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自然数中的平方根：几何与数学的结合

自然数中的平方根：几何与数学的结合
几何画板及绘图内容加载中，请稍等……
2026-05-18 19:28
所属专题 Quadratzahlen und Quadratwurzeln
这个范例展示了自然数中平方根的几何表示，揭示了平方根与自然数之间的关系。核心数学概念包括平方根、实数和几何作图。作图要点涉及使用GeoGebra命令如Sequence、Element、Textfield、Join、If、Segment、Text、PointIn、Corner、Circle、Point、Translate等对象。适合初中或高中数学学习，用于理解平方根的几何意义和作图技巧。

GeoGebra 命令及参数

  1. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {5, 1, 7, 1, 6, 8, 9, 8, 6, 6}
  2. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  3. Textfield(input) → inputBox
  4. Join({Sequence[Segment[Corner[4] + (RasterPixMat * ((k * RasterStepXY), 0)), Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY))], k, 0, RasterMaxX / RasterStepXY], Sequence[Segment[Corner[4] + (RasterPixMat * (0, (k * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], k, 0, RasterMaxY / RasterStepXY]}) → RasterStrecken
  5. Join({Sequence[Text[UnicodeToLetter[65 + k], Midpoint[Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY)), Corner[4] + (RasterPixMat * (((k + 1) * RasterStepXY), RasterMaxY))], false, true, 0, -1], k, 0, RasterMaxX / RasterStepXY - 1], Sequence[Text[("" * k), Midpoint[Corner[4] + (RasterPixMat * (RasterMaxX, ((k - 1) * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], false, true, 1, 0], k, 1, RasterMaxY / RasterStepXY]}) → RasterBeschriftung
  6. Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte
  7. If(step ≟ 3, 7, step ≟ 4, 3, step ≟ 5, 40, step ≟ 6, 10, 28) → actualSqrt
  8. Segment(Origin + ((minX / stepX * disX) - edgeAxisX, 0), Origin + ((maxX / stepX * disX) + edgeAxisX, 0)) → segAxisX
  9. Sequence(m, m, minX, maxX, stepX) → numbersXAxis
  10. Sequence(Origin + ((minX / stepX * disX), 0) + (k * (disX, 0)), k, 0, maxX / stepX - minX / stepX) → PointsXAxis
  11. Sequence(Segment[PointsXAxis(k) + (0, (-disSeg)), PointsXAxis(k) + (0, disSeg)], k, 1, Length[PointsXAxis]) → segXAxis
  12. Sequence(Origin + ((sqrt(k) * disX / stepX), 0), k, 1, 16) → PointsSqrtXAxis
  13. Sequence(Text[((("\textsf{\(" * If[numbersXAxis(k) ≥ 0, "\phantom{\(-\)}", ""]) * If[Element[numbersXAxis, k] ≟ 0, "", ("" * Element[numbersXAxis, k])]) * "\)}"), PointsXAxis(k) + (-0.15, (-disSeg)), false, true, 0, -1], k, 1, (maxX - minX) / stepX + 1) → CapX
  14. If(step ≟ 2, ListSqrtShow, {false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false}) → ListSqrtColor
  15. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(2), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{2}}}}}"), PointsSqrtXAxis(2) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt02
  16. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(5), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{5}}}}}"), PointsSqrtXAxis(5) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt05
  17. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(6), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{6}}}}}"), PointsSqrtXAxis(6) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt06
  18. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(8), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{8}}}}}"), PointsSqrtXAxis(8) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt08
  19. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(10), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{10}}}}}"), PointsSqrtXAxis(10) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt10
