{
  "version": 1,
  "meaning": {
    "en": "Authored subsection coverage and example evidence, not student mastery or universal proof. Teacher reference includes explanations.",
    "zh-Hant": "已編寫的小節覆蓋及示例證據，不代表學生掌握或全稱證明。教師參考包含解釋。"
  },
  "activities": [
    {
      "id": "g10-01-01",
      "chapter": "01",
      "target": {
        "en": "Membership",
        "zh-Hant": "元素與集合"
      },
      "title": {
        "en": "An element or a subset?",
        "zh-Hant": "元素還是子集？"
      },
      "chapterTitle": {
        "en": "Sets",
        "zh-Hant": "集合"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=sets&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-01-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-01-01/"
      },
      "coreClaim": {
        "en": "A contains numbers. A number can be an element; a singleton is a subset exactly when its number belongs to A. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "A 的元素是數；數可以是元素，而單元素集是子集恰好當其數屬於 A。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The universe is {1,…,8}; A is the chosen numeric set. A subset question requires a set as its left operand.",
        "zh-Hant": "全集為 {1,…,8}；A 是選定的數集；子集關係左方必須是集合。"
      },
      "evidence": [
        {
          "label": {
            "en": "A member",
            "zh-Hant": "集合內的數"
          },
          "boundary": false,
          "params": {
            "n": 2,
            "object": "number",
            "set": "even"
          },
          "observation": {
            "en": "The object 2 is an element; subset status is not applicable.",
            "zh-Hant": "物件 2屬於 A；不是集合，不能用子集關係。"
          }
        },
        {
          "label": {
            "en": "Its singleton",
            "zh-Hant": "它的單元素集"
          },
          "boundary": false,
          "params": {
            "n": 2,
            "object": "singleton",
            "set": "even"
          },
          "observation": {
            "en": "The object {2} is not an element; subset status is true.",
            "zh-Hant": "物件 {2}不屬於 A；子集判斷為真。"
          }
        },
        {
          "label": {
            "en": "The empty set",
            "zh-Hant": "空集"
          },
          "boundary": true,
          "params": {
            "n": 3,
            "object": "empty",
            "set": "even"
          },
          "observation": {
            "en": "The object ∅ is not an element; subset status is true.",
            "zh-Hant": "物件 ∅不屬於 A；子集判斷為真。"
          }
        }
      ]
    },
    {
      "id": "g10-01-02",
      "chapter": "01",
      "target": {
        "en": "Subsets and the empty set",
        "zh-Hant": "子集與空集"
      },
      "title": {
        "en": "Containment, equality and the empty set",
        "zh-Hant": "包含、相等與空集"
      },
      "chapterTitle": {
        "en": "Sets",
        "zh-Hant": "集合"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=sets&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-01-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-01-02/"
      },
      "coreClaim": {
        "en": "A⊆B allows equality. A proper subset additionally requires an element of B outside A. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "A⊆B 容許相等；真子集另外要求 B 有元素不在 A。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "a and b are integers from 0 to 8; an upper bound of zero denotes the empty set.",
        "zh-Hant": "a、b 是 0 至 8 的整數；上界為零表示空集。"
      },
      "evidence": [
        {
          "label": {
            "en": "Proper inclusion",
            "zh-Hant": "真包含"
          },
          "boundary": false,
          "params": {
            "a": 3,
            "b": 5
          },
          "observation": {
            "en": "A⊆B: true; A⊊B: true.",
            "zh-Hant": "A⊆B：真；A⊊B：真。"
          }
        },
        {
          "label": {
            "en": "Equal sets",
            "zh-Hant": "集合相等"
          },
          "boundary": true,
          "params": {
            "a": 4,
            "b": 4
          },
          "observation": {
            "en": "A⊆B: true; A⊊B: false.",
            "zh-Hant": "A⊆B：真；A⊊B：假。"
          }
        },
        {
          "label": {
            "en": "Both empty",
            "zh-Hant": "兩者皆空"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "b": 0
          },
          "observation": {
            "en": "A⊆B: true; A⊊B: false.",
            "zh-Hant": "A⊆B：真；A⊊B：假。"
          }
        }
      ]
    },
    {
      "id": "g10-01-03",
      "chapter": "01",
      "target": {
        "en": "Set operations and universe",
        "zh-Hant": "交並補與全集"
      },
      "title": {
        "en": "Operations select different elements",
        "zh-Hant": "不同運算選出哪些元素？"
      },
      "chapterTitle": {
        "en": "Sets",
        "zh-Hant": "集合"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=sets&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-01-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-01-03/"
      },
      "coreClaim": {
        "en": "The decision for each element is respectively OR, AND, A-and-not-B, or not-A in U. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "逐元素的判準分別是「或」、「且」、「在 A 不在 B」或「全集內不在 A」。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "U={1,…,8}; A={1,…,a}; B={b,…,8}. b=9 means B is empty.",
        "zh-Hant": "U={1,…,8}；A={1,…,a}；B={b,…,8}；b=9 表示 B 為空。"
      },
      "evidence": [
        {
          "label": {
            "en": "Intersection",
            "zh-Hant": "交集"
          },
          "boundary": false,
          "params": {
            "a": 5,
            "b": 3,
            "op": "intersection"
          },
          "observation": {
            "en": "Result = {3, 4, 5}",
            "zh-Hant": "結果 = {3, 4, 5}"
          }
        },
        {
          "label": {
            "en": "Difference",
            "zh-Hant": "差集"
          },
          "boundary": false,
          "params": {
            "a": 5,
            "b": 3,
            "op": "difference"
          },
          "observation": {
            "en": "Result = {1, 2}",
            "zh-Hant": "結果 = {1, 2}"
          }
        },
        {
          "label": {
            "en": "Empty A, complement",
            "zh-Hant": "A 為空時的補集"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "b": 9,
            "op": "complement"
          },
          "observation": {
            "en": "Result = {1, 2, 3, 4, 5, 6, 7, 8}",
            "zh-Hant": "結果 = {1, 2, 3, 4, 5, 6, 7, 8}"
          }
        }
      ]
    },
    {
      "id": "g10-01-04",
      "chapter": "01",
      "target": {
        "en": "Inclusion–exclusion",
        "zh-Hant": "重疊分類與容斥"
      },
      "title": {
        "en": "Can these class counts coexist?",
        "zh-Hant": "這組人數可以同時成立嗎？"
      },
      "chapterTitle": {
        "en": "Sets",
        "zh-Hant": "集合"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=sets&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-01-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-01-04/"
      },
      "coreClaim": {
        "en": "The overlap is counted twice in a+b, so subtract it once. Invalid negative categories signal inconsistent data. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "a+b 把交集計了兩次，須減一次；類別出現負數表示資料矛盾。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "Counts are integers; 0≤overlap≤min(a,b), and a+b−overlap≤N.",
        "zh-Hant": "人數為整數；0≤交集≤min(a,b)，且 a+b−交集≤N。"
      },
      "evidence": [
        {
          "label": {
            "en": "A feasible survey",
            "zh-Hant": "可行調查"
          },
          "boundary": false,
          "params": {
            "total": 40,
            "a": 24,
            "b": 21,
            "overlap": 10
          },
          "observation": {
            "en": "Feasible counts sum to 40.",
            "zh-Hant": "可行類別人數合計 40。"
          }
        },
        {
          "label": {
            "en": "No one chooses neither",
            "zh-Hant": "沒有人兩者皆不選"
          },
          "boundary": true,
          "params": {
            "total": 30,
            "a": 20,
            "b": 18,
            "overlap": 8
          },
          "observation": {
            "en": "Feasible counts sum to 30.",
            "zh-Hant": "可行類別人數合計 30。"
          }
        },
        {
          "label": {
            "en": "Impossible margins",
            "zh-Hant": "矛盾人數"
          },
          "boundary": true,
          "params": {
            "total": 30,
            "a": 20,
            "b": 18,
            "overlap": 5
          },
          "observation": {
            "en": "Impossible counts: a category is negative.",
            "zh-Hant": "人數矛盾：有類別為負數。"
          }
        }
      ]
    },
    {
      "id": "g10-02-01",
      "chapter": "02",
      "target": {
        "en": "Implication and converse",
        "zh-Hant": "正逆命題"
      },
      "title": {
        "en": "A counterexample to an implication",
        "zh-Hant": "命題的反例"
      },
      "chapterTitle": {
        "en": "Mathematical logic",
        "zh-Hant": "常用邏輯用語"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=logic&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-02-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-02-01/"
      },
      "coreClaim": {
        "en": "A counterexample needs p true and q false. A case with p false does not refute the implication. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "反例須 p 真而 q 假；p 假的案例不能推翻此蘊含。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "x and c are real numbers. A displayed test at one x is distinct from the claim for every x.",
        "zh-Hant": "x、c 為實數；某個 x 的檢查不同於對所有 x 的主張。"
      },
      "evidence": [
        {
          "label": {
            "en": "Both true",
            "zh-Hant": "兩者皆真"
          },
          "boundary": false,
          "params": {
            "x": 2,
            "c": 1
          },
          "observation": {
            "en": "This input is not a counterexample to p⇒q.",
            "zh-Hant": "這個輸入不是 p⇒q 的反例。"
          }
        },
        {
          "label": {
            "en": "A refuting witness",
            "zh-Hant": "反例"
          },
          "boundary": true,
          "params": {
            "x": -1,
            "c": -2
          },
          "observation": {
            "en": "This input refutes p⇒q: true premise, false conclusion.",
            "zh-Hant": "此輸入推翻 p⇒q：前件真、後件假。"
          }
        },
        {
          "label": {
            "en": "False premise",
            "zh-Hant": "前件為假"
          },
          "boundary": true,
          "params": {
            "x": -2,
            "c": 1
          },
          "observation": {
            "en": "This input is not a counterexample to p⇒q.",
            "zh-Hant": "這個輸入不是 p⇒q 的反例。"
          }
        }
      ]
    },
    {
      "id": "g10-02-02",
      "chapter": "02",
      "target": {
        "en": "Necessary and sufficient conditions",
        "zh-Hant": "充分必要條件"
      },
      "title": {
        "en": "Necessary or sufficient?",
        "zh-Hant": "必要還是充分？"
      },
      "chapterTitle": {
        "en": "Mathematical logic",
        "zh-Hant": "常用邏輯用語"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=logic&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-02-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-02-02/"
      },
      "coreClaim": {
        "en": "Sufficiency is containment of the p-solution set in the q-solution set; necessity reverses this containment. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "充分性是 p 的解集包含於 q 的解集；必要性則把包含方向反過來。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "Both inequalities are strict and share the real-number universe.",
        "zh-Hant": "兩式均為嚴格不等式，全集同為實數。"
      },
      "evidence": [
        {
          "label": {
            "en": "Sufficient only",
            "zh-Hant": "充分不必要"
          },
          "boundary": false,
          "params": {
            "a": 3,
            "b": 1
          },
          "observation": {
            "en": "Sufficient, not necessary",
            "zh-Hant": "充分不必要"
          }
        },
        {
          "label": {
            "en": "Equivalent",
            "zh-Hant": "充要"
          },
          "boundary": true,
          "params": {
            "a": 2,
            "b": 2
          },
          "observation": {
            "en": "Necessary and sufficient",
            "zh-Hant": "充要"
          }
        },
        {
          "label": {
            "en": "Necessary only",
            "zh-Hant": "必要不充分"
          },
          "boundary": false,
          "params": {
            "a": -1,
            "b": 2
          },
          "observation": {
            "en": "Necessary, not sufficient",
            "zh-Hant": "必要不充分"
          }
        }
      ]
    },
    {
      "id": "g10-02-03",
      "chapter": "02",
      "target": {
        "en": "Universal and existential statements",
        "zh-Hant": "全稱與存在"
      },
      "title": {
        "en": "All or at least one?",
        "zh-Hant": "所有還是至少一個？"
      },
      "chapterTitle": {
        "en": "Mathematical logic",
        "zh-Hant": "常用邏輯用語"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=logic&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-02-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-02-03/"
      },
      "coreClaim": {
        "en": "All requires every allowed input; existence requires one. Enlarging the domain can change these conclusions differently. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "全稱要求每個允許輸入；存在只需一個；擴大定義域可使兩者有不同變化。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The finite domain is {−n,…,n}, n≥1. The alternative domain is R.",
        "zh-Hant": "有限域是 {−n,…,n}，n≥1；另一選項為 R。"
      },
      "evidence": [
        {
          "label": {
            "en": "Existence without universality",
            "zh-Hant": "存在真而全稱假"
          },
          "boundary": false,
          "params": {
            "n": 2,
            "k": 1,
            "strict": "no",
            "domain": "finite"
          },
          "observation": {
            "en": "All: false; exists: true. Domain: finite integers.",
            "zh-Hant": "全稱：假；存在：真。域：有限整數集。"
          }
        },
        {
          "label": {
            "en": "Boundary at zero",
            "zh-Hant": "零的邊界"
          },
          "boundary": true,
          "params": {
            "n": 2,
            "k": 0,
            "strict": "yes",
            "domain": "real"
          },
          "observation": {
            "en": "All: false; exists: true. Domain: R.",
            "zh-Hant": "全稱：假；存在：真。域：R。"
          }
        },
        {
          "label": {
            "en": "Finite domain exhausted",
            "zh-Hant": "有限域沒有見證"
          },
          "boundary": true,
          "params": {
            "n": 2,
            "k": 5,
            "strict": "no",
            "domain": "finite"
          },
          "observation": {
            "en": "All: false; exists: false. Domain: finite integers.",
            "zh-Hant": "全稱：假；存在：假。域：有限整數集。"
          }
        }
      ]
    },
    {
      "id": "g10-02-04",
      "chapter": "02",
      "target": {
        "en": "Negation and counterexamples",
        "zh-Hant": "否定與反例"
      },
      "title": {
        "en": "Negating a quantified statement",
        "zh-Hant": "否定帶量詞命題"
      },
      "chapterTitle": {
        "en": "Mathematical logic",
        "zh-Hant": "常用邏輯用語"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=logic&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-02-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-02-04/"
      },
      "coreClaim": {
        "en": "Negation switches all and exists and replaces > by ≤. Equality belongs to the negation. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "否定要交換全稱與存在，並把 > 改為 ≤；等號屬於否定一方。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The domain is the entire real interval [−n,n], with n>0.",
        "zh-Hant": "定義域為完整實區間 [−n,n]，n>0。"
      },
      "evidence": [
        {
          "label": {
            "en": "Negate all",
            "zh-Hant": "否定全稱"
          },
          "boundary": false,
          "params": {
            "n": 2,
            "k": 1,
            "quantifier": "all"
          },
          "observation": {
            "en": "Negation: ∃x: x² ≤ k; True",
            "zh-Hant": "否定: ∃x: x² ≤ k; 真"
          }
        },
        {
          "label": {
            "en": "Equality is essential",
            "zh-Hant": "等號不可少"
          },
          "boundary": true,
          "params": {
            "n": 2,
            "k": 0,
            "quantifier": "all"
          },
          "observation": {
            "en": "Negation: ∃x: x² ≤ k; True",
            "zh-Hant": "否定: ∃x: x² ≤ k; 真"
          }
        },
        {
          "label": {
            "en": "Negate existence at the maximum",
            "zh-Hant": "最大值處否定存在"
          },
          "boundary": true,
          "params": {
            "n": 2,
            "k": 4,
            "quantifier": "exists"
          },
          "observation": {
            "en": "Negation: ∀x: x² ≤ k; True",
            "zh-Hant": "否定: ∀x: x² ≤ k; 真"
          }
        }
      ]
    },
    {
      "id": "g10-03-01",
      "chapter": "03",
      "target": {
        "en": "Signs between roots",
        "zh-Hant": "兩根之間的符號"
      },
      "title": {
        "en": "Does the middle always solve it?",
        "zh-Hant": "一定取兩根之間嗎？"
      },
      "chapterTitle": {
        "en": "Quadratic inequalities",
        "zh-Hant": "一元二次不等式"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=quadratic-inequality&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-03-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-03-01/"
      },
      "coreClaim": {
        "en": "Two distinct factors change sign at their roots; a negative multiplier reverses the product sign. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "兩個不同因子在零點變號；負乘數使乘積符號反轉。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "Roots are real. a=0 is a constant-zero expression and is classified separately.",
        "zh-Hant": "零點為實數；a=0 時式子恆為零，另作分類。"
      },
      "evidence": [
        {
          "label": {
            "en": "Positive coefficient",
            "zh-Hant": "正首項係數"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "r": -2,
            "s": 3
          },
          "observation": {
            "en": "Solution: [-2, 3]",
            "zh-Hant": "解集: [-2, 3]"
          }
        },
        {
          "label": {
            "en": "Negative coefficient",
            "zh-Hant": "負首項係數"
          },
          "boundary": false,
          "params": {
            "a": -1,
            "r": -2,
            "s": 3
          },
          "observation": {
            "en": "Solution: (−∞, -2] ∪ [3, ∞)",
            "zh-Hant": "解集: (−∞, -2] ∪ [3, ∞)"
          }
        },
        {
          "label": {
            "en": "Zero coefficient",
            "zh-Hant": "零首項係數"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "r": -2,
            "s": 3
          },
          "observation": {
            "en": "Solution: (−∞, ∞)",
            "zh-Hant": "解集: (−∞, ∞)"
          }
        }
      ]
    },
    {
      "id": "g10-03-02",
      "chapter": "03",
      "target": {
        "en": "Repeated and absent real roots",
        "zh-Hant": "重根與無根"
      },
      "title": {
        "en": "How many real zeros?",
        "zh-Hant": "有多少個實零點？"
      },
      "chapterTitle": {
        "en": "Quadratic inequalities",
        "zh-Hant": "一元二次不等式"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=quadratic-inequality&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-03-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-03-02/"
      },
      "coreClaim": {
        "en": "The discriminant counts real zeros for a≠0. If a=0, the expression is linear or constant, so use its own rule. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "a≠0 時判別式決定實零點數；a=0 時變為一次或常數，須另判。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "a,b,c are real numbers; quadratic classification requires a≠0.",
        "zh-Hant": "a、b、c 為實數；二次分類要求 a≠0。"
      },
      "evidence": [
        {
          "label": {
            "en": "Two zeros",
            "zh-Hant": "兩零點"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "b": 0,
            "c": -1
          },
          "observation": {
            "en": "Quadratic: two distinct real roots",
            "zh-Hant": "二次式：兩個不同實根"
          }
        },
        {
          "label": {
            "en": "A repeated root",
            "zh-Hant": "重根"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": -2,
            "c": 1
          },
          "observation": {
            "en": "Quadratic: one repeated real root",
            "zh-Hant": "二次式：一個重實根"
          }
        },
        {
          "label": {
            "en": "No real zero",
            "zh-Hant": "沒有實零點"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": 0,
            "c": 1
          },
          "observation": {
            "en": "Quadratic: no real zero",
            "zh-Hant": "二次式：沒有實零點"
          }
        },
        {
          "label": {
            "en": "Linear degeneration",
            "zh-Hant": "退化為一次式"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "b": 2,
            "c": -2
          },
          "observation": {
            "en": "Linear: one zero",
            "zh-Hant": "一次式：一零點"
          }
        }
      ]
    },
    {
      "id": "g10-03-03",
      "chapter": "03",
      "target": {
        "en": "Strict endpoints",
        "zh-Hant": "嚴格與非嚴格端點"
      },
      "title": {
        "en": "Do the endpoints belong?",
        "zh-Hant": "端點應否包括？"
      },
      "chapterTitle": {
        "en": "Quadratic inequalities",
        "zh-Hant": "一元二次不等式"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=quadratic-inequality&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-03-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-03-03/"
      },
      "coreClaim": {
        "en": "A root gives exactly zero, so it belongs precisely when equality is allowed. Repeated roots need their own sign analysis. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "零點的值恰為零，只在容許等號時納入；重根仍須另作符號分析。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "Use a(x−r)(x−s) with real parameters, including a=0.",
        "zh-Hant": "使用實參數 a(x−r)(x−s)，包括 a=0。"
      },
      "evidence": [
        {
          "label": {
            "en": "Strict inequality",
            "zh-Hant": "嚴格不等式"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "r": 1,
            "s": 4,
            "relation": "lt"
          },
          "observation": {
            "en": "Solution: (1, 4)",
            "zh-Hant": "解集: (1, 4)"
          }
        },
        {
          "label": {
            "en": "Include equality",
            "zh-Hant": "包括等號"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "r": 1,
            "s": 4,
            "relation": "le"
          },
          "observation": {
            "en": "Solution: [1, 4]",
            "zh-Hant": "解集: [1, 4]"
          }
        },
        {
          "label": {
            "en": "Repeated root",
            "zh-Hant": "重根"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "r": 2,
            "s": 2,
            "relation": "lt"
          },
          "observation": {
            "en": "Solution: ∅",
            "zh-Hant": "解集: ∅"
          }
        }
      ]
    },
    {
      "id": "g10-03-04",
      "chapter": "03",
      "target": {
        "en": "Contextual domain",
        "zh-Hant": "情境定義域"
      },
      "title": {
        "en": "A profitable value may be impossible",
