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应用运算规则 - 有理数

应用运算规则 - 有理数
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2025-08-28 02:19
所属专题 Addieren und Subtrahieren von ganzen und rationalen Zahlen
本范例演示了如何应用运算规则来处理有理数。它涉及到加减乘除运算，并展示了如何使用 GeoGebra 来可视化这些运算。核心数学概念包括有理数的定义、加减乘除运算的规则以及如何使用这些规则来解决问题。作图要点包括使用 Shuffle、If、Join 等 GeoGebra 命令来生成随机的有理数，并使用 Sequence、Mod 等命令来控制运算的顺序。适合的学段或用途包括初中数学教育，特别是有理数和运算规则的教学。

GeoGebra 命令及参数

  1. Shuffle(Sequence[Random[1, 3], i, 1, 50]) → randomCorrectBox, {1, 1, 3, 2, 3, 2, 3, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 3, 2, 2, 1, 2, 3, 3, 1, 1, 1, 3, 1, 1, 2, 2, 1, 3, 2}
  2. If(checkBox1, 1, checkBox2, 2, checkBox3, 3) → chosenBox
  3. Join(Sequence[Shuffle[1…4], k, 1, 60]) → VRAuswahl2
  4. Sequence(RandomElement[(1…4) \ {VRAuswahl2(k)}], k, 1, 60) → VRAuswahl3, {1, 3, 2, 4, 2, 1, 3, 4, 4, 1, 1, 1, 4, 1, 1, 1, 2, 4, 1, 4, 4, 3, 2, 2, 3, 1, 3, 1, 2, 3, 4, 2, 4, 4, 2, 2, 3, 3, 4, 3, 2, 3, 3, 3, 1, 3, 1, 4, 4, 2, 4, 4, 3, 4, 4, 3, 2, 3, 3, 1}
  5. Mod(questionCount, 28) → fakeIt
  6. If(FakeItTillYouMakeIt((fakeIt * 4) + 1) ∈ {1, 4}, 2, 3) → solutionDrop1
  7. If(FakeItTillYouMakeIt((fakeIt * 4) + 2) ∈ {1, 4}, 2, 3) → solutionDrop2
  8. Flatten(Sequence[Shuffle[1…4], k, 1, 60]) → VRAuswahl1
  9. Shuffle(Sequence[Random[1, 10], i, 1, 50]) → randomZahl11, {7, 2, 9, 4, 7, 9, 9, 7, 2, 7, 3, 5, 5, 1, 9, 5, 3, 5, 5, 9, 8, 9, 6, 10, 3, 7, 10, 7, 6, 2, 1, 2, 6, 7, 5, 5, 7, 9, 3, 6, 7, 4, 8, 2, 9, 8, 8, 5, 5, 7}
  10. Shuffle(Sequence[Random[1, 10], i, 1, 50]) → randomZahl12, {1, 10, 4, 5, 1, 1, 2, 7, 8, 7, 1, 5, 3, 3, 4, 2, 4, 6, 10, 9, 3, 7, 7, 7, 4, 8, 9, 6, 6, 3, 8, 9, 10, 4, 7, 9, 2, 2, 3, 8, 6, 8, 8, 10, 4, 7, 5, 6, 6, 3}
  11. Shuffle(Sequence[Random[1, 10], i, 1, 50]) → randomZahl13, {4, 4, 4, 10, 8, 4, 6, 4, 7, 6, 3, 2, 5, 5, 8, 8, 7, 5, 5, 9, 6, 2, 4, 1, 10, 2, 2, 7, 2, 3, 8, 10, 7, 6, 6, 6, 8, 1, 7, 6, 7, 9, 1, 3, 3, 6, 2, 8, 2, 9}
  12. Shuffle(Sequence[(1 / 10 * i), i, 1, 50]) → randomZahl21, {1.7000000000000002, 5, 0.9, 4.6000000000000005, 2.8000000000000003, 1.9000000000000001, 2.5, 2.3000000000000003, 2.2, 2, 3.1, 1.1, 3.3000000000000003, 2.4000000000000004, 0.8, 0.6000000000000001, 0.1, 2.9000000000000004, 0.2, 4.1000000000000005, 1.2000000000000002, 4.5, 0.5, 1.3, 1.4000000000000001, 4.800000000000001, 1, 3.6, 3.8000000000000003, 1.5, 0.4, 2.1, 4.4, 4.7, 2.7, 3.4000000000000004, 1.8, 0.30000000000000004, 2.6, 3.2, 3.5, 0.7000000000000001, 3.7, 3.9000000000000004, 1.6, 3, 4.2, 4.3, 4.9, 4}
  13. Shuffle(Sequence[(1 / 10 * i), i, 1, 50]) → randomZahl22, {0.4, 1, 4.3, 1.4000000000000001, 3.4000000000000004, 0.30000000000000004, 0.8, 4, 1.5, 0.1, 1.3, 1.1, 2, 4.6000000000000005, 2.4000000000000004, 0.5, 2.5, 3.5, 2.1, 4.1000000000000005, 3.2, 0.2, 3.8000000000000003, 4.800000000000001, 0.7000000000000001, 2.7, 2.6, 3.6, 1.6, 3.3000000000000003, 1.8, 0.6000000000000001, 2.3000000000000003, 0.9, 4.9, 3.7, 4.4, 1.2000000000000002, 1.9000000000000001, 2.8000000000000003, 4.2, 4.5, 3, 3.1, 5, 4.7, 2.2, 2.9000000000000004, 1.7000000000000002, 3.9000000000000004}
