{
  "schemaVersion": "1.0",
  "moduleId": "algebra-manipulation-0580-pilot",
  "status": "prototype-scenarios",
  "pathways": {
    "simplify-expand": {
      "objectiveCode": "C2.2-1,2 / E2.2-1,2",
      "learnerAction": "Collect like terms and expand bracket products, then verify the identity numerically.",
      "objectiveVerbs": [
        "simplify expressions by collecting like terms",
        "expand products of algebraic expressions"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "simplify-expand-mission-01",
          "prompt": "The board shows the term cards for 5x − 2y + 3x + 7y − x. Choose the simplified expression, then type its value at x = 2, y = 1.",
          "model": {
            "kind": "collect",
            "given": "5x − 2y + 3x + 7y − x",
            "point": [
              2,
              1
            ],
            "cardOptions": [
              "7x + 5y",
              "12x",
              "7x − 5y",
              "7x + 5y + 1"
            ],
            "optionCodes": [
              null,
              "am-unlike-merged",
              "am-sign-lost",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "7x + 5y",
            "answer": "19"
          },
          "hint": "Collect LIKE terms only; expand with the grid — every term times every term, signs carried. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-unlike-merged"
        },
        {
          "id": "simplify-expand-mission-02",
          "prompt": "Expand 4x(3x − 2y). Choose the expanded expression, then type its value at x = 1, y = 2.",
          "model": {
            "kind": "expand-single",
            "given": "4x(3x − 2y)",
            "point": [
              1,
              2
            ],
            "cardOptions": [
              "12x^2 − 8xy",
              "12x^2 − 2y",
              "12x^2 + 8xy",
              "7x^2 − 2y"
            ],
            "optionCodes": [
              null,
              "am-partial-expansion",
              "am-sign-lost",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "12x^2 − 8xy",
            "answer": "-4"
          },
          "hint": "Collect LIKE terms only; expand with the grid — every term times every term, signs carried. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-sign-lost"
        },
        {
          "id": "simplify-expand-mission-03",
          "prompt": "Expand and simplify (x + 4)(x − 3). Choose the expanded expression, then type its value at x = 2.",
          "model": {
            "kind": "expand-double",
            "given": "(x + 4)(x − 3)",
            "point": [
              2,
              0
            ],
            "cardOptions": [
              "x^2 + x − 12",
              "x^2 − 3x",
              "x^2 + x + 12",
              "x^2 + 2x − 12"
            ],
            "optionCodes": [
              null,
              "am-partial-expansion",
              "am-sign-lost",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "x^2 + x − 12",
            "answer": "-6"
          },
          "hint": "Collect LIKE terms only; expand with the grid — every term times every term, signs carried. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-partial-expansion"
        },
        {
          "id": "simplify-expand-mission-04",
          "prompt": "Expand and simplify (2x + 3)(2x + 3). Choose the expanded expression, then type its value at x = 1.",
          "model": {
            "kind": "expand-double",
            "given": "(2x + 3)(2x + 3)",
            "point": [
              1,
              0
            ],
            "cardOptions": [
              "4x^2 + 12x + 9",
              "4x^2 + 6x",
              "4x^2 + 9",
              "4x^2 + 13x + 9"
            ],
            "optionCodes": [
              null,
              "am-partial-expansion",
              "am-square-sum",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "4x^2 + 12x + 9",
            "answer": "25"
          },
          "hint": "Collect LIKE terms only; expand with the grid — every term times every term, signs carried. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-square-sum"
        },
        {
          "id": "simplify-expand-mission-05",
          "prompt": "Expand and simplify (x + 1)(x − 2)(x + 3). Choose the expanded expression, then type its value at x = 2.",
          "model": {
            "kind": "expand-triple",
            "given": "(x + 1)(x − 2)(x + 3)",
            "point": [
              2,
              0
            ],
            "cardOptions": [
              "x^3 + 2x^2 − 5x − 6",
              "x^2 − x − 2",
              "x^3 + 2x^2 + 5x + 6",
              "x^3 + 2x^2 − 5x − 4"
            ],
            "optionCodes": [
              null,
              "am-partial-expansion",
              "am-sign-lost",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "x^3 + 2x^2 − 5x − 6",
            "answer": "0"
          },
