Backward Planning for Worked Examples

This document defines the 9-step backward planning methodology for analyzing a mastery check question and designing a worked example deck.

CRITICAL: Understanding the Input

The problem image is the MASTERY CHECK QUESTION - the actual question students will answer on their exit ticket or assessment.

Your job is to create a worked example that:

1. Teaches the SAME mathematical skill
2. Uses the SAME strategy and steps
3. Uses DIFFERENT context and numbers than the mastery check

Why different context? Students should learn the strategy from the worked example, then apply it independently to the mastery check. If we use the same numbers/context, students can just copy the answer without learning.

CRITICAL: Transcription First

BEFORE doing any analysis, you MUST first transcribe EXACTLY what you see in the image.

This includes:

• All text (problem statement, questions, instructions)
• All numbers and mathematical expressions
• Any diagrams, tables, or visual elements (describe them precisely)
• Answer choices if present
• Any labels, headers, or context provided

Why this matters: If the transcription is wrong, the entire analysis will be wrong. Take extra care to read ALL text, numbers, and visual elements accurately.

Step-by-Step Instructions

The exit ticket is the essential source of truth. Everything in this analysis flows FROM the exit ticket. Follow this backward planning protocol carefully.

STEP 1: Deep Exit Ticket Analysis

This is the foundation - everything else builds on this analysis.

1a: Solve the Problem

• Work through the mastery check step-by-step
• Write out your complete solution with reasoning
• Identify the final answer

1b: Identify Mathematical Structure Be SPECIFIC, not vague:

• "solving two-step equations with variables on both sides"
• NOT "algebra"

Ask yourself:

• What mathematical relationships are present?
• What prior knowledge does this assume?
• What format is the answer expected in?

1c: Articulate What Correct Understanding Looks Like This is the essential piece - what does a student need to understand to get this problem correct?

• What mathematical knowledge must they apply?
• What key insight separates students who get it right from those who don't?
• What does the correct reasoning process look like?

STEP 2: Develop the Big Idea

The Big Idea is a generalizable mathematical principle - NOT something specific to this problem's context.

CRITICAL CONSTRAINT: The Big Idea must transfer beyond the specific problem context. It should NOT reference context-specific details (e.g., "area of a circle is pi*r^2") but instead capture the underlying mathematical structure (e.g., "linear relationships have a constant rate of change; you can identify linearity from a table by checking if the differences are constant"). The Big Idea should apply to ANY problem testing the same mathematical concept.

2a: First Draft (Detailed) Articulate what students need to know, do, or understand to get the exit ticket correct:

• Use bullet sub-points for each key understanding
• Explicitly connect to the exit ticket: "Here is the Big Idea, here is how it shows up in what students must do on the exit ticket, and here is where misconceptions can block them"
• Include mathematical structure and reasoning, not just procedures

Example (from Gr8 Unit 4 Lesson 7 - All/Some/No Solutions):

• A one variable equation of the form ax + b = cx + d has infinite solutions if the coefficients are equal and the constants are equal (a = c and b = d)
• A one variable equation has no solutions if the coefficients are equal but the constants are unequal (a = c but b != d)
• A one variable equation has one solution if the coefficients are unequal (a != c)

2b: Revised Draft (Simplified) Distill to:

• One sentence capturing the core mathematical principle
• 2-4 supporting structural patterns

Example:

The structure of an equation helps us predict how many solutions it will have.

• ax + b = ax + b (same coefficients, same constants): infinite solutions
• ax + b = ax + d (same coefficients, different constants): no solutions
• ax + b = cx + d (different coefficients): one solution

The simplified Big Idea goes into strategyDefinition.bigIdea. The detailed version goes into strategyDefinition.bigIdeaDetailed. The supporting patterns go into strategyDefinition.bigIdeaSupportingPatterns.

STEP 3: Anticipate Misconceptions

This is the most critical step - misconceptions DRIVE the number of worked example steps.

For EACH anticipated misconception, provide:

1. The misconception - what the student incorrectly believes
2. What their work looks like - specific wrong answers/choices they'd make on the exit ticket
3. Root cause - why they make this mistake (what understanding they're missing)

The number of misconceptions you identify here will determine the number of steps in your worked example. Each step should directly address one or more misconceptions. If you identify 2 key stumbling blocks, design 2 steps. If you identify 4, design 4 steps. Aim for 2-5 misconceptions (3 is typical).

Example (Gr8 Unit 4 Lesson 7):

1. Student thinks that a different constant (other than 8) will yield ONE solution
   • What it looks like: For "one value of x," they write "10" (a constant, not a variable term)
   • Root cause: Doesn't understand that differing constants with same coefficients means NO solution
2. Student confuses no solution with infinite solutions
   • What it looks like: For "no values of x," writes 8; for "all values," writes something other than 8
   • Root cause: Can't distinguish the conditions for no vs. infinite solutions
3. Student assumes every equation has one solution
   • What it looks like: For "no values" and "all values," writes "Not possible"
   • Root cause: Has only seen equations with one solution, doesn't believe 0 or infinite solutions exist

STEP 4: Worked Example Design Thinking

Map each misconception to a worked example step. This is where you explicitly connect the design to the exit ticket.