  20. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(11), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{11}}}}}"), PointsSqrtXAxis(11) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt11
  21. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(12), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{12}}}}}"), PointsSqrtXAxis(12) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt12
  22. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(13), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{13}}}}}"), PointsSqrtXAxis(13) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt13
  23. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(14), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{14}}}}}"), PointsSqrtXAxis(14) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt14
  24. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(15), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{15}}}}}"), PointsSqrtXAxis(15) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt15
  25. Textfield(input011) → InputBox011
  26. Textfield(input012) → InputBox012
  27. Textfield(input013) → InputBox013
  28. Textfield(input014) → InputBox014
  29. Textfield(input031) → InputBox031
  30. Textfield(input032) → InputBox032
  31. Textfield(input041) → InputBox041
  32. Textfield(input042) → InputBox042
  33. Textfield(input043) → InputBox043
  34. Textfield(input044) → InputBox044
  35. Textfield(input051) → InputBox051
  36. Textfield(input052) → InputBox052
  37. Textfield(input053) → InputBox053
  38. Textfield(input054) → InputBox054
  39. Textfield(input055) → InputBox055
  40. Textfield(input056) → InputBox056
  41. Textfield(input057) → InputBox057
  42. Textfield(input058) → InputBox058
  43. Textfield(input059) → InputBox059
  44. Textfield(input061) → InputBox061
  45. Textfield(input062) → InputBox062
  46. Textfield(input063) → InputBox063
  47. Textfield(input064) → InputBox064
  48. Textfield(input071) → InputBox071
  49. Textfield(input072) → InputBox072
  50. Textfield(input073) → InputBox073
  51. Textfield(input074) → InputBox074
  52. PointIn(polyAxis(1)) → PosP011
  53. PointIn(polyAxis(1)) → PosP012
  54. PointIn(polyAxis(1)) → PosP013
  55. PointIn(polyAxis(1)) → PosP014
  56. Corner(imgLupe, 3) → PosImgLupe3
  57. Circle(MPLupe, radiusLupe) → circleLupe
  58. Text("\textsf{\bold{Zieh mich!}}", Midpoint[PosImgLupe2, PosImgLupe3] + (-0.35, 0), false, true, 1, 0) → CapZiehMichLupe
  59. Sequence(If[ListSqrtShow(k) ∧ (¬(k ∈ {1, 4, 9, 16})), PointsSqrtXAxis(k)], k, 1, 16) → Points02
  60. Circle(PosP011, radiusFB01) → circleFB011
  61. Circle(PosP012, radiusFB01) → circleFB012
  62. Circle(PosP013, radiusFB01) → circleFB013
  63. Circle(PosP014, radiusFB01) → circleFB014
  64. Text((("\textsf{\sqrt{1}" * If[checkBox01, " = \phantom{ \:}", ""]) * "}"), PosP011 + (-0.12, 0.15), false, true, 0, 1) → Cap011
  65. Text((("\textsf{\sqrt{4}" * If[checkBox01, " = \phantom{ \:}", ""]) * "}"), PosP012 + (-0.12, 0.15), false, true, 0, 1) → Cap012
  66. Text((("\textsf{\sqrt{9}" * If[checkBox01, " = \phantom{ \:}", ""]) * "}"), PosP013 + (-0.12, 0.15), false, true, 0, 1) → Cap013
  67. Text((("\textsf{\sqrt{16}" * If[checkBox01, " = \phantom{ \:}", ""]) * "}"), PosP014 + (-0.12, 0.15), false, true, 0, 1) → Cap014
  68. If(step ≟ 1, PosP011 ≟ PointsXAxis(1) ∧ PosP012 ≟ PointsXAxis(2) ∧ PosP013 ≟ PointsXAxis(3) ∧ PosP014 ≟ PointsXAxis(4), step ≟ 6, correct061 ∧ correct062, true, step ≟ 2, ListSqrtShow ≟ {true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true}, step ≟ 3, correct031 ∧ correct032, step ≟ 4, correct041 ∧ correct042 ∧ correct043 ∧ correct044, step ≟ 5, correct051 ∧ correct052) → clickableBtnNext
