Mathematics · Year 8

Active learning ideas

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.

• National Curriculum Attainment Targets
• KS3: Mathematics - Algebra

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.

What to look for

Present students with the equation 5x - 3 = 2x + 9. Ask them to write down the first step they would take to solve it and justify why. Then, ask them to calculate the value of x.

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

Activity 02

• 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.

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.

What to look for

Give students the equation 3(y + 2) = y + 10. Ask them to solve for y and then substitute their answer back into the original equation to verify their solution. They should show both steps.

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

Activity 03

• 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.

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.

What to look for

Pose the equation 7a + 5 = 3a + 17. Ask students to discuss in pairs: 'Is there more than one correct first step to solve this equation? What are the advantages or disadvantages of starting by subtracting 3a versus subtracting 7a?'

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

Activity 04

• 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.

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.

What to look for

Present students with the equation 5x - 3 = 2x + 9. Ask them to write down the first step they would take to solve it and justify why. Then, ask them to calculate the value of x.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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?'

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

  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.

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

  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.

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Methods used in this brief

• Decision Matrix