Escape the Variable Vault

Common Core standards covered

• 6.EE.B.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
• 6.EE.B.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
• 6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
• 6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

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

Teacher-facing overview

Big idea: Understand solving as finding truth-making values, use variables for unknowns, solve x + p = q and px = q with nonnegative rationals, and represent x > c and x < c constraints on number lines.

Student target: “I can test values as solutions by substitution, solve x + p = q and px = q with rationals, use a variable to model an unknown, and graph x > c constraints as rays.”

Success criteria: I can verify or reject a candidate by substituting it. · I can solve add- and times-locks by the undoing operation. · I can write a lock's equation from its story. · I can graph x > c with the right endpoint style and name three solutions.

Essential question

A lock's equation is a question: which value makes this TRUE? What changes when one lock accepts a single key — and another accepts infinitely many?

Learning target: I can test values as solutions by substitution, solve x + p = q and px = q with rationals, use a variable to model an unknown, and graph x > c constraints as rays.

Success criteria

• I can verify or reject a candidate by substituting it.
• I can solve add- and times-locks by the undoing operation.
• I can write a lock's equation from its story.
• I can graph x > c with the right endpoint style and name three solutions.

Why this matters

6.EE.B is where solving gets its MEANING before its mechanics: B.5 defines a solution as a truth-maker (the substitution test every later check descends from), B.6 makes the variable an honest unknown, B.7 installs the two one-step undo moves that Grade 7's px + q = r will chain, and B.8's infinite-solution rays are the first inequality graphs — open-circle logic that the Goal Line will inherit whole. The vault makes all four physical: keys, locks, and one door that takes a crowd.

Students should already be able to

• Evaluating expressions — The Expression Studio.
• Fraction and decimal operations — Units 3-4.
• Number line graphing — The Harbor Log.

Vocabulary to define explicitly

• Solution (truth-maker)

  A value that makes the equation or inequality TRUE when substituted.

  Example: x = 3 turns x + 5 = 8 into the true 8 = 8.

  Not an example: x = 2 gives 7 = 8 — false; not a solution, no partial credit.

🔒 3 more terms with examples and non-examples — Substitution test, Undoing (inverse operations), Inequality as a condition — are defined in the full lesson. See plans →

Where this shows up outside the classroom

Assessment plan

Formative — for feedback during the lesson

• Keyring verdicts.
• Lock-log checks during the vault run.
• The boundary-test discussion.

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

The task students work on

The vault has five locks and a final door. LOCK 1 (keyring): x + 5 = 8 with keys {2, 3, 4} — test, don't guess. LOCK 2: x + 7 = 19. LOCK 3: 5x = 45. LOCK 4 (fraction): (3/4)x = 6 — the story: three-quarters of the code is 6. LOCK 5 (decimal): x + 2.5 = 7. THE FINAL DOOR: the keypad accepts ANY code greater than 40 — write the condition, graph it, name three codes that work and one that doesn't, and settle whether 40 itself opens the door.

Lock log — every solution substitution-checked.

Anticipated student strategies

Truth-testing defines the goal, undoing achieves it efficiently, and the fraction lock shows why the written method outlives mental facts.

1. Keyring substitution. The student tests each candidate and records true/false verdicts.

   B.5 itself — the definition of solving as truth-hunting. Share first; it grounds everything.

🔒 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

• “x = 2 is 'close enough' for x + 5 = 8.” Substitute: 7 = 8 is false. Equations aren't horseshoes — a solution makes the sentence TRUE, exactly.
• “To solve x + 7 = 19, subtract 7 from the 19 only when it feels right.” — the teacher response is in the full lesson.
• “(3/4)x = 6 means x = 6 × 3/4 = 4.5.” — the teacher response is in the full lesson.
• “The answer to x > 40 is 41.” — 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 4 independent.

1. I doCrack Lock 2 (x + 7 = 19) and Lock 3 (5x = 45), with checks.

🔒 5 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. Solve and check: m + 8 = 20.

🔒 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: Test keys by substitution: a solution makes the equation true. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
2. Step 2: Solve add-locks by subtracting, times-locks by dividing. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
3. Step 3: Crack the fraction and decimal locks. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
4. Step 4: Open the range door: graph every value that works. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”

Differentiation and inclusion

Support: The keyring (substitution set) grounds truth-testing; extend with the fraction lock and the write-your-own-lock station.

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: One lock solved and checked by substitution, plus the inequality door graphed with its endpoint defended.

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

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