Solving Equations with Variables on Both Sides

Active learning helps students grasp the shift from single-sided to two-sided equations by making abstract moves concrete. When variables appear on both sides, students need to physically manipulate terms to see how balance is maintained, which builds confidence and reduces frustration with the new strategy.

• 8th Grade
• Mathematics
• 4 activities
• 20 min–40 min

Learning Objectives

1. Analyze the algebraic steps required to isolate a variable when it appears on both sides of a linear equation.
2. Explain the conditions under which a linear equation yields no solution or infinitely many solutions.
3. Construct a valid step-by-step solution for linear equations with variables on both sides.
4. Evaluate the correctness of a given solution to an equation with variables on both sides.

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Move It or Keep It?

Present an equation like 5x + 3 = 2x + 12. Students decide individually which variable term to move and write their reasoning. Pairs then try both approaches (moving the left term vs. moving the right term) and verify they reach the same answer, confirming that either side works.

Prepare & details

Explain the strategy for collecting variable terms on one side of an equation.

Facilitation Tip: During Think-Pair-Share, circulate and listen for pairs justifying their choice of which variable term to move, not just describing the steps.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Inquiry Circle

⏱ 30 min·Small Groups

Inquiry Circle: What Happens Here?

Groups receive three equations: one with a unique solution, one with no solution, and one with infinite solutions. Without solving, they predict which type each is based on structure. They then solve all three to verify predictions and write a group rule for identifying each case from the equation's form.

Prepare & details

Analyze why some equations have no solution or infinitely many solutions.

Facilitation Tip: In Collaborative Investigation, ask groups to predict the solution type before solving to encourage reasoning over rote calculation.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Gallery Walk

⏱ 35 min·Small Groups

Gallery Walk: Solution Type Sort

Post nine equations around the room. Students rotate in groups, solving each and labeling it as unique solution, no solution, or infinite solutions. Groups compare labels afterward and discuss any equations where group labels disagree.

Prepare & details

Construct a step-by-step solution for an equation with variables on both sides.

Facilitation Tip: For Gallery Walk, enforce that students write their reasoning on the poster, not just the solution, to make thinking visible.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Stations Rotation

⏱ 40 min·Small Groups

Stations Rotation: Set Up and Solve

Stations provide equations at increasing complexity: (1) variables on both sides with no parentheses, (2) variables on both sides with one set of parentheses, (3) equations that lead to no solution or infinite solutions. Students solve and justify each step before moving on.

Prepare & details

Explain the strategy for collecting variable terms on one side of an equation.

Facilitation Tip: During Station Rotation, require students to check their solutions by substituting back into the original equation to reinforce verification.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Teaching This Topic

Teach this topic by first having students experience the disorientation of variables on both sides through structured activities. Experienced teachers avoid telling students which side to move terms to, instead guiding them to notice how negative coefficients or larger terms can complicate calculations. Research shows that allowing students to choose their method fosters deeper understanding and reduces procedural rigidity. Emphasize that the goal is balance, not following a rigid set of rules.

What to Expect

Successful learning looks like students confidently deciding where to move variable terms, explaining their choices, and solving equations accurately while identifying when an equation has no solution or infinite solutions. They should also recognize that their method choice is flexible and not bound by arbitrary rules.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation, watch for students interpreting 3 = 7 as a calculation error rather than a signal that no solution exists.

What to Teach Instead

Ask groups to substitute a random value for x into the original equation to confirm that no x-value satisfies it. Guide them to conclude that the false statement means the equation has no solution, not that they made a mistake.

Common Misconception

During Think-Pair-Share, watch for students insisting that the smaller variable term must be moved to the other side.

What to Teach Instead

Have pairs solve the same equation both ways (moving the smaller term and the larger term) and compare results. Ask them to explain why both methods are valid and lead to the same solution.

Assessment Ideas

Exit Ticket

After Think-Pair-Share, provide students with the equation 4x + 5 = 2x + 11. Ask them to: 1. Write the first step they would take to collect variable terms on one side. 2. Write the next step to isolate the variable. 3. State the final solution.

Quick Check

After Gallery Walk, write two equations on the board: Equation A: 3x - 2 = 3x + 4. Equation B: 5(x + 1) = 5x + 5. Ask students to solve each equation and categorize it as having one solution, no solution, or infinitely many solutions. Collect their work to assess understanding.

Discussion Prompt

After Collaborative Investigation, present the equation 7x - 3 = 7x + 9. Ask students: 'What happens when you try to solve this equation? What does the result tell us about the original equation? Can you think of another equation that behaves the same way?' Listen for explanations that connect the false statement to the concept of no solution.

Extensions & Scaffolding

• Challenge: Provide equations with fractions or decimals, such as 0.5x + 3 = 1.2x - 4.5, and ask students to solve and explain their method choice.
• Scaffolding: Give students equation cards with the first step (move variable term) already completed, so they focus on simplifying and solving.
• Deeper exploration: Have students create their own equations with no solution or infinite solutions and explain how they constructed them using the properties of equality.

Key Vocabulary

Variable Term	An algebraic term that includes a variable, such as 3x or -y.
Constant Term	A term in an algebraic expression that does not contain a variable; it is just a number, such as 5 or -12.
Equivalent Equations	Equations that have the same solution set. Performing operations on both sides of an equation creates equivalent equations.
Identity	An equation that is true for all possible values of the variable, often resulting in a true statement like 0 = 0 after simplification.
Contradiction	An equation that is never true for any value of the variable, often resulting in a false statement like 3 = 7 after simplification.