{
  "id": "LO7",
  "folder": "07_LO7_Argand_Modulus_Argument",
  "zip": "LO7_Argand_Modulus_Argument_SLS_xAPI.zip",
  "title": "Argand Modulus and Argument Observatory",
  "objective": "Representation of complex numbers in the Argand diagram, finding the modulus and argument of a complex number",
  "kind": "polar",
  "accent": "#0070a8",
  "stages": [
    {
      "id": "LO7-S1",
      "level": "Plot",
      "prompt": "Place z = 2 + 2i on the Argand plane.",
      "instruction": "Tap the grid or use arrow keys after focusing the target point.",
      "response": {
        "type": "point",
        "answer": {
          "re": 2,
          "im": 2
        },
        "tolerance": 0.35
      },
      "model": {
        "type": "polar",
        "z": null,
        "guide": {
          "re": 2,
          "im": 2
        },
        "hideResult": true
      },
      "hint": [
        "The real part is the horizontal coordinate.",
        "The imaginary part is the vertical coordinate."
      ],
      "worked": [
        "Move 2 units right from the origin.",
        "Then move 2 units upward.",
        "The point (2,2) represents 2+2i."
      ],
      "misconception": "Swapping the real and imaginary axes or reversing the sign direction."
    },
    {
      "id": "LO7-S2",
      "level": "Find modulus",
      "prompt": "Find |−3 − √3 i|.",
      "instruction": "Choose the exact value.",
      "response": {
        "type": "choice",
        "options": [
          "2√3",
          "3+√3",
          "√6",
          "4"
        ],
        "answer": "2√3"
      },
      "model": {
        "type": "polar",
        "z": {
          "re": -3,
          "im": -1.732
        },
        "showRadius": true,
        "hideLength": true
      },
      "hint": [
        "Use √(a²+b²).",
        "9+3=12."
      ],
      "worked": [
        "The point is (−3,−√3).",
        "Its distance from the origin is √(9+3)=√12.",
        "Therefore the modulus is 2√3."
      ],
      "misconception": "Adding coordinate magnitudes instead of using Pythagoras."
    },
    {
      "id": "LO7-S3",
      "level": "Find argument",
      "prompt": "Find the principal argument of 2 + 2i.",
      "instruction": "Choose the exact angle.",
      "response": {
        "type": "choice",
        "options": [
          "π/4",
          "3π/4",
          "−π/4",
          "π/2"
        ],
        "answer": "π/4"
      },
      "model": {
        "type": "polar",
        "z": {
          "re": 2,
          "im": 2
        },
        "showAngle": true,
        "hideAngle": true
      },
      "hint": [
        "The point is in quadrant I.",
        "tan θ = 2/2 = 1."
      ],
      "worked": [
        "The reference angle satisfies tan θ=1.",
        "In quadrant I the principal angle is positive.",
        "Thus arg(2+2i)=π/4."
      ],
      "misconception": "Using the correct reference angle with the wrong quadrant sign."
    },
    {
      "id": "LO7-S4",
      "level": "Compare distances",
      "prompt": "Which complex number is closest to the origin?",
      "instruction": "Compare squared moduli to avoid unnecessary roots.",
      "response": {
        "type": "choice",
        "options": [
          "−3−√3i",
          "−3",
          "2+2i",
          "1−√3i"
        ],
        "answer": "1−√3i"
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": -3,
            "im": -1.732
          },
          {
            "re": -3,
            "im": 0
          },
          {
            "re": 2,
            "im": 2
          },
          {
            "re": 1,
            "im": -1.732
          }
        ],
        "labels": [
          "A",
          "B",
          "C",
          "D"
        ]
      },
      "hint": [
        "Compare a²+b² for each point.",
        "The squared moduli are 12, 9, 8, and 4."
      ],
      "worked": [
        "Distance is controlled by |z|²=a²+b².",
        "The four squared moduli are 12, 9, 8, and 4.",
        "So 1−√3i is closest."
      ],
      "misconception": "Comparing only real parts, only imaginary parts, or visual quadrant position."
    },
    {
      "id": "LO7-S5",
      "level": "Connect conjugacy",
      "prompt": "For nonzero z,w with principal arguments, |z|=|w| and arg z=−arg w. What follows?",
      "instruction": "Use polar form and state the needed domain condition.",
      "response": {
        "type": "choice",
        "options": [
          "w=z*",
          "w=−z",
          "w=iz",
          "w=z"
        ],
        "answer": "w=z*"
      },
      "model": {
        "type": "mirror",
        "z": {
          "re": 3,
          "im": 2
        },
        "showConjugate": true,
        "showCircle": true
      },
      "hint": [
        "Equal moduli place the points on the same circle.",
        "Opposite arguments reflect across the real axis."
      ],
      "worked": [
        "Write z=r(cosθ+i sinθ).",
        "The conditions give w=r(cos(−θ)+i sin(−θ)).",
        "Hence w=r(cosθ−i sinθ)=z*."
      ],
      "misconception": "Treating opposite arguments as a reflection through the origin rather than across the real axis."
    },
    {
      "id": "LO7-S6",
      "level": "Reconstruct",
      "prompt": "A complex number has modulus 4 and principal argument 2π/3. Which Cartesian form is correct?",
      "instruction": "Use z=r(cosθ+i sinθ).",
      "response": {
        "type": "choice",
        "options": [
          "−2+2√3i",
          "2+2√3i",
          "−2−2√3i",
          "2√3−2i"
        ],
        "answer": "−2+2√3i"
      },
      "model": {
        "type": "polar",
        "radius": 4,
        "angle": 120,
        "hidePoint": true
      },
      "hint": [
        "cos(2π/3)=−1/2.",
        "sin(2π/3)=√3/2."
      ],
      "worked": [
        "Multiply the cosine component by 4: 4(−1/2)=−2.",
        "Multiply the sine component by 4: 4(√3/2)=2√3.",
        "Thus z=−2+2√3i."
      ],
      "misconception": "Using the reference-angle signs from quadrant I instead of quadrant II."
    }
  ]
}