Two-Step Equation Quest

Overview

In this sixth lesson of Algebra Detectives: Solve, Simplify, Share, students move from solving one-step equations to forming and solving two-step linear equations with integer solutions. They compare worked examples, use balance and inverse-operation representations, then collaborate on a mathematical “quest” and present an explanation of a common error.

Learning intentions

• WALT form and solve two-step linear equations with integer solutions.
• WALT identify the operations applied to a variable and undo them in reverse order.
• WALT explain mathematical thinking using equations, representations and precise language.
• WALT check a solution by substituting it back into the original equation.

Success criteria

• I can identify the operations applied to the variable.
• I can undo operations in reverse order and show clear steps.
• I can use a balance model, inverse-operation mat or symbolic working to explain my solution.
• I can verify my answer and explain whether it makes the original equation true.

Curriculum links

• Mathematics and Statistics — algebraic thinking, representing and solving linear relationships.
• Mathematics and Statistics — using mathematical reasoning to justify, communicate and critique solutions.
• Mathematics and Statistics — Mathsteasers: higher-order thinking and challenge for advanced learners.
• Te Mātaiaho mathematical practices: making sense of problems, choosing representations, communicating and reflecting.

Lesson structure (45 minutes)

1. 0–5 min · Hook and retrieval. Open with the hook and retrieval slides and display: “A mystery number is multiplied by 5, then 3 is added. The answer is 18. How could we find the mystery number?” Students independently write a first idea, then recall how to undo one operation and share with a partner.

2. 5–13 min · Worked-example comparison. Teacher presents two solutions to (5x + 3 = 18): one that subtracts 3 then divides by 5, and one that divides by 5 then subtracts 3. Students use mini-whiteboards to decide which is valid, identify the error in the invalid method, and explain why the order matters. Emphasise that inverse operations undo the original operations in reverse order.

3. 13–20 min · Model the balance. Teacher models (5x + 3 = 18) using the Algebra Balance Mat, linking each equation step to maintaining equality: subtract 3 from both sides, then divide both sides by 5. Students record the equation, operation, result and check in the mat, then verify that (x=3) gives (5(3)+3=18). Confer briefly with students who need support with equality or inverse operations.

4. 20–33 min · Team equation quest. Place students in three mixed-readiness teams of three and open the team quest instruction and equation slides. Give each team the two-step equation quest worksheet. Teams solve the core equation (5x+3=18), then complete related equations with integer solutions, recording at least two representations for one equation. They must use the Linear Equations Step Cards as prompts, arrange the solving process in logical order, and prepare a short explanation. Encourage roles of solver, checker and presenter so all students contribute.

5. 33–41 min · Explain the mistake presentations. Each team presents its solution and one deliberately incorrect solution, such as dividing before removing the constant or changing only one side of the equation. Classmates identify the mistake, state the correct inverse operation, and ask one clarifying question. Teacher uses the presentation prompts and discussion slides to reinforce clear mathematical communication and the class values of manaaki, aroha, niwha and atamai.

6. 41–45 min · Check and exit reflection. Display the final prompts on the plenary and exit slides. Students independently solve (4x-5=15), show both inverse-operation steps, and check by substitution on the bottom of the two-step equation quest worksheet. Students finish with: “The first operation I undo is ___ because ___.” Collect worksheets and scan for misconceptions.

Resources

• the Two-Step Equation Quest slide deck
• the two-step equation quest worksheet
• the Algebra Balance Mat
• the Linear Equations Step Cards
• Mini-whiteboards, pens and erasers
• Balance-model counters or linking cubes
• Pencils and coloured pens
• Projector or interactive whiteboard

Assessment

• Listen during the worked-example comparison for whether students understand equality and reverse order.
• Check team worksheets for correctly formed equations, logical inverse operations, clear representations and substitution checks.
• Use the exit reflection to identify students who confuse operation order, apply an inverse operation to only one side, or do not verify solutions.

Differentiation

• Support students with the balance mat, counters, colour-coding for each operation, the step cards and a structured sentence frame: “First I undo ___ by ___ on both sides. Then I ___.”
• During conferencing, rehearse the equation verbally as “five lots of a number plus three” before students write symbols. Accept a balance drawing, table or annotated equation alongside formal working.
• Provide mixed representations—context, balance model and symbolic equation—and pair students strategically so checking and explaining are shared rather than completed by one student.
• Extend advanced learners by asking them to solve equations containing negative constants or variables, such as (3x-7=-1), and create a puzzle with a unique integer solution. They must provide a clue, solution and verification for another team.