Solve, Substitute, Check

Overview

Lesson 3 of 4 in Equation Detectives. Students solve one- and two-step equations using inverse operations, then substitute answers into the original equation to check them. A team-based “Equation Repair Shop” makes common errors visible and gives students practice explaining and correcting mathematical thinking.

Learning intentions

• WALT solve one- and two-step linear equations using efficient inverse operations.
• WALT substitute a proposed value into the original equation.
• WALT decide whether a solution is correct and explain why.
• WALT identify, diagnose and repair common equation-solving errors.

Success criteria

• I can undo operations in the correct order to find the unknown.
• I can show clear steps when solving an equation.
• I can substitute my answer into the original equation and decide whether it is true.
• I can explain and correct an error in someone else’s solution.

Curriculum links

• Number and algebra: use equality, inverse operations and properties of operations to solve equations.
• Number and algebra: represent and reason about unknown quantities using symbols and equations.
• Mathematical and statistical thinking: select efficient strategies, communicate reasoning and evaluate solutions.
• Te Mātaiaho mathematical practices: explore, represent, connect, explain and reflect.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook: “Can you trust the answer?” Open with the hook and lesson introduction. Display a deliberately incorrect solution such as 3x + 4 = 19, so x = 5 and ask students to show thumbs up, down or sideways. Do not confirm the answer immediately; ask, “What evidence would prove it?”

2. 5–15 minutes – Model solve, substitute, check Use the worked-example slides to model x + 7 = 18, then 3x + 4 = 19. Emphasise doing the same operation to both sides, undoing addition or subtraction before multiplication or division, and writing a check: 3(5) + 4 = 19, so 19 = 19. Briefly contrast an efficient inverse-operation method with trial and error.

3. 15–22 minutes – Guided response routine Distribute the solve-and-check worksheet. Complete the first two questions together: one one-step equation and one two-step equation. Students must use the four-part routine: solve, state the value, substitute into the original equation, and write “true” or “false”. Pause to address the misconception that checking means repeating the solving steps.

4. 22–38 minutes – Equation Repair Shop In teams of four, hand out the linear equations step cards as a visual process prompt. Give each team several teacher-prepared error examples on the board or verbally, such as:

• x - 6 = 10, x = 4
• 2x + 5 = 17, 2x = 22, x = 11
• 4x - 3 = 13, x = 4

Teams act as repair technicians: identify the first incorrect step, explain the error, rewrite the solution correctly, and substitute to check. Assign roles of reader, checker, recorder and reporter so all students participate.

5. 38–50 minutes – Independent challenge practice Students complete the remaining questions on the solve-and-check worksheet independently, then compare selected answers with a partner. Include increasing difficulty: one-step equations, two-step equations with positive integers, and equations where the unknown appears after subtraction, such as 20 - 3x = 8. Circulate and ask, “What operation are you undoing?” and “How does your check prove the answer?”

6. 50–56 minutes – Repair Shop share-back Use the discussion and misconception slides to display two repaired solutions. Invite teams to explain the first error rather than simply give the answer. Class members identify whether the correction is supported by substitution and suggest a clearer way to record the working.

7. 56–60 minutes – Exit assessment and preview Return to the hook equation and reveal the correct reasoning. Students complete the final check question on the reflection and exit section: solve 5x - 2 = 23, substitute the answer, and complete the sentence, “Checking is important because…”. Preview the final lesson: writing and solving equations from real situations.

Resources

• the Equation Detectives lesson deck
• the solve-and-check worksheet
• the linear equations step cards
• Whiteboard and markers
• Mini-whiteboards or scrap paper
• Prepared error examples
• Highlighters or coloured pens
• Team role cards or displayed role list

Assessment

• Observe guided practice and team explanations for correct use of inverse operations and equality.
• Check worksheet solutions for accurate working, substitution and interpretation of true or false.
• Use the exit question to identify students who can solve and verify a two-step equation independently.

Differentiation

• Support: provide a worked example, allow students to use the step cards, highlight the operation to undo, and offer equations with smaller positive numbers first.
• Support: pair students strategically and provide sentence frames: “The first error is…”, “I know because…”, and “When I substitute…”.
• EAL/SEN: pre-teach unknown, inverse, substitute, solution and check using the examples; read instructions aloud and accept verbal explanations alongside written working.
• Extension: ask students to create an equation with a deliberate error for another team to repair, or solve and check equations involving brackets such as 2(x + 3) = 14.