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Fractions and Unit (Part 1)

Fractions and Unit (Part 1)
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2020-03-07 07:40
VectorCoordinate GeometryGeometric Construction
This example demonstrates how to understand and represent improper fractions through geometric methods. Core math concepts include the definition of improper fractions, geometric representation of fractions, and unit fractions. It uses commands and objects like Line, Point, Mod, Div, If, Translate, Dilate, Segment, Polygon, Sequence, Midpoint, Vector in GeoGebra for construction and calculation. Suitable for middle school and above learners to grasp the geometric meaning and operations of fractions.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → q
  2. Point(q) → B
  3. Line(A, yAxis) → p
  4. Mod(Numerator, Denominator) → rs2
  5. Div(Numerator, Denominator) → Quot
  6. If(rs2 < nn, rs2, nn - 1) → result
  7. Point(q) → F
  8. Translate(D, Vector[(Quot * Vector[E, F])]) → D'
  9. Dilate(B, 1 / nn, A) → B'
  10. Point(p) → C
  11. Segment(A, B) → a
  12. Segment(A, C) → b
  13. Translate(D, Vector[A, B]) → D'_1
  14. Translate(D, Vector[A, C]) → D'_2
  15. Translate(D'_2, Vector[A, B]) → D''_1
  16. Segment(E, F) → c
  17. Polygon(D, D'_1, D''_1, D'_2) → poly2, d_1, d'_1, d'', d'
  18. Sequence(Polygon[Translate[D, Vector[(i * Vector[E, F])]], Translate[D'_1, Vector[(i * Vector[E, F])]], Translate[D''_1, Vector[(i * Vector[E, F])]], Translate[D'_2, Vector[(i * Vector[E, F])]]], i, 1, Quot - 1) → list1
  19. Translate(D', Vector[A, B]) → D''_2
  20. Translate(D', Vector[A, C]) → D''_3
  21. Translate(D''_3, Vector[A, B]) → D'''_1
  22. Segment(D', D''_2) → d
  23. Segment(D''_2, D'''_1) → e
  24. Segment(D''_3, D'''_1) → f
  25. Segment(D''_3, D') → g
  26. Sequence(Segment[Translate[D''_3, Vector[(i * Vector[A, B'])]], Translate[D', Vector[(i * Vector[A, B'])]]], i, 1, nn - 1) → list2
  27. Translate(D', Vector[(result * Vector[A, B'])]) → D''
  28. Translate(D'', Vector[A, C]) → D'''
  29. Polygon(D', D'', D''', D''_3) → poly3, d'_2, d''_1, d''', d''_3
  30. Segment(G, H) → i
  31. Segment(I, J) → h
  32. Sequence(Segment[Translate[D, Vector[(i * Vector[A, B'])]], Translate[D'_2, Vector[(i * Vector[A, B'])]]], i, 1, Denominator - 1) → list3
  33. Sequence(Sequence[Segment[Translate[Translate[D, Vector[(j * Vector[E, F])]], Vector[(i * Vector[A, B'])]], Translate[Translate[D'_2, Vector[(j * Vector[E, F])]], Vector[(i * Vector[A, B'])]]], i, 1, Denominator - 1], j, 1, Quot - 1) → list4
  34. Mod(Numerator, Denominator) → rest22
  35. Polygon(A, B, M, C) → poly1, a_1, b_1, m_1, c_1
  36. Sequence(Polygon[Translate[D, Vector[(i * Vector[E, F])]], Translate[D'_1, Vector[(i * Vector[E, F])]], Translate[D''_1, Vector[(i * Vector[E, F])]], Translate[D'_2, Vector[(i * Vector[E, F])]]], i, 1, Quot - 1) → liste1
  37. Sequence(Segment[Translate[D''_3, Vector[(i * Vector[A, B'])]], Translate[D', Vector[(i * Vector[A, B'])]]], i, 1, nn - 1) → liste2
  38. Sequence(Segment[Translate[D, Vector[(i * Vector[A, B'])]], Translate[D'_2, Vector[(i * Vector[A, B'])]]], i, 1, Denominator - 1) → liste3
  39. Sequence(Sequence[Segment[Translate[Translate[D, Vector[(j * Vector[E, F])]], Vector[(i * Vector[A, B'])]], Translate[Translate[D'_2, Vector[(j * Vector[E, F])]], Vector[(i * Vector[A, B'])]]], i, 1, Denominator - 1], j, 1, Quot - 1) → liste4
  40. Translate(A, Vector[B, F]) → A'
  41. Translate(B, Vector[B, F]) → B'_1
  42. Translate(M, Vector[B, F]) → M'
  43. Translate(C, Vector[B, F]) → C'
  44. Polygon(A', B'_1, M', C') → poly1', a', b'_1, m', c'
  45. Midpoint(N, O) → R
  46. Midpoint(P, Q) → S
  47. Vector(R, O) → u
  48. Vector(R, N) → v
  49. Vector(S, Q) → w
  50. Vector(S, P) → z
  51. Polygon(T, U, V, W) → poly4, t, u_1, v_1, w_1
  52. Polygon(D, D'_1, D''_1, D'_2) → poly5, d_2, d'_3, d''_2, d'_4
  53. Midpoint(B, A) → Z
  54. Vector(Z, B) → j
  55. Vector(Z, A) → l
  56. Midpoint(A, C) → A_1
  57. Vector(A_1, C) → m
  58. Vector(A_1, A) → n
  59. Midpoint(c') → B_1
  60. Midpoint(B, M) → C_1
  61. Midpoint(B_1, C_1) → D_1
  62. Vector(D_1, B_1) → r
  63. Vector(D_1, C_1) → s
  64. Line(E_1, F_1) → e_1
  65. Segment(G_1, H_1) → f_1
  66. Segment(I_1, J_1) → g_1

AI Drawing Prompt Reference

Draw a coordinate plane with a horizontal baseline and a vertical line through a starting point. Mark a unit segment on the baseline. Based on the given numerator and denominator, compute the whole number part and the remainder. Repeat the unit segment horizontally that many times to form several complete unit rectangles. In the last rectangle, divide the vertical side into equal parts according to the denominator, draw the division lines, and shade the region representing the remainder. The final figure illustrates how an improper fraction consists of whole units plus a fractional part.
The prompt above was compiled from the original drawing commands and is for reference only.