify both sides, move variables, isolate, and check solutions. Small groups spend 8 minutes per station solving progressively harder equations, rotating and comparing answers with a class anchor chart.

Analyze how to simplify expressions on both sides before combining variable terms.

Facilitation Tip

In Station Rotation, provide a checklist at each station so students self-assess their progress and focus on the specific skill for that station.

What to look for

Provide students with the equation 7y - 4 = 2y + 11. Ask them to solve the equation and show all their work, then write one sentence explaining how they checked their answer.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 03

• Problem-Based Learning
• 25 min · Pairs

Error Analysis Hunt: Partner Review

Give pairs sets of solved equations with deliberate errors, like incorrect combining or skipping steps. They identify mistakes, explain fixes, and rewrite correctly, then swap with another pair for peer feedback.

Construct a clear, step-by-step solution for equations with variables on both sides.

Facilitation Tip

For the Error Analysis Hunt, assign pairs the same equation to encourage detailed comparison and shared problem-solving.

What to look for

Pose the question: 'What is the most common mistake students make when solving equations with variables on both sides, and how can we avoid it?' Facilitate a class discussion where students share their insights and strategies.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 04

• Problem-Based Learning
• 30 min · Whole Class

Word Problem Relay: Whole Class Chain

Project a multi-step word problem leading to an equation with variables on both sides. Students line up and solve one step at a time, passing to the next classmate, discussing as a group if stuck.

Explain the strategic steps for isolating the variable when it appears on both sides of an equation.

Facilitation Tip

In the Word Problem Relay, pause after each step to let students articulate their reasoning before moving to the next equation.

What to look for

Present students with the equation 5x + 2 = 3x + 10. Ask them to write down the first step they would take to isolate the variable and explain why they chose that step.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Start with concrete models like the balance scale to show why operations must be equal on both sides. Move students to symbolic work only after they can explain the reasoning behind each step. Avoid rushing to algorithms; instead, build understanding through repeated exposure to varied equations. Research shows that students who name their steps aloud develop stronger retention and transfer skills.

Students will confidently simplify equations, move variables to one side, and constants to the other using clear steps. They will explain each step aloud and verify solutions by substituting back into the original equation. Missteps will be caught and corrected through discussion and modeling.

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Watch Out for These Misconceptions

• During Balance Scale Model: Equation Tiles, watch for students removing tiles from only one side of the scale to 'cancel out' variables.

  Prompt them to remove the same number of variable tiles from both sides, then ask them to explain why this keeps the scale balanced. Have them record the step as an equation on their paper.

• During Station Rotation: Equation Types, watch for students ignoring constants when moving variables, such as subtracting 2x from 3x + 4 = 2x but forgetting to adjust the constant on the other side.

  At the station, model moving both variables and constants together by physically grouping constant tiles and moving them as a unit, then ask students to verbalize the inverse operation for both sides.

• During Error Analysis Hunt: Partner Review, watch for students solving equations without checking their answers, assuming the process is complete after isolating the variable.

  Require students to write the original equation at the top of their paper and substitute their solution back in, highlighting the value in verification. Collect their work to assess whether verification is included.

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Methods used in this brief

• Problem-Based Learning
• Stations Rotation