        "zh-Hant": "獲利數值也可能不可行"
      },
      "chapterTitle": {
        "en": "Quadratic inequalities",
        "zh-Hant": "一元二次不等式"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=quadratic-inequality&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-03-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-03-04/"
      },
      "coreClaim": {
        "en": "The final choice is the intersection of the sign solution, the contextual interval and, when required, the integers. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "最後選擇是符號解集、情境區間及所需整數集的交集。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The profit expression is a mathematical scenario; d≥1 and the integer restriction is explicit.",
        "zh-Hant": "收益式是自編數學情境；d≥1，整數限制須明示。"
      },
      "evidence": [
        {
          "label": {
            "en": "Continuous scale",
            "zh-Hant": "連續規模"
          },
          "boundary": false,
          "params": {
            "a": -1,
            "r": 2,
            "s": 8,
            "d": 10,
            "integer": "no"
          },
          "observation": {
            "en": "Solution: [2, 8]",
            "zh-Hant": "解集: [2, 8]"
          }
        },
        {
          "label": {
            "en": "Capacity binds",
            "zh-Hant": "容量限制"
          },
          "boundary": false,
          "params": {
            "a": -1,
            "r": 2,
            "s": 8,
            "d": 6,
            "integer": "yes"
          },
          "observation": {
            "en": "Solution: {2, 3, 4, 5, 6}",
            "zh-Hant": "解集: {2, 3, 4, 5, 6}"
          }
        },
        {
          "label": {
            "en": "Repeated root is the only choice",
            "zh-Hant": "重根為唯一選擇"
          },
          "boundary": true,
          "params": {
            "a": -1,
            "r": 3,
            "s": 3,
            "d": 6,
            "integer": "yes"
          },
          "observation": {
            "en": "Solution: {3}",
            "zh-Hant": "解集: {3}"
          }
        }
      ]
    },
    {
      "id": "g10-04-01",
      "chapter": "04",
      "target": {
        "en": "Division identity",
        "zh-Hant": "除法恆等式"
      },
      "title": {
        "en": "What does division preserve?",
        "zh-Hant": "除法保留甚麼關係？"
      },
      "chapterTitle": {
        "en": "Polynomial division and remainders",
        "zh-Hant": "多項式除法與餘式定理"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=polynomial-division&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-04-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-04-01/"
      },
      "coreClaim": {
        "en": "Polynomial division gives f=(x−a)q+r, with constant r. The two plotted curves f and (x−a)q differ vertically by r everywhere. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "多項式除法給出 f=(x−a)q+r，r 為常數；圖上 f 與 (x−a)q 在各處的垂直差均為 r。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The divisor is the nonzero monic polynomial x−a; coefficients are real.",
        "zh-Hant": "除式為非零首一多項式 x−a；係數為實數。"
      },
      "evidence": [
        {
          "label": {
            "en": "Nonzero remainder",
            "zh-Hant": "非零餘式"
          },
          "boundary": false,
          "params": {
            "b": -2,
            "c": 0,
            "d": 3,
            "a": 2
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(2))(x^2)+(3)",
            "zh-Hant": "除法恆等式: f(x)=(x−(2))(x^2)+(3)"
          }
        },
        {
          "label": {
            "en": "An exact factor",
            "zh-Hant": "整除"
          },
          "boundary": true,
          "params": {
            "b": 0,
            "c": -1,
            "d": 0,
            "a": 1
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(1))(x^2 +x)+(0)",
            "zh-Hant": "除法恆等式: f(x)=(x−(1))(x^2 +x)+(0)"
          }
        },
        {
          "label": {
            "en": "Divide by x",
            "zh-Hant": "除以 x"
          },
          "boundary": true,
          "params": {
            "b": 2,
            "c": -3,
            "d": 4,
            "a": 0
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(0))(x^2 +2x −3)+(4)",
            "zh-Hant": "除法恆等式: f(x)=(x−(0))(x^2 +2x −3)+(4)"
          }
        }
      ]
    },
    {
      "id": "g10-04-02",
      "chapter": "04",
      "target": {
        "en": "Synthetic division and missing terms",
        "zh-Hant": "綜合除法與缺項"
      },
      "title": {
        "en": "Where did the missing terms go?",
        "zh-Hant": "缺項去了哪裏？"
      },
      "chapterTitle": {
        "en": "Polynomial division and remainders",
        "zh-Hant": "多項式除法與餘式定理"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=polynomial-division&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-04-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-04-02/"
      },
      "coreClaim": {
        "en": "Missing x³ and x terms have coefficient zero. Keeping their slots preserves the powers during multiply-and-add steps. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "缺少的 x³、x 項係數為零；保留位置才能在乘加步驟中保持次方對齊。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The polynomial has degree four even when b or c is zero. Division is by x−a.",
        "zh-Hant": "即使 b 或 c 為零，多項式仍為四次；除式是 x−a。"
      },
      "evidence": [
        {
          "label": {
            "en": "Two internal zero slots",
            "zh-Hant": "兩個內部零格"
          },
          "boundary": false,
          "params": {
            "b": -3,
            "c": 2,
            "a": 1
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(1))(x^3 +x^2 −2x −2)+(0)",
            "zh-Hant": "除法恆等式: f(x)=(x−(1))(x^3 +x^2 −2x −2)+(0)"
          }
        },
        {
          "label": {
            "en": "Only x⁴ remains",
            "zh-Hant": "只剩 x⁴"
          },
          "boundary": true,
          "params": {
            "b": 0,
            "c": 0,
            "a": 2
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(2))(x^3 +2x^2 +4x +8)+(16)",
            "zh-Hant": "除法恆等式: f(x)=(x−(2))(x^3 +2x^2 +4x +8)+(16)"
          }
        },
        {
          "label": {
            "en": "Zero multiplier",
            "zh-Hant": "乘數為零"
          },
          "boundary": true,
          "params": {
            "b": 2,
            "c": -1,
            "a": 0
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(0))(x^3 +2x)+(-1)",
            "zh-Hant": "除法恆等式: f(x)=(x−(0))(x^3 +2x)+(-1)"
          }
        }
      ]
    },
    {
      "id": "g10-04-03",
      "chapter": "04",
      "target": {
        "en": "Remainder theorem",
        "zh-Hant": "餘式定理"
      },
      "title": {
        "en": "Read a remainder from one point",
        "zh-Hant": "由一個點讀出餘式"
      },
      "chapterTitle": {
        "en": "Polynomial division and remainders",
        "zh-Hant": "多項式除法與餘式定理"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=polynomial-division&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-04-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-04-03/"
      },
      "coreClaim": {
        "en": "At x=a the quotient term is multiplied by zero, leaving f(a)=r. The sign of a comes from the complete divisor. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "x=a 時商式項乘零，只剩 f(a)=r；a 的正負須由完整除式讀出。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The divisor must be x−a; division by x+a uses input −a.",
        "zh-Hant": "除式為 x−a；若是 x+a，則代入 −a。"
      },
      "evidence": [
        {
          "label": {
            "en": "Positive a",
            "zh-Hant": "正 a"
          },
          "boundary": false,
          "params": {
            "b": -2,
            "c": 3,
            "a": 2
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(2))(x^2 +2x +2)+(7)",
            "zh-Hant": "除法恆等式: f(x)=(x−(2))(x^2 +2x +2)+(7)"
          }
        },
        {
          "label": {
            "en": "Divisor x+2",
            "zh-Hant": "除式 x+2"
          },
          "boundary": false,
          "params": {
            "b": 0,
            "c": 1,
            "a": -2
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(-2))(x^2 −2x +4)+(-7)",
            "zh-Hant": "除法恆等式: f(x)=(x−(-2))(x^2 −2x +4)+(-7)"
          }
        },
        {
          "label": {
            "en": "Zero remainder",
            "zh-Hant": "零餘式"
          },
          "boundary": true,
          "params": {
            "b": -1,
            "c": 0,
            "a": 1
          },
          "observation": {
            "en": "Division identity: f(x)=(x−(1))(x^2 +x)+(0)",
            "zh-Hant": "除法恆等式: f(x)=(x−(1))(x^2 +x)+(0)"
          }
        }
      ]
    },
    {
      "id": "g10-04-04",
      "chapter": "04",
      "target": {
        "en": "Factor theorem",
        "zh-Hant": "因式定理"
      },
      "title": {
        "en": "Design a factor by changing one coefficient",
        "zh-Hant": "改一個係數設計因式"
      },
      "chapterTitle": {
        "en": "Polynomial division and remainders",
        "zh-Hant": "多項式除法與餘式定理"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=polynomial-division&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-04-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-04-04/"
      },
      "coreClaim": {
        "en": "x−a is a factor iff f(a)=0, so c=−a³−ka. Adjusting c translates the graph vertically. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "x−a 為因式當且僅當 f(a)=0，因此 c=−a³−ka；改 c 使圖像垂直平移。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The leading coefficient stays one; only the constant is repaired.",
        "zh-Hant": "首項係數保持一，只修補常數項。"
      },
      "evidence": [
        {
          "label": {
            "en": "A repair is needed",
            "zh-Hant": "需要修補"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "k": 2,
            "c": 0
          },
          "observation": {
            "en": "Required c=-3; currently a factor: false.",
            "zh-Hant": "所需 c=-3；目前是因式：假。"
          }
        },
        {
          "label": {
            "en": "Already a factor",
            "zh-Hant": "已經是因式"
          },
          "boundary": true,
          "params": {
            "a": -1,
            "k": 2,
            "c": 3
          },
          "observation": {
            "en": "Required c=3; currently a factor: true.",
            "zh-Hant": "所需 c=3；目前是因式：真。"
          }
        },
        {
          "label": {
            "en": "Root at zero",
            "zh-Hant": "零為根"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "k": -2,
            "c": 0
          },
          "observation": {
            "en": "Required c=0; currently a factor: true.",
            "zh-Hant": "所需 c=0；目前是因式：真。"
          }
        }
      ]
    },
    {
      "id": "g10-05-01",
      "chapter": "05",
      "target": {
        "en": "Function mappings",
        "zh-Hant": "對應是否函數"
      },
      "title": {
        "en": "Repair one mapping arrow",
        "zh-Hant": "修補一條對應箭頭"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-01/"
      },
      "coreClaim": {
        "en": "A function gives every allowed input exactly one output. Many-to-one is permitted; a missing arrow or two different outputs from one input violates the rule.",
        "zh-Hant": "函數給每個允許輸入恰好一個輸出。多對一是允許的；漏配或同一輸入有兩個不同輸出均違反條件。"
      },
      "conditions": {
        "en": "The input set is {1,2,3}. Count distinct outputs from each input, not arrows arriving at an output.",
        "zh-Hant": "輸入集合為 {1,2,3}；逐個數其不同輸出，不是數到達某輸出的箭頭。"
      },
      "evidence": [
        {
          "label": {
            "en": "Many-to-one",
            "zh-Hant": "多對一"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "A = {1, 2, 3}; B = {5, 7}; attained range = {5, 7}.",
            "zh-Hant": "A = {1, 2, 3}；B = {5, 7}；實際值域 = {5, 7}。"
          }
        },
        {
          "label": {
            "en": "Missing input 3",
            "zh-Hant": "輸入 3 漏配"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "A = {1, 2, 3}; B = {5, 7}; attained range = {5}.",
            "zh-Hant": "A = {1, 2, 3}；B = {5, 7}；實際值域 = {5}。"
          }
        },
        {
          "label": {
            "en": "Input 2 has two outputs",
            "zh-Hant": "輸入 2 有兩個輸出"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "A = {1, 2, 3}; B = {5, 7}; attained range = {5, 7}.",
            "zh-Hant": "A = {1, 2, 3}；B = {5, 7}；實際值域 = {5, 7}。"
          }
        },
        {
          "label": {
            "en": "Constant output 8",
            "zh-Hant": "固定輸出 8"
          },
          "boundary": false,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "A = {1, 2, 3}; B = {6, 8, 10}; attained range = {8}.",
            "zh-Hant": "A = {1, 2, 3}；B = {6, 8, 10}；實際值域 = {8}。"
          }
        }
      ]
    },
    {
      "id": "g10-05-02",
      "chapter": "05",
      "target": {
        "en": "Three elements of a function",
        "zh-Hant": "函數三要素"
      },
      "title": {
        "en": "Same formula: the same function?",
        "zh-Hant": "公式相同就是同一函數？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-02/"
      },
      "coreClaim": {
        "en": "Formula arithmetic and defined function output are different questions. For x²+1, 0.5 gives arithmetic 1.25, but this is a function value only when 0.5 belongs to the domain.",
        "zh-Hant": "公式算術與函數是否有定義是兩個問題。x²＋1 在 0.5 算得 1.25，但只有定義域允許 0.5 時才是函數值。"
      },
      "conditions": {
        "en": "Compare both the rule and the complete input domain. The range contains attained outputs only.",
        "zh-Hant": "須比較對應法則與完整定義域；值域只包括實際取得的輸出。"
      },
      "evidence": [
        {
          "label": {
            "en": "0.5 in the finite domain",
            "zh-Hant": "有限定義域測 0.5"
          },
          "boundary": true,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Same fixed rule x² + 1. First range: {1, 2, 5}. Second range: [1, 5].",
            "zh-Hant": "同一固定法則 x² + 1。第一個值域：{1, 2, 5}；第二個值域：[1, 5]。"
          }
        },
        {
          "label": {
            "en": "0.5 in the interval",
            "zh-Hant": "區間定義域測 0.5"
          },
          "boundary": false,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Same fixed rule x² + 1. First range: {1, 2, 5}. Second range: [1, 5].",
            "zh-Hant": "同一固定法則 x² + 1。第一個值域：{1, 2, 5}；第二個值域：[1, 5]。"
          }
        },
        {
          "label": {
            "en": "−1 in the first paper domain",
            "zh-Hant": "紙本第一個定義域測 −1"
          },
          "boundary": false,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Same fixed rule x². First range: {0, 1}. Second range: {0, 1}.",
            "zh-Hant": "同一固定法則 x²。第一個值域：{0, 1}；第二個值域：{0, 1}。"
          }
        },
        {
          "label": {
            "en": "−1 in the second paper domain",
            "zh-Hant": "紙本第二個定義域測 −1"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Same fixed rule x². First range: {0, 1}. Second range: {0, 1}.",
            "zh-Hant": "同一固定法則 x²。第一個值域：{0, 1}；第二個值域：{0, 1}。"
          }
        }
      ]
    },
    {
      "id": "g10-05-03",
      "chapter": "05",
      "target": {
        "en": "Intervals",
        "zh-Hant": "區間"
      },
      "title": {
        "en": "Which endpoint belongs?",
        "zh-Hant": "哪個端點屬於集合？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-03/"
      },
      "coreClaim": {
        "en": "A real square root permits zero radicand, giving x ≥ 3. A denominator cannot be zero, giving x ≠ 2. In [0,4), 0 is included and 4 excluded.",
        "zh-Hant": "實數平方根容許被開方數為零，故 x ≥ 3；分母不能為零，故 x ≠ 2。[0,4) 包括 0 而不包括 4。"
      },
      "conditions": {
        "en": "Work over real inputs. Infinity is a direction, never an included endpoint; integers and real intervals are different domains.",
        "zh-Hant": "使用實數輸入；無窮大表示方向，不是可包括的端點；整數集合與實數區間不同。"
      },
      "evidence": [
        {
          "label": {
            "en": "Included endpoint 0",
            "zh-Hant": "包括的端點 0"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Painted set: [0, 4). Infinity is a direction and always uses an open bracket.",
            "zh-Hant": "已塗集合：[0, 4)。無窮大表示方向，必須用圓括號。"
          }
        },
        {
          "label": {
            "en": "Excluded endpoint 4",
            "zh-Hant": "排除的端點 4"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Painted set: [0, 4). Infinity is a direction and always uses an open bracket.",
            "zh-Hant": "已塗集合：[0, 4)。無窮大表示方向，必須用圓括號。"
          }
        },
        {
          "label": {
            "en": "Reciprocal at 2",
            "zh-Hant": "倒數在 2"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Painted set: (−∞, 2) ∪ (2, ∞). Infinity is a direction and always uses an open bracket.",
            "zh-Hant": "已塗集合：(−∞, 2) ∪ (2, ∞)。無窮大表示方向，必須用圓括號。"
          }
        },
        {
          "label": {
            "en": "Square root at 3",
            "zh-Hant": "平方根在 3"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Painted set: [3, ∞). Infinity is a direction and always uses an open bracket.",
            "zh-Hant": "已塗集合：[3, ∞)。無窮大表示方向，必須用圓括號。"
          }
        }
      ]
    },
    {
      "id": "g10-05-04",
      "chapter": "05",
      "target": {
        "en": "Domain and range",
        "zh-Hant": "定義域與值域"
      },
      "title": {
        "en": "Keep every restriction after simplification",
        "zh-Hant": "化簡後仍要保留每項限制"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-04/"
      },
      "coreClaim": {
        "en": "For √(x+1)/(x−2), take x ≥ −1 and x ≠ 2 together. For (x²−1)/(x−1), cancellation gives x+1 only on x ≠ 1. The original graph retains its hole.",
        "zh-Hant": "√(x+1)/(x−2) 須同時滿足 x ≥ −1 及 x ≠ 2；(x²−1)/(x−1) 只在 x ≠ 1 時可約成 x＋1，原圖仍有空缺。"
      },
      "conditions": {
        "en": "The original formula fixes the domain. Finite output sets are not filled in between their elements.",
        "zh-Hant": "定義域由原公式決定；有限值域的元素之間不會自動補滿。"
      },
      "evidence": [
        {
          "label": {
            "en": "Intersect two restrictions",
            "zh-Hant": "兩項限制取交集"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Overlaid masks: 2/2. Current intersection accepts x = 0.",
            "zh-Hant": "已疊加遮罩：2/2。目前交集接受 x = 0。"
          }
        },
        {
          "label": {
            "en": "Root in the denominator at 1",
            "zh-Hant": "根式分母在 1"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Overlaid masks: 2/2. Current intersection rejects x = 1.",
            "zh-Hant": "已疊加遮罩：2/2。目前交集拒絕 x = 1。"
          }
        },
        {
          "label": {
            "en": "Cancel at the hole x=1",
            "zh-Hant": "在空缺 x＝1 約分"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Overlaid masks: 1/1. Current intersection rejects x = 1. Simplified formula: x + 1, retaining original exclusion x ≠ 1.",
            "zh-Hant": "已疊加遮罩：1/1。目前交集拒絕 x = 1。 化簡式：x + 1，保留原限制 x ≠ 1。"
          }
        },
        {
          "label": {
            "en": "Finite range",
            "zh-Hant": "有限值域"
          },
          "boundary": false,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Learner output set: {0, 1, 4}. x = 2; output = —.",
            "zh-Hant": "學生所選輸出集合：{0, 1, 4}。x = 2；輸出 = —。"
          }
        },
        {
          "label": {
            "en": "Interval range",
            "zh-Hant": "區間值域"
          },
          "boundary": false,
          "params": {
            "scenario": "4"
          },
          "observation": {
            "en": "Learner output set: [0, 4]. x = 2; output = —.",
            "zh-Hant": "學生所選輸出集合：[0, 4]。x = 2；輸出 = —。"
          }
        }
      ]
    },
    {
      "id": "g10-05-05",
      "chapter": "05",
      "target": {
        "en": "Tables, graphs and formulas",
        "zh-Hant": "表格圖像公式"
      },
      "title": {
        "en": "Should the points be joined?",
        "zh-Hant": "這些點應該連線嗎？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-05/"
      },
      "coreClaim": {
        "en": "On the real interval [0,4], the affine model gives a line segment. On {0,1,2,3,4}, it gives five separate points. The point (2.5,9) belongs only to the time model.",
        "zh-Hant": "實數區間 [0,4] 的一次模型給線段；{0,1,2,3,4} 給五個離散點。(2.5,9) 只屬於時數模型。"
      },
      "conditions": {
        "en": "Linearity is a stated model assumption. A finite table alone does not establish a unique formula.",
        "zh-Hant": "直線關係是已指定的模型假設；有限表格本身不能確立唯一公式。"
      },
      "evidence": [
        {
          "label": {
            "en": "Continuous hours, joined",
            "zh-Hant": "連續時數並連線"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Fixed domain: [0, 4]. Learner placements: 5/5. Joining: on. The straight-line model is given.",
            "zh-Hant": "固定定義域：[0, 4]。學生已放置：5/5。連線：開啟。題目已給定直線模型。"
          }
        },
        {
          "label": {
            "en": "Whole items, joining is misleading",
            "zh-Hant": "整數件數，連線會誤導"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Fixed domain: {0, 1, 2, 3, 4}. Learner placements: 5/5. Joining: on. The straight-line model is given.",
            "zh-Hant": "固定定義域：{0, 1, 2, 3, 4}。學生已放置：5/5。連線：開啟。題目已給定直線模型。"
          }
        },
        {
          "label": {
            "en": "Whole items, separate points",
            "zh-Hant": "整數件數，保留離散點"
          },
          "boundary": false,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Fixed domain: {0, 1, 2, 3, 4}. Learner placements: 5/5. Joining: off. The straight-line model is given.",
            "zh-Hant": "固定定義域：{0, 1, 2, 3, 4}。學生已放置：5/5。連線：關閉。題目已給定直線模型。"
          }
        }
      ]
    },
    {
      "id": "g10-05-06",
      "chapter": "05",
      "target": {
        "en": "Piecewise boundaries",
        "zh-Hant": "分段邊界"
      },
      "title": {
        "en": "A missing boundary or two outputs?",
        "zh-Hant": "分界漏配還是一個輸入有兩個輸出？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-06/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-06/"
      },
      "coreClaim": {
        "en": "Both open endpoints omit the boundary. Both closed endpoints are harmless if both formulas give the same value there. An overlap giving two distinct values violates uniqueness.",
        "zh-Hant": "兩端都開會漏去分界；兩端都閉但公式值相同仍可定義函數；重疊處有兩個不同值才違反唯一性。"
      },
      "conditions": {
        "en": "For the rental model, the values 6 and 6+4(t−1) agree at t=1. Equality at one point does not justify an interval of overlap.",
        "zh-Hant": "租借模型的 6 與 6＋4(t−1) 在 t＝1 相同；單點相等不代表整段重疊都可行。"
      },
      "evidence": [
        {
          "label": {
            "en": "Source boundary",
            "zh-Hant": "原分界"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Fixed total domain: 0 < t ≤ 4. First branch 0 < t ≤ 1; second branch t > 1, t ≤ 4. At 1: 1 applicable branch(es), 1 distinct output(s).",
            "zh-Hant": "固定總定義域：0 < t ≤ 4。第一段 0 < t ≤ 1；第二段 t > 1，t ≤ 4。輸入 1：1 段適用，1 個不同輸出。"
          }
        },
        {
          "label": {
            "en": "Both ends open",
            "zh-Hant": "兩端均開"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Fixed total domain: 0 < t ≤ 4. First branch 0 < t < 1; second branch t > 1, t ≤ 4. At 1: 0 applicable branch(es), 0 distinct output(s).",
            "zh-Hant": "固定總定義域：0 < t ≤ 4。第一段 0 < t < 1；第二段 t > 1，t ≤ 4。輸入 1：0 段適用，0 個不同輸出。"
          }
        },
        {
          "label": {
            "en": "Both ends closed, same output",
            "zh-Hant": "兩端均閉，輸出相同"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Fixed total domain: 0 < t ≤ 4. First branch 0 < t ≤ 1; second branch t ≥ 1, t ≤ 4. At 1: 2 applicable branch(es), 1 distinct output(s).",
            "zh-Hant": "固定總定義域：0 < t ≤ 4。第一段 0 < t ≤ 1；第二段 t ≥ 1，t ≤ 4。輸入 1：2 段適用，1 個不同輸出。"
