Solving Equations with Variables on Both Sides

Active learning works for equations with variables on both sides because the process of moving terms and balancing scales mirrors the algebraic steps students must take. When students manipulate physical or visual models, they internalize the concept that operations maintain equality, which is harder to grasp through abstract steps alone.

• Year 8
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate the value of the variable in linear equations with variables on both sides.
2. Explain the strategic advantage of collecting variable terms on one side of an equation before isolating the variable.
3. Compare different valid methods for isolating the variable in equations with variables on both sides.
4. Evaluate the correctness of a solution by substituting it back into the original equation.
5. Formulate a linear equation with variables on both sides from a given word problem.

Decision Matrix

⏱ 35 min·Small Groups

Balance Model: Tile Equations

Give groups algebra tiles or paper cutouts for terms. Students build equations on mats, then move tiles to one side while keeping balance, recording steps. Pairs verify by substituting the solution back. Conclude with sharing one insight per group.

Prepare & details

Explain the strategic advantage of collecting variables on one side of an equation.

Facilitation Tip: During Balance Model: Tile Equations, circulate and ask students to explain each tile move aloud to reinforce the connection between physical actions and algebraic steps.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

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

Decision Matrix

⏱ 25 min·Small Groups

Relay Solve: Team Equations

Divide class into teams. Write multi-step equations on cards. First student solves one step, passes to next teammate until complete. Teams race but check each other's work before racing. Debrief fastest accurate team.

Prepare & details

Compare different approaches to isolating the variable in complex equations.

Facilitation Tip: In Relay Solve: Team Equations, stand at the board to model the first step for each team, ensuring they start correctly before handing over control.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

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

Decision Matrix

⏱ 30 min·Pairs

Error Hunt: Faulty Solutions

Provide worksheets with five solved equations containing errors. Pairs identify mistakes, explain why invalid, and correct them. Circulate to prompt discussion on sign changes. Class votes on trickiest error.

Prepare & details

Evaluate the validity of a solution by substituting it back into the original equation.

Facilitation Tip: For Error Hunt: Faulty Solutions, model how to annotate equations with step-by-step justifications before teams begin their hunts.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

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

Decision Matrix

⏱ 20 min·Small Groups

Strategy Sort: Method Match

Prepare cards with equations and strategy labels like 'move x terms first'. Students in small groups match and justify, then test one by solving. Share comparisons whole class.

Prepare & details

Explain the strategic advantage of collecting variables on one side of an equation.

Facilitation Tip: In Strategy Sort: Method Match, listen for pairs debating why one method might be more efficient and highlight these moments for whole-class discussion.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

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

Teaching This Topic

Teachers should emphasize flexibility in choosing which side to collect variables on, as this builds algebraic fluency. Avoid over-teaching a single 'right' method, which can reinforce misconceptions. Research shows that students benefit from seeing multiple valid approaches, so use peer discussions to compare strategies. Always connect visual or physical models back to symbolic algebra to bridge concrete and abstract thinking.

What to Expect

Successful learning looks like students confidently choosing which side to collect variables on, performing inverse operations correctly, and verifying solutions by substitution without prompting. They should explain their reasoning using terms like 'balancing,' 'inverse,' and 'equality' during discussions.

Watch Out for These Misconceptions

Common Misconception

During Strategy Sort: Method Match, watch for students who insist variables must always be moved to the left side.

What to Teach Instead

Use the sorting cards to have them try both methods on the same equation, then compare results. Ask, 'Does the solution change if you move terms to the right side? Why or why not?'

Common Misconception

During Balance Model: Tile Equations, watch for students who incorrectly flip the sign of a term when moving it to the other side.

What to Teach Instead

Have them model the equation with tiles, then physically move the tiles while saying, 'I'm adding 3x to both sides,' to reinforce sign preservation.

Common Misconception

During Relay Solve: Team Equations, watch for students who skip the verification step even after solving.

What to Teach Instead

After teams solve, require them to substitute their answer back into the original equation on the board and explain why it works or doesn’t.

Assessment Ideas

Quick Check

After Balance Model: Tile Equations, present the equation 5x - 3 = 2x + 9. Ask students to write the first step they would take and justify it in one sentence before solving for x.

Exit Ticket

After Error Hunt: Faulty Solutions, give students the equation 3(y + 2) = y + 10. Ask them to solve it and verify their answer by substitution, showing both steps on the exit ticket.

Discussion Prompt

During Strategy Sort: Method Match, pose the equation 7a + 5 = 3a + 17. Ask students to discuss in pairs: 'Is there more than one correct first step? What are the advantages or disadvantages of starting by subtracting 3a versus subtracting 7a?'

Extensions & Scaffolding

• Challenge: Present students with an equation like 2(x + 3) = 3(x - 1) + 5 and ask them to solve it two different ways, then compare the efficiency of each method.
• Scaffolding: Provide equation cards with pre-written steps where students fill in the missing operations or justifications.
• Deeper: Ask students to create their own equation with variables on both sides that requires two steps to solve, then trade with a partner to solve and verify.

Key Vocabulary

Variable	A symbol, usually a letter, that represents an unknown quantity in an algebraic expression or equation.
Coefficient	The numerical factor that multiplies a variable in an algebraic term.
Constant term	A term in an algebraic expression that does not contain a variable; its value is fixed.
Isolate the variable	To perform operations on an equation so that the variable stands alone on one side of the equals sign.
Equality	The state of being equal; maintaining balance on both sides of the equals sign through inverse operations.