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二项式平方公式展开计算：用输入幂完成展开

二项式平方公式展开计算：用输入幂完成展开
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2025-01-09 19:34
函数数列变换几何作图函数图像
所属专题 Binomische Formeln
该范例用于演示如何使用“二项式平方公式”（第2个二项式公式）把形如(a+b)^2的多项式展开并进行对照检查。核心数学概念包括二项式公式的推导与应用、幂的代数展开、代数表达式化简/等价变形，以及通过正确/错误反馈巩固公式使用规范。作图与交互要点是：用户在文本输入框中输入幂或含变量的表达式，系统利用 Expand 对目标代数式进行展开；随后用 Text 生成显示文本，并用 Sequence、Join 等组织多步内容。根据题目选择（可由 SelectedIndex、If、Element 控制不同情形），程序把展开结果与标准答案进行匹配，使用 Case、CaseList 与 VarList/ParList（以及对应对象 var1、var2、par1、par2）管理参数与符号；再通过 ReplaceAll 与 Invert 进行替换与文本处理，最终由 Textfield 呈现输入与计算过程，并用 attemptCount、questionCount、correctCount 等统计表现。适合7–8年级（代数与表达式单元）的课堂练习、作业自测与公式应用教学。

GeoGebra 命令及参数

  1. Expand(TaskFunc) → solFunc
  2. Text(inputFunc) → txtInput
  3. Sequence(Shuffle[1…3], k, 1, 100) → OrderList1, {{1, 3, 2}, {1, 3, 2}, {1, 2, 3}, {2, 1, 3}, {1, 2, 3}, {3, 1, 2}, {1, 3, 2}, {1, 3, 2}, {3, 2, 1}, {2, 1, 3}, {2, 3, 1}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {2, 1, 3}, {1, 3, 2}, {3, 1, 2}, {3, 2, 1}, {1, 2, 3}, {2, 3, 1}, {2, 3, 1}, {2, 1, 3}, {3, 2, 1}, {2, 1, 3}, {1, 3, 2}, {3, 1, 2}, {2, 3, 1}, {3, 2, 1}, {2, 3, 1}, {3, 2, 1}, {1, 3, 2}, {1, 2, 3}, {3, 2, 1}, {2, 1, 3}, {2, 3, 1}, {1, 2, 3}, {3, 1, 2}, {2, 3, 1}, {2, 1, 3}, {3, 1, 2}, {3, 1, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}, {1, 3, 2}, {2, 3, 1}, {1, 2, 3}, {3, 2, 1}, {1, 2, 3}, {3, 2, 1}, {2, 1, 3}, {2, 1, 3}, {1, 2, 3}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}, {1, 3, 2}, {2, 1, 3}, {3, 2, 1}, {3, 2, 1}, {2, 1, 3}, {2, 3, 1}, {1, 3, 2}, {1, 3, 2}, {2, 3, 1}, {3, 1, 2}, {1, 3, 2}, {3, 1, 2}, {2, 3, 1}, {3, 2, 1}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}, {1, 3, 2}, {3, 2, 1}, {3, 1, 2}, {1, 2, 3}, {3, 2, 1}, {1, 3, 2}, {3, 2, 1}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {2, 3, 1}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {2, 1, 3}, {3, 2, 1}, {3, 1, 2}, {2, 3, 1}, {1, 3, 2}, {3, 1, 2}, {3, 1, 2}, {2, 1, 3}, {3, 1, 2}, {2, 1, 3}}
  4. Sequence(Shuffle[1…3], k, 1, 100) → OrderList2, {{2, 3, 1}, {2, 3, 1}, {1, 3, 2}, {1, 3, 2}, {3, 1, 2}, {3, 1, 2}, {1, 3, 2}, {2, 1, 3}, {1, 3, 2}, {3, 1, 2}, {2, 3, 1}, {1, 3, 2}, {3, 2, 1}, {3, 2, 1}, {3, 1, 2}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {3, 1, 2}, {3, 1, 2}, {2, 3, 1}, {2, 3, 1}, {2, 1, 3}, {3, 2, 1}, {3, 2, 1}, {1, 3, 2}, {3, 1, 2}, {1, 3, 2}, {1, 2, 3}, {2, 3, 1}, {2, 1, 3}, {2, 3, 1}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}, {2, 1, 3}, {2, 3, 1}, {1, 3, 2}, {3, 1, 2}, {1, 2, 3}, {3, 1, 2}, {3, 1, 2}, {3, 1, 2}, {3, 2, 1}, {3, 2, 1}, {3, 1, 2}, {3, 1, 2}, {2, 1, 3}, {3, 2, 1}, {1, 3, 2}, {3, 1, 2}, {1, 2, 3}, {3, 1, 2}, {1, 3, 2}, {1, 3, 2}, {2, 3, 1}, {2, 3, 1}, {3, 1, 2}, {3, 1, 2}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {3, 1, 2}, {3, 2, 1}, {2, 3, 1}, {1, 2, 3}, {2, 1, 3}, {1, 3, 2}, {3, 2, 1}, {1, 2, 3}, {3, 2, 1}, {1, 2, 3}, {1, 3, 2}, {2, 3, 1}, {2, 3, 1}, {3, 2, 1}, {2, 3, 1}, {2, 1, 3}, {2, 1, 3}, {1, 3, 2}, {2, 1, 3}, {3, 2, 1}, {2, 3, 1}, {3, 2, 1}, {1, 2, 3}, {2, 3, 1}, {2, 1, 3}, {3, 1, 2}, {3, 1, 2}, {3, 2, 1}, {2, 3, 1}, {3, 1, 2}, {2, 3, 1}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 2, 1}, {3, 1, 2}, {2, 3, 1}}