          }
        },
        {
          "label": {
            "en": "Overlap with distinct outputs",
            "zh-Hant": "重疊處輸出不同"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Fixed total domain: 0 < t ≤ 4. First branch 0 < t ≤ 1.5; second branch t ≥ 1, t ≤ 4. At 1.25: 2 applicable branch(es), 2 distinct output(s).",
            "zh-Hant": "固定總定義域：0 < t ≤ 4。第一段 0 < t ≤ 1.5；第二段 t ≥ 1，t ≤ 4。輸入 1.25：2 段適用，2 個不同輸出。"
          }
        }
      ]
    },
    {
      "id": "g10-05-07",
      "chapter": "05",
      "target": {
        "en": "Composition order",
        "zh-Hant": "複合次序"
      },
      "title": {
        "en": "Swap two function machines",
        "zh-Hant": "交換兩部函數機器"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-07/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-07/"
      },
      "coreClaim": {
        "en": "The orders give 2x+3 and 2(x+3)=2x+6, differing by 3. With square roots, order can also change which inputs are allowed.",
        "zh-Hant": "兩次序給 2x＋3 及 2(x＋3)＝2x＋6，相差 3；有平方根時，次序亦可改變允許輸入。"
      },
      "conditions": {
        "en": "Every intermediate result must belong to the next machine’s domain. Stop when a stage is undefined.",
        "zh-Hant": "每個中間結果都須屬於下一部機器的定義域；某步未定義時必須停止。"
      },
      "evidence": [
        {
          "label": {
            "en": "Double, then add",
            "zh-Hant": "先乘 2 再加 3"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Input 2; order: f(u) = 2u → g(u) = u + 3.",
            "zh-Hant": "輸入 2；次序：f(u) = 2u → g(u) = u + 3。"
          }
        },
        {
          "label": {
            "en": "Add, then double",
            "zh-Hant": "先加 3 再乘 2"
          },
          "boundary": false,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Input 2; order: g(u) = u + 3 → f(u) = 2u.",
            "zh-Hant": "輸入 2；次序：g(u) = u + 3 → f(u) = 2u。"
          }
        },
        {
          "label": {
            "en": "Square-root pair at 0",
            "zh-Hant": "根式對在 0"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Input 0; order: g(u) = u − 1 → f(u) = √u.",
            "zh-Hant": "輸入 0；次序：g(u) = u − 1 → f(u) = √u。"
          }
        },
        {
          "label": {
            "en": "Square-root pair at 1",
            "zh-Hant": "根式對在 1"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Input 1; order: g(u) = u − 1 → f(u) = √u.",
            "zh-Hant": "輸入 1；次序：g(u) = u − 1 → f(u) = √u。"
          }
        }
      ]
    },
    {
      "id": "g10-05-08",
      "chapter": "05",
      "target": {
        "en": "Attained extrema",
        "zh-Hant": "最值"
      },
      "title": {
        "en": "Can the largest value be reached?",
        "zh-Hant": "最大值真的取得了嗎？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-08/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-08/"
      },
      "coreClaim": {
        "en": "On (0,2), every allowed x has a larger allowed input (x+2)/2. Thus 2 is a bound but no maximum is attained. On (0,2], the maximum is 2.",
        "zh-Hant": "在 (0,2)，每個允許的 x 都有較大的允許輸入 (x＋2)/2，故 2 是上界但不是取得的最大值；在 (0,2]，最大值為 2。"
      },
      "conditions": {
        "en": "Extrema depend on the complete stated domain; local peaks or a finite graph window do not establish global extrema.",
        "zh-Hant": "最值依賴完整的指定定義域；局部峰谷或有限圖窗不能確立全域最值。"
      },
      "evidence": [
        {
          "label": {
            "en": "Water: local versus global",
            "zh-Hant": "水位：局部與全域"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Domain [0, 6]. Cursor input 2: output 5.",
            "zh-Hant": "定義域 [0, 6]。游標輸入 2：輸出 5。"
          }
        },
        {
          "label": {
            "en": "Open interval: build a larger value",
            "zh-Hant": "開區間：構造更大值"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Domain (0, 2). Cursor input 1.75: output 1.75.",
            "zh-Hant": "定義域 (0, 2)。游標輸入 1.75：輸出 1.75。"
          }
        },
        {
          "label": {
            "en": "Close the right endpoint",
            "zh-Hant": "包括右端點"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Domain (0, 2]. Cursor input 1: output 1.",
            "zh-Hant": "定義域 (0, 2]。游標輸入 1：輸出 1。"
          }
        }
      ]
    },
    {
      "id": "g10-05-09",
      "chapter": "05",
      "target": {
        "en": "Monotonicity",
        "zh-Hant": "單調性"
      },
      "title": {
        "en": "Decreasing on each piece or the whole domain?",
        "zh-Hant": "各段遞減等於整個定義域遞減嗎？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-09/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-09/"
      },
      "coreClaim": {
        "en": "1/x decreases separately on (−∞,0) and (0,∞). Across zero, −1<1 but f(−1)=−1<f(1)=1, which refutes decrease on the whole domain.",
        "zh-Hant": "1/x 分別在 (−∞,0) 及 (0,∞) 遞減；跨零時 −1<1 但 f(−1)＝−1<f(1)＝1，反駁整個定義域遞減。"
      },
      "conditions": {
        "en": "For strict decrease, every allowed pair x₁<x₂ must satisfy f(x₁)>f(x₂). x=0 is not allowed.",
        "zh-Hant": "嚴格遞減要求每對允許的 x₁<x₂ 都有 f(x₁)>f(x₂)；x＝0 不允許。"
      },
      "evidence": [
        {
          "label": {
            "en": "Both inputs positive",
            "zh-Hant": "兩輸入為正"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "x₁ = 1, x₂ = 2. f(x₂) − f(x₁) = -0.5.",
            "zh-Hant": "x₁ = 1，x₂ = 2。f(x₂)−f(x₁) = -0.5。"
          }
        },
        {
          "label": {
            "en": "Both inputs negative",
            "zh-Hant": "兩輸入為負"
          },
          "boundary": false,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "x₁ = -2, x₂ = -1. f(x₂) − f(x₁) = -0.5.",
            "zh-Hant": "x₁ = -2，x₂ = -1。f(x₂)−f(x₁) = -0.5。"
          }
        },
        {
          "label": {
            "en": "Cross zero",
            "zh-Hant": "跨越零"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "x₁ = -1, x₂ = 1. f(x₂) − f(x₁) = 2.",
            "zh-Hant": "x₁ = -1，x₂ = 1。f(x₂)−f(x₁) = 2。"
          }
        },
        {
          "label": {
            "en": "Forbidden input zero",
            "zh-Hant": "不允許的輸入零"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "x₁ = 0, x₂ = 2. One or both inputs are outside the domain.",
            "zh-Hant": "x₁ = 0，x₂ = 2。至少一個輸入不屬於定義域。"
          }
        }
      ]
    },
    {
      "id": "g10-05-10",
      "chapter": "05",
      "target": {
        "en": "Parity",
        "zh-Hant": "奇偶性"
      },
      "title": {
        "en": "Symmetry needs the matching input",
        "zh-Hant": "對稱需要配對輸入"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-10/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-10/"
      },
      "coreClaim": {
        "en": "Odd/even definitions require a domain symmetric about zero. x² on [0,3] fails that requirement. On a symmetric domain, x² is even, x³ odd, and the zero function both.",
        "zh-Hant": "奇偶性的定義要求定義域關於零對稱；[0,3] 上的 x² 不符合。對稱定義域上，x² 為偶、x³ 為奇，零函數兩者皆是。"
      },
      "conditions": {
        "en": "Check domain symmetry before the identities f(−x)=f(x) or f(−x)=−f(x).",
        "zh-Hant": "檢查 f(−x)＝f(x) 或 f(−x)＝−f(x) 前，先檢查定義域對稱。"
      },
      "evidence": [
        {
          "label": {
            "en": "Square on symmetric domain",
            "zh-Hant": "對稱定義域上的平方"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Domain ℝ. f(2) = 4; actual f(-2) = 4.",
            "zh-Hant": "定義域 ℝ。f(2) = 4；實際 f(-2) = 4。"
          }
        },
        {
          "label": {
            "en": "Square on [0,3]",
            "zh-Hant": "[0,3] 上的平方"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Domain [0, 3]. f(2) = 4; actual f(-2) = Undefined.",
            "zh-Hant": "定義域 [0, 3]。f(2) = 4；實際 f(-2) = 未定義。"
          }
        },
        {
          "label": {
            "en": "Shifted line",
            "zh-Hant": "平移直線"
          },
          "boundary": true,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Domain ℝ. f(2) = 3; actual f(-2) = -1.",
            "zh-Hant": "定義域 ℝ。f(2) = 3；實際 f(-2) = -1。"
          }
        },
        {
          "label": {
            "en": "Zero function",
            "zh-Hant": "零函數"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Domain ℝ. f(2) = 0; actual f(-2) = 0.",
            "zh-Hant": "定義域 ℝ。f(2) = 0；實際 f(-2) = 0。"
          }
        }
      ]
    },
    {
      "id": "g10-05-11",
      "chapter": "05",
      "target": {
        "en": "Basic power functions",
        "zh-Hant": "基本冪函數"
      },
      "title": {
        "en": "Five powers, different forbidden inputs",
        "zh-Hant": "五種冪函數，不同的禁用輸入"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-11/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-11/"
      },
      "coreClaim": {
        "en": "Real √x requires x≥0; 1/x requires x≠0. The integer positive powers x, x² and x³ accept every real input.",
        "zh-Hant": "實數 √x 要求 x≥0；1/x 要求 x≠0；正整數冪 x、x²、x³ 接受每個實數。"
      },
      "conditions": {
        "en": "These are the five specified real-valued rules, not a claim that all real powers have the same domain.",
        "zh-Hant": "此處只討論指定的五種實函數，並非所有實指數冪都有相同定義域。"
      },
      "evidence": [
        {
          "label": {
            "en": "Negative versus positive input",
            "zh-Hant": "負與正輸入"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Probe a = -1; b = 2. 0/5 cards placed. The window shows only part of each graph.",
            "zh-Hant": "探針 a＝-1；b＝2。已放置 0/5 張卡；圖窗只顯示各圖像的一部分。"
          }
        },
        {
          "label": {
            "en": "Zero versus negative input",
            "zh-Hant": "零與負輸入"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Probe a = 0; b = -1. 0/5 cards placed. The window shows only part of each graph.",
            "zh-Hant": "探針 a＝0；b＝-1。已放置 0/5 張卡；圖窗只顯示各圖像的一部分。"
          }
        },
        {
          "label": {
            "en": "Positive fractional input",
            "zh-Hant": "正分數輸入"
          },
          "boundary": false,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Probe a = 0.25; b = 2. 0/5 cards placed. The window shows only part of each graph.",
            "zh-Hant": "探針 a＝0.25；b＝2。已放置 0/5 張卡；圖窗只顯示各圖像的一部分。"
          }
        }
      ]
    },
    {
      "id": "g10-05-12",
      "chapter": "05",
      "target": {
        "en": "Power comparisons",
        "zh-Hant": "冪函數比較"
      },
      "title": {
        "en": "When does the larger exponent give less?",
        "zh-Hant": "較大指數何時反而較小？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-12/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-12/"
      },
      "coreClaim": {
        "en": "x³−x²=x²(x−1). Since x²>0 for x>0, the sign is determined by x−1: less below 1, equal at 1, greater above 1.",
        "zh-Hant": "x³−x²＝x²(x−1)；x>0 時 x²>0，故符號由 x−1 決定：小於 1 時較小，等於 1 時相同，大於 1 時較大。"
      },
      "conditions": {
        "en": "The comparison here assumes x>0 and compares powers of the same base. Comparing two bases is a different question.",
        "zh-Hant": "此比較假設 x>0，並使用同一底數；比較不同底數是另一個問題。"
      },
      "evidence": [
        {
          "label": {
            "en": "Base 0.5",
            "zh-Hant": "底數 0.5"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Positive inputs only. x² = 0.25; x³ = 0.125; difference = 0.125.",
            "zh-Hant": "只使用正輸入。x²＝0.25；x³＝0.125；差＝0.125。"
          }
        },
        {
          "label": {
            "en": "Boundary base 1",
            "zh-Hant": "邊界底數 1"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Positive inputs only. x² = 1; x³ = 1; difference = 0.",
            "zh-Hant": "只使用正輸入。x²＝1；x³＝1；差＝0。"
          }
        },
        {
          "label": {
            "en": "Base 2",
            "zh-Hant": "底數 2"
          },
          "boundary": false,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Positive inputs only. x² = 4; x³ = 8; difference = -4.",
            "zh-Hant": "只使用正輸入。x²＝4；x³＝8；差＝-4。"
          }
        }
      ]
    },
    {
      "id": "g10-05-13",
      "chapter": "05",
      "target": {
        "en": "Substitution",
        "zh-Hant": "換元"
      },
      "title": {
        "en": "Undo the input rule in the right order",
        "zh-Hant": "用正確次序還原輸入"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-13/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-13/"
      },
      "coreClaim": {
        "en": "Set u=2x+1, so x=(u−1)/2. Substitution yields f(u)=6(u−1)/2+5=3u+2. Undoing the input is not inverting f.",
        "zh-Hant": "設 u＝2x＋1，則 x＝(u−1)/2；代入得 f(u)＝6(u−1)/2＋5＝3u＋2。還原輸入不等於求 f 的反函數。"
      },
      "conditions": {
        "en": "The affine input rule is invertible over real numbers. In the discount context, eligibility is checked using the original price.",
        "zh-Hant": "一次輸入規則在實數上可逆；折扣情境的資格按原價判定。"
      },
      "evidence": [
        {
          "label": {
            "en": "Subtract 1, then halve",
            "zh-Hant": "先減 1 再除 2"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "u = 3. Your cards propose how to obtain x from the whole input 2x + 1.",
            "zh-Hant": "u＝3。你排列的卡片表示由完整輸入 2x＋1 求 x 的構想。"
          }
        },
        {
          "label": {
            "en": "Wrong inverse order",
            "zh-Hant": "錯誤的還原次序"
          },
          "boundary": true,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "u = 3. Your cards propose how to obtain x from the whole input 2x + 1.",
            "zh-Hant": "u＝3。你排列的卡片表示由完整輸入 2x＋1 求 x 的構想。"
          }
        },
        {
          "label": {
            "en": "Discount then subtract",
            "zh-Hant": "九折再減 20"
          },
          "boundary": false,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Original price = 100. Original-price eligibility met.",
            "zh-Hant": "原價＝100。符合原價資格。"
          }
        },
        {
          "label": {
            "en": "Reverse discount order",
            "zh-Hant": "交換折扣次序"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Original price = 100. Original-price eligibility met.",
            "zh-Hant": "原價＝100。符合原價資格。"
          }
        }
      ]
    },
    {
      "id": "g10-05-14",
      "chapter": "05",
      "target": {
        "en": "Model choice",
        "zh-Hant": "模型選擇"
      },
      "title": {
        "en": "Cheapest at this quantity or most within budget?",
        "zh-Hant": "指定冊數最便宜，還是預算內印最多？"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-14/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-14/"
      },
      "coreClaim": {
        "en": "P(n)=20+3n. Q(n)=5n for n≤20 and 2n+60 for n>20. Ties occur at integer counts 10 and 40. With budget 160, P allows 46 booklets and Q allows 50.",
        "zh-Hant": "P(n)＝20＋3n；Q 在 n≤20 為 5n，n>20 為 2n＋60。整數同價點為 10、40；預算 160 時 P 可印 46 冊、Q 可印 50 冊。"
      },
      "conditions": {
        "en": "Printing counts are integers 1–80. Rental time uses 0<t≤40 and different cost rules; t=0 is excluded.",
        "zh-Hant": "印刷冊數為 1 至 80 的整數；租借時間使用 0<t≤40 及不同費用規則，排除 t＝0。"
      },
      "evidence": [
        {
          "label": {
            "en": "Tie at 10 booklets",
            "zh-Hant": "10 冊同價"
          },
          "boundary": false,
          "params": {
            "scenario": "0"
          },
          "observation": {
            "en": "Integer quantities only. Budget 200: P permits 60 booklets; Q permits 70. P = 50; Q = 50; P − Q = 0",
            "zh-Hant": "只限整數冊數。預算 200：P 可印 60 冊；Q 可印 70 冊。 P = 50; Q = 50; P − Q = 0"
          }
        },
        {
          "label": {
            "en": "30 booklets",
            "zh-Hant": "30 冊"
          },
          "boundary": false,
          "params": {
            "scenario": "1"
          },
          "observation": {
            "en": "Integer quantities only. Budget 200: P permits 60 booklets; Q permits 70. P = 110; Q = 120; P − Q = -10",
            "zh-Hant": "只限整數冊數。預算 200：P 可印 60 冊；Q 可印 70 冊。 P = 110; Q = 120; P − Q = -10"
          }
        },
        {
          "label": {
            "en": "50 booklets",
            "zh-Hant": "50 冊"
          },
          "boundary": false,
          "params": {
            "scenario": "2"
          },
          "observation": {
            "en": "Integer quantities only. Budget 200: P permits 60 booklets; Q permits 70. P = 170; Q = 160; P − Q = 10",
            "zh-Hant": "只限整數冊數。預算 200：P 可印 60 冊；Q 可印 70 冊。 P = 170; Q = 160; P − Q = 10"
          }
        },
        {
          "label": {
            "en": "Budget 160",
            "zh-Hant": "預算 160"
          },
          "boundary": true,
          "params": {
            "scenario": "3"
          },
          "observation": {
            "en": "Integer quantities only. Budget 160: P permits 46 booklets; Q permits 50. P = 50; Q = 50; P − Q = 0",
            "zh-Hant": "只限整數冊數。預算 160：P 可印 46 冊；Q 可印 50 冊。 P = 50; Q = 50; P − Q = 0"
          }
        },
        {
          "label": {
            "en": "Rental at excluded time zero",
            "zh-Hant": "租借的禁用時間零"
          },
          "boundary": true,
          "params": {
            "scenario": "4"
          },
          "observation": {
            "en": "Time is continuous on 0 < t ≤ 40. t = 0 is outside the stated domain; both costs are undefined.",
            "zh-Hant": "時間在 0 < t ≤ 40 連續變化。 t＝0 不屬於所述定義域；兩個費用均未定義。"
          }
        }
      ]
    },
    {
      "id": "g10-05-15",
      "chapter": "05",
      "target": {
        "en": "Squaring and extraneous roots",
        "zh-Hant": "平方與增根"
      },
      "title": {
        "en": "Squaring can create a false root",
        "zh-Hant": "平方可能產生增根"
      },
      "chapterTitle": {
        "en": "Functions and their properties",
        "zh-Hant": "函數概念與性質"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=function-properties&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-05-15/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-05-15/"
      },
      "coreClaim": {
        "en": "Squaring gives x²=x+2, with candidates −1 and 2. The original square root is nonnegative, so its right side must satisfy x≥0. Substitution rejects −1 and accepts 2.",
        "zh-Hant": "平方得 x²＝x＋2，候選根為 −1、2；原根式非負，右邊須有 x≥0，代回拒絕 −1、接受 2。"
      },
      "conditions": {
        "en": "Real radicand requires x≥−2; equality with the principal square root additionally requires x≥0.",
        "zh-Hant": "實數被開方數要求 x≥−2；與主平方根相等另要求 x≥0。"
      },
      "evidence": [
        {
          "label": {
            "en": "Accepted root 2",
            "zh-Hant": "接受的根 2"
          },
          "boundary": false,
          "params": {
            "x": 2
          },
          "observation": {
            "en": "At x=2: √(x+2)=2, x=2; original equation holds, squared equation holds.",
            "zh-Hant": "x＝2：√(x＋2)＝2，x＝2；原方程成立，平方後方程成立。"
          }
        },
        {
          "label": {
            "en": "Extraneous candidate −1",
            "zh-Hant": "增根候選 −1"
          },
          "boundary": true,
          "params": {
            "x": -1
          },
          "observation": {
            "en": "At x=-1: √(x+2)=1, x=-1; original equation fails, squared equation holds.",
            "zh-Hant": "x＝-1：√(x＋2)＝1，x＝-1；原方程不成立，平方後方程成立。"
          }
        },
        {
          "label": {
            "en": "Outside radical domain −3",
            "zh-Hant": "超出根式定義域 −3"
          },
          "boundary": true,
          "params": {
            "x": -3
          },
          "observation": {
            "en": "At x=-3, x+2<0: the real radical is undefined.",
            "zh-Hant": "x＝-3 時 x＋2<0，實數根式未定義。"
          }
        },
        {
          "label": {
            "en": "Radicand boundary −2",
            "zh-Hant": "被開方數邊界 −2"
          },
          "boundary": true,
          "params": {
            "x": -2
          },
          "observation": {
            "en": "At x=-2: √(x+2)=0, x=-2; original equation fails, squared equation fails.",
            "zh-Hant": "x＝-2：√(x＋2)＝0，x＝-2；原方程不成立，平方後方程不成立。"
          }
        }
      ]
    },
    {
      "id": "g10-06-01",
      "chapter": "06",
      "target": {
        "en": "Negative exponents",
        "zh-Hant": "負指數"
      },
      "title": {
        "en": "A negative exponent is a reciprocal",
        "zh-Hant": "負指數表示倒數"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-01/"
      },
      "coreClaim": {
        "en": "Reducing the exponent by one divides by the nonzero base. Thus a⁰=1 and a⁻ⁿ=1/aⁿ for a≠0. The minus sign in the exponent does not negate the value.",
        "zh-Hant": "指數每減一，便除以非零底數；故 a≠0 時 a⁰＝1、a⁻ⁿ＝1/aⁿ，指數的負號不是把值取負。"
      },
      "conditions": {
        "en": "The exponent here is an integer. Zero to a negative exponent and 0⁰ are not defined in this task.",
        "zh-Hant": "此處指數為整數；本課不定義零的負整數冪及 0⁰。"
      },
      "evidence": [
        {
          "label": {
            "en": "Reciprocal 2⁻³",
            "zh-Hant": "倒數 2⁻³"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "n": -3
          },
          "observation": {
            "en": "2^-3 = 0.125. A negative exponent takes a reciprocal.",
            "zh-Hant": "2^-3＝0.125；負指數表示取倒數。"
          }
        },
        {
          "label": {
            "en": "Odd negative exponent",
            "zh-Hant": "奇數負指數"
          },
          "boundary": false,
          "params": {
            "a": -2,
            "n": -3
          },
          "observation": {
            "en": "-2^-3 = -0.125. A negative exponent takes a reciprocal.",
            "zh-Hant": "-2^-3＝-0.125；負指數表示取倒數。"
          }
        },
        {
          "label": {
            "en": "Zero base forbidden",
            "zh-Hant": "禁用零底數"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "n": -1
          },
          "observation": {
            "en": "Zero to a nonpositive exponent is undefined here.",
            "zh-Hant": "本課零的非正整數冪未定義。"
          }
        },
        {
          "label": {
            "en": "Zero exponent",
            "zh-Hant": "零指數"
          },
          "boundary": true,
          "params": {
            "a": 2,
            "n": 0
          },
          "observation": {
            "en": "2^0 = 1. A negative exponent takes a reciprocal.",
            "zh-Hant": "2^0＝1；負指數表示取倒數。"
          }
        }
      ]
    },
    {
      "id": "g10-06-02",
      "chapter": "06",
      "target": {
        "en": "Fractional exponents",
        "zh-Hant": "分數指數"
      },
      "title": {
        "en": "A fractional exponent names a root",
        "zh-Hant": "分數指數表示根式"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-02/"
      },
      "coreClaim": {
        "en": "For positive a, a^(p/q)=(q√a)ᵖ with integer p and positive integer q. Here 16^(3/4)=2³=8. Negative p then takes a reciprocal.",
        "zh-Hant": "對 a>0，整數 p、正整數 q 滿足 a^(p/q)＝(q 次根 a)ᵖ；故 16^(3/4)＝2³＝8，p 為負時再取倒數。"
      },
      "conditions": {
        "en": "Use a>0 so all rational exponents have one consistent real definition. This activity rejects a≤0 rather than extending selected odd-root cases.",
        "zh-Hant": "採 a>0，使所有有理指數有一致的實數定義；本活動拒絕 a≤0，不擴展個別奇次根情況。"
      },
      "evidence": [
        {
          "label": {
            "en": "Fourth root cubed",
            "zh-Hant": "四次根再立方"
          },
          "boundary": false,
          "params": {
            "a": 16,
            "p": 3,
            "q": 4
          },
          "observation": {
            "en": "The positive 4th root is 2; raising it to 3 gives 8.",
            "zh-Hant": "正 4 次根為 2，再取 3 次方得 8。"
          }
        },
        {
          "label": {
            "en": "Negative numerator",
            "zh-Hant": "負分子"
          },
          "boundary": false,
          "params": {
            "a": 16,
            "p": -3,
            "q": 4
          },
          "observation": {
