One-Variable Equation Solving

Overview

Students will solve one-variable linear equations and justify each step using equality. The lesson builds toward explaining why solution methods work and includes attention to how coefficients and operations affect the solution.

Learning intentions

Students will be able to:

• Solve simple linear equations in one variable, including equations with coefficients represented by letters.
• Explain each step in solving an equation by referencing the equality of numbers at the previous step.
• Solve and check linear equations using inverse operations and justified transformations.
• Recognize common solution errors and correct them by reasoning about equivalence.

Success criteria

• I can isolate the variable to solve a linear equation in one variable.
• I can explain why a step is valid (e.g., “Both sides stay equal because we do the same operation to both sides.”).
• I can solve an equation with a letter coefficient by applying the same reasoning as numeric coefficients.
• I can check my solution and decide if it is consistent with the original equation.

Curriculum links

• Algebra — Reasoning with Equations and Inequalities: Explain each step by reasoning from the equality of numbers asserted at the previous step.
• Algebra — Reasoning with Equations and Inequalities: Solve linear equations in one variable, including equations with coefficients represented by letters.
• Algebra — Creating Equations (connection): Use equation-solving reasoning as a foundation for later problem translation.

Lesson structure (90 minutes)

1. 0–8 min · Retrieval hook. Teacher writes two quick equations on the board (no solving yet):

• “If (x=4), does (2x+1=9) work?”
• “Starting from (x+3=10), what operation could undo +3?” Students do a think-pair-share to explain their reasoning and identify the inverse operation.

2. 8–20 min · Guided example: equality justification. Teacher models solving (x+7=15) and narrates step-by-step justification: subtract 7 from both sides, then check. Students write the solution with a short justification sentence after each line (e.g., “Because both sides were equal, subtracting 7 from both sides keeps them equal.”).

3. 20–35 min · Partner practice: linear equations (numeric coefficients). Teacher assigns 6 equations on a “solve and justify” sheet (mix of forms like (3x=24), (x/5=6), (2(x-1)=10) but only requiring linear simplification). Students work in pairs: solve each equation, then underline the operation that preserves equality and write a brief justification.

4. 35–50 min · Whole-class debrief and error analysis. Teacher displays two incorrect solution samples (e.g., dividing only one side, sign error when distributing, forgetting to check). Students vote on what went wrong and justify the correction using equality reasoning; teacher records “valid step rules” as a class anchor chart.

5. 50–64 min · Letter coefficients: solve for (x). Teacher introduces one-variable equation with a letter coefficient, such as:

• (ax=12) where (a\neq 0) (students solve to (x=\frac{12}{a}))
• and a second example: (2x+b=14) given a specific (b) value (e.g., (b=4) so they can solve numerically while showing the general idea). Students solve and then write a justification sentence for the key step (divide both sides by the coefficient of (x), with (a\neq 0)).

6. 64–78 min · Check-your-work station rotations. Teacher sets up three quick stations (10 minutes each, with 4 minutes overlap buffer):

• Station A: Solve 2 equations independently, then check solutions by substitution.
• Station B: Given a proposed solution, decide if it works and explain why.
• Station C: Match solution steps to the correct equality justification statement. Students rotate and record one justification they used at each station.

7. 78–90 min · Exit ticket and closure. Teacher collects an exit ticket with:

• Solve: (5x-3=22) (show steps).
• Explain: Write one sentence justifying why one transformation used is valid. Students complete quietly; teacher leads a 2-minute closure emphasizing “same operation to both sides” and “check by substitution.”

Resources

• “Solve and Justify” equation worksheet (6 numeric coefficient problems)
• Error analysis cards (2–3 incorrect solution examples)
• Letter coefficient practice sheet (2 problems)
• Station rotation task cards (A, B, C)
• Class anchor chart paper: “Equality-preserving moves”
• Dry-erase boards or notebooks for quick solution writing
• Timer for station rotation management
• Exit tickets (one per student)

Assessment

• Formative checks: teacher circulates during partner practice, listening for justification sentences tied to equality.
• Formative checks: during error analysis, students must explain the specific incorrect equality step and the corrected move.
• Exit ticket: evaluate both correctness (solving) and reasoning (a clear equality-based justification).

Differentiation

• Support: provide sentence starters for justifications (e.g., “I subtracted ___ from both sides because…” “To undo multiplication by ___, I divided both sides by ___.”).
• Support: offer a 3-step scaffold card: Simplify → Isolate variable → Check by substitution.
• Extension: for a subset of students, add an extra equation that requires two-step isolation (e.g., (3(2x-1)=15)) and ask for an additional sentence explaining how distribution affects equality.
• EAL/SEN considerations: allow students to use worked examples from anchor chart; encourage diagramming “both sides” with brackets to show equality being preserved.