  5. Sequence(Shuffle[1…3], k, 1, 100) → OrderList3, {{1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 1, 2}, {3, 1, 2}, {2, 3, 1}, {2, 3, 1}, {2, 3, 1}, {1, 2, 3}, {1, 2, 3}, {2, 1, 3}, {1, 3, 2}, {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {3, 2, 1}, {3, 2, 1}, {3, 1, 2}, {1, 2, 3}, {1, 3, 2}, {1, 3, 2}, {2, 1, 3}, {1, 3, 2}, {1, 2, 3}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}, {3, 2, 1}, {2, 1, 3}, {2, 1, 3}, {1, 3, 2}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}, {3, 1, 2}, {2, 1, 3}, {1, 2, 3}, {2, 3, 1}, {2, 3, 1}, {3, 2, 1}, {2, 3, 1}, {1, 2, 3}, {2, 3, 1}, {3, 1, 2}, {2, 3, 1}, {2, 1, 3}, {1, 2, 3}, {3, 1, 2}, {1, 3, 2}, {2, 3, 1}, {2, 1, 3}, {1, 2, 3}, {1, 3, 2}, {3, 1, 2}, {2, 3, 1}, {1, 3, 2}, {3, 2, 1}, {1, 3, 2}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}, {1, 2, 3}, {1, 2, 3}, {3, 2, 1}, {3, 2, 1}, {2, 3, 1}, {1, 2, 3}, {1, 2, 3}, {3, 1, 2}, {2, 1, 3}, {3, 1, 2}, {3, 1, 2}, {3, 2, 1}, {1, 2, 3}, {1, 2, 3}, {2, 3, 1}, {1, 3, 2}, {2, 3, 1}, {2, 3, 1}, {2, 1, 3}, {2, 1, 3}, {2, 3, 1}, {1, 2, 3}, {2, 3, 1}, {3, 2, 1}, {1, 2, 3}, {1, 2, 3}, {1, 3, 2}, {3, 1, 2}, {3, 2, 1}, {1, 2, 3}, {3, 2, 1}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3}}
  6. Join({{"$\textsf{\phantom{144\:b² }\vphantom{gb²}}$"}, {(("$\textsf{" * DDList1(Order1(1))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList1(Order1(2))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList1(Order1(3))) * "\vphantom{gb²}}$")}}) → dropdown1
  7. Join({{"$\textsf{\phantom{144\:b² }\vphantom{gb²}}$"}, {(("$\textsf{" * DDList3(Order3(1))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList3(Order3(2))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList3(Order3(3))) * "\vphantom{gb²}}$")}}) → dropdown3
  8. Join({{"$\textsf{\phantom{144\:b² }\vphantom{gb²}}$"}, {(("$\textsf{" * DDList2(Order2(1))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList2(Order2(2))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList2(Order2(3))) * "\vphantom{gb²}}$")}}) → dropdown2
  9. SelectedIndex(dropdown1) → indexdd1
  10. SelectedIndex(dropdown2) → indexdd2
  11. SelectedIndex(dropdown3) → indexdd3
  12. SelectedIndex(dropdownSign1) → indexddS1
  13. SelectedIndex(dropdownSign2) → indexddS2
  14. If(Case ≠ 3, indexdd2 - 1 ≟ IndexOf[1, Order2], true) → correctdd2
  15. If(Case ≠ 3, indexddS2 ≟ 2, true) → correctddS2
  16. If(step ≟ 1, correctExtend ∧ correctValue, correctdd1 ∧ correctdd2 ∧ correctdd3 ∧ correctddS1 ∧ correctddS2) → correct
  17. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {10, 4, 7, 4, 5, 2, 2, 1, 6, 1}
  18. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  19. If(step ≟ 1, Defined[inputFunc], indexdd1 ≠ 1 ∧ If[Case ≟ 3, true, indexdd2 ≠ 1] ∧ indexdd3 ≠ 1 ∧ indexddS1 ≠ 1 ∧ If[Case ≟ 3, true, indexddS2 ≠ 1]) → clickableBtnValidate
  20. Textfield(inputFunc) → inputBox
  21. If(CaseVar ≟ 2, "", var1) → txtVar1
  22. If(CaseVar ≟ 1, "", var2) → txtVar2
  23. Invert(pixMat) → invPixMat
  24. If(Case ≟ 3, PosDDS1 + (1.42, 0), PosDDS2 + (1.42, 0)) → PosDD3
  25. ReplaceAll(Text[Expand[TaskFunc]], "+", "-") → txtHint3

AI 绘图提示语参考

绘制一个关于二项式平方展开的交互式练习。首先，在输入框中输入表达式 (a+b)^2 的展开结果，然后选择三个随机排列的选项，确保它们与展开结果匹配。接着，选择正确的符号选项，最后验证你的答案是否正确。注意，每次尝试后，系统会根据你的输入给出反馈。
以上提示语为根据原始绘图命令整理而来，仅供参考。