What stays true from one line to the next?

When solving 2x + 6 = 18, a correct answer alone does not show whether a student understands equality. Write 2x + 6 = 18 → 2x = 12 → x = 6 and ask the student to explain that the first step subtracts 6 from both sides and the next divides both sides by 2. Then substitute 6 into the original equation: 2 × 6 + 6 = 18. This equation and its numbers are an illustrative classroom example.

Connect one relationship in words, a model, and symbols

The same example can be described as “two equal groups plus six single items make eighteen,” or drawn as two equal groups and six singles balanced against 18. Use the model to show which amount is represented by 2x and which by 6; it should reveal the relationship, not decorate the page. Ask what remains unchanged when students move from the drawing to the equation. Check whether they can explain the equal sign as a relationship between two quantities, rather than a signal to write an answer.

Use an incorrect line to reveal the reasoning

Show a worked error in which 6 is subtracted only from the right, producing 2x + 6 = 12. Ask which side changed and whether the new equation is still true; students can test a value or compare the two sides. Have them correct the line and explain why the same operation must be applied to both sides. The explanation matters more than merely copying the corrected answer.

Compare strategies and justify a choice

After students can recognize the structure, compare two routes for 2(x + 3) = 18: expand first to get 2x + 6 = 18, or divide both sides by 2 to get x + 3 = 9. Both lead to x = 6. Ask when each route is convenient and why every step preserves equality. The goal is a reasoned choice, not one memorized strategy for every problem.

The What Works Clearinghouse guide for algebra in grades 6–12 rates analyzing solved problems and using the structure of algebraic representations as minimal evidence, and intentionally choosing among alternative strategies as moderate evidence. These ratings describe the evidence supporting each recommendation; they are not a ranking of which approach works best for every student. The guide also says to build strategic choice after students have some fluency with algebra procedures and strategies.

Use a brief check to plan the next lesson

Record more than whether the answer is right: did the student name the operation, explain why it was applied to both sides, and represent the same relationship another way? If the calculation is right but the explanation is missing, use a model-to-equation prompt. If only one side changes, revisit equality before adding more procedures. Check the idea again with different numbers rather than treating one response as a complete diagnosis.

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