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希腊风角度沙拉：用正切与均值统计构造角度几何

希腊风角度沙拉：用正切与均值统计构造角度几何
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2023-12-20 17:28
多边形角三角函数概率统计
所属专题 Wiederholung Winkel: Bezeichnung, Arten, Messen & Zeichnen
该范例通过“多边形角度构造+正切刻画+数值检验/统计”的方式，展示一组角度样本在平面图形中的分布与显著性评估效果。核心数学概念包括：角（Angle）的几何关系、正切函数（Tangens）与角度/斜率的对应、以及算术平均（Arithmetisches Mittel）与假设检验或显著性检验（Hypothesentest oder Signifikanztest）中的统计思路。作图要点：利用 GeoGebra 的 Polygon 生成平面多边形框架；用 Max、If 与 Textfield 设定条件与可交互参数；通过 Shuffle、Sequence、Element 对选项或点进行随机抽取与序列化；用 Append、Remove、Length 管理样本集合规模；再配合 CountIf 统计满足条件的元素数量，并结合 Text、SpeechBubbleRight 输出文字说明；PointIn 用于判断点是否落在指定区域（例如 allowedRect 系列对象）内，从而形成“角度/区域命中”的计数依据。适合高中阶段（解析几何/三角函数/统计初步）或开展综合活动课：既能理解角与tan的关系，也能直观看到随机采样与显著性检验的图形化含义。

