Pan-Balance Puzzles

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 using the balance model, including one distributive step, and recognize impossible and always-true balances.

Student target: “I can solve equations by removing the same things from both sides of a balance, and I can recognize a balance that is impossible and one that is level no matter what.”

Success criteria: I can translate between a balance picture (bags and blocks) and an equation. · I can remove equal bags or blocks from both sides until one bag stands alone. · I can open a two-bag group into its separate bags and blocks (the distributive step). · I can explain what an impossible balance and an always-level balance each mean.

Essential question

A scale stays level only if you treat both sides identically. What does that one rule let you discover about a bag you can't open?

Learning target: I can solve equations by removing the same things from both sides of a balance, and I can recognize a balance that is impossible and one that is level no matter what.

Success criteria

• I can translate between a balance picture (bags and blocks) and an equation.
• I can remove equal bags or blocks from both sides until one bag stands alone.
• I can open a two-bag group into its separate bags and blocks (the distributive step).
• I can explain what an impossible balance and an always-level balance each mean.

Why this matters

The balance model gives every algebraic move a physical reason: you subtract from both sides because the scale demands it, not because a rule said so. This is the same content as Balance the Books at a gentler ramp — smaller numbers, pictures first, the weird cases (no solution, every solution) introduced as broken and boring scales rather than audit verdicts. Students who own the balance can walk into the audit office next.

Students should already be able to

• Adding and subtracting small whole numbers.
• The idea that a letter stands for an unknown number.
• Two-step equations, even shakily (7.EE.B.4).

Vocabulary to define explicitly

• Bag (the variable)

  A sealed bag holding some unknown number of blocks — the x.

  Example: Two bags and 3 blocks: 2x + 3.

  Not an example: The bag is not worth 1 automatically — its value is exactly what the puzzle determines.

🔒 3 more terms with examples and non-examples — Keeping the balance, The two-bag group, Impossible and always-level balances — 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 removals: paired or one-sided.
• Ledger lines during puzzling — the bundle column especially.
• The special-scale declarations.

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

The task students work on

The Puzzle Guild posts balance puzzles: sealed bags (every bag on a card holds the SAME number of blocks) and loose blocks sitting on level pans. Guild rules: you may remove bags or blocks only in matching amounts from both sides, and you may open tied bundles fully. Solve by getting one bag alone — then declare what the bag holds, and prove the pans still level.

Guild ledger — record each puzzle's balance as an equation, the bag's value, and the check.

Anticipated student strategies

Physical stripping first, guessing second so its failure on the broken scale is felt, symbols last as the compressed version of stripping. The whole lesson is one idea: both sides, identically, until the bag stands alone.

1. Strip both sides. The student removes one bag from each side, then three blocks from each side, narrating: 'a bag and 3 balance 8… so a bag is 5.'

   The model working as intended. Share first and keep the narration — the words ARE the algebra.

🔒 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

• “Remove blocks from the heavy-looking side only.” One-sided removal tilts the scale — the equality is destroyed. Every removal happens twice, once per pan.
• “2(x + 3) opens to 2 bags and 3 blocks.” — the teacher response is in the full lesson.
• “Puzzle 4 means I made a mistake.” — the teacher response is in the full lesson.
• “'Any amount' can't be an answer.” — 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 doSolve the balance 2x + 3 = x + 8.

🔒 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. Solve the balance 4x + 1 = x + 13, showing your paired removals.

🔒 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 equations as balanced pans. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
2. Step 2: Remove equal amounts from both sides. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
3. Step 3: Solve puzzles with bags on both sides. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
4. Step 4: Meet the impossible balance and the always-level balance. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”

Differentiation and inclusion

Support: Every puzzle has a drawn or described balance; numbers stay small and positive. Extend with a two-bag group 2(x + 3) and with writing puzzles for partners.

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: Two puzzles solved with the removal steps shown, plus one impossible-balance explanation.

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

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