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Cash Register Change via Random Polygon Construction: Conditional Probability &

Cash Register Change via Random Polygon Construction: Conditional Probability &
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2021-03-26 21:48
PolygonCalculusLimitProbability & StatisticsGeometric Construction
This example models a “cash register” change-making process using randomized geometric construction. It demonstrates how coin/amount outcomes vary under constraints, and how visual regions and intersection points can represent feasible change configurations. The key mathematical ideas include conditional probability (counting outcomes that satisfy given change/amount conditions), limits and interest calculation (tracking how computed quantities evolve over steps or iterations), and a secant/intersection viewpoint implemented through intersection-driven geometry. In GeoGebra, the construction is built with Point to define anchor centers (e.g., pennycenter), Polygon to form region shapes, Shuffle together with Random and RandomElement to randomize choices or ordering, Intersect to compute intersection points that drive the logic, and If to filter or classify configurations based on conditional rules. Element is used to reference randomly selected objects for consistent downstream computation. Named objects such as A, B, C… along with indexed points (e.g., I_4, Z_3, W_3) and groups like R_i, S_i are used to encode layers of coin-size/quantity and to produce the dynamic “register” visualization. Suitable for high school to early university audiences in math visualization, probabilistic modeling, and geometry-based interactive demonstrations.

GeoGebra Commands & Parameters

  1. Point(xAxis) → C_4
  2. pennycenter(R_{11}, pennysize) → T
  3. pennycenter(R_{13}, pennysize) → U
  4. pennycenter(R_{14}, pennysize) → V
  5. pennycenter(R_{15}, pennysize) → W
  6. pennycenter(R_{16}, pennysize) → Z
  7. pennycenter(R_{10}, pennysize) → A_1
  8. pennycenter(R_{17}, pennysize) → B_1
  9. pennycenter(R_{18}, pennysize) → C_1
  10. pennycenter(R_{19}, pennysize) → D_1
  11. pennycenter(R_1, pennysize) → E_1
  12. pennycenter(R_2, pennysize) → F_1
  13. pennycenter(R_3, pennysize) → G_1
  14. pennycenter(R_4, pennysize) → H_1
  15. pennycenter(R_5, pennysize) → I_1
  16. pennycenter(R_6, pennysize) → J_1
  17. pennycenter(R_7, pennysize) → K_1
  18. pennycenter(R_8, pennysize) → L_1
  19. pennycenter(R_9, pennysize) → M_1
  20. pennycenter(R_{12}, pennysize) → N_1
  21. pennycenter(O_1, nickelsize) → Q_1
  22. pennycenter(O_2, nickelsize) → T_1
  23. pennycenter(O_3, nickelsize) → U_1
  24. pennycenter(O_4, nickelsize) → V_1
  25. pennycenter(O_5, nickelsize) → W_1
  26. pennycenter(O_6, nickelsize) → Z_1
  27. pennycenter(O_7, nickelsize) → A_2
  28. pennycenter(O_8, nickelsize) → B_2
  29. pennycenter(O_9, nickelsize) → C_2
  30. pennycenter(O_{10}, nickelsize) → D_2
  31. pennycenter(O_{11}, nickelsize) → E_2
  32. pennycenter(O_{12}, nickelsize) → F_2
  33. pennycenter(O_{13}, nickelsize) → G_2
  34. pennycenter(O_{14}, nickelsize) → H_2
  35. pennycenter(O_{15}, nickelsize) → I_2
  36. pennycenter(O_{16}, nickelsize) → J_2
  37. pennycenter(O_{17}, nickelsize) → K_2
  38. pennycenter(O_{18}, nickelsize) → L_2
  39. pennycenter(O_{19}, nickelsize) → M_2
  40. pennycenter(O_{20}, nickelsize) → N_2
  41. Polygon(J_3, K_3, C_4, D_4, E_4, F_4, G_4, I_3, L_3, M_3, A, V_3, W_3, Z_3, A_4, B_4) → poly4, h, r, s, t, f_1, g_1, h_1, i_1, i, j, k, l, m, n, p, q
  42. pennycenter(E_6, dimesize) → D_6
  43. pennycenter(E_7, dimesize) → F_6
  44. pennycenter(E_8, dimesize) → G_6
  45. pennycenter(E_9, dimesize) → H_6