GeoGebra 命令及参数

  1. Polygon(allowedRect4, allowedRect3, allowedRect2, allowedRect1) → allowedRect, a_1, b, c, d
  2. Max(widthsOptionsR) → widthR
  3. Max(heightsOptionsR) → heightR
  4. Max(heightR, Max[heightsOptionsL]) → heightL
  5. Max(widthsOptionsL) → widthL
  6. If(isShuffleOptionsLechts, Shuffle[1…numOptionsR], 1…numOptionsR) → indizesOptionsRechts
  7. If(isShuffleOptionsLinks, Shuffle[1…numOptionsL], 1…numOptionsL) → indizesOptionsLinks
  8. Textfield(input1) → InputBox1
  9. Textfield(input2) → InputBox2
  10. Polygon(PtopLeftCornerFeedback, PtopRightCornerFeedback, PbottomRightCornerFeedback, PbottomLeftCornerFeedback) → rectangleCorrect, rectangleCorrectS3, rectangleCorrectS4, rectangleCorrectS2, rectangleCorrectS1
  11. Shuffle(Sequence[Random[0, 10], i, 1, 100]) → randomNosForX, {9, 8, 8, 5, 3, 6, 5, 7, 6, 8, 4, 2, 4, 1, 9, 2, 5, 6, 9, 8, 1, 9, 0, 5, 5, 1, 9, 4, 4, 4, 10, 3, 3, 3, 1, 3, 7, 9, 7, 9, 6, 10, 10, 2, 9, 6, 5, 0, 9, 7, 9, 10, 7, 6, 7, 8, 6, 3, 8, 10, 0, 7, 6, 4, 2, 6, 1, 3, 2, 6, 0, 3, 7, 7, 7, 2, 2, 1, 6, 10, 0, 3, 8, 2, 6, 2, 2, 4, 2, 7, 3, 10, 4, 7, 6, 1, 8, 8, 3, 2}
  12. Element(randomNosForX, Mod[questionCount, 100] + 1) → numX
  13. Shuffle(Sequence[Random[0, 10], i, 1, 100]) → randomNosForY, {6, 3, 10, 2, 2, 9, 2, 3, 9, 3, 10, 3, 0, 5, 8, 2, 8, 8, 4, 7, 1, 2, 6, 0, 8, 9, 0, 6, 0, 5, 5, 8, 10, 7, 3, 9, 2, 7, 2, 8, 3, 8, 1, 4, 5, 8, 7, 8, 9, 8, 5, 4, 5, 2, 8, 5, 6, 2, 3, 8, 8, 9, 9, 7, 2, 9, 0, 8, 5, 3, 1, 10, 5, 4, 6, 4, 4, 2, 5, 10, 5, 2, 4, 6, 8, 2, 6, 4, 7, 4, 2, 4, 2, 7, 4, 7, 8, 4, 8, 9}
  14. Element(randomNosForY, Mod[questionCount, 100] + 1) → numY
  15. Append(true, Sequence[If[randomNosForX(i) ≟ randomNosForX(i - 1), false, true], i, 2, 100]) → stayX
  16. Append(true, Sequence[If[randomNosForY(i) ≟ randomNosForY(i - 1), false, true], i, 2, 100]) → stayY
  17. Remove(Sequence[If[stayX(i), randomNosForX(i), 0], i, 1, 100], Sequence[0, i, 1, 100]) → randomNosForXwithoutRepeat
  18. Remove(Sequence[If[stayY(i), randomNosForY(i), 0], i, 1, 100], Sequence[0, i, 1, 100]) → randomNosForYwithoutRepeat
  19. Element(randomNosForXwithoutRepeat, Mod[questionCount, Length[randomNosForXwithoutRepeat]] + 1) → numXwithoutRepeat
  20. Element(randomNosForYwithoutRepeat, Mod[questionCount, Length[randomNosForYwithoutRepeat]] + 1) → numYwithoutRepeat
  21. Sequence(1) → positionsOptionsRechts
  22. Sequence(Element[optionsRAll, indizesOptionsRechts(i)], i, 1, numOptionsR) → optionsRActual
  23. Length(optionsRActual) → a
  24. Sequence(Element[optionsLAll, indizesOptionsLinks(i)], i, 1, numOptionsL) → optionsLActual
  25. Sequence(allOptionsL(indizesOptionsLinks(i)), i, 1, numOptionsL) → optionsL
  26. Sequence(allOptionsR(indizesOptionsLinks(i)), i, 1, numOptionsL) → optionsR
  27. Polygon((-10, -2), (10, -2), (10, 10), (-10, 10)) → q1, q_1, r_1, s_1, t_1
  28. Polygon(E, F, G, H) → q2, e_1, f_2, g_2, h_2
  29. Polygon(I, J, K, L) → q3, i_2, j_2, k_1, l_1
  30. SpeechBubbleRight(O, 5, 2) → poly1
  31. Text(optionsLActual(1), CenterOptionL1, true, true, 0, 0) → opL1
  32. Text(optionsLActual(2), CenterOptionL2, true, true, 0, 0) → opL2
  33. Text(optionsLActual(3), CenterOptionL3, true, true, 0, 0) → opL3
  34. Text(optionsLActual(4), CenterOptionL4, true, true, 0, 0) → opL4
  35. Text(optionsLActual(5), CenterOptionL5, true, true, 0, 0) → opL5
  36. PointIn(allowedRect) → CenterOptionR1
  37. Text(optionsRActual(1), CenterOptionR1, true, true, 0, 0) → opR1
  38. Polygon(CenterOptionR1, CenterOptionR1 - (widthR / 2, heightR / 2), CenterOptionR1 + (widthR / 2, ((-heightR)) / 2), CenterOptionR1 + (widthR / 2, heightR / 2), CenterOptionR1 - (widthR / 2, ((-heightR)) / 2), CenterOptionR1 - (widthR / 2, heightR / 2)) → rectRMove1, centeroptionr1, f, g, h, i, j
  39. PointIn(allowedRect) → CenterOptionR2
