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用代数瓦片建模配方法：完成平方的代数策略

用代数瓦片建模配方法：完成平方的代数策略
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2024-04-03 07:31
函数数列几何作图函数图像
这个范例通过代数瓦片的直观拼摆，展示如何把二次表达式配成完全平方，从而引导学生理解“完成平方”的代数策略。核心数学概念包括：二次函数/二次方程、配方法、代数恒等变形，以及用同样数量的正负项保持等式平衡。作图与交互要点是用随机生成的题目参数来驱动情境：程序先用 RandomUniform 生成随机数，再用 Take、Sequence、Max、Min、If 等逻辑确定题目中所需项的数量与排布；同时用 Corner、PixelsToPoint、centerBox、Text 等将界面中的元素定位与标注；为构造或移动瓦片/图元，使用 Vector、Join、Zip、Flatten、Translate 等几何与对象组织/变换操作；最终用 Element 与各种脚本对象（如 scriptFixAllText、scriptEmptyInputBoxes、scriptResetApp）控制作答区域、提示/答案显示（showanswer、showhint）以及计数统计（tries、countCorrect）。适合9-12年级课堂教学、探究式学习与差异化练习，用于理解并练习“用代数瓦片完成平方”的思想与步骤。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.6701327566780267, 0.6696444842365005, 0.5259546813852927, 0.34940432864623294, 0.15925969234486, 0.48692867137716483, 0.8114439458968032, 0.7678107285783305, 0.6159506454758904, 0.21449722663377002, 0.011000711186750311, 0.5805051189243565, 0.17910342927723494, 0.7097248746435941, 0.21157901792289321, 0.35434427416458486, 0.5207202170612151, 0.38706390416171044, 0.008425368798318641, 0.6814963727635439, 0.5685331676059519, 0.9139357291113133, 0.7596528585003112, 0.9373853736914356, 0.39108271029508834, 0.5320202108681036, 0.535104510463396, 0.03965300278995254, 0.7042905231091791, 0.6626407342139718, 0.6508093374293212, 0.5717144161560295, 0.6105136277734062, 0.6139963960525399, 0.45587163178868106, 0.5308805254508726, 0.6176963985890018, 0.7992325926479024, 0.5238586654936883, 0.15460380921453729, 0.7975555082110618, 0.677769385812476, 0.335663939701255, 0.3291758587251298, 0.5677393131913874, 0.2679215356243891, 0.5454228580509537, 0.21316407896001327, 0.5935958344284721, 0.48664783931207567, 0.025344776791282213, 0.07302113243459031, 0.48056461409311985, 0.22225669536195722, 0.21930007537863772, 0.7306349677713748, 0.048276431246294244, 0.008073144448070354, 0.8386628238827663, 0.09210887818067978, 0.5122001595937969, 0.869437767371328, 0.5072592558568431, 0.16376941228049396, 0.4881197081360499, 0.6729601726431405, 0.6453096462783551, 0.31529723515655456, 0.27885487770724526, 0.5056907103468128, 0.17497547336175662, 0.9409593493472015, 0.07145821157362801, 0.6528011293644177, 0.15253592838380758, 0.8681931626790238, 0.48662575241007006, 0.5054805418149233, 0.30431753083949603, 0.2682512542174228, 0.2602532033710515, 0.41605576689200285, 0.285287792508395, 0.6262626718706746, 0.04994938347908016, 0.47483494047359764, 0.47069814029797324, 0.7579004251389142, 0.08688805871067211, 0.4228517163713596, 0.522323863718066, 0.14686608946452162, 0.8456338712293238, 0.6965519715354358, 0.031698105297252525, 0.40161195549019707, 0.8815227828670839, 0.7845179917683117, 0.3732741446142837, 0.8011169014689244, 0.8168072827255961, 0.532565432102175, 0.4006751143386899, 0.8581170610216416, 0.8130953001955047, 0.09197550389732742, 0.4891015905736811, 0.09396275762285186, 0.9386182221208361, 0.5062411845163243, 0.4665683178228655, 0.012233300726354973, 0.42455539393564834, 0.7645076002614412, 0.1459272501674251, 0.23838680819942004, 0.3250248870875467, 0.19487807060615786, 0.338609437122015, 