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长方体体积与表面积：用单位立方体分解并分析边长变化

长方体体积与表面积：用单位立方体分解并分析边长变化
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2023-04-05 15:32
立体几何几何作图滑块动画
所属专题 Quader und Würfel - Volumen
该范例通过可操作的长方体模型演示：将长方体分解为合适的单位立方体来确定体积，并进一步观察边长改变时表面积的变化规律。核心数学概念包括长方体（Quader）的体积与公式、用“单位立方体”进行体积测量/验证、以及长度变化对表面积（Oberflächeninhalte）的影响。作图与交互要点是先构造多个盒体与其标注位置，再用中点与拼接/条件逻辑实现布局与文字提示：使用 Midpoint 计算相关点，Shuffle 用于对象随机或变换选择，Join 将元素合并，If 根据输入条件切换/显示结果，Polygon 绘制多边形外观或辅助结构，Textfield 让学生输入或调节参数；随后通过 Element 引用对象并 Append/Remove 管理动态添加与删除的元素；用 Segment 表示必要的边或测量线，整体使用 Box（如 Box1、Box2、Box3…）及其中点标记（MPBox1…）与布局辅助（LayoutHelper、LayoutList、layout、FixedPos 等）来组织多个情境步骤。适合5-6年级及同等入门阶段的立体几何探究、体积/表面积概念建构与公式理解练习。

