{
  "id": "LO4",
  "folder": "04_LO4_Conjugate",
  "zip": "LO4_Conjugate_SLS_xAPI.zip",
  "title": "Conjugate Mirror Studio",
  "objective": "Conjugate of a complex number",
  "kind": "conjugate",
  "accent": "#b21e5b",
  "stages": [
    {
      "id": "LO4-S1",
      "level": "Recognise",
      "prompt": "Find the conjugate of 3 − 4i.",
      "instruction": "Enter its components.",
      "response": {
        "type": "complex",
        "answer": {
          "re": 3,
          "im": 4
        }
      },
      "model": {
        "type": "mirror",
        "z": {
          "re": 3,
          "im": -4
        },
        "showConjugate": false
      },
      "hint": [
        "Keep the real component.",
        "Reflect the imaginary component across the real axis."
      ],
      "worked": [
        "Conjugation maps a + bi to a − bi.",
        "The real coordinate stays 3 and −4 becomes 4.",
        "So (3 − 4i)* = 3 + 4i."
      ],
      "misconception": "Negating both components instead of reflecting only across the real axis."
    },
    {
      "id": "LO4-S2",
      "level": "Construct",
      "prompt": "Place the conjugate of −2 + 5i on the Argand plane.",
      "instruction": "Tap the grid or use arrow keys after focusing the target point.",
      "response": {
        "type": "point",
        "answer": {
          "re": -2,
          "im": -5
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "mirror",
        "z": {
          "re": -2,
          "im": 5
        },
        "showConjugate": false
      },
      "hint": [
        "A mirror image has the same horizontal coordinate.",
        "Its vertical coordinate is the opposite."
      ],
      "worked": [
        "Start at (−2,5).",
        "Reflect across the real axis to (−2,−5).",
        "The conjugate is −2 − 5i."
      ],
      "misconception": "Reflecting across the imaginary axis, which would change the real component."
    },
    {
      "id": "LO4-S3",
      "level": "Apply to a sum",
      "prompt": "Given z = −6 + i, find (z + 3)*.",
      "instruction": "Simplify inside the brackets before conjugating.",
      "response": {
        "type": "complex",
        "answer": {
          "re": -3,
          "im": -1
        }
      },
      "model": {
        "type": "components",
        "expression": "(−6 + i + 3)*",
        "real": "−3",
        "imag": "opposite of +1"
      },
      "hint": [
        "z + 3 = −3 + i.",
        "Conjugation changes +i to −i."
      ],
      "worked": [
        "First combine real terms: z+3=−3+i.",
        "Reflect its imaginary component: +1 becomes −1.",
        "Thus (z+3)*=−3−i."
      ],
      "misconception": "Conjugating z but forgetting that the whole bracket is conjugated after simplification."
    },
    {
      "id": "LO4-S4",
      "level": "Apply to an imaginary shift",
      "prompt": "Given z = −6 + i, find (z + 3i)*.",
      "instruction": "Combine imaginary terms, then conjugate.",
      "response": {
        "type": "complex",
        "answer": {
          "re": -6,
          "im": -4
        }
      },
      "model": {
        "type": "components",
        "expression": "(−6 + i + 3i)*",
        "real": "−6",
        "imag": "opposite of +4"
      },
      "hint": [
        "i + 3i = 4i.",
        "Then reverse the sign of the imaginary component."
      ],
      "worked": [
        "Inside the bracket, z+3i=−6+4i.",
        "Conjugation preserves −6 and maps 4i to −4i.",
        "The result is −6−4i."
      ],
      "misconception": "Treating 3i as a real shift or changing the real sign."
    },
    {
      "id": "LO4-S5",
      "level": "Connect modulus",
      "prompt": "For z = 2 − 3i, what is zz*?",
      "instruction": "Use the geometric meaning as well as multiplication.",
      "response": {
        "type": "number",
        "answer": 13,
        "label": "zz*"
      },
      "model": {
        "type": "mirror",
        "z": {
          "re": 2,
          "im": -3
        },
        "showConjugate": true,
        "showCircle": true
      },
      "hint": [
        "zz* = a² + b².",
        "It is also |z|²."
      ],
      "worked": [
        "z*=2+3i.",
        "(2−3i)(2+3i)=2²+3².",
        "Therefore zz*=13=|z|², a non-negative real number."
      ],
      "misconception": "Using a²−b² instead of the sum of squares produced by conjugates."
    },
    {
      "id": "LO4-S6",
      "level": "Generalise",
      "prompt": "Which identity is always true?",
      "instruction": "Check how conjugation distributes.",
      "response": {
        "type": "choice",
        "options": [
          "(z+w)* = z* + w*",
          "(zw)* = z*w",
          "z+z* is purely imaginary",
          "z−z* is always real"
        ],
        "answer": "(z+w)* = z* + w*"
      },
      "model": {
        "type": "components",
        "expression": "z=a+bi, w=c+di",
        "real": "a+c",
        "imag": "b+d"
      },
      "hint": [
        "Write both numbers in component form.",
        "Conjugate the whole sum and compare."
      ],
      "worked": [
        "z+w=(a+c)+(b+d)i.",
        "Its conjugate is (a+c)−(b+d)i.",
        "This equals (a−bi)+(c−di)=z*+w*."
      ],
      "misconception": "Assuming conjugation changes multiplication into division or confusing the real/imaginary nature of z±z*."
    }
  ]
}