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用面积模型分解并计算多位数除以一位数

用面积模型分解并计算多位数除以一位数
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2023-12-07 05:54
多边形数列变换几何作图
该范例用面积模型把“多位数 ÷ 一位数”转化为可视化的分块分解，从而读出商与余数。核心数学概念包括整除与带余除法、位值分解、商的确定与余数的范围（余数小于除数）。作图与互动要点：通过随机生成被除数、除数与部分算式元素（例如 dividendNum、divisorDigit、quotientInteger、remainder），并用多边形区域（如 partial1~partial4、A1~A5、B2~B5、A2~A4 等）表示分割后的面积块；再结合表格/文本反馈结果（ansQuotient、remainder、quotientInteger），将正确商与余数分别显示为 correctA~correctF 等对象。构造摘要中用到的关键 GeoGebra 命令/对象包括：RandomUniform（生成随机数）、Sequence（生成位置/索引序列）、Take/Element（从序列中取值）、Polygon（绘制面积块多边形）、Vector与Translate（按向量平移排版分块）、Text/Textfield/TableText（显示题目与计算结果）、TableText（结构化呈现信息），以及通过 Corner、PixelsToPoint 等工具完成几何与界面定位。适合 4-5 年级学生用于带余除法的直观理解，也适合作业练习与课堂演示。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.5793528192684818, 0.10650413291991515, 0.8581270496547331, 0.01340248834199562, 0.180473274950375, 0.2163374086248917, 0.011587676324896986, 0.500058846374919, 0.8751894862127395, 0.05222898133163495, 0.010821789189874847, 0.6581508810380622, 0.55070051072857, 0.5268537138836512, 0.8597906106805178, 0.34785710589294155, 0.7986169781235344, 0.9113338332543482, 0.010578847892572468, 0.6803864297422059, 0.35543128205516794, 0.2838718204329495, 0.5710456324279973, 0.5732410474212533, 0.7787019136967149, 0.51676976978688, 0.9118558120717436, 0.8643664014519479, 0.15763963313225604, 0.9238597987870502, 0.6090938337717103, 0.5737515380042838, 0.2582961341730574, 0.6943058358902674, 0.726894750255301, 0.8704466109853368, 0.16600796059005005, 0.04111054798771206, 0.22869646756863024, 0.6858067992333712, 0.14113640274676886, 0.7234709467043959, 0.755825540805241, 0.4300510153773872, 0.9884704362242419, 0.7713750641194602, 0.4181729796654501, 0.19894026804049972, 0.3059236468859951, 0.40851713650189203, 0.4236994029070828, 0.050535453874081745, 0.10435710687732296, 0.20573183819159627, 0.01142848967322035, 0.21858535681681893, 0.9774986625877343, 0.17170886849370634, 0.3764455060797552, 0.0891257136650579, 0.8890056587149792, 0.8007471108214785, 0.8652122621000811, 0.46033649084893463, 0.16273151307273842, 0.6787429946195181, 0.16524223885090122, 0.27260908728581446, 0.7822077778280919, 0.8884809508760336, 0.6348499673834215, 0.0961164538485748, 0.39624199088336926, 0.7936043121957138, 0.8562956945724843, 0.006959668215232024, 0.6793578836514407, 0.8273932142832208, 0.4391915034003624, 0.15017936793609327, 0.6685681324795193, 0.8276610336392993, 0.38051749347512537, 0.06857676450541263, 0.651670835206276, 0.2027048591195743, 0.5030166680487334, 0.24037928722089597, 0.14689999939799397, 0.4753051825251781, 0.3750939604825941, 0.5763159467382271, 0.8093360104780463, 0.3422685625444212, 0.6726409919309012, 0.527633989501868, 0.9459960174895343, 0.8759424313491689, 0.9464390212953533, 0.29099291413772876, 0.628975251441494, 0.9215039755414787, 0.17796386683504936, 0.1840135690510708, 