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Fraction Names and Geometric Construction

Fraction Names and Geometric Construction
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2016-07-19 00:47
TrigonometryProbability & StatisticsOptimizationGeometric Construction
This example demonstrates the representation and naming of fractions in geometric figures through dynamic construction. It involves core mathematical concepts such as fractions, optimization problems, geometry (trapezoid), random variables, and fractal geometry. The construction utilizes GeoGebra commands like Sequence, Random, IndexOf, If, Sum, Polygon, Point, and Segment. It is suitable for high school students and educators exploring the intersection of geometry and probability & statistics.

GeoGebra Commands & Parameters

  1. Sequence(Sample[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, 12, false], i, 1, 10) → matriu1
  2. Random(1, 10) → fila
  3. IndexOf(1, Element[matriu1, fila]) → E1
  4. IndexOf(2, Element[matriu1, fila]) → E2
  5. IndexOf(3, Element[matriu1, fila]) → E3
  6. IndexOf(4, Element[matriu1, fila]) → E4
  7. IndexOf(5, Element[matriu1, fila]) → E5
  8. IndexOf(6, Element[matriu1, fila]) → E6
  9. IndexOf(7, Element[matriu1, fila]) → E7
  10. IndexOf(8, Element[matriu1, fila]) → E8
  11. IndexOf(9, Element[matriu1, fila]) → E9
  12. IndexOf(10, Element[matriu1, fila]) → E10
  13. IndexOf(11, Element[matriu1, fila]) → E11
  14. IndexOf(12, Element[matriu1, fila]) → E12
  15. If(Mod[E1, 3] ≟ 1, E1 + 1, If[Mod[E1, 3] ≟ 2, E1 - 1, E1 - 2]) → F1
  16. If(Mod[E2, 3] ≟ 1, E2 + 1, If[Mod[E2, 3] ≟ 2, E2 - 1, E2 - 2]) → F2
  17. If(Mod[E3, 3] ≟ 1, E3 + 1, If[Mod[E3, 3] ≟ 2, E3 - 1, E3 - 2]) → F3
  18. If(Mod[E4, 3] ≟ 1, E4 + 1, If[Mod[E4, 3] ≟ 2, E4 - 1, E4 - 2]) → F4
  19. If(Mod[E5, 3] ≟ 1, E5 + 1, If[Mod[E5, 3] ≟ 2, E5 - 1, E5 - 2]) → F5
  20. If(Mod[E6, 3] ≟ 1, E6 + 1, If[Mod[E6, 3] ≟ 2, E6 - 1, E6 - 2]) → F6
  21. If(Mod[E7, 3] ≟ 1, E7 + 1, If[Mod[E7, 3] ≟ 2, E7 - 1, E7 - 2]) → F7
  22. If(Mod[E8, 3] ≟ 1, E8 + 1, If[Mod[E8, 3] ≟ 2, E8 - 1, E8 - 2]) → F8
  23. If(Mod[E9, 3] ≟ 1, E9 + 1, If[Mod[E9, 3] ≟ 2, E9 - 1, E9 - 2]) → F9