GeoGebra 命令及参数

  1. Midpoint(Corner[Box1, 2], Corner[Box1, 1]) → MPBox1
  2. Midpoint(Corner[Box4, 1], Corner[Box4, 2]) → MPBox4
  3. Midpoint(Corner[Box3, 1], Corner[Box3, 2]) → MPBox3
  4. Midpoint(Corner[Box2, 2], Corner[Box2, 1]) → MPBox2
  5. Shuffle(1…5) → LayoutHelper, {4, 5, 3, 1, 2}
  6. Join({{4}, LayoutHelper \ {4}}) → LayoutList
  7. If(layout ≟ 1, MPBox2, If[layout ≟ 2, MPBox3, If[layout ≟ 3 ∨ layout ≟ 4 ∨ layout ≟ 5, MPBox4]]) → MP1
  8. Midpoint(Corner[Box5, 1], Corner[Box5, 2]) → MPBox5
  9. If(layout ≟ 1, MPBox4, If[layout ≟ 2, MPBox5, If[layout ≟ 3, MPBox2, If[layout ≟ 4, MPBox2, If[layout ≟ 5, MPBox1]]]]) → MP3
  10. Midpoint(Corner[Box6, 1], Corner[Box6, 2]) → MPBox6
  11. If(layout ≟ 1, MPBox3, If[layout ≟ 2, MPBox2, If[layout ≟ 3, MPBox6, If[layout ≟ 4, MPBox5, If[layout ≟ 5, MPBox2]]]]) → MP2
  12. Polygon(M, I, O, N) → VRegal, m, i, o, n
  13. If(layout ≟ 1, 40, If[layout ≟ 2, 15, If[layout ≟ 3, 50, If[layout ≟ 4, 3, If[layout ≟ 5, 30]]]]) → B1E1
  14. If(layout ≟ 1, 40, If[layout ≟ 2, 10, If[layout ≟ 3, 50, If[layout ≟ 4, 4, If[layout ≟ 5, 40]]]]) → B1E2
  15. If(layout ≟ 1, 40, If[layout ≟ 2, 10, If[layout ≟ 3, 20, If[layout ≟ 4, 7, If[layout ≟ 5, 45]]]]) → B1E3
  16. If(layout ≟ 1, 30, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 5, If[layout ≟ 5, 30]]]]) → B2E1
  17. If(layout ≟ 1, 40, If[layout ≟ 2, 10, If[layout ≟ 3, 50, If[layout ≟ 4, 6, If[layout ≟ 5, 30]]]]) → B2E2
  18. If(layout ≟ 1, 20, If[layout ≟ 2, 10, If[layout ≟ 3, 20, If[layout ≟ 4, 9, If[layout ≟ 5, 30]]]]) → B2E3
  19. If(layout ≟ 1, 14, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 60]]]]) → B3E1
  20. If(layout ≟ 1, 60, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 30]]]]) → B3E2
  21. If(layout ≟ 1, 40, If[layout ≟ 2, 30, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 18]]]]) → B3E3
  22. Textfield(Input1) → InputBox1
  23. Textfield(Input2) → InputBox2
  24. Polygon(PtopLeftCornerFeedback, PtopRightCornerFeedback, PbottomRightCornerFeedback, PbottomLeftCornerFeedback) → rectangleCorrect, ptopleftcornerfeedback, ptoprightcornerfeedback, pbottomrightcornerfeedback, pbottomleftcornerfeedback
  25. Shuffle(Sequence[Random[0, 10], i, 1, 100]) → randomNosForX, {6, 3, 4, 1, 3, 8, 0, 8, 3, 0, 7, 7, 8, 10, 4, 7, 9, 1, 1, 8, 4, 7, 4, 1, 9, 1, 0, 1, 7, 8, 0, 3, 9, 5, 8, 9, 3, 10, 5, 3, 1, 1, 3, 2, 8, 8, 2, 5, 4, 9, 9, 2, 5, 2, 3, 7, 6, 6, 7, 6, 3, 2, 9, 1, 3, 8, 0, 2, 9, 0, 3, 5, 4, 3, 0, 3, 6, 9, 2, 1, 0, 1, 2, 0, 1, 5, 3, 4, 9, 9, 6, 8, 7, 9, 7, 7, 5, 1, 9, 4}
  26. Element(randomNosForX, Mod[questionCount, 100] + 1) → numX
  27. Shuffle(Sequence[Random[0, 10], i, 1, 100]) → randomNosForY, {5, 0, 5, 0, 2, 9, 10, 9, 6, 2, 0, 10, 10, 9, 7, 4, 0, 10, 10, 9, 10, 3, 1, 7, 6, 1, 6, 4, 3, 9, 7, 9, 4, 3, 2, 8, 10, 8, 9, 1, 8, 4, 4, 7, 7, 9, 6, 5, 0, 8, 4, 7, 2, 3, 3, 4, 1, 4, 1, 9, 10, 6, 1, 4, 6, 4, 2, 2, 6, 0, 7, 8, 2, 10, 10, 7, 6, 5, 4, 4, 4, 7, 1, 6, 9, 0, 3, 3, 2, 6, 8, 4, 5, 4, 6, 7, 7, 7, 6, 1}
  28. Element(randomNosForY, Mod[questionCount, 100] + 1) → numY
  29. Append(true, Sequence[If[randomNosForX(i) ≟ randomNosForX(i - 1), false, true], i, 2, 100]) → stayX
  30. Append(true, Sequence[If[randomNosForY(i) ≟ randomNosForY(i - 1), false, true], i, 2, 100]) → stayY
  31. Remove(Sequence[If[stayX(i), randomNosForX(i), 0], i, 1, 100], Sequence[0, i, 1, 100]) → randomNosForXwithoutRepeat
  32. Remove(Sequence[If[stayY(i), randomNosForY(i), 0], i, 1, 100], Sequence[0, i, 1, 100]) → randomNosForYwithoutRepeat
  33. Element(randomNosForXwithoutRepeat, Mod[questionCount, Length[randomNosForXwithoutRepeat]] + 1) → numXwithoutRepeat
  34. Element(randomNosForYwithoutRepeat, Mod[questionCount, Length[randomNosForYwithoutRepeat]] + 1) → numYwithoutRepeat
  35. If(layout ≟ 1, 30, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 5, If[layout ≟ 5, 30]]]]) → B2E1_{1}
  36. If(layout ≟ 1, 60, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 30]]]]) → B3E2_{1}
  37. If(layout ≟ 1, 40, If[layout ≟ 2, 50, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 18]]]]) → B3E3_{1}
  38. Textfield(Input3) → InputBox3
  39. Segment(MP1, J) → f
  40. Segment(MP2, J_{1}) → f_{1}
  41. Segment(MP3, J_{2}) → f_{2}
  42. Segment((4, 2), (4, 4)) → vRegal1
  43. Segment((7, 4), (7, 2)) → vRegal2
  44. Polygon(M, M_2, L_2, I) → VRegal1, g, h, j, k
  45. Polygon(L_2, M_2, O_2, N_2) → VRegal2, l, p, q, r
  46. Polygon(N_2, O_2, N, O) → VRegal3, s, t, a, b
  47. Polygon(PtopLeftCornerFeedback_{1}, PtopRightCornerFeedback_{1}, PbottomRightCornerFeedback_{1}, PbottomLeftCornerFeedback_{1}) → rectangleCorrect_{1}, ptopleftcornerfeedback_{1}, ptoprightcornerfeedback_{1}, pbottomrightcornerfeedback_{1}, pbottomleftcornerfeedback_{1}
  48. Polygon(PtopLeftCornerFeedback_{2}, PtopRightCornerFeedback_{2}, PbottomRightCornerFeedback_{2}, PbottomLeftCornerFeedback_{2}) → rectangleCorrect_{2}, ptopleftcornerfeedback_{2}, ptoprightcornerfeedback_{2}, pbottomrightcornerfeedback_{2}, pbottomleftcornerfeedback_{2}
  49. If(Step ≟ 2, 2, Step ≟ 3, 1) → attemptCountHelper

AI 绘图提示语参考

画一个立方体的体积分析图，包含六个盒子和六个中点点，盒子之间的关系和尺寸变化，体积和表面积的变化，包含一些辅助线和标签，图形中包含一些随机数和动态变化的元素，要求图形清晰、准确、易于理解，包含一些交互元素，允许用户输入参数和查看结果，图形中包含一些辅助线和标签，用于辅助理解和分析。
以上提示语为根据原始绘图命令整理而来，仅供参考。