0.35005677622991943, 0.0004175773761937196, 0.7979345647634726, 0.9596801784666498, 0.7671281532418056, 0.5335080381066812, 0.15018651660737234, 0.13215143829155207, 0.5924352243964468, 0.5422186052791577, 0.5487149676336195, 0.20653267078163415, 0.6628194799004878, 0.7720444174777024, 0.4720187536009705, 0.5593257771290063, 0.6463700069031745, 0.5749653233993658, 0.967997821761309, 0.9315821204300584, 0.094940513058349, 0.6101759341731344, 0.7881235345923577, 0.4557389604380481, 0.6814956090620007, 0.06515948700767693, 0.6371485169132075, 0.9357401144666838, 0.019485878613899537, 0.3135110850166412, 0.7001993099515914, 0.4819195703513505, 0.1362852786065438, 0.20848469304754827, 0.19377230362340236, 0.19829001933727125, 0.124745392394947, 0.05849420122271827, 0.004018941493784478, 0.7057265334503312, 0.7261308192610645, 0.9627418576795753, 0.21475876293036886, 0.2100737957823506, 0.8040955879791455, 0.5217786293493667, 0.21889946697904342, 0.809429582381504, 0.9429453542117031, 0.42911908457070075, 0.6503702592588814, 0.6197688450130963, 0.23420081356496178, 0.6560396496080079, 0.2992905632687006, 0.03267309739159956, 0.47343736196329195, 0.43238175822066655, 0.25089855760503826, 0.8232163704684747, 0.6545222317246305, 0.5915633069666761, 0.4387547257202103, 0.15003384029277092, 0.5407042760008814, 0.12806890683642394, 0.61491229990349, 0.5132614007052617, 0.5715332955931061, 0.030446744084582287, 0.40411314398083237, 0.1793837756561156, 0.2812944301841019, 0.3022816764231499, 0.9256191898116755, 0.38506618638841567, 0.348540072204955, 0.5677463664116559, 0.8650301663692042, 0.47476907445474287, 0.407385353626585}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. Corner(5) → layoutAppSize
  4. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, (xMin, yMin), (xMax, yMax), (800, 500)) → OnboardingTextOffset
  5. Element({-10, -8, -6, 6, 8, 10}, 1 + floor((currentRandomSeeds(1) * 6))) → coeffB
  6. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (120, 250)) → AnchorSquared
  7. centerBox(xHeight, xHeight, AnchorSquared) → squaredBox
  8. Element(Remove[Sequence[-10, 10], {0, absCoeffB / 2^(2)}], 1 + floor((currentRandomSeeds(2) * 19))) → coeffC
  9. Sequence(AnchorX + ((A * (xWidth + gap)), 0), A, 0, abs(absCoeffB) - 1) → anchorsX
  10. Max(Min[1, 1 + animateControl - ceil(animateControl)], 0) → animateCurrent
  11. If(floor(animateControl) < absCoeffB, 0, If[floor(animateControl) ≟ absCoeffB, animateControl - absCoeffB, 1]) → animateStage3A
  12. If(floor(animateControl) < absCoeffB, 0, If[floor(animateControl) ≟ absCoeffB, animateControl - absCoeffB, 1]) → animateStage3
  13. Vector(AnchorStage3 - AnchorStage1) → vShift
  14. Sequence(AnchorUnitStart + (A * (xWidth + gap, 0)) + (animateStage3A * vShift), A, 0, abs(absCoeffC)) → anchorUnits
  15. Sequence(AnchorSquared + ((0.5 * (xHeight + xWidth)) + gap + (A * (xWidth + gap)), 0), A, 0, floor(abs(absCoeffB) / 2) - 1) → anchorsXStage2A
  16. Sequence(AnchorSquared - (0, (0.5 * (xHeight + xWidth)) + gap + (A * (xWidth + gap))), A, 0, floor(abs(absCoeffB) / 2) - 1) → anchorsXStage2B
  17. Join({anchorsXStage2A, anchorsXStage2B}) → anchorsXStage2
