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Completing the Square with Algebra Tiles: Choose an Expression

Completing the Square with Algebra Tiles: Choose an Expression
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2024-04-03 07:29
FunctionGeometric ConstructionSlider AnimationGraphing
This interactive GeoGebra lesson demonstrates the process of completing the square by letting students choose their own algebraic expression and then observing how algebra tiles model the step-by-step transformation. It focuses on rewriting a quadratic expression into an equivalent completed-square form, highlighting how terms and coefficients are adjusted to preserve equality. Key concepts include completing the square for solving quadratic equations, coefficient matching, algebraic equivalence, and understanding why adding a constant to finish the square requires adding the corresponding negative/compensating quantity as well. Construction and interaction are driven by programmatic parameter generation and UI control: randomization is used to select values (RandomUniform, Sequence, Max, If), various geometric and text objects are composed and positioned to build the visual model (Text, Segment, Vector, PixelsToPoint/PointToPixels, Translate, Join, Zip, Flatten, Take, Corner, Element), and app behavior is managed with scripting elements to reset the app and toggle hints/answers (showanswer, showhint, scriptResetApp, scriptFixAllText, scriptEmptyInputBoxes, button controls). Best suited for grades 9–12, classroom exploration, and guided practice of completing-the-square problem solving.

GeoGebra Commands & Parameters

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.8548254949126582, 0.8243255808539557, 0.7053537997663649, 0.5946431173553146, 0.1933366589400638, 0.5815387405495404, 0.7558949230858008, 0.4748264926115612, 0.4010455984507947, 0.7489214548777103, 0.9792670669916175, 0.044054862893809305, 0.019248765430733804, 0.22022913856092435, 0.249495865385833, 0.6319422221139752, 0.4593472751067925, 0.25020255148237586, 0.953566225962684, 0.7128823629942047, 0.0773307093276081, 0.42172646570247124, 0.4326254416052949, 0.21090037978669407, 0.8557573425616123, 0.9117201558238112, 0.9131135776330537, 0.23989808060489587, 0.681077783419194, 0.6263475956985084, 0.17503712671902794, 0.8112439515476332, 0.7240599301966252, 0.6119069270930962, 0.7356085429981243, 0.39127327189274, 0.7506736914291409, 0.3107177830470188, 0.5321673543985873, 0.9640667064196731, 0.1400806698260495, 0.8114716068108491, 0.958679272971693, 0.6498658285873696, 0.6653482484311628, 0.2908436901047139, 0.5551784752751239, 0.06666220593902177, 0.8711705791006473, 0.16768779621947907, 0.9792735629738891, 0.10186119548006432, 0.9612275066796402, 0.4628423673190306, 0.48328151128174984, 0.8324749744971062, 0.5058503031475466, 0.9464749522030766, 0.8396847387174243, 0.24603946179637454, 0.6683874758378703, 0.2809740428377824, 0.5814243301498342, 0.5303769754356413, 0.2160758852721784, 0.5006703006822191, 0.43204886196595793, 0.6084424517050515, 0.31832219721378885, 0.6001275403049549, 0.9639315611727436, 0.14479338227510524, 0.12208690193692093, 0.7203898980502166, 0.1006487383495816, 0.08271380204135219, 0.050537245096163044, 0.8074067785354367, 0.07218189843969866, 0.44570066450300133, 0.037298095962063314, 0.29887736994835506, 0.9288608178486034, 