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Area Models for Improper Fractions – Two Fractions

Area Models for Improper Fractions – Two Fractions
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2020-08-18 22:31
CirclePolygonSequenceTransformationGeometric Construction
This example uses area models to visually demonstrate the magnitude and addition/subtraction relationships of two improper fractions. Core mathematical concepts include fraction representation, conversion between improper fractions and mixed numbers, the correspondence between area and quantity, and conversion among percentages, decimals, and fractions. Construction employs GeoGebra commands Rotate, CircleSector, Polygon, If, Sequence, and Segment to create sectors, rectangles, and segmented shading objects such as circle1sector, rect1shadePartial, frac1Num1, and frac1Denom1. It is suitable for upper‑primary to lower‑secondary fraction instruction, teacher‑created lesson materials, or student‑driven exploration.

GeoGebra Commands & Parameters

  1. Rotate(Sector1a, angle1, Center1) → Sector1b
  2. CircleSector(Center1, Sector1a, Sector1b) → circle1sector
  3. Rotate(Sector1a, (360 / denom1)°, Center1) → Sector1c
  4. Polygon((x(Center1) - 0.09, y(Center1) - 0.07), (x(Center1) - 0.09 + (0.18 * (val1 - floor(val1))), y(Center1) - 0.07), (x(Center1) - 0.09 + (0.18 * (val1 - floor(val1))), y(Center1) + 0.07), (x(Center1) - 0.09, y(Center1) + 0.07)) → rect1shadePartial, g, h, i, j
  5. If(val1 ≤ 5, Sequence[Circle[(i, yRow1), CircleDiameters], i, 0.1, Element[Position, If[val1 ≠ round(val1), floor(val1), ceil(val1)]], 0.2], Sequence[Circle[(i, yRow1), CircleDiameters], i, 0.1, Element[Position, 5], 0.2]) → circle1fill1
  6. If(val1 ≤ 5, Sequence[Circle[(i, yRow1), CircleDiameters], i, 0.1, Element[Position, ceil(val1)], 0.2], Sequence[Circle[(i, yRow1), CircleDiameters], i, 0.1, Element[Position, 5], 0.2]) → circle1outline1
  7. If(val1 ≤ 10, Sequence[Circle[(i, yRow2), CircleDiameters], i, 0.1, Element[Position, floor(val1)], 0.2], Sequence[Circle[(i, yRow2), CircleDiameters], i, 0.1, Element[Position, 5], 0.2]) → circle1fill2
  8. Sequence(Sequence[Segment[Rotate[(k, yRow2), (360 / denom1 * i°), (k, yRow2)], Rotate[(k, yRow2) + (0, CircleDiameters), (360 * (i / denom1)°), (k, yRow2)]], i, 0, denom1], k, 0.1, Element[Position, ceil(val1)], 0.2) → circle1div2
  9. If(val1 ≤ 10, Sequence[Circle[(i, yRow2), CircleDiameters], i, 0.1, Element[Position, ceil(val1)], 0.2], Sequence[Circle[(i, yRow2), CircleDiameters], i, 0.1, Element[Position, 10], 0.2]) → circle1outline2
  10. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow1 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow1 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow1 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow1 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val1 ≤ 5, ceil(val1), 5]], 0.2) → rect1outline1Seq
  11. Sequence(Sequence[Segment[(k - 0.09 + (0.18 / denom1 * i), yRow1 - 0.07), (k - 0.09 + (0.18 / denom1 * i), yRow1 + 0.07)], i, 1, denom1 - 1], k, 0.1, If[val1 ≥ 5, 0.9, Element[Position, ceil(val1)]], 0.2) → rect1div1
  12. Sequence(Sequence[Segment[(k - 0.09 + (0.18 / denom1 * i), yRow2 - 0.07), (k - 0.09 + (0.18 / denom1 * i), yRow2 + 0.07)], i, 1, denom1 - 1], k, 0.1, If[val1 ≥ 10, 0.9, Element[Position, ceil(val1)]], 0.2) → rect1div2
  13. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow1 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow1 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow1 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow1 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val1 ≤ 5, floor(val1), 5]], 0.2) → rect1shade1Seq
  14. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow2 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow2 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow2 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow2 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val1 < 10, floor(val1), 10]], 0.2) → rect1shade2Seq
  15. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow2 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow2 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow2 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow2 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val1 < 10, ceil(val1), 10]], 0.2) → rect1outline2Seq
  16. Rotate(Sector2a, angle2, Center2) → Sector2b
  17. Rotate(Sector2a, (360 / denom2)°, Center2) → Sector2c
  18. CircleSector(Center2, Sector2a, Sector2b) → circle2sector