  18. Zip(Vector[B - A], A, anchorsX, B, anchorsXStage2) → vMove
  19. Min(ceil(animateControl), absCoeffB) → animateIndexB
  20. If(animateIndexB ≤ absCoeffB / 2, 0, pi / 2) → rotateAngle
  21. Translate(Rotate[centerBox[xWidth, xHeight, anchorsX(animateIndexB)], (animateCurrent * rotateAngle), anchorsX(animateIndexB)], (animateCurrent * vMove(animateIndexB))) → moveBoxB
  22. Min(If[isNegativeC, 0, (absCoeffB / 2)^(2)], absCoeffC) → numUnitsMove
  23. Flatten(Sequence[Sequence[(x(anchorsXStage2A(i)), y(anchorsXStage2B(j))), i, 1, absCoeffB / 2], j, 1, absCoeffB / 2]) → anchorUnitsStage2
  24. If(floor(animateControl) < absCoeffB + 2, 0, If[floor(animateControl) ≟ absCoeffB + 2, animateControl - absCoeffB - 2, 1]) → animateStage4
  25. Flatten(Sequence[Sequence[anchorUnits(1) + ((A * (xWidth + gap)), ((-K) * (xWidth + gap))), A, 0, absCoeffB / 2 - 1], K, 1, absCoeffB / 2]) → anchorUnitsStage3
  26. If(floor(animateControl) < absCoeffB + 3, 0, If[floor(animateControl) ≟ absCoeffB + 3, animateControl - absCoeffB - 3, 1]) → animateStage5
  27. Translate(anchorUnitsStage3, Vector[((1 - animateStage5) * Vector[AnchorSubtractStage3 - AnchorStage3])]) → anchorsSubtractStage3
  28. Zip(Vector[B - A], A, anchorUnitsStage3, B, anchorUnitsStage2) → vMoveUnits
  29. Max(Min[ceil(animateControl) - absCoeffB - 4, numUnitsMove], 0) → animateIndexStage6
  30. Zip(centerBox[xWidth, xWidth, A], A, anchorUnitsStage2) → unitsStage2Frames
  31. Zip(Vector[B - A], A, anchorsSubtractStage3, B, anchorUnits) → vMoveSubtract
  32. Translate(centerBox[xWidth, xWidth, anchorsSubtractStage3(animateIndexStage6)], (animateCurrent * vMoveSubtract(animateIndexStage6))) → moveSubtractBox
  33. centerBox(squareWidth, squareWidth, CenterSquare) → squareFrame
  34. Text("x", AnchorSquared - (xHeight / 2, 0), true, true, -1, 0) → lblSideX
  35. Zip(centerBox[xWidth, xWidth, A], A, anchorUnitsStage3) → unitsStage4
  36. Zip(centerBox[xWidth, xWidth, A + (animateStage5 * B)], A, anchorUnitsStage3, Vector[B], vMoveUnits) → unitsStage5
  37. Sequence(centerBox[xWidth, xWidth, anchorUnits(A)], A, 1, absCoeffC) → unitsStage1
  38. Zip(centerBox[xWidth, xWidth, A], A, Take[anchorsSubtractStage3, animateIndexStage6 + 1]) → subtractStage3
  39. If(floor(animateControl) < absCoeffB + 4 + numUnitsMove, 0, If[floor(animateControl) ≟ absCoeffB + 4 + numUnitsMove, animateControl - absCoeffB - 4 - numUnitsMove, 1]) → animateStage7
  40. Take(unitsStage1, 1, If[animateStage7 ≟ 0, animateIndexStage6 - 1, numUnitsMove]) → unitsSubtractStage5
  41. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (If[animateControl ≤ 4, 310, 16], 60)) → PExpr
  42. Element({("" * finalConstant), "", ("+" * finalConstant)}, sgn(finalConstant) + 2) → finalConstantLatex
  43. If(absCoeffC < 0, ("" * absCoeffC), If[absCoeffC ≟ 0, "", ("+" * absCoeffC)]) → coeffCLatex
  44. Text(fullExpression, PExpr, true, true, If[animateControl ≤ 4, 0, 1], -1) → displayExpr
  45. Segment((((-xWidth)) / 2, ((-xHeight)) / 2), (xWidth / 2, xHeight / 2)) → templateSlashX
  46. Sequence(centerBox[xWidth, xHeight, anchorsX(A)], A, animateIndexB + 1, abs(absCoeffB)) → xBlocksStart
