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Fraction Division 4: 1 ÷ a/b (Quotitive) by Mario Villani

Fraction Division 4: 1 ÷ a/b (Quotitive) by Mario Villani
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2014-10-01 19:57
FunctionGeometric ConstructionSlider Animation
This example demonstrates the graphical interpretation of 1 ÷ a/b = b/a and visually illustrates [i]x[/i] ÷ a/b = [i]x[/i] × b/a in the context of quotitive (measurement) division. Key math concepts include fraction division, quotitive division, fraction multiplication, and proportional relationships. It is constructed using GeoGebra commands like Point, Min, Max, Segment, Polygon, Sequence, If, involving objects such as a, b, gcd, A, s, f_1, etc. Suitable for middle school and above mathematics education to enhance understanding of fraction division concepts.

GeoGebra Commands & Parameters

  1. Point(Segment[(x(E) + 0.1, y(E)), (x(E) + 1, y(E))]) → G
  2. Point(Segment[(x(D) + 0.1, y(D)), (x(D) + 0.5, y(D))]) → F
  3. Min(round((x(G) - x(E)) / 0.1), 2 * a') → b'
  4. Max(Sequence[If[(Mod[a * a', i] ≤ 0) ∧ (Mod[b', i] ≤ 0), i, 1], i, 1, Min[a * a', b']]) → gcd
  5. Segment((0, 1), (x(A), y(A))) → f_1
  6. Point(f_1) → B
  7. Segment(A, (x(A) + a, y(A))) → l
  8. Segment((x(A), y(A) + w), (x(A) + a, y(A) + w)) → m
  9. Segment(A, (x(A), y(A) + w)) → n
  10. Segment((x(A) + 9, y(A)), (x(A) + 9, y(A) + w)) → p
  11. Segment((x(A) + 8, y(A)), (x(A) + 8, y(A) + w)) → q
  12. Segment((x(A) + 7, y(A)), (x(A) + 7, y(A) + w)) → r
  13. Segment((x(A) + 6, y(A)), (x(A) + 6, y(A) + w)) → t
  14. Segment((x(A) + 5, y(A)), (x(A) + 5, y(A) + w)) → a_1
  15. Segment((x(A) + 4, y(A)), (x(A) + 4, y(A) + w)) → b_1
  16. Segment((x(A) + 3, y(A)), (x(A) + 3, y(A) + w)) → c_1
  17. Segment((x(A) + 2, y(A)), (x(A) + 2, y(A) + w)) → d_1
  18. Segment((x(A) + 10, y(A)), (x(A) + 10, y(A) + w)) → g_1
  19. Segment((x(A) + 11, y(A)), (x(A) + 11, y(A) + w)) → h_1
  20. Segment((x(A) + 12, y(A)), (x(A) + 12, y(A) + w)) → i_1
  21. Segment((x(B) + b' / a', y(B)), (x(B) + a, y(B))) → f
  22. Point(f) → C
  23. Polygon(B, (floor(a' * x(C) / b') * b' / a', y(B)), (floor(a' * x(C) / b') * b' / a', y(B) + 1), (x(B), y(B) + 1)) → poly2, poly2_1, poly2_2, poly2_3, poly2_4
  24. Segment((x(A) + 1, y(A)), (x(A) + 1, y(A) + w)) → e
  25. Segment(E, (x(E) + 1, y(E))) → j_1
  26. Segment(D, F) → l_1
  27. Polygon(B, (x(B) + 1, y(B)), (x(B) + 1, y(B) + 1), (x(B), y(B) + 1)) → poly1, poly1_1, poly1_2, poly1_3, poly1_4
  28. Sequence(Segment[(k / a', y(B)), (k / a', y(B) + 1)], k, 1, a' - 1) → list2
  29. Sequence(Segment[(k / a', y(B)), (k / a', y(B) + 1)], k, 1, floor(a' * x(C) / b') * b' - 1) → list3
  30. Segment((floor(a' * x(C) / b') * b' / a', y(B)), C) → h
  31. Segment((floor(a' * x(C) / b') * b' / a', y(B) + 1), (x(C), y(C) + 1)) → i
  32. Sequence(Segment[(k / a', y(B)), (k / a', y(B) + 1)], k, 1, floor(a' * x(C)) - 1) → list4
  33. Sequence(Segment[(k * b' / a', y(B)), (k * b' / a', y(B) + 1)], k, 1, floor(a' * x(C) / b') - 1) → list1
  34. Polygon((floor(a' * x(C) / b') * b' / a', y(B)), C, (x(C), y(C) + 1), (floor(a' * x(C) / b') * b' / a', y(B) + 1)) → poly3, poly3_1, poly3_2, poly3_3, poly3_4
  35. Polygon(B, (b' / a', y(B)), (b' / a', y(B) + 1), (x(B), y(B) + 1)) → poly4, poly4_1, poly4_2, poly4_3, poly4_4
  36. Segment((floor(a' * x(C) / b') * b' / a', y(B)), (floor(a' * x(C) / b') * b' / a' + b' / a', y(B))) → b
  37. Segment((floor(a' * x(C) / b') * b' / a', y(B) + 1), (floor(a' * x(C) / b') * b' / a' + b' / a', y(B) + 1)) → c
  38. Segment((floor(a' * x(C) / b') * b' / a' + b' / a', y(B)), (floor(a' * x(C) / b') * b' / a' + b' / a', y(B) + 1)) → d
  39. Sequence(Segment[(floor(a' * x(C) / b') * b' / a' + k / a', y(B)), (floor(a' * x(C) / b') * b' / a' + k / a', y(B) + 1)], k, 1, b' - 1) → list5
  40. Segment(E, (Min[x(G), x(E) + 0.2 * a'], y(G))) → m_1
  41. Segment(D, (x(D) + 0.5, y(D))) → e_1
  42. Polygon(A, (x(A) + a, y(A)), (x(A) + a, y(A) + a), (x(A), y(A) + a)) → poly5, a_2, k_1, n_1, p_1
  43. Segment((x(C), Min[y(B) + 1, 2.5]), (x(C), 2.5)) → k
  44. Polygon((x(B) + 1, y(B)), (x(B) + 2, y(B)), (x(B) + 2, y(B) + 1), (x(B) + 1, y(B) + 1)) → poly6, poly6_4, poly6_1, poly6_2, poly6_3
  45. Sequence(Segment[(k / a' + 1, y(B)), (k / a' + 1, y(B) + 1)], k, 1, a' - 1) → list7
  46. Sequence(Segment[(j / a', y(A)), (j / a', y(A) + 1)], j, 1, a' - 1) → list6
  47. If(b' < a', text40_1, If[(b' > a') ∧ (b' ≠ 2 * a'), text40_3, text40_2]) → text40

AI Drawing Prompt Reference

Create a visual representation to demonstrate the concept of fraction division, showing that 1 divided by a/b equals b/a, and illustrating that x divided by a/b equals x multiplied by b/a. The diagram includes a rectangle and a square, whose relationships represent fraction division and multiplication. Start by drawing a rectangle, then add a square on top, with the side length of the square proportional to one side of the rectangle. Moving the sliders for a and b allows observation of the changes in the diagram and helps understand the meaning of fraction division.
The prompt above was compiled from the original drawing commands and is for reference only.