            "en": "The positive 4th root is 2; raising it to -3 gives 0.125.",
            "zh-Hant": "正 4 次根為 2，再取 -3 次方得 0.125。"
          }
        },
        {
          "label": {
            "en": "Unit base",
            "zh-Hant": "單位底數"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "p": 3,
            "q": 4
          },
          "observation": {
            "en": "The positive 4th root is 1; raising it to 3 gives 1.",
            "zh-Hant": "正 4 次根為 1，再取 3 次方得 1。"
          }
        },
        {
          "label": {
            "en": "Nonpositive base rejected",
            "zh-Hant": "拒絕非正底數"
          },
          "boundary": true,
          "params": {
            "a": -1,
            "p": 1,
            "q": 2
          },
          "observation": {
            "en": "This rational-power model requires a>0, integer p and positive integer q.",
            "zh-Hant": "本有理冪模型要求 a>0、整數 p 及正整數 q。"
          }
        }
      ]
    },
    {
      "id": "g10-06-03",
      "chapter": "06",
      "target": {
        "en": "Fixed increments and multipliers",
        "zh-Hant": "固定增量與倍率"
      },
      "title": {
        "en": "Equal additions or equal ratios?",
        "zh-Hant": "固定增量還是固定倍率？"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-03/"
      },
      "coreClaim": {
        "en": "Fixed increments give L(n)=100+20n. Fixed ratios give E(n)=100×1.2ⁿ. They agree after one round but differ after two: 140 and 144.",
        "zh-Hant": "固定增量給 L(n)＝100＋20n，固定倍率給 E(n)＝100×1.2ⁿ；一輪後相同，兩輪後分別為 140、144。"
      },
      "conditions": {
        "en": "Rounds n are nonnegative integers. A constant ratio is an assumption. A negative linear-model prediction is not a meaningful remaining quantity.",
        "zh-Hant": "輪數 n 為非負整數；固定比率是模型假設。一次模型若預測負數，不能當成實際剩餘量。"
      },
      "evidence": [
        {
          "label": {
            "en": "Two growth rounds",
            "zh-Hant": "兩輪增長"
          },
          "boundary": false,
          "params": {
            "r": 1.2,
            "n": 2
          },
          "observation": {
            "en": "Round 2: fixed increment 140, fixed ratio 144.",
            "zh-Hant": "第 2 輪：固定增量模型 140，固定倍率模型 144。"
          }
        },
        {
          "label": {
            "en": "Three decay rounds",
            "zh-Hant": "三輪衰減"
          },
          "boundary": false,
          "params": {
            "r": 0.8,
            "n": 3
          },
          "observation": {
            "en": "Round 3: fixed increment 40, fixed ratio 51.2.",
            "zh-Hant": "第 3 輪：固定增量模型 40，固定倍率模型 51.2。"
          }
        },
        {
          "label": {
            "en": "No change ratio 1",
            "zh-Hant": "不變倍率 1"
          },
          "boundary": true,
          "params": {
            "r": 1,
            "n": 3
          },
          "observation": {
            "en": "Round 3: fixed increment 100, fixed ratio 100.",
            "zh-Hant": "第 3 輪：固定增量模型 100，固定倍率模型 100。"
          }
        },
        {
          "label": {
            "en": "Initial round zero",
            "zh-Hant": "起始輪數零"
          },
          "boundary": true,
          "params": {
            "r": 1.2,
            "n": 0
          },
          "observation": {
            "en": "Round 0: fixed increment 100, fixed ratio 100.",
            "zh-Hant": "第 0 輪：固定增量模型 100，固定倍率模型 100。"
          }
        }
      ]
    },
    {
      "id": "g10-06-04",
      "chapter": "06",
      "target": {
        "en": "Base and monotonicity",
        "zh-Hant": "底數與單調性"
      },
      "title": {
        "en": "Crossing the base 1 reverses growth",
        "zh-Hant": "底數越過 1，增減方向改變"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-04/"
      },
      "coreClaim": {
        "en": "For a>1, aˣ is increasing; for 0<a<1 it is decreasing. Every positive base gives a⁰=1. At a=1 the rule is constant, not an exponential function in this course.",
        "zh-Hant": "a>1 時 aˣ 遞增；0<a<1 時遞減；每個正底數都有 a⁰＝1；a＝1 為常數，本課不把它稱為指數函數。"
      },
      "conditions": {
        "en": "The real exponential family requires a>0 and a≠1. Its domain is all real numbers and its range is (0,∞).",
        "zh-Hant": "實指數函數要求 a>0 且 a≠1，定義域為實數，值域為 (0,∞)。"
      },
      "evidence": [
        {
          "label": {
            "en": "Growth base 2",
            "zh-Hant": "增長底數 2"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "x": 1
          },
          "observation": {
            "en": "a=2: increasing; a^1=2.",
            "zh-Hant": "a＝2：遞增；a^1＝2。"
          }
        },
        {
          "label": {
            "en": "Decay base 0.5",
            "zh-Hant": "衰減底數 0.5"
          },
          "boundary": false,
          "params": {
            "a": 0.5,
            "x": 1
          },
          "observation": {
            "en": "a=0.5: decreasing; a^1=0.5.",
            "zh-Hant": "a＝0.5：遞減；a^1＝0.5。"
          }
        },
        {
          "label": {
            "en": "Constant boundary 1",
            "zh-Hant": "常數邊界 1"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "x": 1
          },
          "observation": {
            "en": "a=1: constant; a^1=1. Base 1 is the constant boundary, excluded from this course’s exponential family.",
            "zh-Hant": "a＝1：常數；a^1＝1。底數 1 是常數邊界，不屬於本課指數函數族。"
          }
        },
        {
          "label": {
            "en": "Nonpositive base",
            "zh-Hant": "非正底數"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "x": 1
          },
          "observation": {
            "en": "A real exponential model requires a positive base.",
            "zh-Hant": "實指數模型要求正底數。"
          }
        }
      ]
    },
    {
      "id": "g10-06-05",
      "chapter": "06",
      "target": {
        "en": "Logarithms and positive arguments",
        "zh-Hant": "對數意義與真數限制"
      },
      "title": {
        "en": "A logarithm asks for an exponent",
        "zh-Hant": "對數是在問指數"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-05/"
      },
      "coreClaim": {
        "en": "logₐb is the exponent y satisfying aʸ=b. Thus log₂8=3 and log₂(1/4)=−2. A positive exponential never equals zero or a negative number.",
        "zh-Hant": "logₐb 是滿足 aʸ＝b 的指數 y；故 log₂8＝3、log₂(1/4)＝−2；正底數的指數值不會等於零或負數。"
      },
      "conditions": {
        "en": "A real logarithm requires a>0, a≠1 and argument b>0. A negative logarithm value is allowed.",
        "zh-Hant": "實對數要求 a>0、a≠1 及真數 b>0；對數值本身可以為負。"
      },
      "evidence": [
        {
          "label": {
            "en": "Positive logarithm",
            "zh-Hant": "正對數值"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "b": 8
          },
          "observation": {
            "en": "2^3 = 8, so log_2(8)=3.",
            "zh-Hant": "2^3＝8，故 log_2(8)＝3。"
          }
        },
        {
          "label": {
            "en": "Negative logarithm",
            "zh-Hant": "負對數值"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "b": 0.25
          },
          "observation": {
            "en": "2^-2 = 0.25, so log_2(0.25)=-2.",
            "zh-Hant": "2^-2＝0.25，故 log_2(0.25)＝-2。"
          }
        },
        {
          "label": {
            "en": "Zero argument",
            "zh-Hant": "零真數"
          },
          "boundary": true,
          "params": {
            "a": 2,
            "b": 0
          },
          "observation": {
            "en": "Require a>0, a≠1 and positive argument b.",
            "zh-Hant": "須 a>0、a≠1 及正真數 b。"
          }
        },
        {
          "label": {
            "en": "Forbidden base 1",
            "zh-Hant": "禁用底數 1"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": 8
          },
          "observation": {
            "en": "Require a>0, a≠1 and positive argument b.",
            "zh-Hant": "須 a>0、a≠1 及正真數 b。"
          }
        }
      ]
    },
    {
      "id": "g10-06-06",
      "chapter": "06",
      "target": {
        "en": "Inverse reflection",
        "zh-Hant": "反函數鏡射"
      },
      "title": {
        "en": "Swap coordinates for inverse graphs",
        "zh-Hant": "交換坐標看反函數圖像"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-06/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-06/"
      },
      "coreClaim": {
        "en": "The relation y=aˣ is equivalent to x=logₐy. Swapping coordinates reflects the graph in y=x. Domain and range exchange: real inputs/positive outputs become positive inputs/real outputs.",
        "zh-Hant": "y＝aˣ 等價於 x＝logₐy；交換坐標使圖像關於 y＝x 對稱；實數輸入及正輸出交換為正輸入及實數輸出。"
      },
      "conditions": {
        "en": "Require a>0,a≠1. The inverse graph has no point with input zero, even when its output is negative.",
        "zh-Hant": "須 a>0、a≠1；反函數沒有輸入為零的點，即使其輸出可以為負。"
      },
      "evidence": [
        {
          "label": {
            "en": "Point (2,4)",
            "zh-Hant": "點 (2,4)"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "x": 2
          },
          "observation": {
            "en": "(2,4) reflects to (4,2).",
            "zh-Hant": "(2,4) 反射至 (4,2)。"
          }
        },
        {
          "label": {
            "en": "Point with negative exponent",
            "zh-Hant": "負指數的點"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "x": -2
          },
          "observation": {
            "en": "(-2,0.25) reflects to (0.25,-2).",
            "zh-Hant": "(-2,0.25) 反射至 (0.25,-2)。"
          }
        },
        {
          "label": {
            "en": "Decreasing inverse pair",
            "zh-Hant": "遞減的互逆函數對"
          },
          "boundary": false,
          "params": {
            "a": 0.5,
            "x": -1
          },
          "observation": {
            "en": "(-1,2) reflects to (2,-1).",
            "zh-Hant": "(-1,2) 反射至 (2,-1)。"
          }
        },
        {
          "label": {
            "en": "Base 1 has no inverse",
            "zh-Hant": "底數 1 沒有反函數"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "x": 1
          },
          "observation": {
            "en": "An invertible exponential needs a>0 and a≠1.",
            "zh-Hant": "可逆指數函數須 a>0 且 a≠1。"
          }
        }
      ]
    },
    {
      "id": "g10-06-07",
      "chapter": "06",
      "target": {
        "en": "Logarithm laws and counterexamples",
        "zh-Hant": "對數法則與反例"
      },
      "title": {
        "en": "Products become sums of logarithms",
        "zh-Hant": "乘積化成對數的和"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-07/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-07/"
      },
      "coreClaim": {
        "en": "For positive u,v, logₐ(uv)=logₐu+logₐv. There is no corresponding sum rule. With u=2,v=4, the right side is 3 while log₂6 differs.",
        "zh-Hant": "u,v>0 時 logₐ(uv)＝logₐu＋logₐv；沒有相同的加法規則。u＝2、v＝4 時右邊為 3，log₂6 卻不同。"
      },
      "conditions": {
        "en": "Require a>0,a≠1 and u,v>0 separately. A positive product of two negative numbers does not make their separate real logarithms exist.",
        "zh-Hant": "須 a>0、a≠1，並逐一要求 u,v>0；兩個負數乘積為正，不代表它們各自有實對數。"
      },
      "evidence": [
        {
          "label": {
            "en": "Accidental equality at 2 and 2",
            "zh-Hant": "2 與 2 的偶然相同"
          },
          "boundary": true,
          "params": {
            "a": 2,
            "u": 2,
            "v": 2
          },
          "observation": {
            "en": "Product rule: 2=2. Sum claim: 2 happens to equal 2; one accidental equality is not a general law.",
            "zh-Hant": "乘積公式：2＝2；相加猜想：2 偶然等於 2；單次相同不是一般法則。"
          }
        },
        {
          "label": {
            "en": "Counterexample at 2 and 4",
            "zh-Hant": "2 與 4 的反例"
          },
          "boundary": true,
          "params": {
            "a": 2,
            "u": 2,
            "v": 4
          },
          "observation": {
            "en": "Product rule: 3=3. Sum claim: 2.5849625 differs from 3; one accidental equality is not a general law.",
            "zh-Hant": "乘積公式：3＝3；相加猜想：2.5849625 不同於 3；單次相同不是一般法則。"
          }
        },
        {
          "label": {
            "en": "Positive quotient",
            "zh-Hant": "正數商"
          },
          "boundary": false,
          "params": {
            "a": 2,
            "u": 8,
            "v": 2
          },
          "observation": {
            "en": "Product rule: 4=4. Sum claim: 3.3219281 differs from 4; one accidental equality is not a general law.",
            "zh-Hant": "乘積公式：4＝4；相加猜想：3.3219281 不同於 4；單次相同不是一般法則。"
          }
        },
        {
          "label": {
            "en": "Positive product, forbidden factors",
            "zh-Hant": "乘積為正但因數禁用"
          },
          "boundary": true,
          "params": {
            "a": 2,
            "u": -2,
            "v": -4
          },
          "observation": {
            "en": "Each argument u,v must be positive, even when their product is positive.",
            "zh-Hant": "u、v 各自均須為正，即使乘積為正亦然。"
          }
        }
      ]
    },
    {
      "id": "g10-06-08",
      "chapter": "06",
      "target": {
        "en": "Equations and growth thresholds",
        "zh-Hant": "方程與增長閾值"
      },
      "title": {
        "en": "The threshold between two integer rounds",
        "zh-Hant": "兩個整數輪次之間的臨界值"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-08/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-08/"
      },
      "coreClaim": {
        "en": "For growth r>1 and target above 100, the real crossing is n*=log(target/100)/log r, but the first integer round is the least integer ≥n*. For decay 0<r<1, use the first round at or below the target.",
        "zh-Hant": "r>1 且目標大於 100 時，實數臨界為 n*＝log(目標/100)/log r，首次整數輪次須取不小於 n* 的最小整數；0<r<1 衰減時改問首次不超過目標。"
      },
      "conditions": {
        "en": "n is a nonnegative integer and the target is positive. The question is at-or-above for growth and at-or-below for decay. At r=1 only a target already met at n=0 is reachable.",
        "zh-Hant": "n 為非負整數且目標為正；增長問不少於，衰減問不多於；r＝1 時只有 n＝0 已達到的目標可行。"
      },
      "evidence": [
        {
          "label": {
            "en": "Growth to 150",
            "zh-Hant": "增長至 150"
          },
          "boundary": false,
          "params": {
            "r": 1.2,
            "target": 150
          },
          "observation": {
            "en": "First round: 3. Previous value 144; at round 3: 172.8.",
            "zh-Hant": "首次輪數：3；前一輪 144；第 3 輪為 172.8。"
          }
        },
        {
          "label": {
            "en": "Decay to 50",
            "zh-Hant": "衰減至 50"
          },
          "boundary": false,
          "params": {
            "r": 0.8,
            "target": 50
          },
          "observation": {
            "en": "First round: 4. Previous value 51.2; at round 4: 40.96.",
            "zh-Hant": "首次輪數：4；前一輪 51.2；第 4 輪為 40.96。"
          }
        },
        {
          "label": {
            "en": "Target at the initial value",
            "zh-Hant": "目標等於起始值"
          },
          "boundary": true,
          "params": {
            "r": 1.2,
            "target": 100
          },
          "observation": {
            "en": "First round: 0. at round 0: 100.",
            "zh-Hant": "首次輪數：0；第 0 輪為 100。"
          }
        },
        {
          "label": {
            "en": "Constant cannot reach 150",
            "zh-Hant": "常數無法到達 150"
          },
          "boundary": true,
          "params": {
            "r": 1,
            "target": 150
          },
          "observation": {
            "en": "The constant model cannot reach this target in the stated direction.",
            "zh-Hant": "常數模型無法依指定方向達到此目標。"
          }
        },
        {
          "label": {
            "en": "Nonpositive target rejected",
            "zh-Hant": "拒絕非正目標"
          },
          "boundary": true,
          "params": {
            "r": 0.8,
            "target": 0
          },
          "observation": {
            "en": "The multiplier and target must both be positive.",
            "zh-Hant": "倍率與目標均須為正。"
          }
        }
      ]
    },
    {
      "id": "g10-06-09",
      "chapter": "06",
      "target": {
        "en": "Inequality direction",
        "zh-Hant": "不等式方向"
      },
      "title": {
        "en": "A decreasing base reverses the inequality",
        "zh-Hant": "遞減底數會反轉不等號"
      },
      "chapterTitle": {
        "en": "Exponential and logarithmic functions",
        "zh-Hant": "指數函數與對數函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=exponential&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-06-09/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-06-09/"
      },
      "coreClaim": {
        "en": "For a>1, logₐx<logₐb gives 0<x<b and aˣ<aᵇ gives x<b. For 0<a<1, both functions decrease and both comparisons instead give x>b. Equality x=b is excluded.",
        "zh-Hant": "a>1 時，logₐx<logₐb 給 0<x<b，aˣ<aᵇ 給 x<b；0<a<1 時兩種函數均遞減，兩種比較均改為 x>b，並排除 x＝b。"
      },
      "conditions": {
        "en": "Require a>0,a≠1. Logarithmic arguments x,b must be positive; exponential inputs x,b may be any real numbers. Base 1 is excluded here.",
        "zh-Hant": "須 a>0、a≠1；對數的真數 x、b 須為正，指數函數的輸入 x、b 可為任何實數；本課排除底數 1。"
      },
      "evidence": [
        {
          "label": {
            "en": "Increasing logarithm",
            "zh-Hant": "遞增對數"
          },
          "boundary": false,
          "params": {
            "family": "log",
            "a": 2,
            "x": 2,
            "b": 4
          },
          "observation": {
            "en": "Solution: 0 < x < 4. Current input satisfies the strict inequality.",
            "zh-Hant": "解集：0 < x < 4；目前輸入符合嚴格不等式。"
          }
        },
        {
          "label": {
            "en": "Decreasing logarithm, same inputs",
            "zh-Hant": "遞減對數，相同輸入"
          },
          "boundary": true,
          "params": {
            "family": "log",
            "a": 0.5,
            "x": 2,
            "b": 4
          },
          "observation": {
            "en": "Solution: x > 4. Current input does not satisfy the strict inequality.",
            "zh-Hant": "解集：x > 4；目前輸入不符合嚴格不等式。"
          }
        },
        {
          "label": {
            "en": "Equality boundary",
            "zh-Hant": "等號邊界"
          },
          "boundary": true,
          "params": {
            "family": "log",
            "a": 0.5,
            "x": 4,
            "b": 4
          },
          "observation": {
            "en": "Solution: x > 4. Current input does not satisfy the strict inequality.",
            "zh-Hant": "解集：x > 4；目前輸入不符合嚴格不等式。"
          }
        },
        {
          "label": {
            "en": "Forbidden logarithm input zero",
            "zh-Hant": "對數的禁用零輸入"
          },
          "boundary": true,
          "params": {
            "family": "log",
            "a": 2,
            "x": 0,
            "b": 4
          },
          "observation": {
            "en": "Require a>0,a≠1; logarithms additionally need both arguments positive.",
            "zh-Hant": "須 a>0、a≠1；對數另要求兩個真數為正。"
          }
        },
        {
          "label": {
            "en": "Increasing exponential at negative input",
            "zh-Hant": "負輸入的遞增指數"
          },
          "boundary": false,
          "params": {
            "family": "exponential",
            "a": 2,
            "x": -1,
            "b": 0
          },
          "observation": {
            "en": "Solution: x < 0. Current input satisfies the strict inequality.",
            "zh-Hant": "解集：x < 0；目前輸入符合嚴格不等式。"
          }
        },
        {
          "label": {
            "en": "Decreasing exponential, same inputs",
            "zh-Hant": "遞減指數，相同輸入"
          },
          "boundary": true,
          "params": {
            "family": "exponential",
            "a": 0.5,
            "x": -1,
            "b": 0
          },
          "observation": {
            "en": "Solution: x > 0. Current input does not satisfy the strict inequality.",
            "zh-Hant": "解集：x > 0；目前輸入不符合嚴格不等式。"
          }
        }
      ]
    },
    {
      "id": "g10-07-01",
      "chapter": "07",
      "target": {
        "en": "Radians and arc length",
        "zh-Hant": "弧度與弧長"
      },
      "title": {
        "en": "What does one radian measure?",
        "zh-Hant": "一弧度量度甚麼？"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-01/"
      },
      "coreClaim": {
        "en": "A radian angle is the ratio s/r. A full turn has angle 2π, independently of radius; arc length is s=rθ when θ is in radians.",
        "zh-Hant": "弧度是弧長與半徑的比 s/r。無論半徑，一周都是 2π；θ 用弧度時 s＝rθ。"
      },
      "conditions": {
        "en": "Use r>0 and the magnitude of a turn for arc length; a zero turn has zero arc.",
        "zh-Hant": "半徑須 r＞0；弧長取轉角的大小，零轉角弧長為零。"
      },
      "evidence": [
        {
          "label": {
            "en": "Unit radius",
            "zh-Hant": "單位半徑"
          },
          "boundary": false,
          "params": {
            "angle": 90,
            "r": 1
          },
          "observation": {
            "en": "The radian ratio is independent of radius.",
            "zh-Hant": "弧度比值不受半徑影響。"
          }
        },
        {
          "label": {
            "en": "Double radius",
            "zh-Hant": "半徑加倍"
          },
          "boundary": false,
          "params": {
            "angle": 90,
            "r": 2
          },
          "observation": {
            "en": "The radian ratio is independent of radius.",
            "zh-Hant": "弧度比值不受半徑影響。"
          }
        },
        {
          "label": {
            "en": "Zero turn",
            "zh-Hant": "零轉角"
          },
          "boundary": true,
          "params": {
            "angle": 0,
            "r": 2
          },
          "observation": {
            "en": "The radian ratio is independent of radius.",
            "zh-Hant": "弧度比值不受半徑影響。"
          }
        }
      ]
    },
    {
      "id": "g10-07-02",
      "chapter": "07",
      "target": {
        "en": "Unit circle",
        "zh-Hant": "單位圓"
      },
      "title": {
        "en": "One circle, two coordinates",
        "zh-Hant": "一個圓，兩個坐標"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-02/"
      },
      "coreClaim": {
        "en": "The unit-circle point is (cosθ,sinθ). A full turn returns to the same point, while opposite points negate both coordinates.",
        "zh-Hant": "單位圓上的點為 (cosθ,sinθ)。轉一周回到同一點；相對的點兩個坐標均反號。"
      },
      "conditions": {
        "en": "Angles may be negative or exceed360°; these are directed turns, not new radii.",
        "zh-Hant": "角可為負或超過 360°；這些是有向轉角，半徑仍為 1。"
      },
      "evidence": [
        {
          "label": {
            "en": "First quadrant",
            "zh-Hant": "第一象限"
          },
          "boundary": false,
          "params": {
            "angle": 30
          },
          "observation": {
            "en": "The point is (cosθ,sinθ).",
            "zh-Hant": "圓點是 (cosθ,sinθ)。"
          }
        },
        {
          "label": {
            "en": "Second quadrant",
            "zh-Hant": "第二象限"
          },
          "boundary": false,
          "params": {
            "angle": 150
          },
          "observation": {
            "en": "The point is (cosθ,sinθ).",
            "zh-Hant": "圓點是 (cosθ,sinθ)。"
          }
        },
        {
          "label": {
            "en": "Axis boundary",
            "zh-Hant": "坐標軸邊界"
          },
          "boundary": true,
          "params": {
            "angle": 270
          },
          "observation": {
            "en": "The point is (cosθ,sinθ).",
            "zh-Hant": "圓點是 (cosθ,sinθ)。"
          }
        }
      ]
    },
    {
      "id": "g10-07-03",
      "chapter": "07",
      "target": {
        "en": "Related-angle identities",
        "zh-Hant": "誘導公式"
      },
      "title": {
