MathPlotAI Logo MathPlotAI

Modeling Completing the Square with Algebra Tiles (Visual Strategy)

Modeling Completing the Square with Algebra Tiles (Visual Strategy)
Loading the math board and drawing, please wait…
2024-04-03 07:31
FunctionSequenceGeometric ConstructionGraphing
This example demonstrates completing the square using algebra tiles as a visual model, showing students how a quadratic expression can be rearranged into a perfect square plus a remainder. The key math concepts include solving quadratic equations by completing the square, using algebraic equivalence transformations, and understanding why adding a constant term requires adding the same amount of the corresponding negative terms to preserve equality. Construction and interaction are built from controlled, parameter-driven generation: the app uses RandomUniform to create task-specific coefficients, Take/Sequence/Max/Min and If to determine how many tile pieces are needed and how they are arranged. Geometry/UI placement and labeling rely on Corner, PixelsToPoint, centerBox, and Text, while assembling and manipulating objects uses Element together with Vector, Join, Zip, Flatten, and Translate. Answer workflows are handled via showanswer/showhint and scripts that reset and prepare input areas (e.g., scriptResetApp, scriptEmptyInputBoxes). Suitable for grades 9–12 for lesson exploration and practice focused on the strategy behind completing the square.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Create a visual model for completing the square using algebra tiles. A large square in the center represents the quadratic term, with its side labeled 'x'. Based on random coefficients, rectangular tiles representing the linear term are placed to the right and below the square, and they can rotate and move to fill the square. Meanwhile, small square tiles representing the constant term are placed outside and gradually move in to complete the square. Throughout the process, an expression updates in a corner, showing the algebraic steps. All tiles are color-coded, with a grid background for easy observation of area changes.
The prompt above was compiled from the original drawing commands and is for reference only.