  24. If(Mod[E10, 3] ≟ 1, E10 + 1, If[Mod[E10, 3] ≟ 2, E10 - 1, E10 - 2]) → F10
  25. If(Mod[E11, 3] ≟ 1, E11 + 1, If[Mod[E11, 3] ≟ 2, E11 - 1, E11 - 2]) → F11
  26. If(Mod[E12, 3] ≟ 1, E12 + 1, If[Mod[E12, 3] ≟ 2, E12 - 1, E12 - 2]) → F12
  27. IndexOf(10, Element[matriu1, fila]) → o
  28. If(Mod[E1, 3] ≟ 1, E1 + 2, If[Mod[E1, 3] ≟ 2, E1 + 1, E1 - 1]) → G1
  29. If(Mod[E2, 3] ≟ 1, E2 + 2, If[Mod[E2, 3] ≟ 2, E2 + 1, E2 - 1]) → G2
  30. If(Mod[E3, 3] ≟ 1, E3 + 2, If[Mod[E3, 3] ≟ 2, E3 + 1, E3 - 1]) → G3
  31. If(Mod[E4, 3] ≟ 1, E4 + 2, If[Mod[E4, 3] ≟ 2, E4 + 1, E4 - 1]) → G4
  32. If(Mod[E5, 3] ≟ 1, E5 + 2, If[Mod[E5, 3] ≟ 2, E5 + 1, E5 - 1]) → G5
  33. If(Mod[E6, 3] ≟ 1, E6 + 2, If[Mod[E6, 3] ≟ 2, E6 + 1, E6 - 1]) → G6
  34. If(Mod[E7, 3] ≟ 1, E7 + 2, If[Mod[E7, 3] ≟ 2, E7 + 1, E7 - 1]) → G7
  35. If(Mod[E8, 3] ≟ 1, E8 + 2, If[Mod[E8, 3] ≟ 2, E8 + 1, E8 - 1]) → G8
  36. If(Mod[E9, 3] ≟ 1, E9 + 2, If[Mod[E9, 3] ≟ 2, E9 + 1, E9 - 1]) → G9
  37. If(Mod[E10, 3] ≟ 1, E10 + 2, If[Mod[E10, 3] ≟ 2, E10 + 1, E10 - 1]) → G10
  38. If(Mod[E11, 3] ≟ 1, E11 + 2, If[Mod[E11, 3] ≟ 2, E11 + 1, E11 - 1]) → G11
  39. If(Mod[E12, 3] ≟ 1, E12 + 2, If[Mod[E12, 3] ≟ 2, E12 + 1, E12 - 1]) → G12
  40. If(C1 ≟ Cell[4, E1] ∨ C1 ≟ Cell[4, F1] ∨ C1 ≟ Cell[4, G1], 1, 0) → H1
  41. If(C2 ≟ Cell[4, E2] ∨ C2 ≟ Cell[4, F2] ∨ C2 ≟ Cell[4, G2], 1, 0) → H2
  42. If(C3 ≟ Cell[4, E3] ∨ C3 ≟ Cell[4, F3] ∨ C3 ≟ Cell[4, G3], 1, 0) → H3
  43. If(C4 ≟ Cell[4, E4] ∨ C4 ≟ Cell[4, F4] ∨ C4 ≟ Cell[4, G4], 1, 0) → H4
  44. If(C5 ≟ Cell[4, E5] ∨ C5 ≟ Cell[4, F5] ∨ C5 ≟ Cell[4, G5], 1, 0) → H5
  45. If(C6 ≟ Cell[4, E6] ∨ C6 ≟ Cell[4, F6] ∨ C6 ≟ Cell[4, G6], 1, 0) → H6
  46. If(C7 ≟ Cell[4, E7] ∨ C7 ≟ Cell[4, F7] ∨ C7 ≟ Cell[4, G7], 1, 0) → H7
  47. If(C8 ≟ Cell[4, E8] ∨ C8 ≟ Cell[4, F8] ∨ C8 ≟ Cell[4, G8], 1, 0) → H8
  48. If(C9 ≟ Cell[4, E9] ∨ C9 ≟ Cell[4, F9] ∨ C9 ≟ Cell[4, G9], 1, 0) → H9
  49. If(C10 ≟ Cell[4, E10] ∨ C10 ≟ Cell[4, F10] ∨ C10 ≟ Cell[4, G10], 1, 0) → H10
  50. If(C11 ≟ Cell[4, E11] ∨ C11 ≟ Cell[4, F11] ∨ C11 ≟ Cell[4, G11], 1, 0) → H11
  51. If(C12 ≟ Cell[4, E12] ∨ C12 ≟ Cell[4, F12] ∨ C12 ≟ Cell[4, G12], 1, 0) → H12
  52. Sum(H1:H12) → H13
  53. Random(1, 10) → u
  54. Polygon(S_1, Z_1, W_1, T_1) → Polígon15, s_3, z_1, w_1, t_3