          "hint": "Collect LIKE terms only; expand with the grid — every term times every term, signs carried. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-sign-lost"
        }
      ]
    },
    "factorise": {
      "objectiveCode": "C2.2-3 / E2.2-3,4,5",
      "learnerAction": "Factorise fully or complete the square, then verify the identity numerically.",
      "objectiveVerbs": [
        "factorise by extracting common factors",
        "factorise quadratics and related forms",
        "complete the square"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "factorise-mission-01",
          "prompt": "Factorise fully 12x^2 + 18xy. Choose the full factorisation, then type the value of 12x^2 + 18xy at x = 1, y = 1.",
          "model": {
            "kind": "common",
            "given": "12x^2 + 18xy",
            "point": [
              1,
              1
            ],
            "cardOptions": [
              "6x(2x + 3y)",
              "6(2x^2 + 3xy)",
              "2x(12x^2 + 18xy)",
              "6x(18x + 18y)"
            ],
            "optionCodes": [
              null,
              "am-letter-dropped",
              "am-not-fully",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "6x(2x + 3y)",
            "answer": "30"
          },
          "hint": "Factorise FULLY (number and letters), name the pattern, and complete the square by halving. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-letter-dropped"
        },
        {
          "id": "factorise-mission-02",
          "prompt": "Factorise 25x^2 − 4. Choose the factorisation, then type the value of 25x^2 − 4 at x = 3.",
          "model": {
            "kind": "diff-squares",
            "given": "25x^2 − 4",
            "point": [
              3,
              0
            ],
            "cardOptions": [
              "(5x + 2)(5x − 2)",
              "(5x − 2)^2",
              "(5x + 2)^2",
              "(25x + 2)(25x − 2)"
            ],
            "optionCodes": [
              null,
              "am-wrong-pattern",
              "am-wrong-pattern",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "(5x + 2)(5x − 2)",
            "answer": "221"
          },
          "hint": "Factorise FULLY (number and letters), name the pattern, and complete the square by halving. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-wrong-pattern"
        },
        {
          "id": "factorise-mission-03",
          "prompt": "Factorise x^2 + 7x + 12. Choose the factorisation, then type the value of x^2 + 7x + 12 at x = 2.",
          "model": {
            "kind": "trinomial",
            "given": "x^2 + 7x + 12",
            "point": [
              2,
              0
            ],
            "cardOptions": [
              "(x + 3)(x + 4)",
              "(x − 3)(x − 4)",
              "(x + 12)(x + 1)",
              "(x + 3)(x + 5)"
            ],
            "optionCodes": [
              null,
              "am-sign-lost",
              "am-pair-sum",
              "am-arithmetic"
            ]
          },
          "expected": {
            "card": "(x + 3)(x + 4)",
            "answer": "30"
          },
          "hint": "Factorise FULLY (number and letters), name the pattern, and complete the square by halving. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-pair-sum"
        },
        {
          "id": "factorise-mission-04",
          "prompt": "Write x^2 + 10x + 18 in completed-square form. Choose the correct form, then type the value of x^2 + 10x + 18 at x = 1.",
          "model": {
            "kind": "complete-square",
            "given": "x^2 + 10x + 18",
            "point": [
              1,
              0
            ],
            "cardOptions": [
              "(x + 5)^2 − 7",
              "(x + 10)^2 + -82",
              "(x + 5)^2 + 18",
              "(x − 5)^2 + -7"
            ],
            "optionCodes": [
              null,
              "am-half-not",
              "am-half-not",
              "am-sign-lost"
            ]
          },
          "expected": {
            "card": "(x + 5)^2 − 7",
            "answer": "29"
          },
          "hint": "Factorise FULLY (number and letters), name the pattern, and complete the square by halving. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-half-not"
        },
        {
          "id": "factorise-mission-05",
          "prompt": "Factorise fully x^3 + x^2 − 6x. Choose the full factorisation, then type the value of x^3 + x^2 − 6x at x = 2.",
          "model": {
            "kind": "cubic-common",
            "given": "x^3 + x^2 − 6x",
            "point": [
              2,
              0
            ],
            "cardOptions": [
              "x(x − 2)(x + 3)",
              "x(x^2 + x − 6)",
              "x(x + 2)(x − 3)",
              "(x^2 + x − 6)"
            ],
            "optionCodes": [
              null,
              "am-not-fully",
              "am-sign-lost",
              "am-letter-dropped"
            ]
          },