        "en": "Reflect an angle without recalculating",
        "zh-Hant": "反射角，毋須重新計算"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-03/"
      },
      "coreClaim": {
        "en": "Reflection in the y-axis maps(cosθ,sinθ) to(−cosθ,sinθ), giving the supplementary-angle formulas.",
        "zh-Hant": "關於 y 軸反射把 (cosθ,sinθ) 變成 (−cosθ,sinθ)，因而得到補角誘導公式。"
      },
      "conditions": {
        "en": "The identities hold for every real angle; matching plotted decimals is only a check.",
        "zh-Hant": "恆等式對所有實角成立；圖上小數相同只作檢查。"
      },
      "evidence": [
        {
          "label": {
            "en": "30° reflection",
            "zh-Hant": "30° 反射"
          },
          "boundary": false,
          "params": {
            "angle": 30
          },
          "observation": {
            "en": "Reflection keeps y and reverses x.",
            "zh-Hant": "反射保持 y，反轉 x。"
          }
        },
        {
          "label": {
            "en": "Obtuse input",
            "zh-Hant": "鈍角輸入"
          },
          "boundary": false,
          "params": {
            "angle": 120
          },
          "observation": {
            "en": "Reflection keeps y and reverses x.",
            "zh-Hant": "反射保持 y，反轉 x。"
          }
        },
        {
          "label": {
            "en": "Fixed point",
            "zh-Hant": "反射不動點"
          },
          "boundary": true,
          "params": {
            "angle": 90
          },
          "observation": {
            "en": "Reflection keeps y and reverses x.",
            "zh-Hant": "反射保持 y，反轉 x。"
          }
        }
      ]
    },
    {
      "id": "g10-07-04",
      "chapter": "07",
      "target": {
        "en": "Same-angle identities",
        "zh-Hant": "同角關係"
      },
      "title": {
        "en": "A square loses a sign",
        "zh-Hant": "平方會遺失符號"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-04/"
      },
      "coreClaim": {
        "en": "The sum of squares equals1, but solving for cosine requires its quadrant sign. Opposite signs have the same square.",
        "zh-Hant": "平方和等於 1，但由此求餘弦仍要看象限。正負相反的數有相同平方。"
      },
      "conditions": {
        "en": "The identity uses the same angle; square roots alone return a nonnegative value.",
        "zh-Hant": "恆等式中的兩項必須屬於同一角；平方根符號本身取非負值。"
      },
      "evidence": [
        {
          "label": {
            "en": "Positive cosine",
            "zh-Hant": "餘弦為正"
          },
          "boundary": false,
          "params": {
            "angle": 30
          },
          "observation": {
            "en": "The square gives the magnitude; the quadrant supplies the sign.",
            "zh-Hant": "平方只能決定大小，正負仍要看象限。"
          }
        },
        {
          "label": {
            "en": "Negative cosine",
            "zh-Hant": "餘弦為負"
          },
          "boundary": false,
          "params": {
            "angle": 150
          },
          "observation": {
            "en": "The square gives the magnitude; the quadrant supplies the sign.",
            "zh-Hant": "平方只能決定大小，正負仍要看象限。"
          }
        },
        {
          "label": {
            "en": "Zero cosine",
            "zh-Hant": "餘弦為零"
          },
          "boundary": true,
          "params": {
            "angle": 90
          },
          "observation": {
            "en": "The square gives the magnitude; the quadrant supplies the sign.",
            "zh-Hant": "平方只能決定大小，正負仍要看象限。"
          }
        }
      ]
    },
    {
      "id": "g10-07-05",
      "chapter": "07",
      "target": {
        "en": "Circle to sine and cosine",
        "zh-Hant": "由圓到正弦餘弦圖"
      },
      "title": {
        "en": "Unroll the circle into a wave",
        "zh-Hant": "把圓展開成波形"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-05/"
      },
      "coreClaim": {
        "en": "The horizontal graph coordinate records angle; graph height copies the circle y-coordinate. Cosine similarly copies x.",
        "zh-Hant": "圖像的橫坐標記錄角，縱坐標抄錄圓上的 y；餘弦圖則抄錄圓上的 x。"
      },
      "conditions": {
        "en": "The graph axis uses radians even when the control reports degrees; a full turn is2π.",
        "zh-Hant": "圖像橫軸用弧度，即使控制顯示度數；一周為 2π。"
      },
      "evidence": [
        {
          "label": {
            "en": "Quarter turn",
            "zh-Hant": "四分之一周"
          },
          "boundary": false,
          "params": {
            "angle": 90
          },
          "observation": {
            "en": "Green copies the circle y-coordinate; gold copies x.",
            "zh-Hant": "綠線抄錄圓上 y，金線抄錄 x。"
          }
        },
        {
          "label": {
            "en": "Half turn",
            "zh-Hant": "半周"
          },
          "boundary": false,
          "params": {
            "angle": 180
          },
          "observation": {
            "en": "Green copies the circle y-coordinate; gold copies x.",
            "zh-Hant": "綠線抄錄圓上 y，金線抄錄 x。"
          }
        },
        {
          "label": {
            "en": "Full-turn boundary",
            "zh-Hant": "一周邊界"
          },
          "boundary": true,
          "params": {
            "angle": 360
          },
          "observation": {
            "en": "Green copies the circle y-coordinate; gold copies x.",
            "zh-Hant": "綠線抄錄圓上 y，金線抄錄 x。"
          }
        }
      ]
    },
    {
      "id": "g10-07-06",
      "chapter": "07",
      "target": {
        "en": "Tangent and exclusions",
        "zh-Hant": "正切與禁值"
      },
      "title": {
        "en": "Where does tangent break?",
        "zh-Hant": "正切在哪裏中斷？"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-06/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-06/"
      },
      "coreClaim": {
        "en": "Tangent is sinθ/cosθ. At90°+180°k its denominator is0, so these inputs are excluded, not assigned a very large value.",
        "zh-Hant": "正切為 sinθ/cosθ。在 90°＋180°k 分母為 0，必須排除，並非給它一個很大的值。"
      },
      "conditions": {
        "en": "θ≠90°+180°k for integer k. The basic period is180°.",
        "zh-Hant": "θ≠90°＋180°k，k 為整數；基本周期為 180°。"
      },
      "evidence": [
        {
          "label": {
            "en": "Before the pole",
            "zh-Hant": "漸近線之前"
          },
          "boundary": false,
          "params": {
            "angle": 85
          },
          "observation": {
            "en": "The quotient is defined at this input.",
            "zh-Hant": "此輸入的商有定義。"
          }
        },
        {
          "label": {
            "en": "At the pole",
            "zh-Hant": "漸近線位置"
          },
          "boundary": true,
          "params": {
            "angle": 90
          },
          "observation": {
            "en": "The denominator is zero: there is no tangent value here.",
            "zh-Hant": "分母為零，此處沒有正切值。"
          }
        },
        {
          "label": {
            "en": "After the pole",
            "zh-Hant": "漸近線之後"
          },
          "boundary": false,
          "params": {
            "angle": 95
          },
          "observation": {
            "en": "The quotient is defined at this input.",
            "zh-Hant": "此輸入的商有定義。"
          }
        }
      ]
    },
    {
      "id": "g10-07-07",
      "chapter": "07",
      "target": {
        "en": "Angle sums and differences",
        "zh-Hant": "和差角"
      },
      "title": {
        "en": "Can sine distribute over addition?",
        "zh-Hant": "正弦能分配到加法嗎？"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-07/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-07/"
      },
      "coreClaim": {
        "en": "The sum-angle identity is sin(α+β)=sinαcosβ+cosαsinβ. Sine is not an additive function.",
        "zh-Hant": "和角公式是 sin(α＋β)＝sinαcosβ＋cosαsinβ，正弦並非可直接相加。"
      },
      "conditions": {
        "en": "All angles must use the same unit. A successful special case such as β=0 is not a general proof.",
        "zh-Hant": "角必須使用同一單位；β＝0 等特殊例子成立，不代表一般成立。"
      },
      "evidence": [
        {
          "label": {
            "en": "Two30° turns",
            "zh-Hant": "兩個 30°"
          },
          "boundary": false,
          "params": {
            "alpha": 30,
            "beta": 30
          },
          "observation": {
            "en": "The product identity agrees; the simple additive claim generally fails.",
            "zh-Hant": "乘積恆等式吻合，直接相加的主張一般不成立。"
          }
        },
        {
          "label": {
            "en": "Unequal turns",
            "zh-Hant": "不同轉角"
          },
          "boundary": false,
          "params": {
            "alpha": 30,
            "beta": 60
          },
          "observation": {
            "en": "The product identity agrees; the simple additive claim generally fails.",
            "zh-Hant": "乘積恆等式吻合，直接相加的主張一般不成立。"
          }
        },
        {
          "label": {
            "en": "A misleading special case",
            "zh-Hant": "容易誤判的特例"
          },
          "boundary": true,
          "params": {
            "alpha": 30,
            "beta": 0
          },
          "observation": {
            "en": "The product identity agrees; the simple additive claim generally fails.",
            "zh-Hant": "乘積恆等式吻合，直接相加的主張一般不成立。"
          }
        }
      ]
    },
    {
      "id": "g10-07-08",
      "chapter": "07",
      "target": {
        "en": "Double angles",
        "zh-Hant": "倍角"
      },
      "title": {
        "en": "Double an angle, not its sine",
        "zh-Hant": "角加倍，不是正弦值加倍"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-08/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-08/"
      },
      "coreClaim": {
        "en": "Putting α=β=θ into the sum formula gives sin2θ=2sinθcosθ and cos2θ=cos²θ−sin²θ.",
        "zh-Hant": "在和角公式取 α＝β＝θ，便得 sin2θ＝2sinθcosθ，以及 cos2θ＝cos²θ−sin²θ。"
      },
      "conditions": {
        "en": "The identities hold for every real θ; 2sinθ alone is generally different.",
        "zh-Hant": "恆等式對所有實角成立，單獨 2sinθ 通常不同。"
      },
      "evidence": [
        {
          "label": {
            "en": "45°",
            "zh-Hant": "45°"
          },
          "boundary": false,
          "params": {
            "angle": 45
          },
          "observation": {
            "en": "The product identity agrees; the simple additive claim generally fails.",
            "zh-Hant": "乘積恆等式吻合，直接相加的主張一般不成立。"
          }
        },
        {
          "label": {
            "en": "30°",
            "zh-Hant": "30°"
          },
          "boundary": false,
          "params": {
            "angle": 30
          },
          "observation": {
            "en": "The product identity agrees; the simple additive claim generally fails.",
            "zh-Hant": "乘積恆等式吻合，直接相加的主張一般不成立。"
          }
        },
        {
          "label": {
            "en": "Zero angle",
            "zh-Hant": "零角"
          },
          "boundary": true,
          "params": {
            "angle": 0
          },
          "observation": {
            "en": "The product identity agrees; the simple additive claim generally fails.",
            "zh-Hant": "乘積恆等式吻合，直接相加的主張一般不成立。"
          }
        }
      ]
    },
    {
      "id": "g10-07-09",
      "chapter": "07",
      "target": {
        "en": "Amplitude",
        "zh-Hant": "振幅"
      },
      "title": {
        "en": "Amplitude and a reflected wave",
        "zh-Hant": "振幅與反轉的波形"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-09/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-09/"
      },
      "coreClaim": {
        "en": "For A sinx, the range is[−|A|,|A|]. Negative A reflects the wave; its amplitude is |A|. A=0 gives a constant.",
        "zh-Hant": "A sinx 的值域為 [−|A|,|A|]。A 為負使波形反轉，振幅為 |A|；A＝0 時成為常數。"
      },
      "conditions": {
        "en": "For a nonconstant sine wave A≠0. The zero case has range{0} and no least positive period.",
        "zh-Hant": "非恆定正弦波須 A≠0；A＝0 時值域為 {0}，沒有最小正周期。"
      },
      "evidence": [
        {
          "label": {
            "en": "Positive coefficient",
            "zh-Hant": "正係數"
          },
          "boundary": false,
          "params": {
            "A": 2
          },
          "observation": {
            "en": "The grey reference is sinx; compare one parameter at a time.",
            "zh-Hant": "灰線是 sinx，請每次只比較一項參數。"
          }
        },
        {
          "label": {
            "en": "Negative coefficient",
            "zh-Hant": "負係數"
          },
          "boundary": false,
          "params": {
            "A": -2
          },
          "observation": {
            "en": "The grey reference is sinx; compare one parameter at a time.",
            "zh-Hant": "灰線是 sinx，請每次只比較一項參數。"
          }
        },
        {
          "label": {
            "en": "No oscillation",
            "zh-Hant": "沒有振動"
          },
          "boundary": true,
          "params": {
            "A": 0
          },
          "observation": {
            "en": "This is a constant: do not apply the nonconstant basic-period formula.",
            "zh-Hant": "這是常數，不能套用非恆定波形的基本周期公式。"
          }
        }
      ]
    },
    {
      "id": "g10-07-10",
      "chapter": "07",
      "target": {
        "en": "Period",
        "zh-Hant": "周期"
      },
      "title": {
        "en": "Frequency squeezes the horizontal axis",
        "zh-Hant": "頻率壓縮橫軸"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-10/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-10/"
      },
      "coreClaim": {
        "en": "The angle must change by2π to repeat. A nonzero ω therefore gives basic period2π/|ω|; negative ω reverses direction.",
        "zh-Hant": "函數內的角改變 2π 才重複；ω 非零時基本周期為 2π/|ω|，負 ω 則反轉方向。"
      },
      "conditions": {
        "en": "This basic-period formula requires nonzero frequency and a nonzero sine coefficient. ω=0 gives a constant.",
        "zh-Hant": "此基本周期公式要求頻率及正弦係數非零；ω＝0 時為常數。"
      },
      "evidence": [
        {
          "label": {
            "en": "Unit frequency",
            "zh-Hant": "單位頻率"
          },
          "boundary": false,
          "params": {
            "omega": 1
          },
          "observation": {
            "en": "The grey reference is sinx; compare one parameter at a time.",
            "zh-Hant": "灰線是 sinx，請每次只比較一項參數。"
          }
        },
        {
          "label": {
            "en": "Double frequency",
            "zh-Hant": "頻率加倍"
          },
          "boundary": false,
          "params": {
            "omega": 2
          },
          "observation": {
            "en": "The grey reference is sinx; compare one parameter at a time.",
            "zh-Hant": "灰線是 sinx，請每次只比較一項參數。"
          }
        },
        {
          "label": {
            "en": "Zero frequency",
            "zh-Hant": "零頻率"
          },
          "boundary": true,
          "params": {
            "omega": 0
          },
          "observation": {
            "en": "This is a constant: do not apply the nonconstant basic-period formula.",
            "zh-Hant": "這是常數，不能套用非恆定波形的基本周期公式。"
          }
        }
      ]
    },
    {
      "id": "g10-07-11",
      "chapter": "07",
      "target": {
        "en": "Phase",
        "zh-Hant": "相位"
      },
      "title": {
        "en": "Phase is divided by frequency",
        "zh-Hant": "相位要除以頻率"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-11/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-11/"
      },
      "coreClaim": {
        "en": "Factoring ωx+φ=ω(x+φ/ω) shows horizontal shift−φ/ω. Phase and horizontal displacement are different quantities.",
        "zh-Hant": "把 ωx＋φ 寫成 ω(x＋φ/ω)，可見水平位移為 −φ/ω；相位與水平位移並非同一量。"
      },
      "conditions": {
        "en": "ω≠0 is required for a horizontal-shift interpretation; convert φ to radians before plotting against radian x.",
        "zh-Hant": "水平平移的解讀要求 ω≠0；若 x 用弧度，φ 亦須先換成弧度。"
      },
      "evidence": [
        {
          "label": {
            "en": "Frequency1",
            "zh-Hant": "頻率 1"
          },
          "boundary": false,
          "params": {
            "omega": 1,
            "phase": 90
          },
          "observation": {
            "en": "The shift is −φ/ω, not just −φ.",
            "zh-Hant": "位移為 −φ/ω，不只是 −φ。"
          }
        },
        {
          "label": {
            "en": "Frequency2",
            "zh-Hant": "頻率 2"
          },
          "boundary": false,
          "params": {
            "omega": 2,
            "phase": 90
          },
          "observation": {
            "en": "The shift is −φ/ω, not just −φ.",
            "zh-Hant": "位移為 −φ/ω，不只是 −φ。"
          }
        },
        {
          "label": {
            "en": "Zero frequency",
            "zh-Hant": "零頻率"
          },
          "boundary": true,
          "params": {
            "omega": 0,
            "phase": 90
          },
          "observation": {
            "en": "This is a constant: do not apply the nonconstant basic-period formula.",
            "zh-Hant": "這是常數，不能套用非恆定波形的基本周期公式。"
          }
        }
      ]
    },
    {
      "id": "g10-07-12",
      "chapter": "07",
      "target": {
        "en": "Periodic models",
        "zh-Hant": "周期情境建模"
      },
      "title": {
        "en": "A wheel needs a starting position",
        "zh-Hant": "摩天輪需要起點"
      },
      "chapterTitle": {
        "en": "Trigonometric functions",
        "zh-Hant": "三角函數"
      },
      "kind": "core",
      "sourceLesson": "/math-studio/?lesson=trigonometry&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-07-12/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-07-12/"
      },
      "coreClaim": {
        "en": "With an upward start at centre height, h(t)=H+Rsin(2πt/T). Changing the starting phase changes the height at the same time.",
        "zh-Hant": "從中心高度向上開始時，h(t)＝H＋Rsin(2πt/T)；改變起始相位，同一時間的高度也會改變。"
      },
      "conditions": {
        "en": "Use T>0,R>0 and consistent time units. A physical wheel above ground additionally needs H≥R.",
        "zh-Hant": "須 T＞0、R＞0，時間單位一致；整個輪在地面以上還需 H≥R。"
      },
      "evidence": [
        {
          "label": {
            "en": "Upward quarter turn",
            "zh-Hant": "向上四分之一周"
          },
          "boundary": false,
          "params": {
            "R": 5,
            "H": 6,
            "T": 40,
            "t": 10,
            "phase": 0
          },
          "observation": {
            "en": "Check starting position, direction and units before trusting the model.",
            "zh-Hant": "採用模型前先檢查起點、方向及單位。"
          }
        },
        {
          "label": {
            "en": "Three-quarter turn",
            "zh-Hant": "四分之三周"
          },
          "boundary": false,
          "params": {
            "R": 5,
            "H": 6,
            "T": 40,
            "t": 30,
            "phase": 0
          },
          "observation": {
            "en": "Check starting position, direction and units before trusting the model.",
            "zh-Hant": "採用模型前先檢查起點、方向及單位。"
          }
        },
        {
          "label": {
            "en": "Invalid zero period",
            "zh-Hant": "無效零周期"
          },
          "boundary": true,
          "params": {
            "R": 5,
            "H": 6,
            "T": 0,
            "t": 10,
            "phase": 0
          },
          "observation": {
            "en": "Need R>0,T>0 and H≥R for an above-ground wheel.",
            "zh-Hant": "地面以上的摩天輪須 R＞0、T＞0、H≥R。"
          }
        }
      ]
    },
    {
      "id": "g10-08-01",
      "chapter": "08",
      "target": {
        "en": "Vector translation",
        "zh-Hant": "向量平移"
      },
      "title": {
        "en": "Move the arrow without changing the vector",
        "zh-Hant": "移動箭頭，向量有改變嗎？"
      },
      "chapterTitle": {
        "en": "Plane vectors",
        "zh-Hant": "平面向量"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=vectors&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-08-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-08-01/"
      },
      "coreClaim": {
        "en": "A free vector is determined by magnitude and direction, not by its position. Translation preserves its component differences.",
        "zh-Hant": "自由向量由大小及方向決定，不由位置決定；平移不改變終點與起點的坐標差。"
      },
      "conditions": {
        "en": "Use directed displacement. The zero vector has magnitude0 but no defined direction.",
        "zh-Hant": "須使用有向位移；零向量的模為 0，但方向未定。"
      },
      "evidence": [
        {
          "label": {
            "en": "Origin start",
            "zh-Hant": "由原點開始"
          },
          "boundary": false,
          "params": {
            "x": 0,
            "y": 0,
            "dx": 3,
            "dy": 2
          },
          "observation": {
            "en": "Position changed; components and magnitude determine the vector.",
            "zh-Hant": "位置可變，分量及模決定向量。"
          }
        },
        {
          "label": {
            "en": "Translated start",
            "zh-Hant": "起點平移"
          },
          "boundary": false,
          "params": {
            "x": 2,
            "y": -1,
            "dx": 3,
            "dy": 2
          },
          "observation": {
            "en": "Position changed; components and magnitude determine the vector.",
            "zh-Hant": "位置可變，分量及模決定向量。"
          }
        },
        {
          "label": {
            "en": "Zero vector",
            "zh-Hant": "零向量"
          },
          "boundary": true,
          "params": {
            "x": 2,
            "y": -1,
            "dx": 0,
            "dy": 0
          },
          "observation": {
            "en": "The zero vector has no defined direction.",
            "zh-Hant": "零向量沒有確定方向。"
          }
        }
      ]
    },
    {
      "id": "g10-08-02",
      "chapter": "08",
      "target": {
        "en": "Vector addition and subtraction",
        "zh-Hant": "向量加減"
      },
      "title": {
        "en": "A detour and its final displacement",
        "zh-Hant": "繞路與最後位移"
      },
      "chapterTitle": {
        "en": "Plane vectors",
        "zh-Hant": "平面向量"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=vectors&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-08-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-08-02/"
      },
      "coreClaim": {
        "en": "Vector addition joins arrows head-to-tail and adds components. Subtraction adds the opposite arrow; path length is a different quantity.",
        "zh-Hant": "向量相加把箭頭首尾相接，分量相加；相減則加上相反向量，路程是另一個量。"
      },
      "conditions": {
        "en": "All arrows use the same coordinate scale. Resultant magnitude need not equal the sum of magnitudes.",
        "zh-Hant": "所有箭頭須使用同一坐標尺度；合向量的模未必等於兩模之和。"
      },
      "evidence": [
        {
          "label": {
            "en": "Add displacements",
            "zh-Hant": "相加位移"
          },
          "boundary": false,
          "params": {
            "ax": 2,
            "ay": 1,
            "bx": -1,
            "by": 3,
            "operation": "add"
          },
          "observation": {
            "en": "The resultant joins the original start to the final endpoint.",
            "zh-Hant": "合位移連接最初起點與最後終點。"
          }
        },
        {
          "label": {
            "en": "Subtract displacements",
            "zh-Hant": "相減位移"
          },
          "boundary": false,
          "params": {
            "ax": 2,
            "ay": 1,
            "bx": -1,
            "by": 3,
            "operation": "subtract"
          },
          "observation": {
            "en": "The resultant joins the original start to the final endpoint.",
            "zh-Hant": "合位移連接最初起點與最後終點。"
          }
        },
        {
          "label": {
            "en": "Return to the start",
            "zh-Hant": "回到起點"
          },
          "boundary": true,
          "params": {