0.42439902191420087, 0.6896362098523268, 0.6215230501707465, 0.5565393779616057, 0.21427122679580707, 0.46845170118385215, 0.17170096187505016, 0.7673836731664069, 0.3585690264945313, 0.08121044438137291, 0.19206961147246707, 0.614491005855408, 0.27613409585400245, 0.9275011922377127, 0.9440433949068943, 0.3680329043196984, 0.20505028554345803, 0.025116133320607924, 0.4936124980666168, 0.9709867664381447, 0.8362729740167038, 0.8478469857440444, 0.8084857617239417, 0.2802143117227001, 0.31828934959517796, 0.430865919916097, 0.248905207791013, 0.0629598136919596, 0.885540105402717, 0.620227961444579, 0.932065040708396, 0.6590302949834729, 0.6492217200363375, 0.8363418011428274, 0.35267497340813575, 0.4076990955269075, 0.48591427994693764, 0.8630044621275698, 0.8307528221167815, 0.4805910575818067, 0.49724410648479206, 0.7337802150724506, 0.4322136569180921, 0.07746181871746725, 0.12249106861462976, 0.05612774885335334, 0.18824284279645132, 0.5408758221949993, 0.1856894124684392, 0.800739652768382, 0.49579751300086994, 0.038316124231533896, 0.2793470410089653, 0.7390567545286253, 0.13387615011763396, 0.5562269972352019, 0.828421814679051, 0.6158551035505464, 0.9284282431053783, 0.719893802729458, 0.5006448235416101, 0.17617652765255387, 0.3329459718070196, 0.48590649925356255, 0.582668876951735, 0.5666842362460727, 0.2818872933220244, 0.04023292795518566, 0.41584681891390385, 0.050773233814683655, 0.3701246080913177, 0.6046442361705109, 0.8807082781511177, 0.06445001620018902, 0.45359582493239675, 0.4073895518221113, 0.505403965633555, 0.6961204681954214, 0.2942543712042644, 0.05060837642367526, 0.740525836952263, 0.26798410745801465, 0.3862722296359855, 0.7869988921213069, 0.6307874542033728, 0.06973132621279067, 0.39024489604158674, 0.7517269738043657, 0.0730624348256431, 0.956676386857855, 0.25787974374077854, 0.6624649132442717, 0.5402301991432201, 0.10433103719877292, 0.49131898716282163, 0.12079080173243076}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. Element({200, 300, 400, 500, 600, 700}, 1 + floor((currentRandomSeeds(10) * 6))) → partial1
  4. Element({100, 50}, 1 + floor((currentRandomSeeds(9) * 2))) → partial2
  5. Element({{1, 1, 2, 2}, {1, 2, 1, 2}, {1, 2, 2, 1}, {2, 1, 1, 2}, {2, 1, 2, 1}, {2, 2, 1, 1}}, 1 + floor((currentRandomSeeds(2) * 6))) → positions
  6. 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
  7. Corner(5) → layoutAppSize
  8. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  9. Textfield(studentA) → inputA
  10. Textfield(studentB) → inputB
  11. Textfield(studentC) → inputC
  12. Textfield(studentD) → inputD
  13. Textfield(studentE) → inputE
  14. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), AnchorPX) → AnchorModel
  15. Sequence(AnchorModel + ((boxWidth * A), 0), A, 0, numBoxes - 1) → boxAnchors
  16. Zip(cornerBox[boxWidth, boxHeight, A], A, Take[boxAnchors, 1, numBoxes]) → allBoxes