  69. Text((("\textsf{\footnotesize{" * If[step ≟ 3, "\bold{\textcolor{db6114}{", ListSqrtColor(7), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{7}}}}}"), PointsSqrtXAxis(7) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt07
  70. Text((("\textsf{\footnotesize{" * If[step ≟ 4, "\bold{\textcolor{db6114}{", ListSqrtColor(3), (("\bold{\textcolor{" * colorSqrt) * "}{"), "{{"]) * "\sqrt{3}}}}}"), PointsSqrtXAxis(3) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt03
  71. Circle(PointsXAxis(2) + (0, (-disSeg) - 0.375), radiusBG) → circleBG02
  72. Circle(PointsXAxis(3) + (0, (-disSeg) - 0.375), radiusBG) → circleBG03
  73. Circle(PointsXAxis(4) + (0, (-disSeg) - 0.375), radiusBG) → circleBG04
  74. Circle(PointsXAxis(1) + (0, (-disSeg) - 0.375), radiusBG) → circleBG01
  75. Point(Line[PosAxisStart, yAxis]) → PosAxisStart05
  76. Translate(segXAxis, Vector[PosAxisStart, PosAxisStart05]) → segXAxis05
  77. Translate(segAxisX, Vector[PosAxisStart, PosAxisStart05]) → segAxisX05
  78. Translate(PointsXAxis, Vector[PosAxisStart, PosAxisStart05]) → PointsXAxis05
  79. Text((("\textsf{\footnotesize{\sqrt{" * If[step ≟ 5, 25, step ≟ 6, 4, 16]) * "}}}"), PointsXAxis05(1) + (-0.1, disSeg), false, true, 0, 1) → txtAxis051
  80. Text((("\textsf{\footnotesize{\sqrt{" * If[step ≟ 5, 36, step ≟ 6, 9, 25]) * "}}}"), PointsXAxis05(2) + (-0.1, disSeg), false, true, 0, 1) → txtAxis052
  81. Text((("\textsf{\footnotesize{\sqrt{" * If[step ≟ 5, 49, step ≟ 6, 16, 36]) * "}}}"), PointsXAxis05(3) + (-0.1, disSeg), false, true, 0, 1) → txtAxis053
  82. Text((("\textsf{\footnotesize{\sqrt{" * If[step ≟ 5, 64, step ≟ 6, 25, 49]) * "}}}"), PointsXAxis05(4) + (-0.1, disSeg), false, true, 0, 1) → txtAxis054
  83. Text((("\textsf{\footnotesize{\sqrt{" * actualSqrt) * "}}}"), Point05 + (-0.1, disSeg), false, true, 0, 1) → txtAxis055
  84. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(1), "\bold{\textcolor{black}{", "{{"]) * "\sqrt{1}}}}}"), PointsSqrtXAxis(1) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt01
  85. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(4), "\bold{\textcolor{black}{", "{{"]) * "\sqrt{4}}}}}"), PointsSqrtXAxis(4) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt04
  86. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(9), "\bold{\textcolor{black}{", "{{"]) * "\sqrt{9}}}}}"), PointsSqrtXAxis(9) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt09
  87. Text((("\textsf{\footnotesize{" * If[ListSqrtColor(16), "\bold{\textcolor{black}{", "{{"]) * "\sqrt{16}}}}}"), PointsSqrtXAxis(16) + (-0.1, disSeg), false, true, 0, 1) → txtSqrt16
  88. Text((("\textsf{{\bold{\textcolor{black}{" * If[step ≟ 5 ∧ correct052 ∨ step ≟ 6 ∧ correct062 ∨ step ≟ 7 ∧ correct072, (ceil(sqrt(actualSqrt)) * ""), "?"]) * "}}}}"), PointsXAxis05(3) + (0, (-disSeg)), false, true, 0, -1) → txtAxis057
  89. Text((("\textsf{{\bold{\textcolor{black}{" * If[step ≟ 5 ∧ correct051 ∨ step ≟ 6 ∧ correct061 ∨ step ≟ 7 ∧ correct071, (floor(sqrt(actualSqrt)) * ""), "?"]) * "}}}}"), PointsXAxis05(2) + (0, (-disSeg)), false, true, 0, -1) → txtAxis056
  90. If(step < 5, (2.75, 6.5), (3.75, 7.1)) → PosHintFit
  91. If(step < 5, (5.7, 6.5), (5.7, 7.1)) → PosHintNatural
  92. If(step ≟ 1, (6.55, 3.6), step ≟ 4 ∧ correct041 ∧ correct042, (6.85, 2.75), (12.6, 8.25)) → PosBtnShowSolution