            "ax": 2,
            "ay": 1,
            "bx": -2,
            "by": -1,
            "operation": "add"
          },
          "observation": {
            "en": "The resultant joins the original start to the final endpoint.",
            "zh-Hant": "合位移連接最初起點與最後終點。"
          }
        }
      ]
    },
    {
      "id": "g10-08-03",
      "chapter": "08",
      "target": {
        "en": "Scaling and collinearity",
        "zh-Hant": "數乘與共線"
      },
      "title": {
        "en": "A negative scalar reverses direction",
        "zh-Hant": "負倍數反轉方向"
      },
      "chapterTitle": {
        "en": "Plane vectors",
        "zh-Hant": "平面向量"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=vectors&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-08-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-08-03/"
      },
      "coreClaim": {
        "en": "λa has magnitude |λ||a|; λ<0 reverses direction. A zero product has no defined direction.",
        "zh-Hant": "λa 的模為 |λ||a|；λ＜0 時方向反轉，零乘積則沒有確定方向。"
      },
      "conditions": {
        "en": "Direction comparisons require both vectors to be nonzero. Scalar multiples are collinear algebraically, including a zero product.",
        "zh-Hant": "比較方向要求兩向量非零；代數上的數乘關係亦包括零乘積。"
      },
      "evidence": [
        {
          "label": {
            "en": "Enlarge",
            "zh-Hant": "放大"
          },
          "boundary": false,
          "params": {
            "ax": 2,
            "ay": 1,
            "lambda": 2
          },
          "observation": {
            "en": "Magnitude uses |λ|; direction comparison excludes zero.",
            "zh-Hant": "模使用 |λ|，比較方向須排除零。"
          }
        },
        {
          "label": {
            "en": "Reverse",
            "zh-Hant": "反向"
          },
          "boundary": false,
          "params": {
            "ax": 2,
            "ay": 1,
            "lambda": -2
          },
          "observation": {
            "en": "Magnitude uses |λ|; direction comparison excludes zero.",
            "zh-Hant": "模使用 |λ|，比較方向須排除零。"
          }
        },
        {
          "label": {
            "en": "Zero scalar",
            "zh-Hant": "零倍數"
          },
          "boundary": true,
          "params": {
            "ax": 2,
            "ay": 1,
            "lambda": 0
          },
          "observation": {
            "en": "Magnitude uses |λ|; direction comparison excludes zero.",
            "zh-Hant": "模使用 |λ|，比較方向須排除零。"
          }
        }
      ]
    },
    {
      "id": "g10-08-04",
      "chapter": "08",
      "target": {
        "en": "Basis and coordinates",
        "zh-Hant": "基底與坐標"
      },
      "title": {
        "en": "Two directions need to be independent",
        "zh-Hant": "兩個方向要互不共線"
      },
      "chapterTitle": {
        "en": "Plane vectors",
        "zh-Hant": "平面向量"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=vectors&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-08-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-08-04/"
      },
      "coreClaim": {
        "en": "Two noncollinear vectors form a basis and give unique coefficients. Collinear vectors cover only a line, and may give no solution or nonunique solutions.",
        "zh-Hant": "兩個不共線向量形成基底，係數唯一；共線時只能覆蓋一條線，可能無解或不唯一。"
      },
      "conditions": {
        "en": "The determinant e₁x e₂y−e₁y e₂x must be nonzero for a plane basis.",
        "zh-Hant": "平面基底要求行列式 e₁x e₂y−e₁y e₂x 非零。"
      },
      "evidence": [
        {
          "label": {
            "en": "Standard basis",
            "zh-Hant": "標準基底"
          },
          "boundary": false,
          "params": {
            "e1x": 1,
            "e1y": 0,
            "e2x": 0,
            "e2y": 1,
            "x": 3,
            "y": 2
          },
          "observation": {
            "en": "The independent directions give one coefficient pair.",
            "zh-Hant": "兩個獨立方向給出唯一係數。"
          }
        },
        {
          "label": {
            "en": "Oblique basis",
            "zh-Hant": "斜基底"
          },
          "boundary": false,
          "params": {
            "e1x": 1,
            "e1y": 1,
            "e2x": -1,
            "e2y": 1,
            "x": 3,
            "y": 2
          },
          "observation": {
            "en": "The independent directions give one coefficient pair.",
            "zh-Hant": "兩個獨立方向給出唯一係數。"
          }
        },
        {
          "label": {
            "en": "Dependent directions",
            "zh-Hant": "方向相關"
          },
          "boundary": true,
          "params": {
            "e1x": 1,
            "e1y": 0,
            "e2x": 2,
            "e2y": 0,
            "x": 3,
            "y": 2
          },
          "observation": {
            "en": "Dependent directions do not form a plane basis.",
            "zh-Hant": "相關方向不能形成平面基底。"
          }
        }
      ]
    },
    {
      "id": "g10-08-05",
      "chapter": "08",
      "target": {
        "en": "Dot product and perpendicularity",
        "zh-Hant": "內積與垂直"
      },
      "title": {
        "en": "What does the sign of a dot product tell us?",
        "zh-Hant": "內積的正負告訴我們甚麼？"
      },
      "chapterTitle": {
        "en": "Plane vectors",
        "zh-Hant": "平面向量"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=vectors&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-08-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-08-05/"
      },
      "coreClaim": {
        "en": "For nonzero vectors, a·b=|a||b|cosθ is positive,zero or negative for acute,right or obtuse angles. With a zero vector the angle is undefined.",
        "zh-Hant": "兩非零向量有 a·b＝|a||b|cosθ，銳角、直角、鈍角分別對應正、零、負；若有零向量，夾角未定。"
      },
      "conditions": {
        "en": "Both vectors must be nonzero before concluding perpendicularity from a zero dot product.",
        "zh-Hant": "由零內積推論垂直前，必須確認兩向量皆非零。"
      },
      "evidence": [
        {
          "label": {
            "en": "Acute angle",
            "zh-Hant": "銳角"
          },
          "boundary": false,
          "params": {
            "lengthA": 3,
            "lengthB": 2,
            "angle": 60
          },
          "observation": {
            "en": "Positive dot product: acute angle.",
            "zh-Hant": "內積為正：銳角。"
          }
        },
        {
          "label": {
            "en": "Right angle",
            "zh-Hant": "直角"
          },
          "boundary": false,
          "params": {
            "lengthA": 3,
            "lengthB": 2,
            "angle": 90
          },
          "observation": {
            "en": "Both nonzero: zero dot product means perpendicular.",
            "zh-Hant": "兩者非零，零內積可判定垂直。"
          }
        },
        {
          "label": {
            "en": "Zero vector",
            "zh-Hant": "零向量"
          },
          "boundary": true,
          "params": {
            "lengthA": 0,
            "lengthB": 2,
            "angle": 90
          },
          "observation": {
            "en": "Zero dot product here does not establish a right angle.",
            "zh-Hant": "這裏內積為零，仍不能建立直角。"
          }
        }
      ]
    },
    {
      "id": "g10-08-06",
      "chapter": "08",
      "target": {
        "en": "Distance and midpoint",
        "zh-Hant": "兩點距離與中點"
      },
      "title": {
        "en": "Translate a midpoint and a distance",
        "zh-Hant": "平移中點與距離"
      },
      "chapterTitle": {
        "en": "Plane vectors",
        "zh-Hant": "平面向量"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=vectors&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-08-06/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-08-06/"
      },
      "coreClaim": {
        "en": "A common translation preserves endpoint differences and distance. The midpoint receives the same translation as the endpoints.",
        "zh-Hant": "共同平移不改變端點之差及距離；中點與端點同樣平移。"
      },
      "conditions": {
        "en": "Distance uses the Euclidean metric with the same unit on both axes; coincident endpoints have distance0.",
        "zh-Hant": "距離按兩軸相同單位的歐氏度量計算；兩點重合時距離為 0。"
      },
      "evidence": [
        {
          "label": {
            "en": "Original endpoints",
            "zh-Hant": "原端點"
          },
          "boundary": false,
          "params": {
            "ax": -1,
            "ay": 0,
            "bx": 3,
            "by": 2,
            "tx": 0,
            "ty": 0
          },
          "observation": {
            "en": "The midpoint translates once; the distance remains fixed.",
            "zh-Hant": "中點平移一次，距離保持不變。"
          }
        },
        {
          "label": {
            "en": "Translate both",
            "zh-Hant": "共同平移"
          },
          "boundary": false,
          "params": {
            "ax": -1,
            "ay": 0,
            "bx": 3,
            "by": 2,
            "tx": 2,
            "ty": 1
          },
          "observation": {
            "en": "The midpoint translates once; the distance remains fixed.",
            "zh-Hant": "中點平移一次，距離保持不變。"
          }
        },
        {
          "label": {
            "en": "Coincident endpoints",
            "zh-Hant": "兩點重合"
          },
          "boundary": true,
          "params": {
            "ax": 1,
            "ay": 1,
            "bx": 1,
            "by": 1,
            "tx": 2,
            "ty": 1
          },
          "observation": {
            "en": "The midpoint translates once; the distance remains fixed.",
            "zh-Hant": "中點平移一次，距離保持不變。"
          }
        }
      ]
    },
    {
      "id": "g10-09-01",
      "chapter": "09",
      "target": {
        "en": "Sine rule",
        "zh-Hant": "正弦定理"
      },
      "title": {
        "en": "A triangle and three equal ratios",
        "zh-Hant": "三角形中的三個相等比"
      },
      "chapterTitle": {
        "en": "Solving triangles",
        "zh-Hant": "解三角形"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solve-triangles&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-09-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-09-01/"
      },
      "coreClaim": {
        "en": "The sine rule gives a/sinA=b/sinB=c/sinC=2R. Holding angles fixes shape; changing one side scales every side and the common ratio.",
        "zh-Hant": "正弦定理給出 a/sinA＝b/sinB＝c/sinC＝2R。角固定形狀，一邊改變便使所有邊及共同比按同倍縮放。"
      },
      "conditions": {
        "en": "Angles A,B,C are positive and sum to180°; opposite side labels must match.",
        "zh-Hant": "A、B、C 必須為正且和為 180°；邊與其對角標記要配對。"
      },
      "evidence": [
        {
          "label": {
            "en": "30–60 triangle",
            "zh-Hant": "30–60 三角形"
          },
          "boundary": false,
          "params": {
            "A": 30,
            "B": 60,
            "a": 3
          },
          "observation": {
            "en": "The three ratios are equal; the opposite labels matter.",
            "zh-Hant": "三個比相等，但必須配對對邊與對角。"
          }
        },
        {
          "label": {
            "en": "Double scale",
            "zh-Hant": "尺度加倍"
          },
          "boundary": false,
          "params": {
            "A": 30,
            "B": 60,
            "a": 6
          },
          "observation": {
            "en": "The three ratios are equal; the opposite labels matter.",
            "zh-Hant": "三個比相等，但必須配對對邊與對角。"
          }
        },
        {
          "label": {
            "en": "Angles sum to180°",
            "zh-Hant": "角和已達 180°"
          },
          "boundary": true,
          "params": {
            "A": 90,
            "B": 90,
            "a": 3
          },
          "observation": {
            "en": "Positive sides and three positive angles summing to180° are required.",
            "zh-Hant": "須有正邊長，三個正內角之和為 180°。"
          }
        }
      ]
    },
    {
      "id": "g10-09-02",
      "chapter": "09",
      "target": {
        "en": "Cosine rule",
        "zh-Hant": "餘弦定理"
      },
      "title": {
        "en": "Pythagoras with a correction term",
        "zh-Hant": "勾股定理多了一項修正"
      },
      "chapterTitle": {
        "en": "Solving triangles",
        "zh-Hant": "解三角形"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solve-triangles&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-09-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-09-02/"
      },
      "coreClaim": {
        "en": "The cosine rule is c²=a²+b²−2abcosC. The correction vanishes at90°, is negative for acute C and positive for obtuse C.",
        "zh-Hant": "餘弦定理為 c²＝a²＋b²−2abcosC；C＝90° 時修正項消失，銳角時減少平方和，鈍角時增加。"
      },
      "conditions": {
        "en": "a,b>0 and0°<C<180° for a nondegenerate triangle; C must be their included angle.",
        "zh-Hant": "非退化三角形要求 a、b＞0 且 0°＜C＜180°；C 必須是兩邊的夾角。"
      },
      "evidence": [
        {
          "label": {
            "en": "Acute included angle",
            "zh-Hant": "銳夾角"
          },
          "boundary": false,
          "params": {
            "a": 5,
            "b": 7,
            "C": 60
          },
          "observation": {
            "en": "The correction term depends on the included angle.",
            "zh-Hant": "修正項由夾角決定。"
          }
        },
        {
          "label": {
            "en": "Right angle",
            "zh-Hant": "直角"
          },
          "boundary": false,
          "params": {
            "a": 5,
            "b": 7,
            "C": 90
          },
          "observation": {
            "en": "The correction term depends on the included angle.",
            "zh-Hant": "修正項由夾角決定。"
          }
        },
        {
          "label": {
            "en": "Degenerate boundary",
            "zh-Hant": "退化邊界"
          },
          "boundary": true,
          "params": {
            "a": 5,
            "b": 7,
            "C": 180
          },
          "observation": {
            "en": "Boundary angle: a degenerate figure, not a proper triangle.",
            "zh-Hant": "邊界角形成退化圖形，並非真正三角形。"
          }
        }
      ]
    },
    {
      "id": "g10-09-03",
      "chapter": "09",
      "target": {
        "en": "SSA solution counts",
        "zh-Hant": "SSA 零一兩解"
      },
      "title": {
        "en": "The same SSA data can make two triangles",
        "zh-Hant": "同一組 SSA 資料可有兩個三角形"
      },
      "chapterTitle": {
        "en": "Solving triangles",
        "zh-Hant": "解三角形"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solve-triangles&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-09-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-09-03/"
      },
      "coreClaim": {
        "en": "The sine rule fixes sinB=b sinA/a. Both B and180°−B may satisfy it, but each must leave C=180°−A−B positive.",
        "zh-Hant": "正弦定理給出 sinB＝b sinA/a。B 與 180°−B 都可能符合，但每個候選都要使 C＝180°−A−B 為正。"
      },
      "conditions": {
        "en": "All sides and angles must be positive. A sine value above1 means no triangle; a zero remaining angle is excluded.",
        "zh-Hant": "邊與角必須為正；正弦值大於 1 表示無解，餘角為零須排除。"
      },
      "evidence": [
        {
          "label": {
            "en": "Too short:0",
            "zh-Hant": "太短：零解"
          },
          "boundary": false,
          "params": {
            "A": 30,
            "b": 10,
            "a": 4
          },
          "observation": {
            "en": "0 nondegenerate triangle(s); each candidate needs A+B<180°.",
            "zh-Hant": "0 個非退化三角形；每個候選均須 A＋B＜180°。"
          }
        },
        {
          "label": {
            "en": "Tangent:1",
            "zh-Hant": "相切：一解"
          },
          "boundary": true,
          "params": {
            "A": 30,
            "b": 10,
            "a": 5
          },
          "observation": {
            "en": "1 nondegenerate triangle(s); each candidate needs A+B<180°.",
            "zh-Hant": "1 個非退化三角形；每個候選均須 A＋B＜180°。"
          }
        },
        {
          "label": {
            "en": "Ambiguous:2",
            "zh-Hant": "雙解情況"
          },
          "boundary": false,
          "params": {
            "A": 30,
            "b": 10,
            "a": 6
          },
          "observation": {
            "en": "2 nondegenerate triangle(s); each candidate needs A+B<180°.",
            "zh-Hant": "2 個非退化三角形；每個候選均須 A＋B＜180°。"
          }
        },
        {
          "label": {
            "en": "Equal sides:1",
            "zh-Hant": "兩邊相等：一解"
          },
          "boundary": true,
          "params": {
            "A": 30,
            "b": 10,
            "a": 10
          },
          "observation": {
            "en": "1 nondegenerate triangle(s); each candidate needs A+B<180°.",
            "zh-Hant": "1 個非退化三角形；每個候選均須 A＋B＜180°。"
          }
        }
      ]
    },
    {
      "id": "g10-09-04",
      "chapter": "09",
      "target": {
        "en": "Area and included angle",
        "zh-Hant": "面積與夾角"
      },
      "title": {
        "en": "The same two sides, different areas",
        "zh-Hant": "兩邊相同，面積可以不同"
      },
      "chapterTitle": {
        "en": "Solving triangles",
        "zh-Hant": "解三角形"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solve-triangles&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-09-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-09-04/"
      },
      "coreClaim": {
        "en": "Area is ab sinC/2. It is greatest at90°; supplementary included angles have the same area but generally different third sides.",
        "zh-Hant": "面積為 ab sinC/2，在 90° 最大；互補夾角面積相同，但第三邊通常不同。"
      },
      "conditions": {
        "en": "a,b>0 and0°<C<180°. Endpoint angles give zero area and a degenerate figure.",
        "zh-Hant": "須 a、b＞0 且 0°＜C＜180°；端點角面積為零，是退化圖形。"
      },
      "evidence": [
        {
          "label": {
            "en": "30°",
            "zh-Hant": "30°"
          },
          "boundary": false,
          "params": {
            "a": 5,
            "b": 7,
            "C": 30
          },
          "observation": {
            "en": "Area follows the altitude, not only the two side lengths.",
            "zh-Hant": "面積隨高改變，並非只由兩邊長決定。"
          }
        },
        {
          "label": {
            "en": "90° maximum",
            "zh-Hant": "90° 最大值"
          },
          "boundary": false,
          "params": {
            "a": 5,
            "b": 7,
            "C": 90
          },
          "observation": {
            "en": "Area follows the altitude, not only the two side lengths.",
            "zh-Hant": "面積隨高改變，並非只由兩邊長決定。"
          }
        },
        {
          "label": {
            "en": "Zero-area boundary",
            "zh-Hant": "零面積邊界"
          },
          "boundary": true,
          "params": {
            "a": 5,
            "b": 7,
            "C": 0
          },
          "observation": {
            "en": "Boundary angle: a degenerate figure, not a proper triangle.",
            "zh-Hant": "邊界角形成退化圖形，並非真正三角形。"
          }
        }
      ]
    },
    {
      "id": "g10-09-05",
      "chapter": "09",
      "target": {
        "en": "Inaccessible distances",
        "zh-Hant": "不可到達距離測量"
      },
      "title": {
        "en": "Measure a distance without crossing the river",
        "zh-Hant": "不用過河也能量距離"
      },
      "chapterTitle": {
        "en": "Solving triangles",
        "zh-Hant": "解三角形"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solve-triangles&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-09-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-09-05/"
      },
      "coreClaim": {
        "en": "A baseline c with base angles A,B fixes C=180°−A−B. The sine rule gives AC=c sinB/sinC; its perpendicular river width is AC sinA.",
        "zh-Hant": "基線 c 及底角 A、B 給出 C＝180°−A−B；正弦定理得 AC＝c sinB/sinC，垂直河寬則為 AC sinA。"
      },
      "conditions": {
        "en": "c>0,A>0,B>0,A+B<180°. Assume a straight baseline, a common horizontal plane and accurate sighting angles.",
        "zh-Hant": "須 c＞0、A＞0、B＞0、A＋B＜180°；假設基線直、觀測在同一水平面及角度準確。"
      },
      "evidence": [
        {
          "label": {
            "en": "Unequal sighting angles",
            "zh-Hant": "不同觀測角"
          },
          "boundary": false,
          "params": {
            "baseline": 20,
            "A": 45,
            "B": 60
          },
          "observation": {
            "en": "The sightline AP and perpendicular width are different measurements.",
            "zh-Hant": "視線 AP 與垂直河寬是不同的量。"
          }
        },
        {
          "label": {
            "en": "Symmetric sighting",
            "zh-Hant": "對稱觀測"
          },
          "boundary": false,
          "params": {
            "baseline": 20,
            "A": 45,
            "B": 45
          },
          "observation": {
            "en": "The sightline AP and perpendicular width are different measurements.",
            "zh-Hant": "視線 AP 與垂直河寬是不同的量。"
          }
        },
        {
          "label": {
            "en": "No remaining angle",
            "zh-Hant": "沒有剩餘內角"
          },
          "boundary": true,
          "params": {
            "baseline": 20,
            "A": 90,
            "B": 90
          },
          "observation": {
            "en": "Need a positive baseline and A+B<180°, with both angles positive.",
            "zh-Hant": "須基線為正，兩角皆為正且 A＋B＜180°。"
          }
        }
      ]
    },
    {
      "id": "g10-10-01",
      "chapter": "10",
      "target": {
        "en": "Classifying solids",
        "zh-Hant": "柱錐台球分類"
      },
      "title": {
        "en": "Classify by structure, not by viewpoint",
        "zh-Hant": "按結構分類，不按外觀"
      },
      "chapterTitle": {
        "en": "Solid geometry",
        "zh-Hant": "立體幾何初步"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solid&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-10-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-10-01/"
      },
      "coreClaim": {
        "en": "Structure determines the class: parallel congruent bases describe prisms or cylinders; a common apex describes pyramids or cones; cutting off a smaller cone gives a frustum. A sphere has no flat base.",
        "zh-Hant": "結構決定類別：平行全等底面對應柱體；共頂點對應錐體；截去小錐得到台體；球沒有平底面。"
      },
      "conditions": {
        "en": "Use nondegenerate solids. A curved surface is not a polygonal face, and the apparent outline changes with viewpoint.",
        "zh-Hant": "使用非退化立體；曲面不是多邊形面，畫面輪廓會隨視角改變。"
      },
      "evidence": [
        {
          "label": {
            "en": "Prism",
            "zh-Hant": "棱柱"
          },
          "boundary": false,
          "params": {
            "shape": "prism",
            "view": 25
          },
          "observation": {
            "en": "Rotation changes the image, not the structural classification.",
            "zh-Hant": "旋轉改變圖像，不改變結構分類。"
          }
        },
        {
          "label": {
            "en": "Cone",
            "zh-Hant": "圓錐"
          },
          "boundary": false,
          "params": {
            "shape": "cone",
            "view": 65
          },
          "observation": {
            "en": "Rotation changes the image, not the structural classification.",
            "zh-Hant": "旋轉改變圖像，不改變結構分類。"
          }
        },
        {
          "label": {
            "en": "No base or apex",
            "zh-Hant": "沒有底面與頂點"
          },
          "boundary": true,
          "params": {
            "shape": "sphere",
            "view": 25
          },
          "observation": {
            "en": "Rotation changes the image, not the structural classification.",
            "zh-Hant": "旋轉改變圖像，不改變結構分類。"
          }
        }
      ]
    },
    {
      "id": "g10-10-02",
      "chapter": "10",
      "target": {
        "en": "Oblique projection and true length",
        "zh-Hant": "斜二測與真長"
      },
      "title": {
        "en": "A drawing does not preserve every length",