For each misconception from Step 3:

• Describe which step will address it and HOW
• If a misconception needs its own dedicated step, give it one
• If two misconceptions can be addressed in the same step, group them

Write a brief design rationale explaining:

• Why the worked example is structured this way
• How the steps connect back to what students need for the exit ticket
• What discovery questions ("What do you notice?") will guide students toward the Big Idea

Example:

I want to create problems where students work with equations that have none, one, and infinite solutions so they can compare how they are different. I want them to notice both the structure of the equation and the structure of the solution.

• Step 1 addresses misconception 3 (assumes one solution): Show equations with ONE solution so students see the baseline
• Step 2 addresses misconception 1 (different constant = one solution): Show equations with NO solutions so students see what happens when coefficients match but constants differ
• Step 3 addresses misconception 2 (confuses no/infinite): Show equations with INFINITE solutions so students can compare with Step 2

STEP 5: Define ONE Clear Strategy

The strategy thread runs through ALL visuals.

5a: Name the Strategy Give it a clear, memorable name:

• "Balance and Isolate"
• "Find the Unit Rate"
• "Plot and Connect"

5b: State it in One Sentence Student-facing explanation:

• "To solve this, we [VERB] the [OBJECT] to find [GOAL]"

5c: Identify Moves (2-5, determined by misconceptions) Each move addresses one or more misconceptions from Step 3. The number of moves should match the misconception-to-step mapping from Step 4. Each move: [Action verb] -> [What it accomplishes] -> [Which misconception it addresses]

5d: Define Consistent Language These step verbs MUST:

• Be the EXACT same throughout all visuals
• Appear on every visual header ("STEP 1: [VERB]")
• Be referenced in CFU questions

5e: Define Discovery Questions For each step, define a "What do you notice?" style question that guides students toward the Big Idea:

• "What do you notice about the structure of the equations?"
• "What pattern do you see in the solutions?"
• "How does this compare to what we saw in the previous step?"

STEP 6: Create Three Scenarios

ALL must use DIFFERENT contexts from the mastery check:

DO:

• Match context to grade level interests (gaming, social media, sports, STEM)
• Keep mathematical difficulty identical
• Give each scenario a visual anchor (icon/theme)

DO NOT:

• Use the same context as the mastery check
• Use the same numbers as the mastery check
• Change the problem type between scenarios

IMPORTANT: Scenario Graph Plans If the problem requires a coordinate graph, create a graphPlan for EACH scenario with that scenario's specific equations and values:

• Each scenario has different numbers, so each needs its own equations, scale, keyPoints, and annotations
• The graph structure (number of lines, annotation type) stays the same across scenarios
• Only the specific values change based on each scenario's numbers

IMPORTANT: Scenario 1 Diagram Evolution Scenario 1 is the worked example, so it MUST have its own diagramEvolution with its specific numbers in the ASCII art:

• The diagramEvolution shows how the visual develops step-by-step for Scenario 1's context/numbers
• Scenarios 2 and 3 do NOT need diagramEvolution (they are practice problems)

STEP 7: Determine Visual Type

CRITICAL: ALL graphics/diagrams MUST use math-ui primitives where available. See diagram-patterns.md for the mapping from diagram types to math-ui primitives.

• text-only: No graphics needed (rare - pure text/equation problems)
• table: Simple data tables with highlighting
• svg-visual: ALL other graphics - use math-ui primitives:
  • Coordinate planes and graphs -> CoordinatePlane primitive
  • Tape diagrams (bar models) -> BarDecomposition primitive
  • Hanger diagrams (balance equations) -> HangerDiagram primitive
  • Aligned equations -> AlignedEquationStack primitive
  • Non-graph diagrams -> see diagram-patterns.md for primitive mappings

For non-graph visuals: READ diagram-patterns.md to see the exact visual structure students expect from Illustrative Mathematics curriculum and which math-ui primitives to use.

STEP 8: Visual Planning (REQUIRED if Visual Type is "svg-visual")

IF you selected "svg-visual" above, you MUST plan your visual now.

For coordinate-graph subtype, complete graph planning:

8a: List Your Equations Write out every line/equation that will appear:

Line 1: y = [equation] (e.g., y = 5x)
Line 2: y = [equation] (e.g., y = 5x + 20)

8b: Calculate Key Data Points (REQUIRED in graphPlan.keyPoints) For EACH line, calculate y at key x values. These MUST be included in the keyPoints array:

• Y-intercepts (where line crosses y-axis)
• Solution points (the answer to the problem)
• Any point specifically asked about in the problem
• Points used for slope triangles or annotations

8c: Determine Scale (<=10 ticks on each axis)

• X_MAX: rightmost x-value needed (common: 4, 5, 6, 8, 10)
• Y_MAX: use scale tables to get exactly 9-10 ticks

8d: Plan Annotations What mathematical relationship to show?

