{
  "id": "LO8",
  "folder": "08_LO8_Argand_Geometric_Effects",
  "zip": "LO8_Argand_Geometric_Effects_SLS_xAPI.zip",
  "title": "Argand Transformation Workshop",
  "objective": "Geometrical effects of conjugation, negation, addition, subtraction, and multiplication by i",
  "kind": "transformations",
  "accent": "#8a3ffc",
  "stages": [
    {
      "id": "LO8-S1",
      "level": "Conjugate",
      "prompt": "Place the image of z=2+3i under conjugation.",
      "instruction": "Tap the grid or use arrow keys to position the target.",
      "response": {
        "type": "point",
        "answer": {
          "re": 2,
          "im": -3
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "transform",
        "z": {
          "re": 2,
          "im": 3
        },
        "operation": "conjugate"
      },
      "hint": [
        "Conjugation is reflection across the real axis.",
        "Keep x=2 and reverse y=3."
      ],
      "worked": [
        "Start at (2,3).",
        "Reflect vertically across the real axis.",
        "The image is (2,−3), representing 2−3i."
      ],
      "misconception": "Reflecting across the imaginary axis or through the origin."
    },
    {
      "id": "LO8-S2",
      "level": "Negate",
      "prompt": "Place the image of z=2+3i under negation.",
      "instruction": "Negation is a half-turn about the origin.",
      "response": {
        "type": "point",
        "answer": {
          "re": -2,
          "im": -3
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "transform",
        "z": {
          "re": 2,
          "im": 3
        },
        "operation": "negate"
      },
      "hint": [
        "Multiply both components by −1.",
        "A half-turn changes both coordinates."
      ],
      "worked": [
        "The vector (2,3) is reversed.",
        "Its endpoint moves to (−2,−3).",
        "Therefore −z=−2−3i."
      ],
      "misconception": "Changing only the imaginary sign, which performs conjugation instead."
    },
    {
      "id": "LO8-S3",
      "level": "Rotate by i",
      "prompt": "Place iz when z=2+3i.",
      "instruction": "Multiplication by i is a 90° anticlockwise rotation.",
      "response": {
        "type": "point",
        "answer": {
          "re": -3,
          "im": 2
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "transform",
        "z": {
          "re": 2,
          "im": 3
        },
        "operation": "multiply_i"
      },
      "hint": [
        "i(a+bi)=−b+ai.",
        "Map (a,b) to (−b,a)."
      ],
      "worked": [
        "i(2+3i)=2i+3i².",
        "Since i²=−1, this is −3+2i.",
        "The point rotates to (−3,2) without changing its modulus."
      ],
      "misconception": "Rotating clockwise or swapping coordinates without changing one sign."
    },
    {
      "id": "LO8-S4",
      "level": "Add vectors",
      "prompt": "Let z=−2+3i and w=−1−2i. Place z+w.",
      "instruction": "Use the parallelogram or head-to-tail rule.",
      "response": {
        "type": "point",
        "answer": {
          "re": -3,
          "im": 1
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "vector",
        "z": {
          "re": -2,
          "im": 3
        },
        "w": {
          "re": -1,
          "im": -2
        },
        "operation": "add"
      },
      "hint": [
        "Add real coordinates and imaginary coordinates separately.",
        "The vertical components are 3+(−2)."
      ],
      "worked": [
        "Horizontal: −2+(−1)=−3.",
        "Vertical: 3+(−2)=1.",
        "The resultant is −3+i, at (−3,1)."
      ],
      "misconception": "Adding vector lengths instead of components or losing the negative sign of w."
    },
    {
      "id": "LO8-S5",
      "level": "Subtract vectors",
      "prompt": "Let z=−2+3i and w=−1−2i. Place z−w.",
      "instruction": "Add the opposite of w.",
      "response": {
        "type": "point",
        "answer": {
          "re": -1,
          "im": 5
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "vector",
        "z": {
          "re": -2,
          "im": 3
        },
        "w": {
          "re": -1,
          "im": -2
        },
        "operation": "subtract"
      },
      "hint": [
        "Compute −2−(−1).",
        "Compute 3−(−2)."
      ],
      "worked": [
        "Horizontal: −2+1=−1.",
        "Vertical: 3+2=5.",
        "So z−w=−1+5i, the vector from w to z."
      ],
      "misconception": "Subtracting only one component or using z+w."
    },
    {
      "id": "LO8-S6",
      "level": "Synthesize",
      "prompt": "|z|=3 and 0<arg z<π/2. Points P,Q,R represent z, 2iz, and (1+2i)z. What is the area of quadrilateral OPRQ?",
      "instruction": "Recognise the parallelogram generated by z and 2iz.",
      "response": {
        "type": "number",
        "answer": 18,
        "label": "area"
      },
      "model": {
        "type": "transform",
        "z": {
          "re": 2.4,
          "im": 1.8
        },
        "operation": "parallelogram",
        "scaleVector": 2
      },
      "hint": [
        "R=z+2iz, so OPRQ is a parallelogram.",
        "Multiplication by i rotates 90°; |2iz|=2|z|."
      ],
      "worked": [
        "The adjacent side vectors are z and 2iz.",
        "They are perpendicular with lengths 3 and 6.",
        "Area = 3×6=18 square units."
      ],
      "misconception": "Using the perimeter, forgetting the factor 2, or treating the vectors as non-perpendicular."
    }
  ]
}