积极 AI Logo 积极 AI

比较汽车比赛中的变化率：斜率与线性关系

比较汽车比赛中的变化率：斜率与线性关系
几何画板及绘图内容加载中，请稍等……
2023-03-03 11:36
函数解析几何几何作图滑块动画
本范例通过可视化展示不同速度下汽车在赛道上的位置变化，帮助学生理解斜率作为变化率的含义。核心数学概念包括线性关系、斜率比较和坐标几何。作图要点涉及使用GeoGebra命令如RandomUniform、Sequence和Polygon来构造赛道和汽车模型。适合6-8年级学生用于代数和函数学习，适用于课堂练习和在线教学资源。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.07536995706471185, 0.32824398293471024, 0.5405278112233036, 0.6649148474939078, 0.5006162004100416, 0.7149908206683631, 0.888918252052941, 0.8786544653798103, 0.10978001909606723, 0.6298269886342113, 0.9498566092712807, 0.9279592129695333, 0.6174705258993755, 0.1551642949034372, 0.2957008027515774, 0.05758673623316867, 0.9001759058417332, 0.19903558280325603, 0.7294353040385582, 0.6914133513232606, 0.9701593479656597, 0.8559646729748696, 0.3761454681992267, 0.17980763395486749, 0.02932263874741608, 0.5113984128222984, 0.2576343850162359, 0.9716646786294414, 0.16366187698215062, 0.9045126101951195, 0.4333801691474773, 0.423603190732944, 0.026508621833590058, 0.923881330036718, 0.8721936337858348, 0.07584882750367412, 0.15100525293034534, 0.26236404698933385, 0.559786061134826, 0.28853675021938935, 0.7692067463582213, 0.5416183292026325, 0.007993882608907925, 0.8035620506347639, 0.7168321728520424, 0.9728853970986978, 0.8816316300887643, 0.20569143162605644, 0.19900726682394748, 0.42912425367352064, 0.5088565020597612, 0.1624206440395982, 0.5833455624976838, 0.3441846712064137, 0.5705277623525934, 0.4650211251006965, 0.7724085334088369, 0.647512665621895, 0.1633313939072731, 0.4433659451547565, 0.2079667183935607, 0.9178719518767109, 0.9104141836310834, 0.3044590179054461, 0.5045513745448905, 0.7067424908083919, 0.03193487397797327, 0.5591131547715676, 0.5995426273831705, 0.8914230819498179, 0.12628852066181256, 0.24293143017971278, 0.18148326635593448, 0.3894167855599936, 0.09079771225720888, 0.05664258956095468, 0.647438218642499, 0.189419355316947, 0.59838902182481, 0.09692592292940654, 0.15403791302935002, 0.13015702156358225, 0.46418491411457735, 0.9421315773144102, 0.6553499707037717, 0.20643722711579168, 0.823041357000883, 0.2177619364693272, 0.963916588739453, 0.38207637628196206, 0.035847723816624905, 0.6242259188186552, 0.21917631131623364, 0.24186111993750073, 0.16888120411406188, 0.8642122064813994, 0.52005675455585, 0.6393754614789219, 0.31666907278532364, 0.2190485605325656, 0.6157810630287772, 0.8243928931450714, 0.17466271074517734, 0.8126237248647271, 0.5908742198177606, 0.022583999788699814, 0.9135632709365417, 0.2378447781637362, 0.33698573375064605, 0.6115315313389861, 0.24236463414097356, 0.5141865441726815, 0.9754645605616517, 0.590471667910253, 0.9120907873772066, 0.4756617439715817, 0.37062664701241566, 0.5267961101208722, 0.6768042258775492, 0.506498713368133, 0.5543500128506502, 0.9301518450200118, 0.932749501505285, 0.37696097390219485, 0.4380127628160008, 0.33496607463587275, 0.5304705981178562, 0.8840759756154087, 0.9078572364218642, 0.49440141859885084, 0.8684025848163209, 