  19. Sequence(Sequence[Segment[Rotate[(k, yRow3), (360 / denom2 * i°), (k, yRow3)], Rotate[(k, yRow3) + (0, CircleDiameters), (360 * (i / denom2)°), (k, yRow3)]], i, 0, denom2], k, 0.1, If[val2 ≥ 5, 0.9, Element[Position, ceil(val2)]], 0.2) → circle2div1
  20. Sequence(Sequence[Segment[Rotate[(k, yRow4), (360 / denom2 * i°), (k, yRow4)], Rotate[(k, yRow4) + (0, CircleDiameters), (360 * (i / denom2)°), (k, yRow4)]], i, 0, denom2], k, 0.1, Element[Position, ceil(val2)], 0.2) → circle2div2
  21. If(val2 ≤ 5, Sequence[Circle[(i, yRow3), CircleDiameters], i, 0.1, Element[Position, If[val2 ≠ round(val2), floor(val2), ceil(val2)]], 0.2], Sequence[Circle[(i, yRow3), CircleDiameters], i, 0.1, Element[Position, 5], 0.2]) → circle2fill1
  22. If(val2 ≤ 10, Sequence[Circle[(i, yRow4), CircleDiameters], i, 0.1, Element[Position, If[val2 ≠ round(val2), floor(val2), ceil(val2)]], 0.2], Sequence[Circle[(i, yRow4), CircleDiameters], i, 0.1, Element[Position, 10], 0.2]) → circle2fill2
  23. Sequence(Sequence[Segment[Rotate[(k, yRow1), (360 / denom1 * i°), (k, yRow1)], Rotate[(k, yRow1) + (0, CircleDiameters), (360 * (i / denom1)°), (k, yRow1)]], i, 0, denom1], k, 0.1, If[val1 ≥ 5, 0.9, Element[Position, ceil(val1)]], 0.2) → circle1div1
  24. If(val2 ≤ 5, Sequence[Circle[(i, yRow3), CircleDiameters], i, 0.1, Element[Position, ceil(val2)], 0.2], Sequence[Circle[(i, yRow3), CircleDiameters], i, 0.1, Element[Position, 5], 0.2]) → circle2outline1
  25. If(val1 ≤ 10, Sequence[Circle[(i, yRow4), CircleDiameters], i, 0.1, Element[Position, ceil(val2)], 0.2], Sequence[Circle[(i, yRow4), CircleDiameters], i, 0.1, Element[Position, 10], 0.2]) → circle2outline2
  26. Sequence(Sequence[Segment[(k - 0.09 + (0.18 / denom2 * i), yRow3 - 0.07), (k - 0.09 + (0.18 / denom2 * i), yRow3 + 0.07)], i, 1, denom2 - 1], k, 0.1, If[val2 ≥ 5, 0.9, Element[Position, ceil(val2)]], 0.2) → rect2div1
  27. Sequence(Sequence[Segment[(k - 0.09 + (0.18 / denom2 * i), yRow4 - 0.07), (k - 0.09 + (0.18 / denom2 * i), yRow4 + 0.07)], i, 1, denom2 - 1], k, 0.1, If[val2 ≥ 10, 0.9, Element[Position, ceil(val2)]], 0.2) → rect2div2
  28. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow4 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow4 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow4 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow4 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val2 ≤ 10, ceil(val2), 10]], 0.2) → rect2outline2Seq
  29. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow4 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow4 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow4 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow4 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val2 ≤ 10, floor(val2), 10]], 0.2) → rect2shade2Seq
  30. Polygon((x(Center2) - 0.09, y(Center2) - 0.07), (x(Center2) - 0.09 + (0.18 * (val2 - floor(val2))), y(Center2) - 0.07), (x(Center2) - 0.09 + (0.18 * (val2 - floor(val2))), y(Center2) + 0.07), (x(Center2) - 0.09, y(Center2) + 0.07)) → rect2shadePartial, l, p, q, r
  31. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow3 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow3 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow3 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow3 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val2 ≤ 4, ceil(val2), 5]], 0.2) → rect2outline1Seq
  32. Sequence(Polygon[Translate[(Element[Position, 1], 0) + (-0.09, yRow3 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow3 - 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (0.09, yRow3 + 0.07), Vector[(0, 0), (i, 0)]], Translate[(Element[Position, 1], 0) + (-0.09, yRow3 + 0.07), Vector[(0, 0), (i, 0)]]], i, 0, Element[Position, If[val2 ≤ 5, floor(val2), 5]], 0.2) → rect2shade1Seq
  33. Segment((0, 0.285), (-20, 0.285)) → dividerHoriz1
  34. Segment((0, -0.08), (-20, -0.08)) → dividerHoriz2

AI Drawing Prompt Reference

Draw two side‑by‑side area models for improper fractions; each model consists of a circle divided into equal sectors by the denominator, with a shaded sector showing the fractional part, and two rows of rectangles representing whole units — whole rectangles fully shaded and a final rectangle partially shaded according to the remainder. Concentric circles mark unit counts on the circular model, while vertical segments inside the rectangles indicate the denominator divisions. Horizontal lines separate the two fraction models, together illustrating magnitude, addition or subtraction, and conversion to mixed numbers.
The prompt above was compiled from the original drawing commands and is for reference only.