0.33388588938765107, 0.3244793779207024, 0.2649663826371737, 0.6119086922169998, 0.7350204831051192, 0.8481949669838652, 0.24473878660498175, 0.2750812887749394, 0.9208866904783966, 0.12051375903460239, 0.4637453397183963, 0.32321020491069485, 0.4434329389674912, 0.05169485879944131, 0.8489621524034654, 0.1494666518701368, 0.7905402684226256, 0.6311589018312113, 0.5752468688938315, 0.6438547610651368, 0.0391327504690752, 0.959934113274367, 0.24261971074125807, 0.7308252007426114, 0.2533342268736919, 0.9112911647074451, 0.12601290027408807, 0.2439674259941245, 0.9275512321700546, 0.8799421887242275, 0.9686517852555591, 0.08049526322088184, 0.7687275369725027, 0.9135290612567649, 0.9869418563534376, 0.05768375967028661, 0.30856832213859514, 0.0531639836688145, 0.5486304986232309, 0.5675459291182771, 0.11527221933934333, 0.8525085251753861, 0.1270978870697112, 0.1548566092465979, 0.9652373306669274, 0.12729100570757157, 0.6829016990785025, 0.9871364066528167, 0.5683222638123578, 0.9392603320560312, 0.46916012655826334, 0.7234019957848487, 0.7655463850048272, 0.7803624862695742, 0.6700036792665761, 0.13136997642820966, 0.4356033503445888, 0.8244000873154919, 0.11116046954923754, 0.8703699774281334, 0.16623979606697462, 0.3945972506363663, 0.27876108396134813, 0.19947819102265874, 0.9284166873729707, 0.3163423143536776, 0.6166581772674675, 0.08732638969460937, 0.9152227095280532, 0.9849174661191601, 0.9239523478548932, 0.6616716947755308, 0.20375039092575944, 0.21824908907079124, 0.08299986241677385, 0.5431315147793093, 0.08657254017683928, 0.014625085164798811, 0.18394927385308843, 0.16849315692100925, 0.12007200844870991, 0.39492439341264207, 0.30481314097976236, 0.6947014551479902, 0.6596809428751073, 0.48819181860507166, 0.6862270267003541, 0.5850085672655166, 0.01707536720601408, 0.3483650840113017, 0.1773826553124577, 0.6538381969251288, 0.701549029752595, 0.6686238615078947, 0.18414305647891713, 0.5466727047292412, 0.5871554471252861, 0.8981795231978387, 0.2020565423587069, 0.3957145440441569, 0.6787600811808708, 0.2543620371952242, 0.18588339645529334, 0.8312047697859359, 0.5638845538315781, 0.8605845250438718, 0.6333321030962124, 0.8520993554564408, 0.3631526445592118, 0.6981783974843492, 0.4539747834347205, 0.5283100280016995, 0.8802632335770939, 0.6871917891019502, 0.8047689904495958, 0.3599096425469103, 0.6027176562411295}
  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, -4, -2, 2, 4, 6, 8, 10}, slideB) → coeffB
  6. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (120, 250)) → AnchorSquared
  7. centerBox(xHeight, xHeight, AnchorSquared) → squaredBox
  8. Sequence(AnchorX + ((A * (xWidth + gap)), 0), A, 0, abs(absCoeffB) - 1) → anchorsX
  9. Max(Min[1, 1 + animateControl - ceil(animateControl)], 0) → animateCurrent
  10. If(floor(animateControl) < absCoeffB, 0, If[floor(animateControl) ≟ absCoeffB, animateControl - absCoeffB, 1]) → animateStage3A
  11. If(floor(animateControl) < absCoeffB, 0, If[floor(animateControl) ≟ absCoeffB, animateControl - absCoeffB, 1]) → animateStage3
  12. Vector(AnchorStage3 - AnchorStage1) → vShift
  13. Sequence(AnchorUnitStart + (A * (xWidth + gap, 0)) + (animateStage3A * vShift), A, 0, abs(absCoeffC)) → anchorUnits
  14. Sequence(AnchorSquared + ((0.5 * (xHeight + xWidth)) + gap + (A * (xWidth + gap)), 0), A, 0, floor(abs(absCoeffB) / 2) - 1) → anchorsXStage2A
