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天平方程：用操作模型学习等式平衡

天平方程：用操作模型学习等式平衡
几何画板及绘图内容加载中，请稍等……
2016-08-23 22:48
函数最值优化不等式几何作图滑块动画
① 本范例通过一个可交互的天平模型，直观展示方程两边保持平衡的数学现象，学生可拖动砝码或变量块来改变方程两边的表达式。② 核心数学概念包括等式性质、线性方程、变量与常数项的组合，以及通过操作理解移项和合并同类项的原理。③ 作图要点：利用GeoGebra的Element、Polygon、OrthogonalLine、Point、Sum、If、Segment、Line、Intersect等命令构建天平结构、砝码列表和变量块，并通过ProblemPicker和NewProblem随机生成新题目，配合滑块控制表达式上下限。④ 适合初中数学方程教学，也可用于小学高年级的代数启蒙，帮助学生从具体操作过渡到抽象符号运算。

GeoGebra 命令及参数

  1. Element(LeftSideUnitList, ProblemPicker) → LeftUnitFromList
  2. Element(RightSideUnitList, ProblemPicker) → RightUnitFromList
  3. Element(RightSideXList, ProblemPicker) → RightXFromList
  4. Element(LeftSideXList, ProblemPicker) → LeftXFromList
  5. Polygon(FulcrumBottomLeft, FulcrumTop, FulcrumBottomRight) → poly1, fulcrumbottomright, fulcrumbottomleft, fulcrumtop
  6. OrthogonalLine(A_11, fulcrumtop) → b_2
  7. Point(b_2) → RightUnit5
  8. OrthogonalLine(A_12, fulcrumtop) → a_2
  9. Point(a_2) → RightUnit4
  10. OrthogonalLine(A_13, fulcrumtop) → t_1
  11. Point(t_1) → RightUnit3
  12. OrthogonalLine(A_14, fulcrumtop) → s_1
  13. Point(s_1) → RightUnit2
  14. OrthogonalLine(A_15, fulcrumtop) → r_1
  15. Point(r_1) → RightUnit1
  16. Sum({(y(RightUnit1) > ExpressionLowerLimit) ∧ (y(RightUnit1) < ExpressionUpperLimit), (y(RightUnit2) > ExpressionLowerLimit) ∧ (y(RightUnit2) < ExpressionUpperLimit), (y(RightUnit3) > ExpressionLowerLimit) ∧ (y(RightUnit3) < ExpressionUpperLimit), (y(RightUnit4) > ExpressionLowerLimit) ∧ (y(RightUnit4) < ExpressionUpperLimit), (y(RightUnit5) > ExpressionLowerLimit) ∧ (y(RightUnit5) < ExpressionUpperLimit)}) → SumRightSideUnit
  17. OrthogonalLine(A_20, fulcrumtop) → q_1
  18. Point(q_1) → LeftUnit1
  19. OrthogonalLine(A_19, fulcrumtop) → p_1
  20. Point(p_1) → LeftUnit2
  21. OrthogonalLine(A_18, fulcrumtop) → n_1
  22. Point(n_1) → LeftUnit3
  23. OrthogonalLine(A_17, fulcrumtop) → m_1
  24. Point(m_1) → LeftUnit4
  25. OrthogonalLine(A_16, fulcrumtop) → l_1
  26. Point(l_1) → LeftUnit5
  27. Sum({(y(LeftUnit1) > ExpressionLowerLimit) ∧ (y(LeftUnit1) < ExpressionUpperLimit), (y(LeftUnit2) > ExpressionLowerLimit) ∧ (y(LeftUnit2) < ExpressionUpperLimit), (y(LeftUnit3) > ExpressionLowerLimit) ∧ (y(LeftUnit3) < ExpressionUpperLimit), (y(LeftUnit4) > ExpressionLowerLimit) ∧ (y(LeftUnit4) < ExpressionUpperLimit), (y(LeftUnit5) > ExpressionLowerLimit) ∧ (y(LeftUnit5) < ExpressionUpperLimit)}) → SumLeftSideUnit
  28. OrthogonalLine(A_6, fulcrumtop) → k_1
  29. Point(k_1) → RightX5
  30. OrthogonalLine(A_7, fulcrumtop) → j_1
  31. Point(j_1) → RightX4
  32. OrthogonalLine(A_8, fulcrumtop) → g_1
  33. Point(g_1) → RightX3
  34. OrthogonalLine(A_9, fulcrumtop) → c_1
  35. Point(c_1) → RightX2
  36. OrthogonalLine(A_10, fulcrumtop) → b_1
  37. Point(b_1) → RightX1
  38. Sum({(y(RightX1) > ExpressionLowerLimit) ∧ (y(RightX1) < ExpressionUpperLimit), (y(RightX2) > ExpressionLowerLimit) ∧ (y(RightX2) < ExpressionUpperLimit), (y(RightX3) > ExpressionLowerLimit) ∧ (y(RightX3) < ExpressionUpperLimit), (y(RightX4) > ExpressionLowerLimit) ∧ (y(RightX4) < ExpressionUpperLimit), (y(RightX5) > ExpressionLowerLimit) ∧ (y(RightX5) < ExpressionUpperLimit)}) → SumRightSideX