          "expected": {
            "card": "x(x − 2)(x + 3)",
            "answer": "0"
          },
          "hint": "Factorise FULLY (number and letters), name the pattern, and complete the square by halving. Check: both sides must give the same value at the test number.",
          "misconceptionCode": "am-not-fully"
        }
      ]
    },
    "change-subject": {
      "objectiveCode": "C2.5-4 / E2.5-7",
      "learnerAction": "Rearrange a formula to make a named letter the subject, then verify the rearrangement at a test point.",
      "objectiveVerbs": [
        "rearrange formulae to change the subject",
        "solve linear equations by collecting the unknown on one side"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "change-subject-mission-01",
          "prompt": "Make x the subject of 6x + 2 = y. Choose the rearrangement, then type the value of x at y = 20.",
          "model": {
            "kind": "subject-once",
            "given": "6x + 2 = y",
            "subject": "x",
            "make": "x",
            "point": {
              "x": 3,
              "y": 20
            },
            "steps": [
              "Start: 6x + 2 = y.",
              "Undo +2: subtract 2 from both sides → 6x = y − 2.",
              "Undo ×6: divide both sides by 6 → x = (y − 2)/6."
            ],
            "cardOptions": [
              "x = (y − 2)/6",
              "x = (y + 2)/6",
              "x = y/6 − 2",
              "x = (y − 2) × 6"
            ],
            "optionCodes": [
              null,
              "am-inverse-wrong",
              "am-both-sides",
              "am-inverse-wrong"
            ]
          },
          "expected": {
            "card": "x = (y − 2)/6",
            "answer": "3"
          },
          "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.",
          "misconceptionCode": "am-inverse-wrong"
        },
        {
          "id": "change-subject-mission-02",
          "prompt": "Make k the subject of h = 9 − 4k. Choose the rearrangement, then type the value of k at h = 1.",
          "model": {
            "kind": "subject-once",
            "given": "h = 9 − 4k",
            "subject": "k",
            "make": "k",
            "point": {
              "k": 2,
              "h": 1
            },
            "steps": [
              "Start: h = 9 − 4k.",
              "Undo −4k: add 4k to both sides → h + 4k = 9.",
              "Undo +h: subtract h from both sides → 4k = 9 − h.",
              "Undo ×4: divide both sides by 4 → k = (9 − h)/4."
            ],
            "cardOptions": [
              "k = (9 − h)/4",
              "k = (h − 9)/4",
              "k = 9 − h/4",
              "k = (9 − h) × 4"
            ],
            "optionCodes": [
              null,
              "am-sign-move",
              "am-both-sides",
              "am-inverse-wrong"
            ]
          },
          "expected": {
            "card": "k = (9 − h)/4",
            "answer": "2"
          },
          "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.",
          "misconceptionCode": "am-sign-move"
        },
        {
          "id": "change-subject-mission-03",
          "prompt": "Make x the subject of y = 8x + 2x − 5. Choose the rearrangement, then type the value of x at y = 15.",
          "model": {
            "kind": "subject-collect",
            "given": "y = 8x + 2x − 5",
            "subject": "x",
            "make": "x",
            "point": {
              "x": 2,
              "y": 15
            },
            "steps": [
              "Start: y = 8x + 2x − 5.",
              "Collect the x terms: 8x + 2x = 10x → y = 10x − 5.",
              "Undo −5: add 5 to both sides → y + 5 = 10x.",
              "Undo ×10: divide both sides by 10 → x = (y + 5)/10."
            ],
            "cardOptions": [
              "x = (y + 5)/10",
              "x = (y + 5)/8",
              "x = (y − 5)/10",
              "x = (y + 5)/6"
            ],
            "optionCodes": [
              null,
              "am-collect-first",
              "am-sign-move",
              "am-collect-first"
            ]
          },
          "expected": {
            "card": "x = (y + 5)/10",
            "answer": "2"
          },
          "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.",
          "misconceptionCode": "am-collect-first"
        },
        {
          "id": "change-subject-mission-04",
          "prompt": "Make r the subject of h = 5r². Choose the rearrangement, then type the value of r at h = 20.",
          "model": {
            "kind": "subject-power",
            "given": "h = 5r²",
            "subject": "r",
            "make": "r",
            "point": {
              "r": 2,
              "h": 20
            },
            "steps": [
              "Start: h = 5r².",
              "Undo ×5: divide both sides by 5 → h/5 = r².",
              "Undo the square: square root both sides → r = √(h/5)."