0.3615624346390591, 0.3146203386934635, 0.8231115423689915, 0.7297878958846665, 0.9499546707376534, 0.5177126311927898, 0.7527202903779816, 0.4378047853926212, 0.8818182829735262, 0.8083600827093506, 0.2526877468129239, 0.43651963610892097, 0.3980361853912613, 0.48494053565826073, 0.3601539337535182, 0.03768754829359946, 0.18941502495523055, 0.20725291114596148, 0.7254666654060017, 0.5855247960374899, 0.6533212681927846, 0.3746504807699598, 0.6054024019639545, 0.6133639103269122, 0.9477023991497175, 0.07585027977748975, 0.7535792771349589, 0.07757696012773511, 0.294683091303444, 0.6657994848006024, 0.027239352286472774, 0.5253144359153514, 0.43294736577934273, 0.8672348252926271, 0.0714421201675007, 0.053457071132529455, 0.5123268815096389, 0.8115151866049974, 0.8957643601530497, 0.5067147268550152, 0.6954118084937855, 0.7349375586863393, 0.40484135149891726, 0.24303116088779542, 0.16958514245000209, 0.9730658986690702, 0.012512169579103383, 0.33436588332083994, 0.058652175648664984, 0.28660720703498044, 0.019947809328005366, 0.3405235106172481, 0.8761964754292657, 0.8880634142394255, 0.8809419989080765, 0.7829883382738959, 0.842939660462526, 0.42876796760368263, 0.9125367005827982, 0.7946965793860303, 0.15771667379396972, 0.47098456878771006, 0.3963306589058199, 0.9203404500026036, 0.1361993165804028, 0.49950136658915695, 0.22518499387510751, 0.6201484529369897, 0.553210415481495}
  2. Sequence(xMin + px16 + (n * (boxWidth + px16)), n, 0, 4) → xStarts, {-1.7447999999999997, 8.1189, 17.982599999999998, 27.8463, 37.71}
  3. makeBox((xStarts(1), dimensions(4) - (56 * pixelsH)), boxWidth, (boxRatio * (dimensions(4) - dimensions(2)))) → car1Box
  4. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  5. If(randomSpeed ≟ 1, "fastest", "slowest") → textRandomSpeed
  6. Sequence((A * 5), A, 4, 12) → possibleRates, {20, 25, 30, 35, 40, 45, 50, 55, 60}
  7. Element(possibleRates, 1 + floor((currentRandomSeeds(1) * Length[possibleRates]))) → randomRate1
  8. Element(Remove[possibleRates, {randomRate1}], 1 + floor((currentRandomSeeds(2) * (Length[possibleRates] - 1)))) → randomRate2
  9. Element(Remove[possibleRates, {randomRate1, randomRate2}], 1 + floor((currentRandomSeeds(3) * (Length[possibleRates] - 2)))) → randomRate3
  10. Element(Remove[possibleRates, {randomRate1, randomRate2, randomRate3}], 1 + floor((currentRandomSeeds(4) * (Length[possibleRates] - 3)))) → randomRate4
  11. IndexOf(Max[{randomRate1, randomRate2, randomRate3, randomRate4}], {randomRate1, randomRate2, randomRate3, randomRate4}) → maxIndex
  12. IndexOf(Min[{randomRate1, randomRate2, randomRate3, randomRate4}], {randomRate1, randomRate2, randomRate3, randomRate4}) → minIndex
  13. If(randomSpeed ≟ 1, maxIndex, minIndex) → correctIndex
  14. Corner(5) → layoutAppSize
  15. Polygon({(-4.5, 105), (5.9, 105), (5.9, -10), (-4.5, -10)}) → graphArea
  16. VerticalText("Miles") → labelY
  17. Segment((0, 00), (0, 105)) → yAxisSeg
  18. Segment((0, 0), (5.9, 0)) → xAxisSeg