  15. Sequence(AnchorSquared - (0, (0.5 * (xHeight + xWidth)) + gap + (A * (xWidth + gap))), A, 0, floor(abs(absCoeffB) / 2) - 1) → anchorsXStage2B
  16. Join({anchorsXStage2A, anchorsXStage2B}) → anchorsXStage2
  17. Zip(Vector[B - A], A, anchorsX, B, anchorsXStage2) → vMove
  18. Min(ceil(animateControl), absCoeffB) → animateIndexB
  19. If(animateIndexB ≤ absCoeffB / 2, 0, pi / 2) → rotateAngle
  20. Translate(Rotate[centerBox[xWidth, xHeight, anchorsX(animateIndexB)], (animateCurrent * rotateAngle), anchorsX(animateIndexB)], (animateCurrent * vMove(animateIndexB))) → moveBoxB
  21. Min(If[isNegativeC, 0, (absCoeffB / 2)^(2)], absCoeffC) → numUnitsMove
  22. Flatten(Sequence[Sequence[(x(anchorsXStage2A(i)), y(anchorsXStage2B(j))), i, 1, absCoeffB / 2], j, 1, absCoeffB / 2]) → anchorUnitsStage2
  23. If(floor(animateControl) < absCoeffB + 2, 0, If[floor(animateControl) ≟ absCoeffB + 2, animateControl - absCoeffB - 2, 1]) → animateStage4
  24. Flatten(Sequence[Sequence[anchorUnits(1) + ((A * (xWidth + gap)), ((-K) * (xWidth + gap))), A, 0, absCoeffB / 2 - 1], K, 1, absCoeffB / 2]) → anchorUnitsStage3
  25. If(floor(animateControl) < absCoeffB + 3, 0, If[floor(animateControl) ≟ absCoeffB + 3, animateControl - absCoeffB - 3, 1]) → animateStage5
  26. Translate(anchorUnitsStage3, Vector[((1 - animateStage5) * Vector[AnchorSubtractStage3 - AnchorStage3])]) → anchorsSubtractStage3
  27. Zip(Vector[B - A], A, anchorUnitsStage3, B, anchorUnitsStage2) → vMoveUnits
  28. Max(Min[ceil(animateControl) - absCoeffB - 4, numUnitsMove], 0) → animateIndexStage6
  29. Zip(centerBox[xWidth, xWidth, A], A, anchorUnitsStage2) → unitsStage2Frames
  30. Zip(Vector[B - A], A, anchorsSubtractStage3, B, anchorUnits) → vMoveSubtract
  31. Translate(centerBox[xWidth, xWidth, anchorsSubtractStage3(animateIndexStage6)], (animateCurrent * vMoveSubtract(animateIndexStage6))) → moveSubtractBox
  32. centerBox(squareWidth, squareWidth, CenterSquare) → squareFrame
  33. Text("x", AnchorSquared - (xHeight / 2, 0), true, true, -1, 0) → lblSideX
  34. Zip(centerBox[xWidth, xWidth, A], A, anchorUnitsStage3) → unitsStage4
  35. Zip(centerBox[xWidth, xWidth, A + (animateStage5 * B)], A, anchorUnitsStage3, Vector[B], vMoveUnits) → unitsStage5
  36. Sequence(centerBox[xWidth, xWidth, anchorUnits(A)], A, 1, absCoeffC) → unitsStage1
  37. Zip(centerBox[xWidth, xWidth, A], A, Take[anchorsSubtractStage3, animateIndexStage6 + 1]) → subtractStage3
  38. If(floor(animateControl) < absCoeffB + 4 + numUnitsMove, 0, If[floor(animateControl) ≟ absCoeffB + 4 + numUnitsMove, animateControl - absCoeffB - 4 - numUnitsMove, 1]) → animateStage7
  39. Take(unitsStage1, 1, If[animateStage7 ≟ 0, animateIndexStage6 - 1, numUnitsMove]) → unitsSubtractStage5
  40. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (If[animateControl ≤ 4, 310, 16], 60)) → PExpr
  41. Element({("" * finalConstant), "", ("+" * finalConstant)}, sgn(finalConstant) + 2) → finalConstantLatex
  42. If(absCoeffC < 0, ("" * absCoeffC), If[absCoeffC ≟ 0, "", ("+" * absCoeffC)]) → coeffCLatex
  43. Text(fullExpression, PExpr, true, true, If[animateControl ≤ 4, 0, 1], -1) → displayExpr
  44. Segment((((-xWidth)) / 2, ((-xHeight)) / 2), (xWidth / 2, xHeight / 2)) → templateSlashX
  45. Sequence(centerBox[xWidth, xHeight, anchorsX(A)], A, animateIndexB + 1, abs(absCoeffB)) → xBlocksStart