  14. Shuffle(Sequence[(1 / 10 * i), i, 1, 50]) → randomZahl23, {3, 1.2000000000000002, 2.9000000000000004, 4.6000000000000005, 0.1, 1.6, 2.8000000000000003, 1, 4.7, 0.9, 4.3, 0.6000000000000001, 1.1, 4.9, 3.8000000000000003, 4.800000000000001, 5, 3.2, 4.5, 0.30000000000000004, 2.4000000000000004, 0.5, 2.2, 1.8, 3.6, 0.8, 0.4, 3.1, 1.5, 3.4000000000000004, 4.2, 3.7, 2.5, 4.1000000000000005, 1.7000000000000002, 2, 4.4, 0.2, 3.5, 1.4000000000000001, 1.9000000000000001, 2.7, 2.6, 2.3000000000000003, 4, 3.3000000000000003, 2.1, 0.7000000000000001, 1.3, 3.9000000000000004}
  15. Flatten(Sequence[Shuffle[1…3], k, 1, 50]) → randomQuestionType
  16. Shuffle(Sequence[Random[1, 2], i, 1, 50]) → random5050, {1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2}
  17. Flatten(Sequence[Shuffle[1…2], k, 1, 50]) → randomQuestionType3
  18. If(beispiel ≟ 1, 1, If[level < 3, randomQuestionType(beispiel), randomQuestionType3(beispiel)]) → questionType
  19. Sequence(RandomElement[(1…4) \ {VRAuswahl2(k), VRAuswahl3(k)}], k, 1, 60) → VRAuswahl4, {4, 4, 1, 3, 4, 2, 2, 3, 2, 3, 3, 4, 2, 3, 4, 2, 3, 1, 3, 3, 1, 4, 1, 3, 1, 4, 4, 2, 3, 2, 3, 4, 1, 3, 1, 1, 4, 2, 1, 2, 1, 1, 2, 2, 3, 4, 3, 1, 3, 3, 3, 2, 2, 2, 1, 2, 4, 2, 1, 2}
  20. Shuffle(Sequence[Random[1, 10], i, 1, 50]) → randomZahl14, {1, 7, 5, 4, 10, 10, 10, 1, 3, 3, 6, 6, 7, 4, 5, 5, 1, 7, 6, 5, 6, 9, 1, 6, 2, 5, 4, 4, 5, 7, 7, 1, 10, 9, 8, 9, 1, 10, 4, 3, 8, 7, 6, 9, 5, 8, 8, 4, 9, 7}
  21. Shuffle(Sequence[(1 / 10 * i), i, 1, 50]) → randomZahl24, {0.6000000000000001, 0.9, 1.5, 4.6000000000000005, 4.4, 0.5, 3.6, 4.9, 1.6, 2.4000000000000004, 3.4000000000000004, 2.5, 4.1000000000000005, 2.3000000000000003, 5, 4.3, 1.7000000000000002, 1.2000000000000002, 4.2, 3, 2, 2.2, 1.9000000000000001, 0.4, 3.8000000000000003, 3.7, 1.1, 1, 3.2, 1.3, 0.1, 4.5, 3.9000000000000004, 4, 2.8000000000000003, 4.800000000000001, 3.1, 3.5, 2.7, 1.4000000000000001, 1.8, 0.7000000000000001, 0.8, 2.9000000000000004, 3.3000000000000003, 4.7, 2.1, 2.6, 0.2, 0.30000000000000004}
  22. ParseToNumber(ReplaceAll[input3Text, ",", "."]) → input3Zahl
  23. ReplaceAll(input3Text, ",", ".") → input3TextOhneKomma
  24. Split(input3TextOhneKomma, {"+", "–", " "}) → allInput3Zahlen
  25. Split(input3Text, {""}) → allInput3
  26. CountIf(x ≟ "+", allInput3) → numOfPlus
  27. CountIf(x ≟ "–", allInput3) → numOfMinus
  28. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {8, 3, 6, 1, 4, 9, 7, 4, 7, 5}
  29. Shuffle(Sequence[möglicheNenner(Random[1, 6]), i, 1, 50]) → randomZahl3Nenner, {3, 8, 6, 4, 3, 4, 6, 4, 8, 6, 8, 8, 3, 8, 5, 8, 5, 3, 2, 2, 2, 2, 4, 3, 2, 2, 4, 6, 3, 2, 8, 8, 3, 8, 5, 5, 2, 2, 8, 5, 6, 4, 8, 2, 8, 5, 3, 5, 2, 6}