  39. OrthogonalLine(A_5, fulcrumtop) → t
  40. Point(t) → LeftX5
  41. OrthogonalLine(A_4, fulcrumtop) → s
  42. Point(s) → LeftX4
  43. OrthogonalLine(A_3, fulcrumtop) → r
  44. Point(r) → LeftX3
  45. OrthogonalLine(A_2, fulcrumtop) → q
  46. Point(q) → LeftX2
  47. OrthogonalLine(A_1, fulcrumtop) → p
  48. Point(p) → LeftX1
  49. Sum({(y(LeftX1) > ExpressionLowerLimit) ∧ (y(LeftX1) < ExpressionUpperLimit), (y(LeftX2) > ExpressionLowerLimit) ∧ (y(LeftX2) < ExpressionUpperLimit), (y(LeftX3) > ExpressionLowerLimit) ∧ (y(LeftX3) < ExpressionUpperLimit), (y(LeftX4) > ExpressionLowerLimit) ∧ (y(LeftX4) < ExpressionUpperLimit), (y(LeftX5) > ExpressionLowerLimit) ∧ (y(LeftX5) < ExpressionUpperLimit)}) → SumLeftSideX
  50. If(BeamLeftTotal > BeamRightTotal, -1, If[BeamLeftTotal ≟ BeamRightTotal, 0, 1]) → BeamLeftYValue
  51. Segment(BeamLeft, BeamRight) → a
  52. Line(BeamLeft, BeamRight) → b
  53. Point(b) → A
  54. Point(b) → B
  55. OrthogonalLine(A, b) → c
  56. OrthogonalLine(BeamLeft, b) → d
  57. OrthogonalLine(BeamRight, a) → e
  58. OrthogonalLine(B, b) → f
  59. Point(c) → C
  60. OrthogonalLine(C, c) → g
  61. Intersect(d, g) → D
  62. Intersect(e, g) → E
  63. Intersect(f, g) → F
  64. Polygon(C, A, BeamLeft, D) → poly2, LeftBasketLeftSide, LeftBasketBottomSide, LeftBasketRightSide, LeftBasketTopSide
  65. Polygon(BeamRight, B, F, E) → poly3, RightBasketBottomSide, RightBasketRightSide, RightBasketTopSide, RightBasketLeftSide
  66. Segment(A, B) → h
  67. Polygon(H, J, M, L) → poly4, h_1, LeftExpressionBottom, m, l
  68. Polygon(N, O, K, I) → poly5, n, RightExpressionBottom, k, i
  69. Point(Circle[FulcrumBottomLeft, -7]) → P
  70. Segment(FulcrumBottomLeft, P) → a_1
  71. OrthogonalLine(A_10, fulcrumtop) → d_1
  72. OrthogonalLine(A_9, fulcrumtop) → e_1
  73. OrthogonalLine(A_9, fulcrumtop) → f_1
  74. OrthogonalLine(A_8, fulcrumtop) → i_1
  75. Sum({(y(LeftX1) > ExpressionLowerLimit) ∧ (y(LeftX1) < ExpressionUpperLimit), (y(LeftX2) > ExpressionLowerLimit) ∧ (y(LeftX2) < ExpressionUpperLimit), (y(LeftX3) > ExpressionLowerLimit) ∧ (y(LeftX3) < ExpressionUpperLimit), (y(LeftX4) > ExpressionLowerLimit) ∧ (y(LeftX4) < ExpressionUpperLimit), (y(LeftX5) > ExpressionLowerLimit) ∧ (y(LeftX5) < ExpressionUpperLimit)}) → LeftSideXTotal
  76. If(BeamLeftYValue ≟ 0, "Equation Balanced!", "Not Balanced") → text7
  77. Element(RightSideXList, 2) → j
  78. Element(RightSideXList, 1) → Test
  79. Segment(S, TRC) → c_2
  80. OrthogonalLine(S, c_2) → d_2
  81. Point(d_2) → T
  82. OrthogonalLine(T, d_2) → e_2
  83. OrthogonalLine(TRC, c_2) → f_2
  84. Intersect(e_2, f_2) → U
  85. Polygon(S, T, U, TRC) → poly6, s_2, t_2, u, trc

AI 绘图提示语参考

绘制一个天平平衡图，中央有三角形支点，水平横梁穿过支点。横梁左右两侧各悬挂五个单位方块和五个未知数方块（用带x的矩形表示），方块可上下移动以表示数量增减。左右两侧的方块数量由随机题目决定，需保持平衡。横梁下方有刻度或限制线，表示表达式数值范围。整体为教学用示意图，强调等式两边平衡关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。