  46. 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
  47. Vector(AnchorStage1 - AnchorStage3) → vSlideSubtract
  48. Append((1000, 1000), Take[anchorUnits, 1, Min[animateIndexStage6, numUnitsMove]]) → anchorsCrosses
  49. Sequence(Translate[crossTemplate, anchorUnits(A)], A, 1, absCoeffC) → possibleCrosses
  50. Flatten(Take[possibleCrosses, 1, If[animateStage7 ≟ 0, animateIndexStage6 - 1, numUnitsMove]]) → displayCrosses
  51. Sequence(centerBox[(1.4 * xWidth), (1.4 * xWidth), anchorUnits(A)], A, 1, absCoeffC) → possibleFades
  52. Take(possibleFades, 1, If[animateStage7 ≟ 0, animateIndexStage6 - 1, numUnitsMove]) → fadeBoxes
  53. Zip(centerBox[xWidth - (2 * borderWidth), xWidth - (2 * borderWidth), A], A, Take[anchorsSubtractStage3, animateIndexStage6 + 1]) → subtractStage3Centers
  54. Translate(centerBox[xWidth - (2 * borderWidth), xWidth - (2 * borderWidth), anchorsSubtractStage3(animateIndexStage6)], (animateCurrent * vMoveSubtract(animateIndexStage6))) → moveSubtractCenter
  55. Zip(centerBox[xWidth, xWidth, A], A, Join[{anchorUnitsStage3, anchorsSubtractStage3}]) → fadeCoverStage4
  56. Sequence(centerBox[xWidth - (2 * gap), xWidth - (2 * gap), anchorUnits(A)], A, 1, absCoeffC) → negativeUnitsStage1
  57. Sequence(centerBox[xWidth - (2 * gap), xHeight - (2 * gap), anchorsX(A)], A, animateIndexB + 1, abs(absCoeffB)) → negativeXStage1
  58. Translate(Rotate[centerBox[xWidth - (2 * gap), xHeight - (2 * gap), anchorsX(animateIndexB)], (animateCurrent * rotateAngle), anchorsX(animateIndexB)], (animateCurrent * vMove(animateIndexB))) → negativeMoveBox
  59. PointToPixels((xMin, yMin), (xMax, yMax), (800, 500), CenterX) → CenterXPX
  60. PointToPixels((xMin, yMin), (xMax, yMax), (800, 500), AnchorUnitStart) → CenterUnitsPX
  61. 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
  62. Text("\vphantom{4}x", AnchorSquared + (0, xHeight / 2), true, true, 0, 1) → lblTopA
  63. Text(("\vphantom{x^2}" * LaTeX[((If[isNegativeB, "-", ""] * absCoeffB) * "x")]), CenterX + (0, ((-xHeight)) / 2), true, true, 0, -1) → captionSlideB
  64. Text(("\vphantom{x^2}" * LaTeX[(If[isNegativeC, "-", ""] * absCoeffC)]), CenterUnit + (0, ((-xHeight)) / 2), true, true, 0, -1) → captionSlideC
  65. Text(("" * coeffB / 2), AnchorSquared - ((0.5 * xHeight), (xHeight * 0.5) + (xWidth * absCoeffB / 4)), true, true, -1, 0) → lblSideB
  66. Text(("" * coeffB / 2), AnchorSquared + ((0.5 * xHeight) + (xWidth * absCoeffB / 4), (0.5 * xHeight)), true, true, 0, 1) → lblTopB
  67. Text("x^2", AnchorSquared + (0, ((-xHeight)) / 2), true, true, 0, -1) → lblSqStage1

AI Drawing Prompt Reference

This is a visual representation of the completing the square method. The graphic consists of a series of algebraic tiles that initially show a quadratic expression. Users can drag and add tiles to gradually transform the expression into a perfect square form. A central area represents the perfect square part, along with labels and text that annotate the current expression and steps. The tiles are arranged in a specific order to demonstrate how to adjust coefficients by adding and moving tiles, ultimately yielding an equivalent perfect square expression. The graphic dynamically illustrates the completing the square process, helping to understand how to add a constant to complete the square.
The prompt above was compiled from the original drawing commands and is for reference only.