  19. Zip(n + "\vphantom{|}&" + (m) + "\vphantom{|}\\\hline", n, randomPointsX2, m, randomPointsY2) → textTable2List
  20. Corner(labelX, 1) → PointBottomCases
  21. Polygon({(xTrackStart, yTrackStart), (xTrackStart, yTrackEnd), (xTrackEnd, yTrackEnd), (xTrackEnd, yTrackStart)}) → trackPolygon
  22. Sequence(Segment[(xTrackStart, yTrackEnd + (n * (yTrackStart - yTrackEnd) / 4)), (xTrackEnd, yTrackEnd + (n * (yTrackStart - yTrackEnd) / 4))], n, 1, 3) → trackLines
  23. Sequence((xTrackStart + 1, yTrackStart - (((2 * n) - 1) * (yTrackStart - yTrackEnd) / 8)), n, 1, 4) → carStart
  24. Sequence((xTrackEnd + 3, yTrackEnd + (((2 * n) - 1) * (yTrackStart - yTrackEnd) / 8)), n, 1, 4) → carEndOriginal
  25. Zip(A + ((750 / 800 * (xMax - xMin)), 0), A, carStart) → carEnd
  26. Zip(A + (Min[1, (animationSlider * R / Min[randomRates])] * (B - A)), A, carStart, B, carEnd, R, randomRates) → movingPoints
  27. Sequence((xTrackEnd - 10, yTrackStart - (((2 * n) - 1) * (yTrackStart - yTrackEnd) / 8)), n, 1, 4) → carRibbonPoints
  28. Sequence(Sequence[Polygon[{(xTrackEnd - ((k + 1) * finishBoxWidth), yTrackStart - (((2 * n) + k) * finishBoxHeight)), (xTrackEnd - ((k + 1) * finishBoxWidth), yTrackStart - (((2 * n) + 1 + k) * finishBoxHeight)), (xTrackEnd - ((k + 2) * finishBoxWidth), yTrackStart - (((2 * n) + 1 + k) * finishBoxHeight)), (xTrackEnd - ((k + 2) * finishBoxWidth), yTrackStart - (((2 * n) + k) * finishBoxHeight))}], n, 0, 7], k, 0, 1) → finishLineBlack, {{3.5582092503764216, 3.558209250375967, 3.558209250375967, 3.5582092503764216, 3.558209250375512, 3.5582092503764216, 3.558209250375512, 3.558209250377331}, {3.558209250375967, 3.558209250375967, 3.558209250375967, 3.5582092503764216, 3.5582092503764216, 3.5582092503764216, 3.5582092503764216, 3.5582092503764216}}
  29. Sequence(Sequence[Polygon[{(xTrackEnd - ((k + 2) * finishBoxWidth), yTrackStart - (((2 * n) - k + 1) * finishBoxHeight)), (xTrackEnd - ((k + 2) * finishBoxWidth), yTrackStart - (((2 * n) + 2 - k) * finishBoxHeight)), (xTrackEnd - ((k + 1) * finishBoxWidth), yTrackStart - (((2 * n) + 2 - k) * finishBoxHeight)), (xTrackEnd - ((k + 1) * finishBoxWidth), yTrackStart - (((2 * n) - k + 1) * finishBoxHeight))}], n, 0, 7], k, 0, 1) → finishLineWhite, {{3.5582092503764216, 3.5582092503764216, 3.5582092503768763, 3.5582092503768763, 3.5582092503764216, 3.5582092503764216, 3.5582092503764216, 3.558209250375512}, {3.5582092503764216, 3.5582092503764216, 3.5582092503768763, 3.558209250375967, 3.5582092503764216, 3.558209250375512, 3.558209250377331, 3.558209250375512}}
  30. Sort(randomRates) → randomRatesSorted
  31. Zip(IndexOf[n, randomRatesSorted], n, randomRates) → indexPlace
  32. Corner(textSize, 4) → PointSizeA_{start}
  33. Vector(PointSizeA_{start}, PointSizeA) → pointSizeA
  34. makeBox((xStarts(2), dimensions(4) - (56 * pixelsH)), boxWidth, (boxRatio * (dimensions(4) - dimensions(2)))) → car2Box
  35. makeBox((xStarts(3), dimensions(4) - (56 * pixelsH)), boxWidth, (boxRatio * (dimensions(4) - dimensions(2)))) → car3Box
