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Fraction Division 2: Partitive Division p/q ÷ b

Fraction Division 2: Partitive Division p/q ÷ b
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2012-07-18 15:20
PolygonFunctionSequenceGeometric ConstructionSlider Animation
In collections Primary School Mathematics Collection Division of Fractions Division of Fractions Division of Fractions Fraction Division Fraction Division
This example illustrates the geometric interpretation of partitive fraction division, showing that p/q ÷ b equals p/q × 1/b. It uses GeoGebra commands such as Segment, Point, Sequence, Polygon, and GCD to construct an interactive visual model. The core concepts include fraction division, reciprocal multiplication, and linear representation of division. Designed for middle and high school students and teachers to intuitively grasp fraction division through dynamic graphics.

GeoGebra Commands & Parameters

  1. Segment(A, (x(A) + 1, y(A))) → l1
  2. Point(Segment[(x(A), y(A)), (x(A) + 1, y(A))]) → P
  3. Segment(B, (x(B) + 1, y(B))) → l2
  4. Point(Segment[(x(B) + 2 / 6, y(B)), (x(B) + 1, y(B))]) → Q
  5. Point(Segment[(x(C) + 1 / 6 - 0.005, y(C)), (x(Q) - 1 / 6, y(C))]) → R
  6. Segment(C, (x(C) + 1, y(C))) → l3
  7. Sequence(Circle[((2.5 * (i - 1)), 0), 1], i, 1, a) → list1
  8. Sequence(Sequence[CircleSector[((f * sin((((2 * k) - 1) * θ / 2))) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2)))), ((f * sin((((2 * k) - 1) * θ / 2))) + sin((k * θ)) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2))) + cos((k * θ))), ((f * sin((((2 * k) - 1) * θ / 2))) + sin(((k - 1) * θ)) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ)))], k, 1, b], m, 1, a) → list4
  9. Sequence(CircleSector[(((-1 * f) * sin((((2 * k) - 1) * θ / 2))) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2)))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ)) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin((k * θ)) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2))) + cos((k * θ)))], k, 1, c) → list5
  10. CircleSector(((2.5 * a), 0), ((2.5 * a), 1), ((2.5 * a) - sin((c * θ)), cos((c * θ)))) → s1
  11. Sequence(Segment[((2.5 * a), 0), ((2.5 * a) - sin((j * θ)), cos((j * θ)))], j, 1, c - 1) → list2
  12. Sequence(Segment[((2.5 * a), 0), ((2.5 * a) - sin((j * θ)), cos((j * θ)))], j, c + 1, b - 1) → list3
  13. CircleSector(((2.5 * a), 0), ((2.5 * a) - sin((c * θ)), cos((c * θ))), ((2.5 * a), 1)) → s2
  14. Segment(D, (x(D) + 1, y(D))) → l4
  15. Point(Segment[(x(D) + 2 / 5, y(D)), (x(D) + 1, y(D))]) → S
  16. Sequence(Sequence[CircleSector[(((-1 * f) * sin((((2 * k) - 1) * θ / 2))) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2)))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ)) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ) + θ / d) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ) + θ / d))], k, 1, b], m, 1, a) → list6
  17. Sequence(CircleSector[(((-1 * f) * sin((((2 * k) - 1) * θ / 2))) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2)))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ)) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ) + θ / d) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ) + θ / d))], k, 1, c) → list7
  18. Sequence(Sequence[Sequence[Segment[(((-1 * f) * sin((((2 * k) - 1) * θ / 2))) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2)))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ) + (n * θ / d)) + (2.5 * (m - 1)), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ) + (n * θ / d)))], n, 1, d - 1], k, 1, b], m, 1, a) → list8
  19. Sequence(Sequence[Segment[(((-1 * f) * sin((((2 * k) - 1) * θ / 2))) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2)))), (((-1 * f) * sin((((2 * k) - 1) * θ / 2))) - sin(((k - 1) * θ) + (n * θ / d)) + (2.5 * a), (f * cos((((2 * k) - 1) * θ / 2))) + cos(((k - 1) * θ) + (n * θ / d)))], n, 1, d - 1], k, 1, c) → list9
  20. Polygon((0.016528925619834656, -3), (1.2165289256198346, -3), (1.2165289256198346, -1.3), (0.016528925619834656, -1.3)) → poly1, e, g, h, i
  21. GCD((a * b) + c - ((floor(((a * b) + c) / ((b * d))) * b) * d), (b * d)) → k
  22. Segment(A, P) → j
  23. Segment(C, R) → l
  24. Segment(B, Q) → m
  25. Segment(D, S) → n