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Fractions as Parts of Squares

Fractions as Parts of Squares
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2014-02-18 21:58
PolygonCoordinate Geometry
This example demonstrates how to visualize fractions by dividing squares into parts. It involves core math concepts like fractions, squares, limits, logic, and combinatorics. The construction uses GeoGebra commands like If and Sequence, along with objects such as A1-A10 and C1-C10 to represent divided sections and calculate centroids. Suitable for middle school students and above to learn fractions, geometric partitioning, and centroid computation, as well as for mathematical education demonstrations.

GeoGebra Commands & Parameters

  1. If(a - ((A1 - 1) * b) > 0, If[a - ((A1 - 1) * b) < b, a - ((A1 - 1) * b), b], 0) → B1
  2. If(a - ((A2 - 1) * b) > 0, If[a - ((A2 - 1) * b) < b, a - ((A2 - 1) * b), b], 0) → B2
  3. If(a - ((A3 - 1) * b) > 0, If[a - ((A3 - 1) * b) < b, a - ((A3 - 1) * b), b], 0) → B3
  4. If(a - ((A4 - 1) * b) > 0, If[a - ((A4 - 1) * b) < b, a - ((A4 - 1) * b), b], 0) → B4
  5. If(a - ((A5 - 1) * b) > 0, If[a - ((A5 - 1) * b) < b, a - ((A5 - 1) * b), b], 0) → B5
  6. If(a - ((A6 - 1) * b) > 0, If[a - ((A6 - 1) * b) < b, a - ((A6 - 1) * b), b], 0) → B6
  7. If(a - ((A7 - 1) * b) > 0, If[a - ((A7 - 1) * b) < b, a - ((A7 - 1) * b), b], 0) → B7
  8. If(a - ((A8 - 1) * b) > 0, If[a - ((A8 - 1) * b) < b, a - ((A8 - 1) * b), b], 0) → B8
  9. If(a - ((A9 - 1) * b) > 0, If[a - ((A9 - 1) * b) < b, a - ((A9 - 1) * b), b], 0) → B9
  10. If(a - ((A10 - 1) * b) > 0, If[a - ((A10 - 1) * b) < b, a - ((A10 - 1) * b), b], 0) → B10
  11. Sequence(Polygon[C1 + ((0, 1 / b) * i), D1 + ((0, 1 / b) * i), D1 + ((0, 1 / b) * (i + 1)), C1 + ((0, 1 / b) * (i + 1))], i, 0, B1 - 1) → E1
  12. Sequence(Polygon[C3 + ((0, 1 / b) * i), D3 + ((0, 1 / b) * i), D3 + ((0, 1 / b) * (i + 1)), C3 + ((0, 1 / b) * (i + 1))], i, 0, B3 - 1) → E3
  13. Sequence(Polygon[C5 + ((0, 1 / b) * i), D5 + ((0, 1 / b) * i), D5 + ((0, 1 / b) * (i + 1)), C5 + ((0, 1 / b) * (i + 1))], i, 0, B5 - 1) → E5
  14. Sequence(Polygon[C7 + ((0, 1 / b) * i), D7 + ((0, 1 / b) * i), D7 + ((0, 1 / b) * (i + 1)), C7 + ((0, 1 / b) * (i + 1))], i, 0, B7 - 1) → E7
  15. Sequence(Polygon[C9 + ((0, 1 / b) * i), D9 + ((0, 1 / b) * i), D9 + ((0, 1 / b) * (i + 1)), C9 + ((0, 1 / b) * (i + 1))], i, 0, B9 - 1) → E9
  16. Sequence(Polygon[C7 + ((1 / b, 0) * i), D1 + ((1 / b, 0) * i), D1 + ((1 / b, 0) * (i + 1)), C7 + ((1 / b, 0) * (i + 1))], i, 0, B2 - 1) → E2
  17. Sequence(Polygon[C9 + ((1 / b, 0) * i), D3 + ((1 / b, 0) * i), D3 + ((1 / b, 0) * (i + 1)), C9 + ((1 / b, 0) * (i + 1))], i, 0, B4 - 1) → E4
  18. Sequence(Polygon[C6 + (0, 1) + ((1 / b, 0) * i), C6 + ((1 / b, 0) * i), C6 + ((1 / b, 0) * (i + 1)), C6 + (0, 1) + ((1 / b, 0) * (i + 1))], i, 0, B6 - 1) → E6
  19. Sequence(Polygon[C8 + (0, 1) + ((1 / b, 0) * i), C8 + ((1 / b, 0) * i), C8 + ((1 / b, 0) * (i + 1)), C8 + (0, 1) + ((1 / b, 0) * (i + 1))], i, 0, B8 - 1) → E8
  20. Sequence(Polygon[C10 + (0, 1) + ((1 / b, 0) * i), C10 + ((1 / b, 0) * i), C10 + ((1 / b, 0) * (i + 1)), C10 + (0, 1) + ((1 / b, 0) * (i + 1))], i, 0, B10 - 1) → E10
  21. Sequence(Polygon[Cell[3, i], Cell[4, i], Cell[4, i + 5], Cell[3, i + 5]], i, 1, 5) → squares1
  22. Sequence(Polygon[Cell[3, i], Cell[4, i], Cell[4, i] + (0, 1), Cell[3, i] + (0, 1)], i, 6, 10) → squares2