        "zh-Hant": "圖上長度未必是真長"
      },
      "chapterTitle": {
        "en": "Solid geometry",
        "zh-Hant": "立體幾何初步"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solid&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-10-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-10-02/"
      },
      "coreClaim": {
        "en": "An oblique representation keeps the chosen front axis and scales/reorients depth. Parallel segments remain parallel, but general lengths and angles do not remain true.",
        "zh-Hant": "斜二測保留選定的正面軸，並縮短及傾斜縱深；平行線仍平行，一般長度及角度卻不保真。"
      },
      "conditions": {
        "en": "The picture uses a specified depth scale q>0 and nonparallel projected axes. A collapsed drawing cannot recover depth.",
        "zh-Hant": "圖須指定正的縱深比例 q，投影兩軸不能平行；壓成一直線後無法由圖恢復縱深。"
      },
      "evidence": [
        {
          "label": {
            "en": "Half-depth square",
            "zh-Hant": "縱深半長正方形"
          },
          "boundary": false,
          "params": {
            "a": 4,
            "b": 4,
            "q": 0.5,
            "angle": 45
          },
          "observation": {
            "en": "The drawn depth is scaled; do not measure it as a real length.",
            "zh-Hant": "圖上縱深已縮短，不能直接量作真實長度。"
          }
        },
        {
          "label": {
            "en": "Full-depth drawing",
            "zh-Hant": "縱深不縮短"
          },
          "boundary": false,
          "params": {
            "a": 4,
            "b": 4,
            "q": 1,
            "angle": 45
          },
          "observation": {
            "en": "The drawn depth is scaled; do not measure it as a real length.",
            "zh-Hant": "圖上縱深已縮短，不能直接量作真實長度。"
          }
        },
        {
          "label": {
            "en": "Collapsed axes",
            "zh-Hant": "投影軸重合"
          },
          "boundary": true,
          "params": {
            "a": 4,
            "b": 4,
            "q": 0.5,
            "angle": 0
          },
          "observation": {
            "en": "The projection has collapsed and lost depth information.",
            "zh-Hant": "投影壓成一直線，遺失縱深資訊。"
          }
        }
      ]
    },
    {
      "id": "g10-10-03",
      "chapter": "10",
      "target": {
        "en": "Nets and surface area",
        "zh-Hant": "展開圖與表面積"
      },
      "title": {
        "en": "A cone height is not its slant height",
        "zh-Hant": "圓錐高不等於母線"
      },
      "chapterTitle": {
        "en": "Solid geometry",
        "zh-Hant": "立體幾何初步"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solid&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-10-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-10-03/"
      },
      "coreClaim": {
        "en": "A cylinder unwraps to a rectangle of width2πr and height h. A cone unwraps to a sector with radius ℓ=√(r²+h²) and arc2πr, so its lateral area isπrℓ.",
        "zh-Hant": "圓柱側面展成寬 2πr、高 h 的矩形；圓錐展成半徑 ℓ＝√(r²＋h²)、弧長 2πr 的扇形，側面積為 πrℓ。"
      },
      "conditions": {
        "en": "r>0,h>0. Total surface includes the base: one for a cone,two for a cylinder. Use square units.",
        "zh-Hant": "須 r＞0、h＞0；總表面積要加底面，圓錐一個、圓柱兩個，單位為平方單位。"
      },
      "evidence": [
        {
          "label": {
            "en": "Cone net",
            "zh-Hant": "圓錐展開圖"
          },
          "boundary": false,
          "params": {
            "shape": "cone",
            "r": 3,
            "h": 4
          },
          "observation": {
            "en": "Use the unwrapped length: slant height for a cone, vertical height for a cylinder.",
            "zh-Hant": "用展開後的長度：圓錐用母線，圓柱用垂直高。"
          }
        },
        {
          "label": {
            "en": "Cylinder net",
            "zh-Hant": "圓柱展開圖"
          },
          "boundary": false,
          "params": {
            "shape": "cylinder",
            "r": 3,
            "h": 4
          },
          "observation": {
            "en": "Use the unwrapped length: slant height for a cone, vertical height for a cylinder.",
            "zh-Hant": "用展開後的長度：圓錐用母線，圓柱用垂直高。"
          }
        },
        {
          "label": {
            "en": "Zero height",
            "zh-Hant": "零高退化"
          },
          "boundary": true,
          "params": {
            "shape": "cone",
            "r": 3,
            "h": 0
          },
          "observation": {
            "en": "Zero height is a degenerate boundary; the area formula has a limit, not an ordinary solid.",
            "zh-Hant": "零高為退化邊界；面積公式有極限值，並非一般立體。"
          }
        }
      ]
    },
    {
      "id": "g10-10-04",
      "chapter": "10",
      "target": {
        "en": "Prism and cone volume",
        "zh-Hant": "柱錐體積與垂直高"
      },
      "title": {
        "en": "The same base and vertical height",
        "zh-Hant": "同底同垂直高"
      },
      "chapterTitle": {
        "en": "Solid geometry",
        "zh-Hant": "立體幾何初步"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solid&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-10-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-10-04/"
      },
      "coreClaim": {
        "en": "Vprism=Sh andVcone=Sh/3 use the perpendicular height. The one-third relation is a geometric theorem, not a result proved by a fill animation.",
        "zh-Hant": "柱體 V＝Sh、錐體 V＝Sh/3，h 是垂直高；三分之一是幾何定理，不能只靠填滿動畫當證明。"
      },
      "conditions": {
        "en": "S>0 and h>0 for a solid; h=0 is degenerate and has zero volume. Use cubic units.",
        "zh-Hant": "立體要求 S＞0、h＞0；h＝0 是退化情況，體積為零，單位為立方單位。"
      },
      "evidence": [
        {
          "label": {
            "en": "Prism",
            "zh-Hant": "柱體"
          },
          "boundary": false,
          "params": {
            "shape": "prism",
            "S": 12,
            "h": 5,
            "view": 25
          },
          "observation": {
            "en": "Use perpendicular height, not a sloping edge.",
            "zh-Hant": "使用垂直高，不用斜邊。"
          }
        },
        {
          "label": {
            "en": "Cone",
            "zh-Hant": "錐體"
          },
          "boundary": false,
          "params": {
            "shape": "cone",
            "S": 12,
            "h": 5,
            "view": 25
          },
          "observation": {
            "en": "Use perpendicular height, not a sloping edge.",
            "zh-Hant": "使用垂直高，不用斜邊。"
          }
        },
        {
          "label": {
            "en": "Zero perpendicular height",
            "zh-Hant": "垂直高為零"
          },
          "boundary": true,
          "params": {
            "shape": "cone",
            "S": 12,
            "h": 0,
            "view": 25
          },
          "observation": {
            "en": "Zero height collapses the solid.",
            "zh-Hant": "零高使立體退化。"
          }
        }
      ]
    },
    {
      "id": "g10-10-05",
      "chapter": "10",
      "target": {
        "en": "Frustum decomposition",
        "zh-Hant": "台體分割"
      },
      "title": {
        "en": "A frustum is a difference of cones",
        "zh-Hant": "圓台是兩個圓錐之差"
      },
      "chapterTitle": {
        "en": "Solid geometry",
        "zh-Hant": "立體幾何初步"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solid&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-10-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-10-05/"
      },
      "coreClaim": {
        "en": "Subtract similar cones to obtain V=πh(R²+Rr+r²)/3. Simply averaging end areas omits the way intermediate cross-sections vary.",
        "zh-Hant": "由相似圓錐相減得 V＝πh(R²＋Rr＋r²)/3；只平均上下底面積，會忽略中間截面的變化。"
      },
      "conditions": {
        "en": "R>0,0≤r≤R,h>0. r=0 gives a cone; r=R gives the cylinder limit.",
        "zh-Hant": "須 R＞0、0≤r≤R、h＞0；r＝0 為圓錐，r＝R 為圓柱極限。"
      },
      "evidence": [
        {
          "label": {
            "en": "A frustum",
            "zh-Hant": "一般圓台"
          },
          "boundary": false,
          "params": {
            "R": 3,
            "r": 1,
            "h": 4,
            "view": 25
          },
          "observation": {
            "en": "The missing cross-term matters; averaging end areas overestimates.",
            "zh-Hant": "交叉項不能漏；平均上下底面積會高估。"
          }
        },
        {
          "label": {
            "en": "Cone boundary",
            "zh-Hant": "圓錐邊界"
          },
          "boundary": true,
          "params": {
            "R": 3,
            "r": 0,
            "h": 4,
            "view": 25
          },
          "observation": {
            "en": "Zero top radius gives a cone.",
            "zh-Hant": "上底半徑為零時成為圓錐。"
          }
        },
        {
          "label": {
            "en": "Cylinder boundary",
            "zh-Hant": "圓柱邊界"
          },
          "boundary": true,
          "params": {
            "R": 3,
            "r": 3,
            "h": 4,
            "view": 25
          },
          "observation": {
            "en": "Equal radii give the cylinder limit.",
            "zh-Hant": "半徑相等時成為圓柱極限。"
          }
        }
      ]
    },
    {
      "id": "g10-10-06",
      "chapter": "10",
      "target": {
        "en": "Sphere measures and scaling",
        "zh-Hant": "球的面積體積與縮放"
      },
      "title": {
        "en": "Double a sphere: four or eight times?",
        "zh-Hant": "球半徑加倍：四倍還是八倍？"
      },
      "chapterTitle": {
        "en": "Solid geometry",
        "zh-Hant": "立體幾何初步"
      },
      "kind": "supplement",
      "sourceLesson": "/math-studio/?lesson=solid&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-10-06/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-10-06/"
      },
      "coreClaim": {
        "en": "A sphere has surface area4πr² and volume4πr³/3. Scaling radius by k changes area by k² and volume by k³.",
        "zh-Hant": "球表面積為 4πr²，體積為 4πr³/3；半徑乘 k，面積乘 k²、體積乘 k³。"
      },
      "conditions": {
        "en": "r>0,k>0 for spheres; k=0 collapses to a point, not a sphere. Area and volume use different units.",
        "zh-Hant": "球要求 r＞0、k＞0；k＝0 壓成一點，已不是球；面積與體積單位不同。"
      },
      "evidence": [
        {
          "label": {
            "en": "Same radius",
            "zh-Hant": "半徑不變"
          },
          "boundary": false,
          "params": {
            "r": 1,
            "k": 1
          },
          "observation": {
            "en": "Area uses the square factor; volume uses the cube factor.",
            "zh-Hant": "面積按平方倍數，體積按立方倍數。"
          }
        },
        {
          "label": {
            "en": "Double radius",
            "zh-Hant": "半徑加倍"
          },
          "boundary": false,
          "params": {
            "r": 1,
            "k": 2
          },
          "observation": {
            "en": "Area uses the square factor; volume uses the cube factor.",
            "zh-Hant": "面積按平方倍數，體積按立方倍數。"
          }
        },
        {
          "label": {
            "en": "Collapse to a point",
            "zh-Hant": "壓成一點"
          },
          "boundary": true,
          "params": {
            "r": 1,
            "k": 0
          },
          "observation": {
            "en": "The scaled object is a point, not a sphere.",
            "zh-Hant": "縮放後只剩一點，已不是球。"
          }
        }
      ]
    },
    {
      "id": "g10-11-01",
      "chapter": "11",
      "target": {
        "en": "Direct variation",
        "zh-Hant": "正變"
      },
      "title": {
        "en": "Which power controls the multiplier?",
        "zh-Hant": "哪個次方決定倍率？"
      },
      "chapterTitle": {
        "en": "Variation",
        "zh-Hant": "變數法"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=variation&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-11-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-11-01/"
      },
      "coreClaim": {
        "en": "The fixed invariant is y/xⁿ=k. Doubling x multiplies y by 2ⁿ when a ratio is defined. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "不變量為 y/xⁿ=k；比例有定義時，x 加倍令 y 乘 2ⁿ。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "Here x>0 and n∈{1,2,3}. If k=0, both outputs vanish and their ratio is undefined.",
        "zh-Hant": "此處 x>0、n∈{1,2,3}；k=0 時兩輸出皆零，比值沒有定義。"
      },
      "evidence": [
        {
          "label": {
            "en": "Linear direct variation",
            "zh-Hant": "一次正變"
          },
          "boundary": false,
          "params": {
            "k": 2,
            "n": 1,
            "x": 1
          },
          "observation": {
            "en": "Output multiplier: 2; identity multiplier 2ⁿ=2.",
            "zh-Hant": "輸出比：2；恆等式倍率 2ⁿ=2。"
          }
        },
        {
          "label": {
            "en": "Square variation",
            "zh-Hant": "平方正變"
          },
          "boundary": false,
          "params": {
            "k": 2,
            "n": 2,
            "x": 1
          },
          "observation": {
            "en": "Output multiplier: 4; identity multiplier 2ⁿ=4.",
            "zh-Hant": "輸出比：4；恆等式倍率 2ⁿ=4。"
          }
        },
        {
          "label": {
            "en": "Zero proportionality constant",
            "zh-Hant": "比例常數為零"
          },
          "boundary": true,
          "params": {
            "k": 0,
            "n": 3,
            "x": 2
          },
          "observation": {
            "en": "Output multiplier: undefined (0/0); identity multiplier 2ⁿ=8.",
            "zh-Hant": "輸出比：無定義（0/0）；恆等式倍率 2ⁿ=8。"
          }
        }
      ]
    },
    {
      "id": "g10-11-02",
      "chapter": "11",
      "target": {
        "en": "Inverse variation",
        "zh-Hant": "反變"
      },
      "title": {
        "en": "A fixed product instead of a fixed ratio",
        "zh-Hant": "固定乘積而非固定比"
      },
      "chapterTitle": {
        "en": "Variation",
        "zh-Hant": "變數法"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=variation&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-11-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-11-02/"
      },
      "coreClaim": {
        "en": "The product xy=k is fixed; for nonzero t the new output is y/t. x=0 is excluded. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "乘積 xy=k 固定；非零 t 下，新輸出為 y/t；x=0 被排除。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "x≠0 and t>0; even if k=0, the original expression k/x remains undefined at zero.",
        "zh-Hant": "x≠0、t>0；即使 k=0，原式 k/x 在零仍沒有定義。"
      },
      "evidence": [
        {
          "label": {
            "en": "Double a positive input",
            "zh-Hant": "正輸入加倍"
          },
          "boundary": false,
          "params": {
            "k": 4,
            "x": 1,
            "t": 2
          },
          "observation": {
            "en": "The fixed product is 4; y(tx)=y(x)/2.",
            "zh-Hant": "固定乘積為 4；y(tx)=y(x)/2。"
          }
        },
        {
          "label": {
            "en": "Negative input",
            "zh-Hant": "負輸入"
          },
          "boundary": false,
          "params": {
            "k": 4,
            "x": -2,
            "t": 2
          },
          "observation": {
            "en": "The fixed product is 4; y(tx)=y(x)/2.",
            "zh-Hant": "固定乘積為 4；y(tx)=y(x)/2。"
          }
        },
        {
          "label": {
            "en": "Excluded zero input",
            "zh-Hant": "排除零輸入"
          },
          "boundary": true,
          "params": {
            "k": 0,
            "x": 0,
            "t": 2
          },
          "observation": {
            "en": "x=0 is excluded even when k=0.",
            "zh-Hant": "即使 k=0，x=0 仍被排除。"
          }
        }
      ]
    },
    {
      "id": "g10-11-03",
      "chapter": "11",
      "target": {
        "en": "Joint variation",
        "zh-Hant": "聯變"
      },
      "title": {
        "en": "Change one factor or both?",
        "zh-Hant": "改一個因子還是兩個？"
      },
      "chapterTitle": {
        "en": "Variation",
        "zh-Hant": "變數法"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=variation&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-11-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-11-03/"
      },
      "coreClaim": {
        "en": "One doubled factor doubles y; two doubled factors multiply y by four. Zero inputs remain legitimate but can make ratios undefined. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "一個因子加倍使 y 加倍；兩者加倍使 y 乘四；零輸入仍合法，但比值可能無定義。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "k,x,z are real; y/(xz) is used only when xz≠0.",
        "zh-Hant": "k、x、z 為實數；y/(xz) 只在 xz≠0 時使用。"
      },
      "evidence": [
        {
          "label": {
            "en": "Two nonzero factors",
            "zh-Hant": "兩個非零因子"
          },
          "boundary": false,
          "params": {
            "k": 2,
            "x": 1,
            "z": 2
          },
          "observation": {
            "en": "Doubling one factor gives 2y; doubling both gives 4y, including zero cases.",
            "zh-Hant": "一個因子加倍得 2y；兩個加倍得 4y，包括零值情況。"
          }
        },
        {
          "label": {
            "en": "A negative factor",
            "zh-Hant": "一個負因子"
          },
          "boundary": false,
          "params": {
            "k": 2,
            "x": -1,
            "z": 2
          },
          "observation": {
            "en": "Doubling one factor gives 2y; doubling both gives 4y, including zero cases.",
            "zh-Hant": "一個因子加倍得 2y；兩個加倍得 4y，包括零值情況。"
          }
        },
        {
          "label": {
            "en": "A zero factor",
            "zh-Hant": "零因子"
          },
          "boundary": true,
          "params": {
            "k": 2,
            "x": 0,
            "z": 2
          },
          "observation": {
            "en": "Doubling one factor gives 2y; doubling both gives 4y, including zero cases.",
            "zh-Hant": "一個因子加倍得 2y；兩個加倍得 4y，包括零值情況。"
          }
        }
      ]
    },
    {
      "id": "g10-11-04",
      "chapter": "11",
      "target": {
        "en": "Partial variation",
        "zh-Hant": "部分變"
      },
      "title": {
        "en": "The fixed part changes the ratio",
        "zh-Hant": "固定部分改變比值"
      },
      "chapterTitle": {
        "en": "Variation",
        "zh-Hant": "變數法"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=variation&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-11-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-11-04/"
      },
      "coreClaim": {
        "en": "A fixed part a gives y(0)=a. Equal x increments give equal output increments, but output ratios are not generally fixed. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "固定部分 a 使 y(0)=a；相等輸入增量給相等輸出增量，但輸出比未必固定。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "x is nonnegative; a and b are model parameters, not measured empirical facts.",
        "zh-Hant": "x 非負；a、b 是模型參數，不是實測事實。"
      },
      "evidence": [
        {
          "label": {
            "en": "A fixed starting amount",
            "zh-Hant": "固定起始量"
          },
          "boundary": false,
          "params": {
            "a": 3,
            "b": 2,
            "x": 2
          },
          "observation": {
            "en": "Doubling difference: y(2x)−2y(x)=-3=−a",
            "zh-Hant": "加倍差: y(2x)−2y(x)=-3=−a"
          }
        },
        {
          "label": {
            "en": "Direct variation boundary",
            "zh-Hant": "正變邊界"
          },
          "boundary": true,
          "params": {
            "a": 0,
            "b": 2,
            "x": 2
          },
          "observation": {
            "en": "Doubling difference: y(2x)−2y(x)=0=−a",
            "zh-Hant": "加倍差: y(2x)−2y(x)=0=−a"
          }
        },
        {
          "label": {
            "en": "Zero input",
            "zh-Hant": "零輸入"
          },
          "boundary": true,
          "params": {
            "a": 3,
            "b": 2,
            "x": 0
          },
          "observation": {
            "en": "Doubling difference: y(2x)−2y(x)=-3=−a",
            "zh-Hant": "加倍差: y(2x)−2y(x)=-3=−a"
          }
        }
      ]
    },
    {
      "id": "g10-12-01",
      "chapter": "12",
      "target": {
        "en": "Greatest common polynomial factor",
        "zh-Hant": "最大公因式"
      },
      "title": {
        "en": "Which factor copies are shared?",
        "zh-Hant": "哪些因式副本是共有的？"
      },
      "chapterTitle": {
        "en": "Polynomial factors and multiples",
        "zh-Hant": "多項式公因式與公倍式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=polynomial-gcd&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-12-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-12-01/"
      },
      "coreClaim": {
        "en": "A common divisor cannot use more copies than either polynomial contains, so it uses the minimum exponent of each factor. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "公因式不能比任一多項式有更多副本，因此各因式取最低次冪。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "Both polynomials are monic and nonzero. Exponent zero contributes the factor 1.",
        "zh-Hant": "兩式首一且非零；零次方提供因式 1。"
      },
      "evidence": [
        {
          "label": {
            "en": "Two shared factors",
            "zh-Hant": "兩個共有因式"
          },
          "boundary": false,
          "params": {
            "m": 2,
            "n": 1,
            "p": 1,
            "q": 3
          },
          "observation": {
            "en": "Monic result: GCD = (x−1)^1(x+1)^1",
            "zh-Hant": "首一結果: GCD = (x−1)^1(x+1)^1"
          }
        },
        {
          "label": {
            "en": "Coprime factors",
            "zh-Hant": "互質因式"
          },
          "boundary": true,
          "params": {
            "m": 2,
            "n": 0,
            "p": 0,
            "q": 2
          },
          "observation": {
            "en": "Monic result: GCD = (x−1)^0(x+1)^0",
            "zh-Hant": "首一結果: GCD = (x−1)^0(x+1)^0"
          }
        },
        {
          "label": {
            "en": "One polynomial is one",
            "zh-Hant": "一式為一"
          },
          "boundary": true,
          "params": {
            "m": 0,
            "n": 0,
            "p": 2,
            "q": 1
          },
          "observation": {
            "en": "Monic result: GCD = (x−1)^0(x+1)^0",
            "zh-Hant": "首一結果: GCD = (x−1)^0(x+1)^0"
          }
        }
      ]
    },
    {
      "id": "g10-12-02",
      "chapter": "12",
      "target": {
        "en": "Least common polynomial multiple",
        "zh-Hant": "最小公倍式"
      },
      "title": {
        "en": "The smallest shared multiple",
        "zh-Hant": "最少需要哪些因式？"
      },
      "chapterTitle": {
        "en": "Polynomial factors and multiples",
        "zh-Hant": "多項式公因式與公倍式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=polynomial-gcd&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-12-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-12-02/"
      },
      "coreClaim": {
        "en": "A common multiple must contain enough copies for both source polynomials, so each exponent is the maximum, not the sum. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "公倍式須有足夠副本供兩原式整除，因此各次冪取最大值而非相加。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "All polynomials are monic and nonzero, with only the displayed distinct factors.",
        "zh-Hant": "所有多項式首一且非零，只含所示兩個不同因式。"
      },
      "evidence": [
        {
          "label": {
            "en": "Shared and unshared copies",
            "zh-Hant": "共有與非共有副本"
          },
          "boundary": false,
          "params": {
            "m": 2,
            "n": 1,
            "p": 1,
            "q": 3
          },
          "observation": {
            "en": "Monic result: LCM = (x−1)^2(x+1)^3",
            "zh-Hant": "首一結果: LCM = (x−1)^2(x+1)^3"
          }
        },
        {
          "label": {
            "en": "Equal polynomials",