  17. Vector((0, 0), ((0.5 * boxWidth), (0.6 * boxHeight))) → vArrow
  18. Vector((0, 0), ((0.9 * boxWidth), (0.15 * boxHeight))) → vOffset
  19. Textfield(studentF) → inputF
  20. Text((((("\textcolor{" * HIGHLIGHTb) * "}{") * divisorDigit) * "}"), AnchorModel + (0, boxHeight / 2), true, true, -1, 0) → textDivisor
  21. Translate(vArrow, boxAnchors(1) + vOffset) → arrow1
  22. Translate(vArrow, boxAnchors(2) + vOffset) → arrow2
  23. Translate(vArrow, boxAnchors(3) + vOffset) → arrow3
  24. TableText({(((("\textcolor{" * HIGHLIGHTa) * "}{") * C1) * "}"), (("" * If[C2 ≥ 0, ("-~" * If[B2 < 1000, "~", ""])]) * If[(¬showsolution) ∧ positions(1) ≟ 2, (("\phantom{" * B2) * "}"), ("" * B2)]), (((("\textcolor{" * HIGHLIGHTa) * "}{") * C2) * "}")}, "vr", 2, (heightPX - 10) / 3) → cell1
  25. TableText({(((("\textcolor{" * HIGHLIGHTa) * "}{") * C2) * "}"), (("" * If[C3 ≥ 0, ("-~" * If[B3 < 1000, "~", ""])]) * If[(¬showsolution) ∧ positions(2) ≟ 2, (("\phantom{" * B3) * "}"), ("" * B3)]), (((("\textcolor{" * HIGHLIGHTa) * "}{") * C3) * "}")}, "vr", 2, (heightPX - 10) / 3) → cell2
  26. TableText({(((("\textcolor{" * HIGHLIGHTa) * "}{") * C3) * "}"), (If[C4 ≥ 0, ("-~~" * If[B4 < 100, "~", ""])] * If[(¬showsolution) ∧ positions(3) ≟ 2, (("\phantom{" * B4) * "}"), ("" * B4)]), (((("\textcolor{" * HIGHLIGHTa) * "}{") * C4) * "}")}, "vr", 2, (heightPX - 10) / 3) → cell3
  27. TableText({(((("\textcolor{" * HIGHLIGHTa) * "}{") * C4) * "}"), (If[C5 ≥ 0, ("-~~" * If[B5 < 100, "~", ""])] * If[(¬showsolution) ∧ positions(4) ≟ 2, (("\phantom{" * B5) * "}"), ("" * B5)]), (((("\textcolor{" * HIGHLIGHTa) * "}{") * C5) * "}")}, "vr", 2, (heightPX - 10) / 3) → cell4
  28. TableText(cell1, cell2, cell3, cell4, "c", widthPX, heightPX) → fullTable
  29. Text(("" * partial1), AnchorModel + (boxWidth / 2, (1.1 * boxHeight)), true, true, 0, 1) → displayPartial1
  30. Text(("" * partial2), boxAnchors(2) + (boxWidth / 2, (1.1 * boxHeight)), true, true, 0, 1) → displayPartial2
  31. Text(("" * partial3), boxAnchors(3) + (boxWidth / 2, (1.1 * boxHeight)), true, true, 0, 1) → displayPartial3
  32. Text(("" * partial4), boxAnchors(4) + (boxWidth / 2, (1.1 * boxHeight)), true, true, 0, 1) → displayPartial4
  33. Zip(Text["\plus", A + (0, (1.1 * boxHeight)), true, true, 0, 1], A, {boxAnchors(2), boxAnchors(3), boxAnchors(4)}) → additionSigns
  34. Textfield(studentA) → inputA1
  35. Textfield(studentB) → inputB1
  36. Textfield(studentC) → inputC1
  37. Textfield(studentD) → inputD1
  38. Zip(Segment[A + ((0.2 * boxHeight), (0.3 * boxHeight)), A + ((0.8 * boxWidth), (0.3 * boxHeight))], A, boxAnchors) → dividerLines

AI 绘图提示语参考

绘制一个多位数除以单位数的面积模型除法演示图形。在画面左侧放置一个大的矩形框，内部用三条垂直分割线将其分为四个横向排列的子矩形，每个子矩形上方标注部分数值（如100、50等），子矩形内部用对角线或交叉线表示该区域的面积。右侧空白处设置五个文本输入框，用于学生输入商和余数。图形上方或中间位置显示除数（单位数），用醒目的颜色标注。整个图形下方或旁边有一个表格，显示部分积的竖式计算过程，表格中部分数字用遮挡效果处理。整体布局要清晰展示位值分解的概念，体现除法作为重复分组的几何意义。
以上提示语为根据原始绘图命令整理而来，仅供参考。