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平方根的性质探究与几何应用

平方根的性质探究与几何应用
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2025-11-26 03:44
所属专题 Quadratzahlen und Quadratwurzeln
本范例通过动态交互演示平方根的基本数学性质。①演示内容：展示平方根函数图像特征及实数范围内的运算规律；②核心数学概念：平方根定义、实数分类、根号运算规则；③作图要点：利用Sequence和Element命令生成平方根数列，通过Polygon和Segment构造相关几何图形，用Mirror实现图形对称变换，借助CircleArc绘制圆弧，Line命令绘制辅助线，Textfield和Join实现文本信息整合，Corner确定界面元素位置；④适用学段：初中数学（实数单元）、高中数学（函数与图像章节），适合教师课堂演示或学生自主探究平方根概念。

GeoGebra 命令及参数

  1. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {1, 8, 4, 2, 1, 8, 3, 3, 7, 8}
  2. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  3. Textfield(input) → inputBox
  4. Textfield(input32) → inputBox32
  5. 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
  6. 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
  7. Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte
  8. Corner(txtLukas, 1) → pBlaseLU
  9. Corner(txtLukas, 2) → pBlaseRU
  10. Corner(txtLukas, 3) → pBlaseRO
  11. Corner(txtLukas, 4) → pBlaseLO
  12. Segment(pBlaseLU, pBlaseRU) → sBlaseLukas_{1}
  13. Segment(pBlaseRU, pBlaseVU) → sBlaseLukas_{2}
  14. Segment(pBlaseVU, pBlaseVH) → sBlaseLukas_{3}
  15. Segment(pBlaseVH, pBlaseVO) → sBlaseLukas
  16. Segment(pBlaseVO, pBlaseRO) → sBlaseLukas_{6}
  17. Segment(pBlaseRO, pBlaseLO) → sBlaseLukas_{4}
  18. Segment(pBlaseLO, pBlaseLU) → sBlaseLukas_{5}
  19. Textfield(input21) → inputBox21
  20. Textfield(input22) → inputBox22
  21. Textfield(input31) → inputBox31
  22. CircleArc(pfeil3_{1}, pfeil3_{2}, pfeil3_{3}) → pfeil3
  23. Line(pMirror3, pMirror3_{1}) → mirror3
  24. Mirror(pfeil3, mirror3) → pfeil3'
  25. Segment(pSpitze3_{1}, pSpitze3) → spitze3
  26. Segment(pSpitze3_{2}, pSpitze3_{3}) → spitze3_{1}
  27. Textfield(input41) → inputBox41
  28. Textfield(input42) → inputBox42
  29. Textfield(input43) → inputBox43
  30. Textfield(input44) → inputBox44
  31. Textfield(input45) → inputBox45
  32. Textfield(input46) → inputBox46
  33. Textfield(input47) → inputBox47
  34. Textfield(input48) → inputBox48
  35. Polygon(pBlaseLU, pBlaseRU, pBlaseVU, pBlaseVH, pBlaseVO, pBlaseRO, pBlaseLO) → poly1, pblaselu, pblaseru, pblasevu, pblasevh, pblasevo, pblasero, pblaselo