• Y-intercept shift (vertical arrow showing difference)
• Parallel lines (same slope label)
• Slope comparison
• Intersection point

STEP 9: Generate Diagram Evolution Preview (REQUIRED for ALL worked examples)

After planning the visual structure, create a step-by-step evolution showing how the diagram develops across the visual sequence.

This is the most important preview for teachers - it shows EXACTLY how the visual will build from initial state through each step of the solution.

9a: Create Initial State Show the diagram as it appears in the problem setup:

• For coordinate graphs: axes with scale labels, but no lines drawn yet
• For tape diagrams: the total bar with the unknown marked
• For hanger diagrams: the initial equation on the balance
• For number lines: the empty line with range marked

9b: Create Step-by-Step Evolution For EACH strategy move, show:

1. The diagram state AFTER that step is completed (building cumulatively on previous steps)
2. What was added/changed from the previous state
3. Use ASCII art that matches the visual type

9c: Match Steps to Strategy Moves The number of steps in diagramEvolution MUST match strategyDefinition.moves.length:

• 2 moves = 2 evolution steps
• 3 moves = 3 evolution steps
• 5 moves = 5 evolution steps

Example for Coordinate Graph (2-line system of equations):

"diagramEvolution": {
  "initialState": "    y\n  50|\n    |\n  40|\n    |\n  30|\n    |\n  20|\n    |\n  10|\n    |\n   0+----+----+----+----+---- x\n        2    4    6    8",
  "steps": [
    {
      "header": "STEP 1: IDENTIFY",
      "ascii": "    y\n  50|              /\n    |            /\n  40|          /\n    |        /\n  30|      /\n    |    /\n  20+--*           <- y-intercept (0, 20)\n    |/\n  10|\n    |\n   0+----+----+----+----+---- x\n        2    4    6    8",
      "changes": ["Draw first line y = 5x + 20 (blue)", "Mark y-intercept at (0, 20)"]
    }
  ]
}

Example for Tape Diagram (division problem):

"diagramEvolution": {
  "initialState": "Total: 24 stickers to share among 4 friends\n\n+----------------------------------------+\n|                 24                     |\n+----------------------------------------+",
  "steps": [
    {
      "header": "STEP 1: PARTITION",
      "ascii": "+----------+----------+----------+----------+\n|    ?     |    ?     |    ?     |    ?     | = 24\n+----------+----------+----------+----------+\n    ^           ^           ^           ^\n Friend 1   Friend 2   Friend 3   Friend 4",
      "changes": ["Divide tape into 4 equal parts", "Mark each part with ? (unknown)"]
    },
    {
      "header": "STEP 2: CALCULATE",
      "ascii": "+----------+----------+----------+----------+\n|    6     |    6     |    6     |    6     | = 24\n+----------+----------+----------+----------+\n\n24 / 4 = 6 stickers per friend",
      "changes": ["Calculate: 24 / 4 = 6", "Fill in each box with 6"]
    }
  ]
}

Completion Checklist (Verify Before Responding)

• problemTranscription contains EXACT verbatim text from image (all text, numbers, diagrams)
• Problem was FULLY solved step-by-step (Step 1a)
• Problem type is SPECIFIC (not vague like "algebra") (Step 1b)
• Big Idea is a generalizable mathematical principle - not context-specific (Step 2)
• Big Idea has TWO DRAFTS: detailed (bigIdeaDetailed) and simplified (bigIdea) (Step 2)
• Anticipated misconceptions are structured with studentWorkExample and rootCause (Step 3)
• Each misconception maps to a worked example step via addressedInStep (Step 4)
• Number of strategy moves matches the misconception-to-step mapping (Step 5)
• Design rationale explains WHY the WE is structured this way (Step 4)
• Discovery questions guide students toward Big Idea (Step 5e)
• Strategy has consistent verbs used throughout all visuals (Step 5d)
• CFU question templates reference strategy verbs
• ALL 3 scenarios use DIFFERENT contexts from the mastery check
• All scenarios use the SAME mathematical structure and strategy
• diagramEvolution is included showing how content develops across visuals:
  • initialState shows the Problem Setup
  • keyElements array explains each element and what it represents
  • steps array has one entry per strategy move
  • Each step's header matches the header ("STEP N: VERB")
  • Steps build cumulatively (each shows previous + new additions)
  • Changes array lists what was added/modified in that step
• IF visualType is "svg-visual":
  • svgSubtype is specified (coordinate-graph, diagram, shape, number-line, or other)
  • IF svgSubtype is "coordinate-graph":
    • graphPlan has equations, keyPoints, scale, and annotations
    • EACH scenario has its own graphPlan with that scenario's specific values
    • All graphPlans have: equations with slope/y-intercept, proper scale, and annotations
    • keyPoints array includes: y-intercepts, solution points, and any points to be labeled
• Scenario 1 has its own diagramEvolution with Scenario 1's specific values in the ASCII art

Output Format

See output-schema.md for the complete JSON output schema.

Return ONLY valid JSON matching that schema. Do not include any explanation or markdown formatting.