{
  "id": "LO2",
  "folder": "02_LO2_Four_Operations",
  "zip": "LO2_Four_Operations_SLS_xAPI.zip",
  "title": "Complex Arithmetic Vector Lab",
  "objective": "Four operations of complex numbers",
  "kind": "operations",
  "accent": "#0b7a75",
  "stages": [
    {
      "id": "LO2-S1",
      "level": "Add",
      "prompt": "Find (2 + 3i) + (−5 + i).",
      "instruction": "Enter the real and imaginary components.",
      "response": {
        "type": "complex",
        "answer": {
          "re": -3,
          "im": 4
        }
      },
      "model": {
        "type": "vector",
        "z": {
          "re": 2,
          "im": 3
        },
        "w": {
          "re": -5,
          "im": 1
        },
        "operation": "add"
      },
      "hint": [
        "Add horizontal components together.",
        "Add vertical components together."
      ],
      "worked": [
        "Real: 2 + (−5) = −3.",
        "Imaginary: 3 + 1 = 4.",
        "The sum is −3 + 4i; geometrically it is the vector resultant."
      ],
      "misconception": "Combining a real component with an imaginary component."
    },
    {
      "id": "LO2-S2",
      "level": "Subtract",
      "prompt": "Find (−2 + 3i) − (5 − i).",
      "instruction": "Treat subtraction as adding the opposite vector.",
      "response": {
        "type": "complex",
        "answer": {
          "re": -7,
          "im": 4
        }
      },
      "model": {
        "type": "vector",
        "z": {
          "re": -2,
          "im": 3
        },
        "w": {
          "re": 5,
          "im": -1
        },
        "operation": "subtract"
      },
      "hint": [
        "Distribute the minus sign to both components of w.",
        "The imaginary calculation is 3 − (−1)."
      ],
      "worked": [
        "(−2 + 3i) − (5 − i) = −2 + 3i − 5 + i.",
        "Real: −2 − 5 = −7; imaginary: 3 + 1 = 4.",
        "So the result is −7 + 4i."
      ],
      "misconception": "Negating only the real component when subtracting a complex number."
    },
    {
      "id": "LO2-S3",
      "level": "Multiply",
      "prompt": "Find (1 + 2i)(5 + 4i).",
      "instruction": "Expand, replace i², then collect components.",
      "response": {
        "type": "complex",
        "answer": {
          "re": -3,
          "im": 14
        }
      },
      "model": {
        "type": "components",
        "expression": "(1 + 2i)(5 + 4i)",
        "real": "1·5 + (2i)(4i)",
        "imag": "1·4 + 2·5"
      },
      "hint": [
        "The last product is 8i².",
        "Use i² = −1 before collecting real terms."
      ],
      "worked": [
        "Expand: 5 + 4i + 10i + 8i².",
        "Replace i² by −1: 5 − 8 + 14i.",
        "The product is −3 + 14i."
      ],
      "misconception": "Treating i² as +1 or failing to combine both cross terms."
    },
    {
      "id": "LO2-S4",
      "level": "Divide",
      "prompt": "Write (1 − 2i)/(3 + 4i) in a + bi form.",
      "instruction": "Use the conjugate of the denominator.",
      "response": {
        "type": "complex",
        "answer": {
          "re": -0.2,
          "im": -0.4
        },
        "tolerance": 0.001
      },
      "model": {
        "type": "fraction",
        "numerator": "1 − 2i",
        "denominator": "3 + 4i",
        "multiplier": "3 − 4i"
      },
      "hint": [
        "Multiply top and bottom by 3 − 4i.",
        "The denominator becomes 3² + 4²."
      ],
      "worked": [
        "(1 − 2i)(3 − 4i) = −5 − 10i.",
        "(3 + 4i)(3 − 4i) = 25.",
        "The quotient is −1/5 − (2/5)i = −0.2 − 0.4i."
      ],
      "misconception": "Conjugating only the denominator or using a² − b² for the denominator product."
    },
    {
      "id": "LO2-S5",
      "level": "Invert",
      "prompt": "Find 1/(3 + 4i).",
      "instruction": "Give decimal components or equivalent fractions.",
      "response": {
        "type": "complex",
        "answer": {
          "re": 0.12,
          "im": -0.16
        },
        "tolerance": 0.001
      },
      "model": {
        "type": "fraction",
        "numerator": "1",
        "denominator": "3 + 4i",
        "multiplier": "3 − 4i"
      },
      "hint": [
        "Use the conjugate 3 − 4i.",
        "The squared modulus is 25."
      ],
      "worked": [
        "Multiply by (3 − 4i)/(3 − 4i).",
        "The numerator becomes 3 − 4i and the denominator becomes 25.",
        "So 1/(3 + 4i) = 3/25 − (4/25)i."
      ],
      "misconception": "Changing the sign of the imaginary part without dividing by the squared modulus."
    },
    {
      "id": "LO2-S6",
      "level": "Verify",
      "prompt": "Is (−3 + √7 i)/2 a root of z² + 3z + 4 = 0?",
      "instruction": "Choose the conclusion supported by substitution.",
      "response": {
        "type": "choice",
        "options": [
          "Yes; substitution gives 0",
          "No; the imaginary part remains",
          "No; its conjugate is the only root",
          "Cannot decide without decimals"
        ],
        "answer": "Yes; substitution gives 0"
      },
      "model": {
        "type": "components",
        "expression": "z² + 3z + 4",
        "real": "substitute exactly",
        "imag": "track √7 terms"
      },
      "hint": [
        "Keep the common denominator 4.",
        "The imaginary terms from z² and 3z cancel."
      ],
      "worked": [
        "For z = (−3 + √7 i)/2, z² = (1 − 3√7 i)/2.",
        "Then 3z = (−9 + 3√7 i)/2.",
        "Adding z² + 3z + 4 gives 0, so it is a root."
      ],
      "misconception": "Using a decimal approximation and mistaking rounding residue for a nonzero result."
    }
  ]
}