  40. Text(optionsRActual(2), CenterOptionR2, true, true, 0, 0) → opR2
  41. Polygon(CenterOptionR2, CenterOptionR2 - (widthR / 2, heightR / 2), CenterOptionR2 + (widthR / 2, ((-heightR)) / 2), CenterOptionR2 + (widthR / 2, heightR / 2), CenterOptionR2 - (widthR / 2, ((-heightR)) / 2), CenterOptionR2 - (widthR / 2, heightR / 2)) → rectRMove2, centeroptionr2, k, l, m, n, p
  42. PointIn(allowedRect) → CenterOptionR3
  43. Text(optionsRActual(3), CenterOptionR3, true, true, 0, 0) → opR3
  44. Polygon(CenterOptionR3, CenterOptionR3 - (widthR / 2, heightR / 2), CenterOptionR3 + (widthR / 2, ((-heightR)) / 2), CenterOptionR3 + (widthR / 2, heightR / 2), CenterOptionR3 - (widthR / 2, ((-heightR)) / 2), CenterOptionR3 - (widthR / 2, heightR / 2)) → rectRMove3, centeroptionr3, q, r, s, t, e
  45. PointIn(allowedRect) → CenterOptionR4
  46. Text(optionsRActual(4), CenterOptionR4, true, true, 0, 0) → opR4
  47. Polygon(CenterOptionR4, CenterOptionR4 - (widthR / 2, heightR / 2), CenterOptionR4 + (widthR / 2, ((-heightR)) / 2), CenterOptionR4 + (widthR / 2, heightR / 2), CenterOptionR4 - (widthR / 2, ((-heightR)) / 2), CenterOptionR4 - (widthR / 2, heightR / 2)) → rectRMove4, centeroptionr4, f_1, g_1, h_1, i_1, j_1
  48. PointIn(allowedRect) → CenterOptionR5
  49. Text(optionsRActual(5), CenterOptionR5, true, true, 0, 0) → opR5
  50. Polygon(CenterOptionR5, CenterOptionR5 - (widthR / 2, heightR / 2), CenterOptionR5 + (widthR / 2, ((-heightR)) / 2), CenterOptionR5 + (widthR / 2, heightR / 2), CenterOptionR5 - (widthR / 2, ((-heightR)) / 2), CenterOptionR5 - (widthR / 2, heightR / 2)) → rectRMove5, centeroptionr5, m_1, n_1, b_1, c_1, d_1
  51. If(CountIf[P ∈ answerRectLarger1(1), P, verticesR1] ≟ 4, 1, -1, CountIf[P ∈ answerRectLarger2(1), P, verticesR1] ≟ 4, 2, CountIf[P ∈ answerRectLarger3(1), P, verticesR1] ≟ 4, 3, CountIf[P ∈ answerRectLarger4(1), P, verticesR1] ≟ 4, 4, CountIf[P ∈ answerRectLarger5(1), P, verticesR1] ≟ 4, 5) → indexAnswerR1
  52. If(CountIf[P ∈ answerRectLarger1(1), P, verticesR2] ≟ 4, 1, -1, CountIf[P ∈ answerRectLarger2(1), P, verticesR2] ≟ 4, 2, CountIf[P ∈ answerRectLarger3(1), P, verticesR2] ≟ 4, 3, CountIf[P ∈ answerRectLarger4(1), P, verticesR2] ≟ 4, 4, CountIf[P ∈ answerRectLarger5(1), P, verticesR2] ≟ 4, 5) → indexAnswerR2
  53. If(CountIf[P ∈ answerRectLarger1(1), P, verticesR3] ≟ 4, 1, -1, CountIf[P ∈ answerRectLarger2(1), P, verticesR3] ≟ 4, 2, CountIf[P ∈ answerRectLarger3(1), P, verticesR3] ≟ 4, 3, CountIf[P ∈ answerRectLarger4(1), P, verticesR3] ≟ 4, 4, CountIf[P ∈ answerRectLarger5(1), P, verticesR3] ≟ 4, 5) → indexAnswerR3
  54. If(CountIf[P ∈ answerRectLarger1(1), P, verticesR4] ≟ 4, 1, -1, CountIf[P ∈ answerRectLarger2(1), P, verticesR4] ≟ 4, 2, CountIf[P ∈ answerRectLarger3(1), P, verticesR4] ≟ 4, 3, CountIf[P ∈ answerRectLarger4(1), P, verticesR4] ≟ 4, 4, CountIf[P ∈ answerRectLarger5(1), P, verticesR4] ≟ 4, 5) → indexAnswerR4
  55. If(CountIf[P ∈ answerRectLarger1(1), P, verticesR5] ≟ 4, 1, -1, CountIf[P ∈ answerRectLarger2(1), P, verticesR5] ≟ 4, 2, CountIf[P ∈ answerRectLarger3(1), P, verticesR5] ≟ 4, 3, CountIf[P ∈ answerRectLarger4(1), P, verticesR5] ≟ 4, 4, CountIf[P ∈ answerRectLarger5(1), P, verticesR5] ≟ 4, 5) → indexAnswerR5
  56. CountIf(ind ≠ -1, ind, indizesAnswerR) → numAnswersDraggedToLeft
  57. If(correct, "\vphantom{✘}✓ Richtig", If[correctpartially, "\vphantom{✓}✘ Teilweise richtig", "\vphantom{✓}✘ Falsch"]) → correctTextTemp
  58. Text((("\Large\textsf{" * correctTextTemp) * "}"), PStartCorrectText, true, true, 1, 0) → correctText

AI 绘图提示语参考

画一个希腊风格的角度沙拉图，包含多个矩形和圆形的选项，表示不同角度的可能性。图中有两个圆形的输入框，用户可以输入数字，表示角度的大小。图中还包含多个矩形区域，表示不同角度的可能性，用户可以通过点击这些区域来选择角度。图中还包含一些圆形的选项，表示不同角度的可能性，用户可以通过点击这些选项来选择角度。图中还包含一些线条和点，表示角度的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。