  55. Polygon(B_2, C_2, B_3, E_3) → Polígon30, b_7, c_7, b_8, e_6
  56. Polygon(H_3, I_3, J_3, K_3) → Polígon32, h_6, i_6, j_6, k_6
  57. Polygon(L_3, M_3, N_3, O_3) → Polígon33, l_6, m_6, n_6, o_3
  58. Polygon(P_3, Q_3, R_3, S_3) → Polígon34, p_6, q_6, r_6, s_6
  59. Polygon(T_3, U_3, V_3, W_3) → Polígon35, t_6, u_3, v_3, w_4
  60. Polygon(Z_3, A_4, B_4, C_4) → Polígon36, z_4, a_7, b_{10}, c_9
  61. Polygon(F_3, G_3, D_4, E_4) → Polígon31, f_6, g_6, d_6, e_7
  62. Polygon(F_4, G_4, H_4, I_4) → Polígon37, f_7, g_7, h_7, i_7
  63. Polygon(J_4, K_4, L_4, M_4) → Polígon38, j_7, k_7, l_7, m_7
  64. Polygon(N_4, O_4, P_4, Q_4) → Polígon39, n_7, o_4, p_7, q_7
  65. Polygon(R_4, S_4, T_4, U_4) → Polígon40, r_7, s_7, t_7, u_4
  66. Polygon(V_4, W_4, Z_4, A_5) → Polígon41, v_4, w_5, z_5, a_8
  67. Point(b_7) → B_5
  68. Point(b_8) → C_5
  69. Point(j_6) → E_5
  70. Point(k_7) → H_5
  71. Point(m_7) → I_5
  72. Point(d_6) → J_5
  73. Point(f_6) → K_5
  74. Point(s_6) → M_5
  75. Point(w_4) → N_5
  76. Point(u_3) → O_5
  77. Point(o_4) → Q_5
  78. Point(u_4) → R_5
  79. Point(s_7) → S_5
  80. Point(a_8) → V_5
  81. Point(w_5) → W_5
  82. Point(a_7) → A_6
  83. Polygon(B_2, C_2, B_3, E_3) → Polígon42, b_9, c_8, b_{11}, e_8
  84. Polygon(P_3, S_3, R_3, Q_3) → Polígon43, p_8, s_8, r_8, q_8
  85. Polygon(T_3, W_3, V_3, U_3) → Polígon44, t_8, w_6, v_2, u_2
  86. Polygon(Z_3, C_4, B_4, A_4) → Polígon45, z_6, c_{10}, b_{12}, a_9
  87. Polygon(V_4, A_5, Z_4, W_4) → Polígon46, v_5, a_{10}, z_7, w_7
  88. Polygon(N_4, Q_4, P_4, O_4) → Polígon47, n_8, q_9, p_9, o_1
  89. Polygon(F_3, G_3, D_4, E_4) → Polígon48, f_8, g_8, d_7, e_9
  90. Polygon(H_3, I_3, J_3, K_3) → Polígon49, h_8, i_8, j_8, k_8
  91. Polygon(L_3, M_3, N_3, O_3) → Polígon50, l_8, m_8, n_9, o_2
  92. Polygon(J_4, M_4, L_4, K_4) → Polígon51, j_9, m_9, l_9, k_9
  93. Polygon(R_4, U_4, T_4, S_4) → Polígon52, r_9, u_5, t_9, s_9
  94. Polygon(F_4, I_4, H_4, G_4) → Polígon53, f_9, i_9, h_9, g_9
  95. Segment(B_6, C_6) → d_8
  96. Segment(D_6, E_6) → d_9
  97. Segment(F_6, G_5) → d_{10}
  98. Segment(G_6, H_6) → e_{10}
  99. Segment(I_6, J_6) → f_{10}
  100. Segment(K_6, H_5) → g_{10}