            "zh-Hant": "兩式相等"
          },
          "boundary": true,
          "params": {
            "m": 2,
            "n": 1,
            "p": 2,
            "q": 1
          },
          "observation": {
            "en": "Monic result: LCM = (x−1)^2(x+1)^1",
            "zh-Hant": "首一結果: LCM = (x−1)^2(x+1)^1"
          }
        },
        {
          "label": {
            "en": "One polynomial is one",
            "zh-Hant": "一式為一"
          },
          "boundary": true,
          "params": {
            "m": 0,
            "n": 0,
            "p": 2,
            "q": 1
          },
          "observation": {
            "en": "Monic result: LCM = (x−1)^2(x+1)^1",
            "zh-Hant": "首一結果: LCM = (x−1)^2(x+1)^1"
          }
        }
      ]
    },
    {
      "id": "g10-12-03",
      "chapter": "12",
      "target": {
        "en": "Euclidean algorithm",
        "zh-Hant": "輾轉相除"
      },
      "title": {
        "en": "Why do remainders preserve the GCD?",
        "zh-Hant": "餘式為何保留最大公因式？"
      },
      "chapterTitle": {
        "en": "Polynomial factors and multiples",
        "zh-Hant": "多項式公因式與公倍式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=polynomial-gcd&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-12-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-12-03/"
      },
      "coreClaim": {
        "en": "The last nonzero Euclidean remainder, made monic, is the GCD. Sometimes B already divides A. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "輾轉相除最後一個非零餘式首一化後是最大公因式；有時 B 已整除 A。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "The displayed polynomials are nonzero; each remainder has smaller degree. The zero remainder terminates the algorithm.",
        "zh-Hant": "所示多項式非零；每步餘式次數下降；零餘式使演算法結束。"
      },
      "evidence": [
        {
          "label": {
            "en": "A linear GCD",
            "zh-Hant": "一次最大公因式"
          },
          "boundary": false,
          "params": {
            "r": 1,
            "u": 1,
            "v": 2
          },
          "observation": {
            "en": "Monic GCD: x −1",
            "zh-Hant": "首一最大公因式: x −1"
          }
        },
        {
          "label": {
            "en": "B divides A",
            "zh-Hant": "B 整除 A"
          },
          "boundary": true,
          "params": {
            "r": 1,
            "u": -1,
            "v": 1
          },
          "observation": {
            "en": "Monic GCD: x^2 −1",
            "zh-Hant": "首一最大公因式: x^2 −1"
          }
        },
        {
          "label": {
            "en": "A root at zero",
            "zh-Hant": "零為共有根"
          },
          "boundary": true,
          "params": {
            "r": 0,
            "u": 0,
            "v": 0
          },
          "observation": {
            "en": "Monic GCD: x^2",
            "zh-Hant": "首一最大公因式: x^2"
          }
        }
      ]
    },
    {
      "id": "g10-13-01",
      "chapter": "13",
      "target": {
        "en": "Distinct linear factors",
        "zh-Hant": "不同一次因式"
      },
      "title": {
        "en": "Two denominators, one identity",
        "zh-Hant": "兩個分母，一條恆等式"
      },
      "chapterTitle": {
        "en": "Partial fractions",
        "zh-Hant": "部分分式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=partial-fractions&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-13-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-13-01/"
      },
      "coreClaim": {
        "en": "After clearing denominators, mx+n=A(x−b)+B(x−a) is a polynomial identity. Substitution there is valid even at a or b. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "消分母後，mx+n=A(x−b)+B(x−a) 是多項式恆等式，在 a、b 處亦可合法代入。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "a≠b; the original fractions require x≠a,b.",
        "zh-Hant": "a≠b；原分式要求 x≠a,b。"
      },
      "evidence": [
        {
          "label": {
            "en": "Distinct linear factors",
            "zh-Hant": "不同一次因式"
          },
          "boundary": false,
          "params": {
            "a": 0,
            "b": -1,
            "m": 0,
            "n": 1
          },
          "observation": {
            "en": "Polynomial part=0; A=1, B=-1; x≠0,-1",
            "zh-Hant": "多項式部分=0; A=1, B=-1; x≠0,-1"
          }
        },
        {
          "label": {
            "en": "A numerator cancellation",
            "zh-Hant": "分子可約去因式"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": -1,
            "m": 1,
            "n": -1
          },
          "observation": {
            "en": "Polynomial part=0; A=0, B=1; x≠1,-1",
            "zh-Hant": "多項式部分=0; A=0, B=1; x≠1,-1"
          }
        },
        {
          "label": {
            "en": "Repeated factors need a new form",
            "zh-Hant": "重複因式須換形式"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": 1,
            "m": 1,
            "n": 2
          },
          "observation": {
            "en": "This distinct-root model requires a≠b. Use the repeated-factor activity instead.",
            "zh-Hant": "此不同根模型要求 a≠b；請改用重因式活動。"
          }
        }
      ]
    },
    {
      "id": "g10-13-02",
      "chapter": "13",
      "target": {
        "en": "Repeated factors",
        "zh-Hant": "重複因式"
      },
      "title": {
        "en": "Why include every repeated power?",
        "zh-Hant": "重因式為何逐次冪列出？"
      },
      "chapterTitle": {
        "en": "Partial fractions",
        "zh-Hant": "部分分式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=partial-fractions&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-13-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-13-02/"
      },
      "coreClaim": {
        "en": "For numerator mx+n, coefficient comparison gives A=m and B=ma+n. Dropping A removes an entire variable term unless m=0. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "分子為 mx+n 時，比較係數得 A=m、B=ma+n；除非 m=0，刪 A 會漏掉變量項。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "x≠a. The repeated-root form is used even if a numerator cancellation is possible.",
        "zh-Hant": "x≠a；即使分子可約去因式，仍使用重根形式。"
      },
      "evidence": [
        {
          "label": {
            "en": "Both powers are needed",
            "zh-Hant": "兩個次冪都需要"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "m": 1,
            "n": 0
          },
          "observation": {
            "en": "Repeated-factor coefficients: A=1, B=1; x≠1",
            "zh-Hant": "重因式係數: A=1, B=1; x≠1"
          }
        },
        {
          "label": {
            "en": "First coefficient is zero",
            "zh-Hant": "一次項係數為零"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "m": 0,
            "n": 2
          },
          "observation": {
            "en": "Repeated-factor coefficients: A=0, B=2; x≠1",
            "zh-Hant": "重因式係數: A=0, B=2; x≠1"
          }
        },
        {
          "label": {
            "en": "Second coefficient is zero",
            "zh-Hant": "二次項係數為零"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "m": 1,
            "n": -1
          },
          "observation": {
            "en": "Repeated-factor coefficients: A=1, B=0; x≠1",
            "zh-Hant": "重因式係數: A=1, B=0; x≠1"
          }
        }
      ]
    },
    {
      "id": "g10-13-03",
      "chapter": "13",
      "target": {
        "en": "Improper fractions",
        "zh-Hant": "先除法的假分式"
      },
      "title": {
        "en": "Divide before decomposing",
        "zh-Hant": "先除法，再分解"
      },
      "chapterTitle": {
        "en": "Partial fractions",
        "zh-Hant": "部分分式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=partial-fractions&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-13-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-13-03/"
      },
      "coreClaim": {
        "en": "The quotient is 1. Only the lower-degree remainder is decomposed into simple fractions. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "商為 1；只有較低次餘式才再拆為簡單分式。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "a≠b and x≠a,b. The numerator and denominator are monic quadratics.",
        "zh-Hant": "a≠b、x≠a,b；分子及分母均為首一二次式。"
      },
      "evidence": [
        {
          "label": {
            "en": "A nonzero remainder",
            "zh-Hant": "非零餘式"
          },
          "boundary": false,
          "params": {
            "p": 2,
            "q": 3,
            "a": 1,
            "b": -1
          },
          "observation": {
            "en": "Polynomial part=1; A=3, B=-1; x≠1,-1",
            "zh-Hant": "多項式部分=1; A=3, B=-1; x≠1,-1"
          }
        },
        {
          "label": {
            "en": "Zero remainder, two holes",
            "zh-Hant": "餘式零但仍有兩禁值"
          },
          "boundary": true,
          "params": {
            "p": 0,
            "q": -1,
            "a": 1,
            "b": -1
          },
          "observation": {
            "en": "Polynomial part=1; A=0, B=0; x≠1,-1",
            "zh-Hant": "多項式部分=1; A=0, B=0; x≠1,-1"
          }
        },
        {
          "label": {
            "en": "A cancelled factor",
            "zh-Hant": "可約去一因式"
          },
          "boundary": true,
          "params": {
            "p": 0,
            "q": 0,
            "a": 0,
            "b": 1
          },
          "observation": {
            "en": "Polynomial part=1; A=0, B=1; x≠0,1",
            "zh-Hant": "多項式部分=1; A=0, B=1; x≠0,1"
          }
        }
      ]
    },
    {
      "id": "g10-13-04",
      "chapter": "13",
      "target": {
        "en": "Excluded original inputs",
        "zh-Hant": "化簡後保留禁值"
      },
      "title": {
        "en": "Does cancellation repair a hole?",
        "zh-Hant": "約分能填補空點嗎？"
      },
      "chapterTitle": {
        "en": "Partial fractions",
        "zh-Hant": "部分分式"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=partial-fractions&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-13-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-13-04/"
      },
      "coreClaim": {
        "en": "Cancellation preserves values only at original allowed inputs. The simplified formula may extend the domain, but the original hole remains excluded. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.",
        "zh-Hant": "約分只保留原容許輸入的值；簡式可以延伸定義域，但原空點仍須排除。 畫面上的有限例子用來形成猜想；下方論證說明它在所列條件下為何成立。"
      },
      "conditions": {
        "en": "a≠b. Original domain excludes a,b; the written simplified formula still excludes b.",
        "zh-Hant": "a≠b；原定義域排除 a、b，所寫簡式仍排除 b。"
      },
      "evidence": [
        {
          "label": {
            "en": "A normal allowed input",
            "zh-Hant": "一般容許輸入"
          },
          "boundary": false,
          "params": {
            "a": 1,
            "b": -1,
            "c": 2,
            "x": 0
          },
          "observation": {
            "en": "Original excludes 1,-1; the written simplified formula excludes -1.",
            "zh-Hant": "原式排除 1、-1；所寫簡式排除 -1。"
          }
        },
        {
          "label": {
            "en": "The cancelled root",
            "zh-Hant": "被約去的根"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": -1,
            "c": 2,
            "x": 1
          },
          "observation": {
            "en": "Original excludes 1,-1; the written simplified formula excludes -1.",
            "zh-Hant": "原式排除 1、-1；所寫簡式排除 -1。"
          }
        },
        {
          "label": {
            "en": "Both singularities are removable",
            "zh-Hant": "兩處均可去"
          },
          "boundary": true,
          "params": {
            "a": 1,
            "b": -1,
            "c": 1,
            "x": -1
          },
          "observation": {
            "en": "Original excludes 1,-1; the written simplified formula excludes -1.",
            "zh-Hant": "原式排除 1、-1；所寫簡式排除 -1。"
          }
        }
      ]
    },
    {
      "id": "g10-14-01",
      "chapter": "14",
      "target": {
        "en": "Skew lines",
        "zh-Hant": "異面直線"
      },
      "title": {
        "en": "Not intersecting does not imply parallel",
        "zh-Hant": "不相交未必平行"
      },
      "chapterTitle": {
        "en": "Parallelism and perpendicularity in space",
        "zh-Hant": "空間中的平行與垂直"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=space-relations&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-14-01/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-14-01/"
      },
      "coreClaim": {
        "en": "Space lines can intersect,be parallel,or be skew. Skew lines are nonparallel and noncoplanar, even if their projections cross.",
        "zh-Hant": "空間兩線可相交、平行或異面。異面線不平行且不共面，即使投影看似相交。"
      },
      "conditions": {
        "en": "The objects are full lines with nonzero directions, not only the visible edge segments.",
        "zh-Hant": "研究的是方向非零的完整直線，不只圖上可見的線段。"
      },
      "evidence": [
        {
          "label": {
            "en": "Parallel",
            "zh-Hant": "平行"
          },
          "boundary": false,
          "params": {
            "relation": "parallel",
            "view": 25
          },
          "observation": {
            "en": "Rotate the view; the mathematical relation stays fixed.",
            "zh-Hant": "旋轉視角，數學關係仍保持不變。"
          }
        },
        {
          "label": {
            "en": "Intersecting",
            "zh-Hant": "相交"
          },
          "boundary": true,
          "params": {
            "relation": "intersect",
            "view": 25
          },
          "observation": {
            "en": "Rotate the view; the mathematical relation stays fixed.",
            "zh-Hant": "旋轉視角，數學關係仍保持不變。"
          }
        },
        {
          "label": {
            "en": "Skew",
            "zh-Hant": "異面"
          },
          "boundary": false,
          "params": {
            "relation": "skew",
            "view": 25
          },
          "observation": {
            "en": "Rotate the view; the mathematical relation stays fixed.",
            "zh-Hant": "旋轉視角，數學關係仍保持不變。"
          }
        }
      ]
    },
    {
      "id": "g10-14-02",
      "chapter": "14",
      "target": {
        "en": "Line–plane parallelism",
        "zh-Hant": "線面平行"
      },
      "title": {
        "en": "A parallel line must lie outside the plane",
        "zh-Hant": "線面平行須在平面外"
      },
      "chapterTitle": {
        "en": "Parallelism and perpendicularity in space",
        "zh-Hant": "空間中的平行與垂直"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=space-relations&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-14-02/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-14-02/"
      },
      "coreClaim": {
        "en": "A line parallel to a direction inside a plane can either lie in that plane or remain outside it. Strict line-plane parallelism requires no intersection.",
        "zh-Hant": "直線方向與面內某線平行，仍可能整條在該面內；嚴格線面平行要求沒有交點。"
      },
      "conditions": {
        "en": "To use the criterion, the line must lie outside the plane and be parallel to a line within it.",
        "zh-Hant": "使用判定時，直線須在平面外，並平行於面內一條直線。"
      },
      "evidence": [
        {
          "label": {
            "en": "Outside parallel",
            "zh-Hant": "面外平行"
          },
          "boundary": false,
          "params": {
            "height": 1,
            "tilt": 0,
            "view": 25
          },
          "observation": {
            "en": "The direction is parallel and every point remains outside.",
            "zh-Hant": "方向平行，且所有點都在面外。"
          }
        },
        {
          "label": {
            "en": "Contained boundary",
            "zh-Hant": "落在面內"
          },
          "boundary": true,
          "params": {
            "height": 0,
            "tilt": 0,
            "view": 25
          },
          "observation": {
            "en": "The outside-plane condition fails.",
            "zh-Hant": "「平面外」的條件不成立。"
          }
        },
        {
          "label": {
            "en": "Tilted intersection",
            "zh-Hant": "傾斜相交"
          },
          "boundary": false,
          "params": {
            "height": 1,
            "tilt": -1,
            "view": 25
          },
          "observation": {
            "en": "A nonzero vertical direction gives one intersection.",
            "zh-Hant": "豎直方向非零，因而有一個交點。"
          }
        }
      ]
    },
    {
      "id": "g10-14-03",
      "chapter": "14",
      "target": {
        "en": "Plane–plane parallelism",
        "zh-Hant": "面面平行"
      },
      "title": {
        "en": "Two test lines must intersect",
        "zh-Hant": "兩條測試線必須相交"
      },
      "chapterTitle": {
        "en": "Parallelism and perpendicularity in space",
        "zh-Hant": "空間中的平行與垂直"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=space-relations&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-14-03/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-14-03/"
      },
      "coreClaim": {
        "en": "Two intersecting lines in one plane, each parallel to the other plane, determine two independent directions. Two parallel test lines determine only one direction and can miss a tilt.",
        "zh-Hant": "面內兩條相交線各自平行另一平面，才提供兩個獨立方向；兩條平行測試線只有一個方向，可能漏掉傾斜。"
      },
      "conditions": {
        "en": "The planes must be distinct. Coincident planes are not strictly parallel in the no-intersection convention used here.",
        "zh-Hant": "兩面須不同；本活動以不相交定義嚴格平行，因此重合不計平行。"
      },
      "evidence": [
        {
          "label": {
            "en": "Parallel planes",
            "zh-Hant": "兩面平行"
          },
          "boundary": false,
          "params": {
            "height": 1,
            "sx": 0,
            "sy": 0,
            "view": 25
          },
          "observation": {
            "en": "Two independent directions and distinct planes are required.",
            "zh-Hant": "須檢查兩個獨立方向，且兩面不同。"
          }
        },
        {
          "label": {
            "en": "One-direction counterexample",
            "zh-Hant": "只測一方向的反例"
          },
          "boundary": true,
          "params": {
            "height": 1,
            "sx": 0,
            "sy": 1,
            "view": 25
          },
          "observation": {
            "en": "Both parallel x test lines pass, yet the planes intersect.",
            "zh-Hant": "兩條平行 x 測試線均通過，兩面仍相交。"
          }
        },
        {
          "label": {
            "en": "Coincident planes",
            "zh-Hant": "兩面重合"
          },
          "boundary": true,
          "params": {
            "height": 0,
            "sx": 0,
            "sy": 0,
            "view": 25
          },
          "observation": {
            "en": "Coincidence is excluded from strict parallelism.",
            "zh-Hant": "重合不算嚴格平行。"
          }
        }
      ]
    },
    {
      "id": "g10-14-04",
      "chapter": "14",
      "target": {
        "en": "Line–plane perpendicularity",
        "zh-Hant": "線面垂直"
      },
      "title": {
        "en": "Perpendicular to two parallel lines is not enough",
        "zh-Hant": "垂直兩條平行線仍不足"
      },
      "chapterTitle": {
        "en": "Parallelism and perpendicularity in space",
        "zh-Hant": "空間中的平行與垂直"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=space-relations&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-14-04/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-14-04/"
      },
      "coreClaim": {
        "en": "A line perpendicular to two intersecting lines of a plane is perpendicular to every plane direction. Two parallel lines repeat the same test and do not control the other direction.",
        "zh-Hant": "直線垂直面內兩條相交線，才能垂直面內所有方向；兩平行線只是重複測同一方向。"
      },
      "conditions": {
        "en": "Use a nonzero candidate direction and two genuinely intersecting plane lines.",
        "zh-Hant": "候選方向須非零，面內兩測試線必須真正相交。"
      },
      "evidence": [
        {
          "label": {
            "en": "Normal direction",
            "zh-Hant": "法向方向"
          },
          "boundary": false,
          "params": {
            "vx": 0,
            "vy": 0,
            "vz": 1,
            "tests": "intersecting",
            "view": 25
          },
          "observation": {
            "en": "Independent tests force the two horizontal components to zero.",
            "zh-Hant": "獨立測試才迫使兩個水平分量同為零。"
          }
        },
        {
          "label": {
            "en": "Parallel-test counterexample",
            "zh-Hant": "平行測試線反例"
          },
          "boundary": true,
          "params": {
            "vx": 0,
            "vy": 1,
            "vz": 0,
            "tests": "parallel",
            "view": 25
          },
          "observation": {
            "en": "Counterexample: repeated parallel tests miss another plane direction.",
            "zh-Hant": "反例：重複平行方向的測試，漏掉另一面內方向。"
          }
        },
        {
          "label": {
            "en": "Zero direction is invalid",
            "zh-Hant": "零方向無效"
          },
          "boundary": true,
          "params": {
            "vx": 0,
            "vy": 0,
            "vz": 0,
            "tests": "intersecting",
            "view": 25
          },
          "observation": {
            "en": "A zero vector does not define a line direction.",
            "zh-Hant": "零向量不能定義直線方向。"
          }
        }
      ]
    },
    {
      "id": "g10-14-05",
      "chapter": "14",
      "target": {
        "en": "Plane–plane perpendicularity",
        "zh-Hant": "面面垂直"
      },
      "title": {
        "en": "Perpendicular planes do not make every line normal",
        "zh-Hant": "兩面垂直，不代表每條線都垂直"
      },
      "chapterTitle": {
        "en": "Parallelism and perpendicularity in space",
        "zh-Hant": "空間中的平行與垂直"
      },
      "kind": "additional",
      "sourceLesson": "/math-studio/?lesson=space-relations&grade=g10",
      "url": {
        "en": "/teaching/senior-math/inquiries/g10-14-05/",
        "zh-Hant": "/zh/teaching/senior-math/inquiries/g10-14-05/"
      },
      "coreClaim": {
        "en": "Perpendicular planes contain some lines normal to the other plane, but the common hinge line lies in both planes. Other directions can be oblique.",
        "zh-Hant": "垂直兩面內有些線垂直另一面，但共同摺痕在兩面之內；其他方向還可傾斜。"
      },
      "conditions": {
        "en": "Use the angle between the planes; at fold0° or180° the planes coincide. A candidate line must be tested separately.",
        "zh-Hant": "使用兩面的夾角；摺角 0° 或 180° 時兩面重合，候選直線仍須另行判定。"
      },
      "evidence": [
        {
          "label": {
            "en": "Across a right-angle fold",
            "zh-Hant": "直角摺面橫向線"
          },
          "boundary": false,
          "params": {
            "fold": 90,
            "direction": 90,
            "view": 25
          },
          "observation": {
            "en": "A plane relation does not replace checking the selected line.",
            "zh-Hant": "平面關係不能代替對選定直線的檢查。"
          }
        },
        {
          "label": {
            "en": "Along the hinge",
            "zh-Hant": "沿摺痕的反例"
          },
          "boundary": true,
          "params": {
            "fold": 90,
            "direction": 0,
            "view": 25
          },
          "observation": {
            "en": "The planes are perpendicular, but this line is not normal to the floor.",
            "zh-Hant": "兩面垂直，但這條線並非地面的法線。"
          }
        },
        {
          "label": {
            "en": "A45° fold",
            "zh-Hant": "45° 摺面"
          },
          "boundary": false,
          "params": {
            "fold": 45,
            "direction": 90,
            "view": 25
          },
          "observation": {
            "en": "A plane relation does not replace checking the selected line.",
            "zh-Hant": "平面關係不能代替對選定直線的檢查。"
          }
        }
      ]
    }
  ]
}