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Proof without words: Division of Fractions

Proof without words: Division of Fractions
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2020-03-02 10:26
VectorFunctionCoordinate GeometryOptimizationGeometric Construction
This example visually demonstrates the principle of fraction division, involving core mathematical concepts such as fraction operations, vectors, geometric constructions, and optimization problems. It utilizes commands and objects like Vector, LCM, DivPoint, Polygon, Segment, Textfield, Sequence, Point, Translate, and Midpoint in GeoGebra. Suitable for middle school mathematics education to help students intuitively understand the geometric meaning of fraction division.

GeoGebra Commands & Parameters

  1. Vector(A, U) → v_1
  2. LCM(b, d) → o
  3. DivPoint(A, D, o) → list7
  4. DivPoint(B, C, o) → list8
  5. Polygon(A, Element[list7, t], Element[list8, t], B) → poly5, q, r, f_1, g_1
  6. Polygon(E, F, G, H) → poly2, e, f_2, g, h
  7. Segment(I, J) → i
  8. Segment(K, L) → j
  9. Textfield(a) → InputBox1
  10. Textfield(b) → InputBox2
  11. Textfield(c) → InputBox3
  12. Textfield(d) → InputBox4
  13. Polygon(A, D, C, B) → poly1, a_1, d_1, c_1, b_1
  14. DivPoint(A, D, b) → list1
  15. DivPoint(B, C, b) → list2
  16. DivPoint(E, F, d) → list3
  17. DivPoint(H, G, d) → list4
  18. Sequence(Segment[Element[list1, i], Element[list2, i]], i, 1, b) → list5
  19. Sequence(Segment[Element[list3, i], Element[list4, i]], i, 1, d) → list6
  20. Polygon(A, Element[list1, a], Element[list2, a], B) → poly3, a_2, k, l, b_2
  21. Polygon(E, Element[list3, c], Element[list4, c], H) → poly4, e_1, m, n, h_1
  22. DivPoint(E, F, o) → list9
  23. DivPoint(H, G, o) → list10
  24. Sequence(Segment[Element[list7, i], Element[list8, i]], i, 1, o) → list11
  25. Sequence(Segment[Element[list9, i], Element[list10, i]], i, 1, o) → list12
  26. Sequence(Polygon[Element[list7, (t * i)], Element[list7, (t * (i + 1))], Element[list8, (t * (i + 1))], Element[list8, (t * i)]], i, 1, p - 1) → list13
  27. Segment(M, N) → f
  28. Segment(O, P) → i_1
  29. Point(e_1) → V
  30. Vector(V, U) → u_1
  31. Translate(poly5, v_1) → poly4'
  32. Translate(poly3, v_1) → poly3'
  33. Sequence(Translate[Element[list11, i], v_1], i, 0, t) → liste1
  34. Translate(k, v_1) → k'
  35. Translate(a_2, v_1) → j_1
  36. Translate(B, v_1) → B'
  37. Translate(A, v_1) → A'
  38. Segment(B', A') → k_1
  39. Translate(l, v_1) → l'
  40. Translate(e_1, u_1) → l_1
  41. Translate(H, u_1) → H'
  42. Translate(m, u_1) → m'
  43. Translate(n, u_1) → n'
  44. Midpoint(j_1) → W
  45. Midpoint(n') → Z
  46. Midpoint(m') → A_1

AI Drawing Prompt Reference

Draw a geometric figure to illustrate the concept of fraction division. Start by creating vector A and U, then calculate the least common multiple o of b and d. Next, find the division points of A, D, B, and C with respect to o, forming polygons. Use these division points and vectors to construct a series of polygons and segments, demonstrating the geometric meaning of fraction division. Finally, complete the figure by translating and finding midpoints.
The prompt above was compiled from the original drawing commands and is for reference only.