Balance the Books

Common Core standards covered

• 8.EE.C.7Solve linear equations in one variable.
• 8.EE.C.7.aGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
• 8.EE.C.7.bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

Teacher-facing overview

Big idea: Solve linear equations in one variable, including those requiring distribution and combining like terms, and classify the number of solutions.

Student target: “I can solve linear equations that need the distributive property and like terms, including rational coefficients, and classify an equation as having one, no, or infinitely many solutions from its simplified form.”

Success criteria: I can expand with the distributive property — including a subtracted group — and collect like terms. · I can solve a linear equation and CHECK the solution in the original. · I can clear fractions by multiplying both sides by a common denominator. · I can read x = a, a = a, or a = b at the end of my work and say what each verdict means.

Essential question

An equation can have exactly one solution, no solution, or every number as a solution. How does the algebra TELL you which — before you ever guess a number?

Learning target: I can solve linear equations that need the distributive property and like terms, including rational coefficients, and classify an equation as having one, no, or infinitely many solutions from its simplified form.

Success criteria

• I can expand with the distributive property — including a subtracted group — and collect like terms.
• I can solve a linear equation and CHECK the solution in the original.
• I can clear fractions by multiplying both sides by a common denominator.
• I can read x = a, a = a, or a = b at the end of my work and say what each verdict means.

Why this matters

This is the last one-variable stop before systems, and the three-verdicts idea is the important cargo: an equation is a claim, and algebra is the audit that returns exactly one of three findings — one number works, every number works, or no number works. The same trichotomy returns next unit as two lines crossing once, being identical, or being parallel. Outside class, the no-solution case is what fraud detection and program verification actually do: show that no possible value makes the books balance.

Students should already be able to

• Solve two-step equations like 3x + 5 = 20 (Grade 7: 7.EE.B.4).
• Add, subtract and multiply negative numbers and simple fractions.
• Substitute a value into an expression to evaluate it.

Vocabulary to define explicitly

• Solution

  A number that makes the equation true when substituted for the variable.

  Example: x = 8 solves 4x − 7 = 2x + 9, because 25 = 25.

  Not an example: x = 8 is not 'the answer' until it survives the check — an unchecked number is a candidate.

🔒 4 more terms with examples and non-examples — Distributive property, Like terms, Identity, Contradiction (no solution) — are defined in the full lesson. See plans →

Where this shows up outside the classroom

Assessment plan

Formative — for feedback during the lesson

• Warm-up check habit: who proves x = 5 without being asked twice?
• Ledger verdicts during casework — especially the argument quality on cards 2 and 3.
• The two-lines preview: listen for who anticipates parallel lines for the tampered case.

Summative — the graded evidence: The exit ticket (one solve, two classifications), scored with the four-criterion rubric.

The task students work on

Your team is an audit firm. Each Ledger Card is a client's account claim written as an equation. The audit protocol: expand, collect, isolate — then deliver one of three verdicts. BALANCED: exactly one value makes it true (report the value, with a check). REDUNDANT: the claim is true for every value — the two sides were the same books twice. TAMPERED: no value can make it true — someone altered an entry, and the algebra proves it.

Anticipated student strategies

Procedure, then intuition's limit, then the picture. The guess-and-check discussion is deliberately second so its failure on the tampered ledger motivates why the algebraic audit matters, and the two-lines view points directly at the next unit.

1. Undo operations from the outside in. For 2(x + 4) = 3x − 1, the student expands to 2x + 8 = 3x − 1, then peels terms off both sides — subtract 2x, add 1 — to reach x = 9.

   The workhorse. Share first, and insist on the check: 2(13) = 26 and 3(9) − 1 = 26.

🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →

Lesson script

🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →

Misconceptions and how to respond

• “2(x + 4) = 2x + 4.” The 2 multiplies everything inside: 2x + 8. Test with x = 1: 2(5) = 10, but 2 + 4 = 6.
• “−2(x − 4) = −2x − 8.” — the teacher response is in the full lesson.
• “0 = 0 means x = 0.” — the teacher response is in the full lesson.
• “10 = 6 means I made a mistake.” — the teacher response is in the full lesson.
• “Multiply only the fraction terms when clearing denominators.” — the teacher response is in the full lesson.
• “When a term crosses the equals sign, it keeps its sign.” — the teacher response is in the full lesson.

🔒 Every misconception ships with a tested classroom response. See plans →

Scaffolded practice

The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 3 independent.

1. I doAudit the claim 2(x + 4) = 3x − 1.

🔒 4 more prompts across all three release tiers are in the full lesson. See plans →

Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.

Exit ticket

1. Audit 6(x − 1) = 2x + 14: solve it, check it, and give the verdict.

🔒 2 more exit-ticket questions are in the full lesson. See plans →

Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.

Printable handouts in the kit

All handouts are in Printable_Handouts.pdf inside the kit download.

Materials and classroom resources

Student experience

Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.

Detailed facilitation notes

1. Step 1: Model solving 2(x + 4) = 3x − 1, narrating each step. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
2. Step 2: Auditors work through six case files, showing all steps. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
3. Step 3: Two cases simplify to 5 = 5 and 5 = 7 — teams decide what each means. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
4. Step 4: Students write one equation of each solution type and trade with another team. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”

Differentiation and inclusion

Support: Use algebra tiles and two-step equations for support; extend to equations with fractional coefficients.

Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.

Extension: Require a second solution, alter a condition, or have students write a challenge for another group.

Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.

Assessment plan

Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.

End evidence: Students solve one multi-step equation and produce one equation with no solution, explaining how they built it.

Mastery: The student completes the task accurately and explains the strategy with limited prompting.

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