            ],
            "cardOptions": [
              "r = √(h/5)",
              "r = √(h)/5",
              "r = √(5h)",
              "r = (h/5)²"
            ],
            "optionCodes": [
              null,
              "am-both-sides",
              "am-inverse-wrong",
              "am-power-order"
            ]
          },
          "expected": {
            "card": "r = √(h/5)",
            "answer": "2"
          },
          "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.",
          "misconceptionCode": "am-both-sides"
        },
        {
          "id": "change-subject-mission-05",
          "prompt": "Make t the subject of s = 6√t. Choose the rearrangement, then type the value of t at s = 18.",
          "model": {
            "kind": "subject-power",
            "given": "s = 6√t",
            "subject": "t",
            "make": "t",
            "point": {
              "t": 9,
              "s": 18
            },
            "steps": [
              "Start: s = 6√t.",
              "Undo ×6: divide both sides by 6 → s/6 = √t.",
              "Undo the square root: square both sides → t = (s/6)²."
            ],
            "cardOptions": [
              "t = (s/6)²",
              "t = √(s/6)",
              "t = (6s)²",
              "t = s/6"
            ],
            "optionCodes": [
              null,
              "am-power-order",
              "am-both-sides",
              "am-inverse-wrong"
            ]
          },
          "expected": {
            "card": "t = (s/6)²",
            "answer": "9"
          },
          "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.",
          "misconceptionCode": "am-power-order"
        }
      ]
    },
    "rational-expressions": {
      "objectiveCode": "E2.3-2",
      "learnerAction": "Factorise and simplify algebraic fractions, then verify the simplification numerically.",
      "objectiveVerbs": [
        "factorise and simplify algebraic fractions",
        "cancel common factors in rational expressions"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "rational-expressions-mission-01",
          "prompt": "Simplify (18x²y)/(12xy²). Choose the simplified fraction, then type its value at x = 6, y = 1.",
          "model": {
            "kind": "rational-mono-cancel",
            "given": "(18x²y)/(12xy²)",
            "num": "18x²y",
            "den": "12xy²",
            "numParts": [
              "18",
              "x²",
              "y"
            ],
            "denParts": [
              "12",
              "x",
              "y²"
            ],
            "cancelPairs": [],
            "steps": [
              "Numbers: 18 and 12 share 6 → divide both by 6: 18 ÷ 6 = 3, 12 ÷ 6 = 2.",
              "Letters: x: 2 − 1 = 1, so x stays on top; y: 2 − 1 = 1, so y stays below.",
              "Cancel: (18x²y)/(12xy²) = (3x)/(2y)."
            ],
            "result": "(3x)/(2y)",
            "point": {
              "x": 6,
              "y": 1
            },
            "cardOptions": [
              "(3x)/(2y)",
              "(3y)/(2x)",
              "(4x)/(2y)",
              "3/2"
            ],
            "optionCodes": [
              null,
              "am-cancel-wrong-pair",
              "am-arithmetic",
              "am-letters-vanished"
            ]
          },
          "expected": {
            "card": "(3x)/(2y)",
            "answer": "9"
          },
          "hint": "Cancel the shared NUMBER factor (divide both by it) and each shared LETTER by subtracting its powers — a letter disappears only when the powers match exactly.",
          "misconceptionCode": "am-cancel-wrong-pair"
        },
        {
          "id": "rational-expressions-mission-02",
          "prompt": "Simplify (20xy²)/(16x²y). Choose the simplified fraction, then type its value at x = 1, y = 4.",
          "model": {
            "kind": "rational-mono-cancel",
            "given": "(20xy²)/(16x²y)",
            "num": "20xy²",
            "den": "16x²y",
            "numParts": [
              "20",
              "x",
              "y²"
            ],
            "denParts": [
              "16",
              "x²",
              "y"
            ],
            "cancelPairs": [],
            "steps": [
              "Numbers: 20 and 16 share 4 → divide both by 4: 20 ÷ 4 = 5, 16 ÷ 4 = 4.",
              "Letters: x: 2 − 1 = 1, so x stays below; y: 2 − 1 = 1, so y stays on top.",
              "Cancel: (20xy²)/(16x²y) = (5y)/(4x)."
