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Solving Equations with Model: Visual Balance and Algebraic Steps

Solving Equations with Model: Visual Balance and Algebraic Steps
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2016-05-15 06:56
FunctionInequalityGeometric ConstructionSlider AnimationGraphing
In collections Expressions and equations Expressions and equations
① This interactive GeoGebra example demonstrates how to build and solve a linear equation using a visual model, turning abstract algebraic operations into tangible graphical actions. ② Core mathematical concepts include equations, properties of equality, transposition and combining like terms, as well as the geometric meaning of roots (solutions); it also involves subtraction, exponential functions, and quadrilaterals as basic elements. ③ Construction highlights: using GeoGebra commands such as Point, Circle, Polygon, Ray, Intersect to create shapes, and CreatePositiveBag, CreatePositiveMarble, CreateNegativeBag, CreateNegativeMarble, CountIf, Random to simulate positive/negative 'bags' and 'marbles' with counting and random generation, thereby modeling the balance of both sides of an equation; objects include n, B series, N series, listNegBags, A, j, etc., allowing users to change quantities interactively to solve. ④ Suitable for middle and high school students learning algebra equations, and for teachers demonstrating in class, helping students understand each step of solving equations and transition from concrete models to abstract symbols.

GeoGebra Commands & Parameters

  1. Point(n) → B
  2. Point(j) → A
  3. Circle(M_6, a) → c
  4. Circle(A_6, b) → d
  5. Point(d) → C_1
  6. Polygon(A_6, C_1, 4) → poly1, e, f, g, h, D_1, E_1
  7. CreatePositiveBag(b, A_1) → poly2
  8. CreatePositiveBag(b, A_5) → poly3
  9. CreatePositiveBag(b, A_3) → poly4
  10. CreatePositiveBag(b, A_4) → poly5
  11. CreatePositiveBag(b, A_2) → poly6
  12. CreatePositiveBag(b, A_7) → poly7
  13. CreatePositiveBag(b, A_8) → poly8
  14. CreatePositiveBag(b, A_9) → poly9
  15. CreatePositiveBag(b, A_10) → poly10
  16. CreatePositiveMarble(a, M_1) → k
  17. CreatePositiveMarble(a, M_4) → p
  18. CreatePositiveMarble(a, M_5) → q
  19. CreatePositiveMarble(a, M_10) → r
  20. CreatePositiveMarble(a, M_9) → s
  21. CreatePositiveMarble(a, M_3) → t
  22. CreatePositiveMarble(a, M_2) → c_1
  23. CreatePositiveMarble(a, M_8) → d_1
  24. CreatePositiveMarble(a, M_7) → e_1
  25. CountIf((x > x(A)) ∧ (x < x(B)), listPosBags) → PositiveBags
  26. CountIf((x > x(A)) ∧ (x < x(B)), listPosMarbles) → PositiveMarbles
  27. Circle(N_4, a) → f_1
  28. CreateNegativeMarble(a, N_1) → g_1
  29. CreateNegativeMarble(a, N_2) → h_1
  30. CreateNegativeMarble(a, N_3) → k_1
  31. CreateNegativeMarble(a, N_5) → p_1
  32. CreateNegativeMarble(a, N_6) → q_1
  33. CreateNegativeMarble(a, N_7) → r_1
  34. CreateNegativeMarble(a, N_8) → s_1
  35. CreateNegativeMarble(a, N_9) → t_1
  36. CreateNegativeMarble(a, N_10) → c_2
  37. CreatePositiveBag(b, B_3) → poly11
  38. CreateNegativeBag(b, B_7) → poly12
  39. CreateNegativeBag(b, B_6) → poly13
  40. CreateNegativeBag(b, B_10) → poly14
