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用隐形立方体计算体积

用隐形立方体计算体积
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2023-03-18 09:19
多边形数列立体几何几何作图滑块动画
该范例通过隐藏立方体单元，引导学生探索棱柱体积的计算方法。它演示了如何在不逐一数出每个立方体的情况下，利用长、宽、高的乘积或底面积乘高来快速求得体积。核心数学概念是体积的度量、立方体单元、棱柱体积公式以及空间推理。作图要点包括使用GeoGebra的RandomUniform生成随机尺寸，通过Prism和Cube命令构建立体图形，利用Textfield和InputBox接收学生输入，并用Sequence、Zip等命令管理随机种子和验证逻辑。适合小学高年级（4-5年级）学生进行体积概念的实践练习，也适用于教师课堂演示或学生自主探究，帮助建立从具体计数到抽象公式的思维过渡。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.1985571235238883, 0.3747460724081399, 0.8201023415293685, 0.413941026343629, 0.9580610092118979, 0.9532396596321169, 0.40903753117592906, 0.8689044514022999, 0.9677671245463821, 0.9248155795513775, 0.22079165652016206, 0.8843200579033657, 0.4032992895551073, 0.0015681815436908609, 0.08038008650236657, 0.29409635413929663, 0.4148793148887635, 0.4460118103389175, 0.22964906666612092, 0.12579896732996187, 0.14186804022180113, 0.21335602119014882, 0.08624711608203661, 0.5604600589825768, 0.8303637266917386, 0.802244118154383, 0.9575371195840408, 0.2584650457890849, 0.4170062032779738, 0.053077092533620585, 0.949603196260776, 0.4638565083370723, 0.10450721393377367, 0.5472149775314659, 0.07434936411556867, 0.6327837752421652, 0.7252387981454076, 0.46045482201497484, 0.7339656328177513, 0.24615253084824062, 0.4584588989102166, 0.140229658544052, 0.2717540673985659, 0.9017932675844289, 0.9044397678435976, 0.8679550544036241, 0.4014618084846333, 0.06766749472845958, 0.47488663098397227, 0.13587006288895706, 0.10385984693741061, 0.8347702182718699, 0.5435959312618279, 0.3215763668669794, 0.9193353555523798, 0.9464628550588227, 0.5342994358288029, 0.33327477990708443, 0.8487325809130822, 0.9599846069427432, 0.21407657158592724, 0.9244303898028499, 0.1315812931383337, 0.4958413794753706, 0.5305905092620262, 0.1982677195198501, 0.9011935832521802, 0.45166029271430064, 0.34971348973526156, 0.1801044268709735, 0.6298895642712049, 0.788923386166371, 0.6668555789901136, 0.129077520472109, 0.9348284730803904, 0.8037587587885737, 0.780496207832065, 0.6402979917490262, 0.5055768099475766, 0.38666733179529045, 0.5400410039620908, 0.4138896851743153, 0.7486365996862828, 0.3207032646890486, 0.21218693765249785, 0.16872619576780615, 0.4916858521978765, 0.2332016333355101, 0.8804768108340585, 0.9159738534413164, 0.6412507255357188, 0.3426603288386718, 0.14166003596190438, 0.6307815670705613, 0.8619327897001307, 0.8998352753375969, 0.6798488085814152, 0.8096284245831337, 0.6031564522829401, 0.8104958956900019, 0.12527062806833547, 0.7959468297670473, 0.31068140663775534, 0.5866140054783586, 0.5086648111123792, 0.907684519708866, 0.7648413062247824, 0.6430235895628105, 0.2750571583771809, 0.8674506589156967, 0.37562431889484343, 0.567789500262776, 0.30176960311640716, 0.4788115600837418, 0.8463030787852123, 0.8266617676814388, 0.3750823058705079, 0.7611587111461637, 0.4023508016956722, 0.26506896019869475, 0.9279633175734371, 0.5003668443324203, 0.5608470213054119, 0.44511701797768155, 0.10771745848236566, 0.2431921459096496, 0.9674809219359044, 0.02001165331462279, 0.29727777902072955, 0.7900308618751064, 0.20360179214567567, 0.8742108510318782, 0.17717476385738629, 0.3732654872287096, 0.2645456316452947, 0.184440183511728, 0.746965650652961, 0.486971086817247, 0.9026836152289247, 0.9444673718634056, 0.40509203160076707, 0.008236653148151873, 0.8742924532054477, 0.24523361483459177, 0.14475718276280639, 0.7107759675511467, 0.29843848164742887, 0.7715783195155869, 0.030474806926029916, 0.17786059707461796, 0.038596611916459106, 0.957369114364521, 0.7754389359049524, 0.5539015391190559, 0.45369208212086065, 0.5223635773174239, 0.7438005441331278, 0.6822394545911281, 