            ],
            "result": "(5y)/(4x)",
            "point": {
              "x": 1,
              "y": 4
            },
            "cardOptions": [
              "(5y)/(4x)",
              "(6y)/(4x)",
              "(5x)/(4y)",
              "5/4"
            ],
            "optionCodes": [
              null,
              "am-arithmetic",
              "am-cancel-wrong-pair",
              "am-letters-vanished"
            ]
          },
          "expected": {
            "card": "(5y)/(4x)",
            "answer": "5"
          },
          "hint": "Cancel the shared NUMBER factor (divide both by it) and each shared LETTER by subtracting its powers — a letter disappears only when the powers match exactly.",
          "misconceptionCode": "am-arithmetic"
        },
        {
          "id": "rational-expressions-mission-03",
          "prompt": "Simplify (x² + 7x + 12)/(x² + x − 6). Choose the simplified fraction, then type its value at x = 8.",
          "model": {
            "kind": "rational-pair-cancel",
            "given": "(x² + 7x + 12)/(x² + x − 6)",
            "num": "x² + 7x + 12",
            "den": "x² + x − 6",
            "numParts": [
              "(x + 4)",
              "(x + 3)"
            ],
            "denParts": [
              "(x − 2)",
              "(x + 3)"
            ],
            "cancelPairs": [
              [
                "(x + 3)",
                "(x + 3)"
              ]
            ],
            "steps": [
              "Factorise the top: x² + 7x + 12 = (x + 4)(x + 3).",
              "Factorise the bottom: x² + x − 6 = (x − 2)(x + 3).",
              "Cancel the shared bracket (x + 3): (x² + 7x + 12)/(x² + x − 6) = (x + 4)/(x − 2)."
            ],
            "result": "(x + 4)/(x − 2)",
            "point": {
              "x": 8
            },
            "cardOptions": [
              "(x + 4)/(x − 2)",
              "(7x + 12)/(x − 6)",
              "(x + 4)/(x + 3)",
              "(x + 3)/(x − 2)"
            ],
            "optionCodes": [
              null,
              "am-cancel-across-sum",
              "am-cancel-wrong-pair",
              "am-cancel-wrong-pair"
            ]
          },
          "expected": {
            "card": "(x + 4)/(x − 2)",
            "answer": "2"
          },
          "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.",
          "misconceptionCode": "am-cancel-across-sum"
        },
        {
          "id": "rational-expressions-mission-04",
          "prompt": "Simplify (x² + 9x + 20)/(x² + 3x − 4). Choose the simplified fraction, then type its value at x = 3.",
          "model": {
            "kind": "rational-pair-cancel",
            "given": "(x² + 9x + 20)/(x² + 3x − 4)",
            "num": "x² + 9x + 20",
            "den": "x² + 3x − 4",
            "numParts": [
              "(x + 5)",
              "(x + 4)"
            ],
            "denParts": [
              "(x − 1)",
              "(x + 4)"
            ],
            "cancelPairs": [
              [
                "(x + 4)",
                "(x + 4)"
              ]
            ],
            "steps": [
              "Factorise the top: x² + 9x + 20 = (x + 5)(x + 4).",
              "Factorise the bottom: x² + 3x − 4 = (x − 1)(x + 4).",
              "Cancel the shared bracket (x + 4): (x² + 9x + 20)/(x² + 3x − 4) = (x + 5)/(x − 1)."