  36. makeBox((xStarts(4), dimensions(4) - (56 * pixelsH)), boxWidth, (boxRatio * (dimensions(4) - dimensions(2)))) → car4Box
  37. Element({"Car A", "Car B", "Car C", "Car D"}, correctIndex) → correctCar
  38. Sequence((xTrackEnd - 5.8, yTrackStart - (((2 * n) - 1) * (yTrackStart - yTrackEnd) / 8)), n, 1, 4) → CarSpeedPoints
  39. If(randomSpeed ≟ 1, minIndex, maxIndex) → incorrectIndex
  40. Sequence(Text[n, (n - 0.175, -2.5), true, true], n, 0, 5) → xNums
  41. Zip(Vector[Midpoint[CarPointA, CarPointC], n], n, movingPoints) → vectorCars
  42. Translate(carAGreen, vectorCars(1)) → carAMoving
  43. Translate(carCPurple, vectorCars(3)) → carCMoving
  44. Translate(carDOrange, vectorCars(4)) → carDMoving
  45. Translate(carBRed, vectorCars(2)) → carBMoving
  46. Sequence(Segment[(n, 0), (n, -1.5)], n, 0, 5) → xAxisSegs
  47. Sequence(Text[(20 * n), ((-If[n < 5, 1.1, 1.3]), (20 * n) + 4), true, true], n, 1, 5) → yNums
  48. If(showanswer ∨ showsolution ∨ validate ∧ correct, If[correctIndex ≟ 1, "#008475", selectedIndex ≟ 1, "#B00020", "#252525"], validate ∧ selectedIndex ≟ 1, "#B00020", selectedIndex ≟ 1, "#6557D2", "#252525") → car1Color
  49. If(showanswer ∨ showsolution ∨ validate ∧ correct, If[correctIndex ≟ 2, "#008475", selectedIndex ≟ 2, "#B00020", "#252525"], validate ∧ selectedIndex ≟ 2, "#B00020", selectedIndex ≟ 2, "#6557D2", "#252525") → car2Color
  50. If(showanswer ∨ showsolution ∨ validate ∧ correct, If[correctIndex ≟ 3, "#008475", selectedIndex ≟ 3, "#B00020", "#252525"], validate ∧ selectedIndex ≟ 3, "#B00020", selectedIndex ≟ 3, "#6557D2", "#252525") → car3Color
  51. If(showanswer ∨ showsolution ∨ validate ∧ correct, If[correctIndex ≟ 4, "#008475", selectedIndex ≟ 4, "#B00020", "#252525"], validate ∧ selectedIndex ≟ 4, "#B00020", selectedIndex ≟ 4, "#6557D2", "#252525") → car4Color
  52. PolyLine({(xTrackEnd, yTrackStart), (xTrackStart, yTrackStart), (xTrackStart, yTrackEnd), (xTrackEnd, yTrackEnd)}) → trackBorder
  53. Zip(Vertex[n, 3], n, {car1Box, car2Box, car3Box, car4Box}) → PointsValidation
  54. Vector(FeedbackB, PointsValidation(correctIndex)) → correctVect
  55. Vector(FeedbackB, PointsValidation(selectedIndex)) → incorrectVect
  56. Translate(picCheck, correctVect) → correctPic
  57. Translate(picCross, incorrectVect) → incorrectPic
  58. Polygon({(xTrackEnd - (1 * finishBoxWidth), yTrackEnd), (xTrackEnd - (3 * finishBoxWidth), yTrackEnd), (xTrackEnd - (3 * finishBoxWidth), yTrackStart), (xTrackEnd - (1 * finishBoxWidth), yTrackStart)}) → finishOutline
  59. Sequence(Segment[(0, (10 * n)), (-0.1, (10 * n))], n, 0, 9) → yAxisSegs
  60. Sequence(x = xMin + (n * abs(xMax - xMin) / 4), n, 1, 4) → linesAlignment

AI 绘图提示语参考

绘制一个赛车比赛场景，展示四辆汽车在直线赛道上的位置变化，以比较不同速度（斜率）。画面包含一个坐标平面，横轴为时间（0到5），纵轴为距离（0到100），赛道区域用矩形表示，并带有水平分隔线。四辆汽车分别用不同颜色（绿、红、紫、橙）的矩形表示，从起点出发，沿赛道向右移动，移动速度由随机生成的不同速率决定。赛道终点处有黑白相间的方格旗图案。图中还包含坐标轴刻度、标签，以及一个滑块用于控制动画进度，显示每辆车的当前速度值。
以上提示语为根据原始绘图命令整理而来，仅供参考。