  41. CreateNegativeBag(b, B_9) → poly15
  42. CreateNegativeBag(b, B_8) → poly16
  43. CreateNegativeBag(b, B_5) → poly17
  44. CreateNegativeBag(b, B_4) → poly18
  45. CreateNegativeBag(b, B_2) → poly19
  46. CreateNegativeBag(b, B_1) → poly20
  47. CountIf((x > x(A)) ∧ (x < x(B)), listNegMarbles) → NegativeMarbles
  48. CountIf((x > x(A)) ∧ (x < x(B)), listNegBags) → NegativeBags
  49. CountIf(x > x(B), listNegMarbles) → NegativeMarblesRight
  50. CountIf(x > x(B), listPosBags) → PositiveBagsRight
  51. Random(-4, 5) → ValueofX
  52. CountIf(x > x(B), listNegBags) → NegativeBagsRight
  53. CountIf(x > x(B), listPosMarbles) → PositiveMarblesRight
  54. CreatePositiveBag(b, A_11) → poly21
  55. CreatePositiveBag(b, A_18) → poly22
  56. CreatePositiveBag(b, A_12) → poly23
  57. CreatePositiveBag(b, A_17) → poly24
  58. CreatePositiveBag(b, A_15) → poly25
  59. CreatePositiveBag(b, A_20) → poly26
  60. CreatePositiveBag(b, A_16) → poly27
  61. CreatePositiveBag(b, A_14) → poly28
  62. CreatePositiveBag(b, A_13) → poly29
  63. CreatePositiveBag(b, A_19) → poly30
  64. CreatePositiveMarble(a, M_11) → d_2
  65. CreatePositiveMarble(a, M_14) → e_2
  66. CreatePositiveMarble(a, M_12) → f_2
  67. CreatePositiveMarble(a, M_15) → g_2
  68. CreatePositiveMarble(a, M_13) → h_2
  69. CreatePositiveMarble(a, M_16) → k_2
  70. CreatePositiveMarble(a, M_17) → p_2
  71. CreatePositiveMarble(a, M_18) → q_2
  72. CreatePositiveMarble(a, M_19) → r_2
  73. CreatePositiveMarble(a, M_20) → s_2
  74. CreateNegativeBag(b, B_11) → poly31
  75. CreateNegativeBag(b, B_12) → poly32
  76. CreateNegativeBag(b, B_13) → poly33
  77. CreateNegativeBag(b, B_14) → poly34
  78. CreateNegativeBag(b, B_15) → poly35
  79. CreateNegativeBag(b, B_16) → poly36
  80. CreateNegativeBag(b, B_17) → poly37
  81. CreateNegativeBag(b, B_18) → poly38
  82. CreateNegativeBag(b, B_19) → poly39
  83. CreateNegativeBag(b, B_20) → poly40
  84. CreateNegativeMarble(a, N_11) → t_2
  85. CreateNegativeMarble(a, N_12) → c_3
  86. CreateNegativeMarble(a, N_13) → d_3
  87. CreateNegativeMarble(a, N_14) → e_3
  88. CreateNegativeMarble(a, N_15) → f_3
  89. CreateNegativeMarble(a, N_16) → g_3
  90. CreateNegativeMarble(a, N_17) → h_3
  91. CreateNegativeMarble(a, N_18) → k_3
  92. CreateNegativeMarble(a, N_19) → p_3
  93. CreateNegativeMarble(a, N_20) → q_3
  94. Random(-4, 4) → SeedBagLeft
  95. Random(-4, 4) → SeedBagRight
  96. Random(-5, 5) → SeedMarbleLeft
  97. Intersect(j, a_1) → C
  98. Intersect(n, a_1) → D
  99. Ray(C, D) → b_1

AI Drawing Prompt Reference

Draw a balance model to represent the solution of a linear equation. Start by creating two points A and n for the two sides of the equation. Then, draw two circles c and d around A and n, and create a point C on d. Next, use A and C to create a quadrilateral and add some squares and circles inside to represent positive and negative 'bags' and 'marbles'. Finally, simulate the distribution of positive and negative numbers through counting and random generation to show the solution of the equation.
The prompt above was compiled from the original drawing commands and is for reference only.