  101. Segment(L_6, M_6) → h_{10}
  102. Segment(N_6, O_6) → i_{10}
  103. Segment(P_6, Q_6) → j_{10}
  104. Segment(R_6, S_6) → k_{10}
  105. Segment(T_6, U_6) → l_{10}
  106. Segment(V_6, W_6) → m_{10}
  107. If(L1 ≟ Cell[13, E1], 1, 0) → N1
  108. If(L2 ≟ Cell[13, E2], 1, 0) → N2
  109. If(L3 ≟ Cell[13, E3], 1, 0) → N3
  110. If(L4 ≟ Cell[13, E4], 1, 0) → N4
  111. If(L5 ≟ Cell[13, E5], 1, 0) → N5
  112. If(L6 ≟ Cell[13, E6], 1, 0) → N6
  113. If(L7 ≟ Cell[13, E7], 1, 0) → N7
  114. If(L8 ≟ Cell[13, E8], 1, 0) → N8
  115. If(L9 ≟ Cell[13, E9], 1, 0) → N9
  116. If(L10 ≟ Cell[13, E10], 1, 0) → N10
  117. If(L11 ≟ Cell[13, E11], 1, 0) → N11
  118. If(L12 ≟ Cell[13, E12], 1, 0) → N12
  119. Sum(N1:N12) → N13
  120. Polygon(P1, Q1, 4) → Polígon54, n_{10}, p_{10}, q_{10}, r_{10}, Z_6, A_7
  121. Polygon(P2, Q2, 4) → Polígon55, s_{10}, t_{10}, a_{11}, c_{11}, B_7, C_7
  122. Polygon(P3, Q3, 4) → Polígon56, d_{11}, e_{11}, f_{11}, g_{11}, D_7, E_7
  123. Polygon(P4, Q4, 4) → Polígon57, h_{11}, i_{11}, j_{11}, k_{11}, F_7, G_7
  124. Polygon(P5, Q5, 4) → Polígon58, l_{11}, m_{11}, n_{11}, p_{11}, H_7, I_7
  125. Polygon(P6, Q6, 4) → Polígon59, q_{11}, r_{11}, s_{11}, t_{11}, J_7, K_7
  126. Polygon(P7, Q7, 4) → Polígon60, a_{12}, c_{12}, d_{12}, e_{12}, L_7, M_7
  127. Polygon(P8, Q8, 4) → Polígon61, f_{12}, g_{12}, h_{12}, i_{12}, N_7, O_7
  128. Polygon(P9, Q9, 4) → Polígon62, j_{12}, k_{12}, l_{12}, m_{12}, P_7, Q_7
  129. Polygon(P10, Q10, 4) → Polígon63, n_{12}, p_{12}, q_{12}, r_{12}, R_7, S_7
  130. Polygon(P11, Q11, 4) → Polígon64, s_{12}, t_{12}, a_{13}, b_{13}, T_7, U_7
  131. Polygon(P12, Q12, 4) → Polígon65, c_{13}, d_{13}, e_{13}, f_{13}, V_7, W_7
  132. If(R1 ≟ Cell[13, E1], 1, 0) → S1
  133. If(R2 ≟ Cell[13, E2], 1, 0) → S2
  134. If(R3 ≟ Cell[13, E3], 1, 0) → S3
  135. If(R4 ≟ Cell[13, E4], 1, 0) → S4
  136. If(R5 ≟ Cell[13, E5], 1, 0) → S5
  137. If(R6 ≟ Cell[13, E6], 1, 0) → S6
  138. If(R7 ≟ Cell[13, E7], 1, 0) → S7
  139. If(R8 ≟ Cell[13, E8], 1, 0) → S8
  140. If(R9 ≟ Cell[13, E9], 1, 0) → S9
  141. If(R10 ≟ Cell[13, E10], 1, 0) → S10
  142. If(R11 ≟ Cell[13, E11], 1, 0) → S11
  143. If(R12 ≟ Cell[13, E12], 1, 0) → S12
  144. Sum(S1:S12) → S13