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Reasoning to Find Area: Decomposing and Rearranging Polygons (IM 6.1.3 Lesson)

Reasoning to Find Area: Decomposing and Rearranging Polygons (IM 6.1.3 Lesson)
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2020-07-15 21:09
TrianglePolygonGeometric Construction
① This GeoGebra example demonstrates how to find the area of irregular polygons by reasoning—decomposing the shape into known figures (triangles, rectangles) and rearranging parts, rather than relying on formulas. ② Core mathematical concepts include polygon area, decomposition and composition, area conservation, and geometric reasoning. ③ Construction highlights: using GeoGebra commands such as Segment and Polygon, along with custom tools (CustomHexagon, CustomTriangle, CustomQuadrilateral) to create shapes; objects include points (A, B, C, D, E, etc.), segments (f, g, h, etc.), and polygons (poly1), allowing dynamic manipulation of vertices to explore area relationships. ④ Suitable for upper elementary to early middle school (around grade 6) mathematics classes, fostering spatial visualization and logical reasoning, and useful as a teacher demonstration tool.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(C, A) → h
  4. Polygon(A, C, B) → t1, b, a, c
  5. Segment(D, E) → i
  6. Segment(E, F) → j
  7. Segment(F, G) → k
  8. Segment(G, D) → l
  9. Polygon(D, E, F, G) → q1, d, e, f_1, g_1
  10. Segment(H, I) → m
  11. Segment(I, J) → n
  12. Segment(J, K) → p
  13. Segment(K, L) → q
  14. Segment(L, H) → r
  15. Polygon(H, I, J, K, L) → poly1, h_1, i_1, j_1, k_1, l_1
  16. Segment(M, N) → s
  17. Segment(N, O) → t
  18. Segment(O, P) → m_1
  19. Segment(P, Q) → n_1
  20. Segment(Q, R) → p_1
  21. Segment(R, M) → q_1
  22. Polygon(R, M, N, O, P, Q) → poly2, r_1, m_2, n_2, o, p_2, q_2
  23. Segment(S, T) → s_1
  24. Segment(T, U) → t_1
  25. Segment(U, V) → a_1
  26. Segment(V, W) → b_1
  27. Segment(W, Z) → c_1
  28. Segment(Z, A_1) → d_1
  29. Segment(A_1, S) → e_1
  30. Polygon(S, T, U, V, W, Z, A_1) → poly3, s_2, t_2, u, v, w, z_1, f_2
  31. Segment(B_1, C_1) → g_2
  32. Segment(C_1, D_1) → h_2
  33. Segment(D_1, E_1) → i_2
  34. Segment(E_1, F_1) → j_2
  35. Segment(F_1, G_1) → k_2
  36. Segment(G_1, H_1) → l_2
  37. Segment(H_1, I_1) → r_2
  38. Segment(I_1, B_1) → a_2
  39. Polygon(B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1) → poly4, b_2, c_2, d_2, e_2, f_3, g_3, h_3, i_3
  40. Segment(J_1, K_1) → j_3
  41. Segment(K_1, L_1) → k_3
  42. Segment(L_1, M_1) → l_3
  43. Segment(M_1, N_1) → m_3
  44. Segment(N_1, O_1) → n_3
  45. Segment(O_1, P_1) → p_3
  46. Segment(P_1, Q_1) → q_3
  47. Segment(Q_1, R_1) → r_3
  48. Segment(R_1, J_1) → s_3
  49. Polygon(R_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1) → poly5, t_3, a_3, b_3, c_3, d_3, e_3, o_1, f_4, g_4
  50. Segment(S_1, T_1) → h_4
  51. Segment(T_1, U_1) → i_4
  52. Segment(U_1, V_1) → j_4
  53. Segment(V_1, W_1) → k_4
  54. Segment(W_1, Z_1) → l_4
  55. Segment(Z_1, A_2) → m_4
  56. Segment(A_2, B_2) → n_4
  57. Segment(B_2, C_2) → p_4
  58. Segment(C_2, D_2) → q_4
  59. Segment(D_2, S_1) → r_4
  60. Polygon(S_1, T_1, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2) → poly6, s_4, e_4, t_4, u_1, v_1, w_1, a_4, b_4, c_4, d_4
  61. CustomHexagon(E_2, F_2, G_2, H_2, I_2, J_2) → f_5, g_5, h_5, i_5, j_5, k_5, poly7
  62. CustomTriangle(S_2, T_2, U_2) → a_5, b_5, c_5, t2
  63. CustomTriangle(T_2, V_2, W_2) → d_5, e_5, f_6, t3
  64. CustomTriangle(W_2, Z_2, A_3) → g_6, h_6, i_6, t4
  65. CustomTriangle(A_3, B_3, U_2) → j_6, k_6, l_6, t5
  66. CustomQuadrilateral(C_3, D_3, E_3, F_3) → q4, m_6, n_6, p_6, q_6
  67. CustomHexagon(G_3, H_3, I_3, J_3, K_3, L_3) → r_6, s_6, t_6, a_6, b_6, c_6, poly8
  68. Point(c_6) → M_3
  69. CustomHexagon(I_3, M_3, N_3, O_3, P_3, Q_3) → d_6, e_6, f_7, g_7, h_7, i_7, poly9
  70. Point(d_6) → S_3
  71. Segment(R_3, S_3) → l_5
  72. Segment(R_3, T_3) → m_5

AI Drawing Prompt Reference

Draw a figure composed of multiple irregular polygons, including a triangle, a quadrilateral, and a hexagon, demonstrating how to calculate area by decomposing and rearranging shapes. Start by drawing a triangle and a quadrilateral, then add a pentagon and a hexagon, and finally overlap some parts to show how moving and combining these shapes can help calculate the total area.
The prompt above was compiled from the original drawing commands and is for reference only.