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Visualizing Equality of Fractions with Geometric Construction

Visualizing Equality of Fractions with Geometric Construction
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2018-05-20 18:06
PolygonVectorTransformation
① This GeoGebra example demonstrates the equality of fractions (e.g., a/b = c/d) through a geometric construction involving a kite-shaped figure and translations, showing that equal fractions correspond to equal ratios of lengths or areas. ② Core mathematical concepts include fraction equality, proportionality, geometric mean, real numbers, and step functions, which help represent discrete values in the construction. ③ The construction uses commands such as Segment, Vector, Translate, Polygon, and FractionText, with objects including points A, B, C, D, E, F, G, H, I, J and their translated copies (e.g., D', A', i', etc.), creating multiple similar shapes via repeated translations and displaying the ratios as fraction text. ④ This is suitable for middle school to high school students learning fractions, ratios, and geometric transformations, and for teachers to demonstrate the visual link between algebraic fraction equality and geometric proportion.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(C, D) → h
  4. Segment(D, A) → i
  5. Vector(E, F) → u
  6. Translate(D, u) → D'
  7. Translate(A, u) → A'
  8. Segment(D', A') → i'
  9. Translate(D', u) → D''
  10. Translate(A', u) → A''
  11. Segment(D'', A'') → i''
  12. Translate(D'', u) → D'''
  13. Translate(A'', u) → A'''
  14. Segment(D''', A''') → i'''
  15. Translate(D''', u) → G
  16. Translate(A''', u) → H
  17. Segment(G, H) → j
  18. Translate(G, u) → G'
  19. Translate(H, u) → H'
  20. Segment(G', H') → j'
  21. Translate(G', u) → G''
  22. Translate(H', u) → H''
  23. Segment(G'', H'') → j''
  24. Translate(G'', u) → G'''
  25. Translate(H'', u) → H'''
  26. Segment(G''', H''') → j'''
  27. Translate(G''', u) → I
  28. Translate(H''', u) → J
  29. Segment(I, J) → k
  30. Translate(I, u) → I'
  31. Translate(J, u) → J'
  32. Segment(I', J') → k'
  33. Translate(I', u) → I''
  34. Translate(J', u) → J''
  35. Segment(I'', J'') → k''
  36. Translate(I'', u) → I'''
  37. Translate(J'', u) → J'''
  38. Segment(I''', J''') → k'''
  39. Translate(A, v) → A'_1
  40. Vector(A, A'_1) → w
  41. Translate(D, v) → D'_1
  42. Vector(D, D'_1) → a
  43. Polygon(A, D, D'_1, A'_1) → q1, a_1, d, d'_1, a'_1
  44. Vector(K, L) → b
  45. Translate(A, b) → A'_2
  46. Translate(D, b) → D'_2
  47. Translate(D'_1, b) → M
  48. Translate(A'_1, b) → N
  49. Translate(A, b) → A'_3
  50. Translate(D, b) → D'_3
  51. Translate(D'_1, b) → O
  52. Translate(A'_1, b) → P
  53. Polygon(A'_3, D'_3, O, P) → q1', a'_3, d'_3, o_1, p
  54. Translate(A, b) → A'_4
  55. Translate(B, b) → B'
  56. Segment(A'_4, B') → f'
  57. Translate(B, b) → B'_1
  58. Translate(C, b) → C'
  59. Segment(B'_1, C') → g'
  60. Translate(C, b) → C'_1
  61. Translate(D, b) → D'_4
  62. Segment(C'_1, D'_4) → h'
  63. Translate(D, b) → D'_5
  64. Translate(A, b) → A'_5
  65. Segment(D'_5, A'_5) → i'_1
  66. Translate(D', b) → D''_1
  67. Translate(A', b) → A''_1
  68. Segment(D''_1, A''_1) → i''_1
  69. Translate(D'', b) → D'''_1
  70. Translate(A'', b) → A'''_1
  71. Segment(D'''_1, A'''_1) → i'''_1
  72. Translate(D''', b) → Q
  73. Translate(A''', b) → R
  74. Segment(Q, R) → l
  75. Translate(G, b) → G'_1
  76. Translate(H, b) → H'_1
  77. Segment(G'_1, H'_1) → j'_1
  78. Translate(G', b) → G''_1
  79. Translate(H', b) → H''_1
  80. Segment(G''_1, H''_1) → j''_1
  81. Translate(G'', b) → G'''_1
  82. Translate(H'', b) → H'''_1
  83. Segment(G'''_1, H'''_1) → j'''_1
  84. Translate(G''', b) → S
  85. Translate(H''', b) → T
  86. Segment(S, T) → m
  87. Translate(I, b) → I'_1
  88. Translate(J, b) → J'_1
  89. Segment(I'_1, J'_1) → k'_1
  90. Translate(I', b) → I''_1
  91. Translate(J', b) → J''_1
  92. Segment(I''_1, J''_1) → k''_1
  93. Translate(I'', b) → I'''_1
  94. Translate(J'', b) → J'''_1
  95. Segment(I'''_1, J'''_1) → k'''_1
  96. Translate(I''', b) → U
  97. Translate(J''', b) → V
  98. Segment(U, V) → q
  99. Segment(A'''_1, D'''_1) → r
  100. Segment(H'_1, G'_1) → s
  101. Segment(H'''_1, G'''_1) → t
  102. Segment(J'_1, I'_1) → e
  103. Segment(J'''_1, I'''_1) → f_1
  104. Segment(R, Q) → g_1
  105. Segment(H'''_1, G'''_1) → h_1
  106. Segment(J''_1, I''_1) → i_1
  107. Segment(J'_1, I'_1) → j_1
  108. Segment(H'''_1, G'''_1) → k_1
  109. Segment(H'_1, G'_1) → l_1
  110. Segment(A'''_1, D'''_1) → m_1
  111. Segment(H'_1, G'_1) → n_1
  112. Segment(H'''_1, G'''_1) → p_1
  113. Segment(J'_1, I'_1) → q_1
  114. Segment(J'''_1, I'''_1) → r_1
  115. FractionText(c) → text2
  116. Segment(D'_1, N) → s_1