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By all means! - Averaging 3 Numbers

By all means! - Averaging 3 Numbers
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2016-12-29 22:45
PolygonTrigonometrySolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates the concept of averaging three numbers through a vivid three-dimensional geometric construction. It visualizes how arithmetic and geometric means can be represented using spatial objects such as points, segments, cubes, and polygons, while incorporating trigonometric functions and geometric transformations. The core mathematical ideas include mean values, cosine functions, and spatial transformations. The construction uses GeoGebra commands like Point, Segment, Cube, and Polygon, with numerous auxiliary objects (X, Y, Z, O, A, B, C, D, etc.) to build a dynamic model that illustrates the averaging process. Suitable for middle school to undergraduate students, this interactive tool helps learners grasp the geometric interpretation of averages and explore the relationship between numbers and shapes in a hands-on manner.

GeoGebra Commands & Parameters

  1. Point(xAxis) → X
  2. Point(yAxis) → Y
  3. Point(zAxis) → Z
  4. Segment(O, X) → f
  5. Segment(X, B) → g
  6. Segment(B, Z) → h
  7. Segment(Z, O) → i
  8. Segment(O, Y) → j
  9. Segment(Y, A) → k
  10. Segment(A, X) → l
  11. Segment(X, O) → m
  12. Segment(Z, C) → n
  13. Segment(C, D) → p
  14. Segment(D, B) → q
  15. Segment(C, Y) → r
  16. Segment(D, A) → s
  17. Point(Circle[E, Distance[O, E], Segment[O, E]]) → F
  18. Point(Circle[G, Distance[O, G], Segment[O, G]]) → H
  19. Cube(O, G, H) → a, I, faceHIML, faceJKLM, edgeIO, edgeGO, edgeGH, edgeHI, edgeJO, edgeIM, edgeJM, edgeJK, J, edgeGK, edgeKL, edgeHL, edgeLM, K, L, M, faceGHIO, faceIMJO, faceGKJO, faceGHLK
  20. Segment(Q, S) → t
  21. Segment(S, N) → b
  22. Segment(N, O) → c
  23. Segment(O, Q) → d
  24. Segment(N, R) → e
  25. Segment(Y, O) → g_1
  26. Segment(O, N) → h_1
  27. Segment(R, T) → i_1
  28. Segment(T, S) → j_1
  29. Segment(U, Q) → l_1
  30. Segment(U, T) → m_1
  31. Segment(U, P) → f_1
  32. Segment(P, R) → k_1
  33. Segment(Z, O) → n_1
  34. Segment(O, Z) → p_1
  35. Segment(O, Q) → q_1
  36. Segment(O, Z) → r_1
  37. Polygon(V, W, D_1, C_1) → poly1, v, w, d_1, c_1
  38. Polygon(W, A_1, E_1, D_1) → poly2, w_1, a_1, e_1, d_2
  39. Polygon(A_1, B_1, F_1, E_1) → poly3, a_2, b_1, f_2, e_2
  40. Point(s_1) → H_1
  41. Polygon(H_1, J_1, F_1, K_1) → poly4, h_2, j_2, f_3, k_2
  42. Polygon(K_1, H_1, I_1, C_1) → poly5, k_3, h_3, i_2, c_2
  43. Segment(H_1, (x(H_1), 0)) → t_1