            ],
            "result": "(x + 5)/(x − 1)",
            "point": {
              "x": 3
            },
            "cardOptions": [
              "(x + 5)/(x − 1)",
              "(x + 5)/(x + 4)",
              "(x + 4)/(x − 1)",
              "(9x + 20)/(3x − 4)"
            ],
            "optionCodes": [
              null,
              "am-cancel-wrong-pair",
              "am-cancel-wrong-pair",
              "am-cancel-across-sum"
            ]
          },
          "expected": {
            "card": "(x + 5)/(x − 1)",
            "answer": "4"
          },
          "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.",
          "misconceptionCode": "am-cancel-wrong-pair"
        },
        {
          "id": "rational-expressions-mission-05",
          "prompt": "Simplify (x² + 11x + 30)/(x² + 2x − 15). Choose the simplified fraction, then type its value at x = 4.",
          "model": {
            "kind": "rational-pair-cancel",
            "given": "(x² + 11x + 30)/(x² + 2x − 15)",
            "num": "x² + 11x + 30",
            "den": "x² + 2x − 15",
            "numParts": [
              "(x + 6)",
              "(x + 5)"
            ],
            "denParts": [
              "(x − 3)",
              "(x + 5)"
            ],
            "cancelPairs": [
              [
                "(x + 5)",
                "(x + 5)"
              ]
            ],
            "steps": [
              "Factorise the top: x² + 11x + 30 = (x + 6)(x + 5).",
              "Factorise the bottom: x² + 2x − 15 = (x − 3)(x + 5).",
              "Cancel the shared bracket (x + 5): (x² + 11x + 30)/(x² + 2x − 15) = (x + 6)/(x − 3)."
            ],
            "result": "(x + 6)/(x − 3)",
            "point": {
              "x": 4
            },
            "cardOptions": [
              "(x + 6)/(x − 3)",
              "(11x + 30)/(2x − 15)",
              "(x + 6)/(x + 5)",
              "(x + 5)/(x − 3)"
            ],
            "optionCodes": [
              null,
              "am-cancel-across-sum",
              "am-cancel-wrong-pair",
              "am-cancel-wrong-pair"
            ]
          },
          "expected": {
            "card": "(x + 6)/(x − 3)",
            "answer": "10"
          },
          "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.",
          "misconceptionCode": "am-cancel-across-sum"
        }
      ]
    },
    "linear-nonlinear": {
      "objectiveCode": "E2.5-5",
      "learnerAction": "Solve one linear and one non-linear equation by substitution, then verify each solution pair.",
      "objectiveVerbs": [
        "solve simultaneous equations, one linear and one non-linear",
        "substitute a linear expression into a quadratic equation"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "linear-nonlinear-mission-01",
          "prompt": "Solve y = x + 4 and y = x² − 3x + 7 together. Choose the pair-set card, then type the value of y when x = 1.",
          "model": {
            "kind": "system-substitute",
            "shape": "parabola",
            "given": "y = x + 4 and y = x² − 3x + 7",
            "linear": "y = x + 4",
            "nonlinear": "y = x² − 3x + 7",
            "substituted": "x² − 4x + 3 = 0",
            "factored": "(x − 1)(x − 3) = 0",
            "pairs": [
              "(1, 5)",
              "(3, 7)"
            ],
            "pairValues": [
              [
                1,
                5
              ],
              [
                3,
                7
              ]
            ],
            "check": {
              "x": 1,
              "y": 5
            },
            "cardOptions": [
              "(1, 5) and (3, 7)",
              "(5, 1) and (7, 3)",
              "(−1, 3) and (−3, 1)",
              "(1, 5)"
            ],
            "optionCodes": [
              null,
              "am-swap-pair",
              "am-root-sign",
              "am-one-root"
            ]
          },
          "expected": {
            "card": "(1, 5) and (3, 7)",
            "answer": "5"
          },
          "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.",
          "misconceptionCode": "am-swap-pair"
        },
        {
          "id": "linear-nonlinear-mission-02",
          "prompt": "Solve y = x + 17 and x² + y² = 169 together. Choose the pair-set card, then type the value of y when x = −12.",
          "model": {
            "kind": "system-substitute",
            "shape": "circle",
            "given": "y = x + 17 and x² + y² = 169",
            "linear": "y = x + 17",
            "nonlinear": "x² + y² = 169",
            "substituted": "x² + 17x + 60 = 0",
            "factored": "(x + 12)(x + 5) = 0",
            "pairs": [
              "(−12, 5)",
              "(−5, 12)"
            ],
            "pairValues": [
              [
                -12,
                5
              ],
              [
                -5,
                12
              ]
            ],