  47. Sequence(centerBox[If[A > abs(absCoeffB) / 2, xHeight, xWidth], If[A > abs(absCoeffB) / 2, xWidth, xHeight], anchorsXStage2(A)], A, 1, Min[ceil(animateControl) - 1, abs(absCoeffB)]) → xBlocksStage2
  48. Vector(AnchorStage1 - AnchorStage3) → vSlideSubtract
  49. Append((1000, 1000), Take[anchorUnits, 1, Min[animateIndexStage6, numUnitsMove]]) → anchorsCrosses
  50. Sequence(Translate[crossTemplate, anchorUnits(A)], A, 1, absCoeffC) → possibleCrosses
  51. Flatten(Take[possibleCrosses, 1, If[animateStage7 ≟ 0, animateIndexStage6 - 1, numUnitsMove]]) → displayCrosses
  52. Sequence(centerBox[(1.4 * xWidth), (1.4 * xWidth), anchorUnits(A)], A, 1, absCoeffC) → possibleFades
  53. Take(possibleFades, 1, If[animateStage7 ≟ 0, animateIndexStage6 - 1, numUnitsMove]) → fadeBoxes
  54. Zip(centerBox[xWidth - (2 * borderWidth), xWidth - (2 * borderWidth), A], A, Take[anchorsSubtractStage3, animateIndexStage6 + 1]) → subtractStage3Centers
  55. Translate(centerBox[xWidth - (2 * borderWidth), xWidth - (2 * borderWidth), anchorsSubtractStage3(animateIndexStage6)], (animateCurrent * vMoveSubtract(animateIndexStage6))) → moveSubtractCenter
  56. Text(("\vphantom{x^2}" * LaTeX[((If[isNegativeB, "-", ""] * absCoeffB) * "x")]), CenterX + (0, ((-xHeight)) / 2), true, true, 0, -1) → captionSlideB
  57. Text("x^2", AnchorSquared + (0, ((-xHeight)) / 2), true, true, 0, -1) → captionSquared
  58. Zip(centerBox[xWidth, xWidth, A], A, Join[{anchorUnitsStage3, anchorsSubtractStage3}]) → fadeCoverStage4
  59. Sequence(centerBox[xWidth - (2 * gap), xWidth - (2 * gap), anchorUnits(A)], A, 1, absCoeffC) → negativeUnitsStage1
  60. Sequence(centerBox[xWidth - (2 * gap), xHeight - (2 * gap), anchorsX(A)], A, animateIndexB + 1, abs(absCoeffB)) → negativeXStage1
  61. Translate(Rotate[centerBox[xWidth - (2 * gap), xHeight - (2 * gap), anchorsX(animateIndexB)], (animateCurrent * rotateAngle), anchorsX(animateIndexB)], (animateCurrent * vMove(animateIndexB))) → negativeMoveBox
  62. PointToPixels((xMin, yMin), (xMax, yMax), (800, 500), CenterX) → CenterXPX
  63. PointToPixels((xMin, yMin), (xMax, yMax), (800, 500), AnchorUnitStart) → CenterUnitsPX
  64. Sequence(centerBox[If[A > abs(absCoeffB) / 2, xHeight - (2 * gap), xWidth - (2 * gap)], If[A > abs(absCoeffB) / 2, xWidth - (2 * gap), xHeight - (2 * gap)], anchorsXStage2(A)], A, 1, Min[ceil(animateControl) - 1, abs(absCoeffB)]) → negativeXStage2
  65. Text(("\vphantom{x^2}" * LaTeX[(If[isNegativeC, "-", ""] * absCoeffC)]), CenterUnit + (0, ((-xHeight)) / 2), true, true, 0, -1) → captionSlideC
  66. Text("\vphantom{4}x", AnchorSquared + (0, xHeight / 2), true, true, 0, 1) → lblTopA
  67. Text(("" * coeffB / 2), AnchorSquared + ((0.5 * xHeight) + (xWidth * absCoeffB / 4), (0.5 * xHeight)), true, true, 0, 1) → lblTopB
  68. Text(("" * coeffB / 2), AnchorSquared - ((0.5 * xHeight), (xHeight * 0.5) + (xWidth * absCoeffB / 4)), true, true, -1, 0) → lblSideB

AI 绘图提示语参考

绘制一个用于演示配方法的代数瓷砖模型。画面中央有一个大正方形区域，代表二次项，其边长标有“x”。根据随机系数，在正方形右侧和下方排列若干矩形瓷砖，代表一次项，这些瓷砖可旋转并移动以填充正方形。同时，在正方形外放置若干小正方形瓷砖，代表常数项，它们会逐步移入以补全正方形。整个过程中，表达式在角落动态更新，显示配方的代数步骤。所有瓷砖均以不同颜色区分，并带有网格背景，便于观察面积变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。