  46. pennycenter(E_{10}, dimesize) → I_6
  47. pennycenter(E_{11}, dimesize) → J_6
  48. pennycenter(E_{12}, dimesize) → K_6
  49. pennycenter(E_{13}, dimesize) → L_6
  50. pennycenter(E_{14}, dimesize) → M_6
  51. pennycenter(E_{15}, dimesize) → N_6
  52. pennycenter(E_{16}, dimesize) → Q_6
  53. pennycenter(E_{17}, dimesize) → T_6
  54. pennycenter(E_{18}, dimesize) → U_6
  55. pennycenter(E_{19}, dimesize) → V_6
  56. pennycenter(E_{21}, dimesize) → W_6
  57. pennycenter(E_{22}, dimesize) → Z_6
  58. pennycenter(E_{23}, dimesize) → A_7
  59. pennycenter(E_{24}, dimesize) → B_7
  60. pennycenter(E, dimesize) → D_7
  61. pennycenter(E_{20}, dimesize) → F_7
  62. pennycenter(G_7, quartersize) → I_7
  63. pennycenter(G_{10}, quartersize) → J_7
  64. pennycenter(G_9, quartersize) → K_7
  65. pennycenter(G_8, quartersize) → L_7
  66. pennycenter(G_{14}, quartersize) → M_7
  67. pennycenter(G_{13}, quartersize) → N_7
  68. pennycenter(G_{12}, quartersize) → Q_7
  69. pennycenter(G_{11}, quartersize) → T_7
  70. pennycenter(G_{18}, quartersize) → U_7
  71. pennycenter(G_{17}, quartersize) → V_7
  72. pennycenter(G_{16}, quartersize) → W_7
  73. pennycenter(G_{15}, quartersize) → Z_7
  74. pennycenter(G_{19}, quartersize) → A_8
  75. pennycenter(G_{20}, quartersize) → B_8
  76. pennycenter(G_{21}, quartersize) → D_8
  77. pennycenter(G_{22}, quartersize) → F_8
  78. pennycenter(G_{23}, quartersize) → I_8
  79. pennycenter(G_{24}, quartersize) → J_8
  80. pennycenter(G_{25}, quartersize) → K_8
  81. pennycenter(G_{26}, quartersize) → L_8
  82. Shuffle(Remove[Flatten[Sequence[Sequence[Point[{i, j}], i, -32, -22, 3], j, 8, 0, -3]], {(-32, 8), (-23, 2)}]) → leftbanklist, {(-29, 5), (-29, 2), (-32, 5), (-26, 8), (-23, 8), (-32, 2), (-26, 5), (-23, 5), (-29, 8), (-26, 2)}
  83. Polygon(C_4, B_4, A_4, Z_3, E_4, D_4) → poly1, c_4, b_4, a_4, z_3, e_4, d_4
  84. Polygon(B, C, D, F) → poly2, b_1, c_1, d_1, f_2
  85. Intersect(z_3, e_4) → J
  86. Polygon(J, F_4, W_3, Z_3) → poly3, j_2, f_4, w_3, z_1
  87. Polygon(F_4, G_4, V_3, W_3) → poly5, f_3, g_4, v_3, w_1
  88. Polygon(G_4, I_3, A, V_3) → poly6, g_2, i_3, a_1, v_1
  89. Polygon(H_4, I_4, N_3, U_3) → poly7, h_4, i_4, n_3, u_3
  90. Polygon(J_3, K_3, L_3, M_3) → poly8, j_3, k_3, l_3, m_3
  91. Polygon(K, L, M, N) → poly9, k_2, l_2, m_2, n_1
  92. Polygon(O, P, Q, Q_2) → poly10, o, p_1, q_1, q_2
  93. RandomElement({"gum", "yo-yo", "bottle of bubbles", "top", "candy", "lollipop", "pencil"}) → buying, "top"
  94. Element({"1¢", "10¢", "20¢", "50¢", "$1.00"}, u) → maxText
  95. Random(Element[{1, 10, 20, 50, 100}, u - 1], Element[{1, 10, 20, 50, 100}, u]) → cost
  96. If(cost ≟ 100, "$1.00", Text[cost] + "¢") → costText
  97. If(Rtotal ≟ 0, "0¢", If[Rtotal < 100, Rtotal + "¢", "$" + Rhundreds + "." + Rtens + Rones]) → text2
  98. If(cost ≟ Rtotal, "You bought it!", If[cost > Rtotal, "You haven't paid enough.", "You paid too much."]) → checkText

AI Drawing Prompt Reference

Draw a planar “cash register change configuration” built from coin-sized anchor centers. First place several penny-sized center positions and, separately, penny/nickel-sized and dime-sized center positions around them, all in the same coordinate plane. Use selected penny-related center points as vertices to form a single complex polygonal outline that covers the feasible region; include additional penny-related centers as more polygon vertices so the shape extends across the lower and left/right directions. The key visual relationships are that each anchor center serves as a polygon vertex or boundary location, creating a region whose geometry is determined by those coin-size centers.
The prompt above was compiled from the original drawing commands and is for reference only.