  30. Sequence(If[randomZahl3Nenner(i) ≟ 2, Random[1, 2], randomZahl3Nenner(i) ≟ 3, Random[1, 4], randomZahl3Nenner(i) ≟ 4, Random[1, 5], randomZahl3Nenner(i) ≟ 5, Random[1, 7], randomZahl3Nenner(i) ≟ 6, Random[1, 8], randomZahl3Nenner(i) ≟ 8, Random[1, 11]], i, 1, 50) → randomZahl3Zähler1
  31. Sequence(If[randomZahl3Nenner(i) ≟ 2, Random[1, 2], randomZahl3Nenner(i) ≟ 3, Random[1, 4], randomZahl3Nenner(i) ≟ 4, Random[1, 5], randomZahl3Nenner(i) ≟ 5, Random[1, 7], randomZahl3Nenner(i) ≟ 6, Random[1, 8], randomZahl3Nenner(i) ≟ 8, Random[1, 11]], i, 1, 50) → randomZahl3Zähler3
  32. Sequence(If[randomZahl3Nenner(i) ≟ 2, Random[1, 2], randomZahl3Nenner(i) ≟ 3, Random[1, 4], randomZahl3Nenner(i) ≟ 4, Random[1, 5], randomZahl3Nenner(i) ≟ 5, Random[1, 7], randomZahl3Nenner(i) ≟ 6, Random[1, 8], randomZahl3Nenner(i) ≟ 8, Random[1, 11]], i, 1, 50) → randomZahl3Zähler2
  33. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  34. 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
  35. 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
  36. Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte
  37. Translate(imgChilliGrau1, Vector[(1 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliGrau2
  38. Translate(imgChilliRot1, Vector[(1 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliRot2
  39. Translate(imgChilliRot1, Vector[(2 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliRot3
  40. If(questionType ≟ 3, (1.48, 6.76), (1.48, 6.6)) → posSol1LO
  41. If(questionType ≟ 3, (1.48, 5.37), (1.48, 5.21)) → posSol2LO
  42. If(questionType ≟ 3, (1.48, 3.96), (1.48, 3.82)) → posSol3LO
  43. If(correctBox ≟ 1, posSol1LO, correctBox ≟ 2, posSol2LO, posSol3LO) → solLO
  44. If(questionType ≟ 3, (4.6, 5.48), questionType ≟ 1, (4.46, 5.64), (4.8, 5.64)) → posSol1RU
  45. If(questionType ≟ 3, (4.6, 4.11), questionType ≟ 1, (4.46, 4.27), (4.8, 4.27)) → posSol2RU
  46. If(questionType ≟ 3, (4.6, 2.72), questionType ≟ 1, (4.46, 2.88), (4.8, 2.88)) → posSol3RU
  47. If(correctBox ≟ 1, posSol1RU, correctBox ≟ 2, posSol2RU, posSol3RU) → solRU
  48. Shuffle(Sequence[Random[1, 3], i, 1, 50]) → randomCorrectBox_{1}, {3, 3, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 1, 2, 3, 2, 2, 3, 3, 1, 1, 2, 1, 1, 3, 1, 2, 3, 2, 2, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 3}
  49. Length(allInput3Zahlen) → längeZahl3
  50. Textfield(input3Text) → inputBox3
  51. LaTeX(ReplaceAll[input3Text, "-", " – "]) → input3TextLayout
  52. LaTeX(ReplaceAll[input3TextLayout, "+", " + "]) → input3TextLayout2
  53. LaTeX(ReplaceAll[input3TextLayout2, " + ", " + "]) → input3TextLayout3
  54. Flatten(Sequence[Shuffle[(1…4) \ {3}], k, 1, 50]) → l2
  55. Mod(randomZahl21(beispiel), 1) → b