            "check": {
              "x": -12,
              "y": 5
            },
            "cardOptions": [
              "(−12, 5) and (−5, 12)",
              "(12, 29) and (5, 22)",
              "(5, −12) and (12, −5)",
              "(−12, 5)"
            ],
            "optionCodes": [
              null,
              "am-root-sign",
              "am-swap-pair",
              "am-one-root"
            ]
          },
          "expected": {
            "card": "(−12, 5) and (−5, 12)",
            "answer": "5"
          },
          "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.",
          "misconceptionCode": "am-root-sign"
        },
        {
          "id": "linear-nonlinear-mission-03",
          "prompt": "Solve y = x − 8 and xy = −12 together. Choose the pair-set card, then type the value of y when x = 2.",
          "model": {
            "kind": "system-substitute",
            "shape": "hyperbola",
            "given": "y = x − 8 and xy = −12",
            "linear": "y = x − 8",
            "nonlinear": "xy = −12",
            "substituted": "x² − 8x + 12 = 0",
            "factored": "(x − 2)(x − 6) = 0",
            "pairs": [
              "(2, −6)",
              "(6, −2)"
            ],
            "pairValues": [
              [
                2,
                -6
              ],
              [
                6,
                -2
              ]
            ],
            "check": {
              "x": 2,
              "y": -6
            },
            "cardOptions": [
              "(2, −6) and (6, −2)",
              "(2, −6)",
              "(−6, 2) and (−2, 6)",
              "(−2, −10) and (−6, −14)"
            ],
            "optionCodes": [
              null,
              "am-one-root",
              "am-swap-pair",
              "am-root-sign"
            ]
          },
          "expected": {
            "card": "(2, −6) and (6, −2)",
            "answer": "-6"
          },
          "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.",
          "misconceptionCode": "am-one-root"
        },
        {
          "id": "linear-nonlinear-mission-04",
          "prompt": "Solve y = 2x − 1 and y = x² − 6x + 14 together. Choose the pair-set card, then type the value of y when x = 3.",
          "model": {
            "kind": "system-substitute",
            "shape": "parabola",
            "given": "y = 2x − 1 and y = x² − 6x + 14",
            "linear": "y = 2x − 1",
            "nonlinear": "y = x² − 6x + 14",
            "substituted": "x² − 8x + 15 = 0",
            "factored": "(x − 3)(x − 5) = 0",
            "pairs": [
              "(3, 5)",
              "(5, 9)"
            ],
            "pairValues": [
              [
                3,
                5
              ],
              [
                5,
                9
              ]
            ],
            "check": {
              "x": 3,
              "y": 5
            },
            "cardOptions": [
              "(3, 5) and (5, 9)",
              "(3, 5) and (5, 10)",
              "(5, 3) and (9, 5)",
              "(−3, −7) and (−5, −11)"
            ],
            "optionCodes": [
              null,
              "am-arithmetic",
              "am-swap-pair",
              "am-root-sign"
            ]
          },
          "expected": {
            "card": "(3, 5) and (5, 9)",
            "answer": "5"
          },
          "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.",
          "misconceptionCode": "am-arithmetic"
        },
        {
          "id": "linear-nonlinear-mission-05",
          "prompt": "Solve y = x − 7 and x² + y² = 25 together. Choose the pair-set card, then type the value of y when x = 3.",
          "model": {
            "kind": "system-substitute",
            "shape": "circle",
            "given": "y = x − 7 and x² + y² = 25",
            "linear": "y = x − 7",
            "nonlinear": "x² + y² = 25",
            "substituted": "x² − 7x + 12 = 0",
            "factored": "(x − 3)(x − 4) = 0",
            "pairs": [
              "(3, −4)",
              "(4, −3)"
            ],
            "pairValues": [
              [
                3,
                -4
              ],
              [
                4,
                -3
              ]
            ],
            "check": {
              "x": 3,
              "y": -4
            },
            "cardOptions": [
              "(3, −4) and (4, −3)",
              "(−4, 3) and (−3, 4)",
              "(−3, −10) and (−4, −11)",
              "(3, −4)"
            ],
            "optionCodes": [
              null,
              "am-swap-pair",
              "am-root-sign",
              "am-one-root"
            ]
          },
          "expected": {
            "card": "(3, −4) and (4, −3)",
            "answer": "-4"
          },
          "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.",
          "misconceptionCode": "am-swap-pair"
        }
      ]
    }
  }
}