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Area Models for Improper Fractions – 2 Fractions

Area Models for Improper Fractions – 2 Fractions
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2018-10-14 00:28
Geometric Construction
This example demonstrates how to use area models to understand improper fractions. The key mathematical concepts include fractions, area comparison, and geometric intuition. Construction involves GeoGebra commands such as Rotate, CircleSector, Polygon, etc. It is suitable for elementary and middle school mathematics teaching to help students grasp the area representation of improper fractions.

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