0.16465715064701453, 0.8265800783164224, 0.03430931317115668, 0.580872823536631, 0.49543699950820413, 0.43134337211745244, 0.3625893252165656, 0.23796073582315264, 0.30365775676850404, 0.31049730815407195, 0.45856587242134966, 0.5349822017158518, 0.19588713682856146, 0.7105402875003487, 0.9432460299078801, 0.8574162577884733, 0.8755719615010006, 0.6246406923556221, 0.6451784488662415, 0.9707267820408741, 0.35832869608720896, 0.24262752318775646, 0.9214785573497183, 0.5632937772436603, 0.69929262627353, 0.12712855810623633, 0.1313035530434435, 0.957727325415719, 0.9706978530116676, 0.1250410609412507, 0.5210967180389424, 0.3677334505016679, 0.28476492628591643, 0.41347193464352655, 0.08079408195864105, 0.08937039040960885, 0.3173142290879695, 0.5601727076223676, 0.41849659539001693, 0.21314154542608715, 0.8413030335415503, 0.8403206196179155}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. Textfield(input1) → InputBox1
  4. Polygon(Zip[PixelsToPoint[Corner[1], Corner[3], Corner[5], A], A, {(pixelsLeft, pixelsTop), (pixelsLeft, y(Corner[5]) - pixelsBottom), Corner[5] - (pixelsRight, pixelsBottom), (x(Corner[5]) - pixelsRight, pixelsTop)}]) → borderBox
  5. Corner(5) → layoutAppSize
  6. Element({If[layoutX ≟ 16, layoutX - 5, 11], layoutX - 5}, layoutCase) → layoutHintSolnX
  7. Polygon((0, 0, -1), (randomNum1, 0, -1), (randomNum1, randomNum2, -1), (0, randomNum2, -1)) → basePoly, edge1a, edge2a, edge3a, edge4a
  8. Prism(basePoly, randomNum3) → startPrism, PointPrism1, edge1, edge2, edge3, edge4, edgeCD, edgeDE, edgeEF, edgeCF, PointPrism2, PointPrism3, PointPrism4, face1, face3, face4, face2, faceCDEF
  9. Cube(Polygon[{(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 0, 0)}], true) → startCube, PointCube1, edge5, edge8, edgeGJ, edgeGH, edge6, edgeHI, edge7, edgeIJ, PointCube2, PointCube3, PointCube4, face6, face5, face7, face8, faceGHIJ
  10. Flatten(Sequence[Sequence[Sequence[(p1, p2, p3), p1, 0, randomNum1 - 1], p2, 0, randomNum2 - 1], p3, 0, randomNum3 - 1]) → cubePoints
  11. Zip(Translate[startCube, Vector[(0, 0, 0), n]], n, cubePoints) → cubeOutlines
  12. Sequence(Segment[(randomNum1, a, -1), (randomNum1, a, -0.7)], a, 1, randomNum2 - 1) → segmentFB
  13. Sequence(Segment[(randomNum1, 0, a), (randomNum1, 0.3, a)], a, 0, randomNum3 - 2) → segmentFL
  14. Sequence(Segment[(randomNum1, randomNum2, a), (randomNum1, randomNum2 - 0.3, a)], a, 0, randomNum3 - 2) → segmentFR
  15. Sequence(Segment[(randomNum1, a, randomNum3 - 1), (randomNum1, a, randomNum3 - 1.3)], a, 1, randomNum2 - 1) → segmentFT
  16. Sequence(Segment[(a, randomNum2, -1), (a, randomNum2, -0.7)], a, 1, randomNum1 - 1) → segmentSB
  17. Sequence(Segment[(randomNum1, randomNum2, a), (randomNum1 - 0.3, randomNum2, a)], a, 0, randomNum3 - 2) → segmentSL
  18. Sequence(Segment[(0, randomNum2, a), (0.3, randomNum2, a)], a, 0, randomNum3 - 2) → segmentSR
  19. Sequence(Segment[(a, randomNum2, randomNum3 - 1), (a, randomNum2, randomNum3 - 1.3)], a, 1, randomNum1 - 1) → segmentST
  20. Sequence(Segment[(0, a, randomNum3 - 1), (0.3, a, randomNum3 - 1)], a, 1, randomNum2 - 1) → segmentTB
  21. Sequence(Segment[(randomNum1, a, randomNum3 - 1), (randomNum1 - 0.3, a, randomNum3 - 1)], a, 1, randomNum2 - 1) → segmentTF
  22. Sequence(Segment[(a, 0, randomNum3 - 1), (a, 0.3, randomNum3 - 1)], a, 1, randomNum1 - 1) → segmentTL
  23. Sequence(Segment[(a, randomNum2, randomNum3 - 1), (a, randomNum2 - 0.3, randomNum3 - 1)], a, 1, randomNum1 - 1) → segmentTR
  24. CornerThreeD(-1, 11) → CurrentViewDirection

AI 绘图提示语参考

画一个棱柱，使用随机生成的尺寸。首先画出底面多边形，然后画出棱柱的侧面和顶面。底面多边形由随机生成的边长组成，棱柱的高度也是随机的。侧面由一系列平行于底面的矩形组成，顶面和底面保持平行。确保所有边和面都正确连接，形成一个完整的棱柱。
以上提示语为根据原始绘图命令整理而来，仅供参考。