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Exploring Proportional Relationships via Dilation and Linear Links

Exploring Proportional Relationships via Dilation and Linear Links
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2021-04-07 02:56
VectorFunctionTransformationGeometric ConstructionGraphing
In collections Proportional Relationships
This GeoGebra example demonstrates how proportional relationships emerge visually from geometric constructions, and how those geometric correspondences connect to linear relationships. It focuses on ratios, directed segments and points of division (including constant scale behavior), and interprets the results through linear functions/linear equations. The construction uses Dilate to perform scaling (a geometric transformation) and Vector/Segment to represent direction and lengths of corresponding segments. Conditional logic (If) and interactive labeling tools such as Textfield and input boxes update key points and numeric annotations when parameters change. Random is used to generate test configurations so learners can see the proportional pattern under different inputs. Suitable for middle and early high school learning, as an interactive lesson for ratios and the geometric meaning of linear behavior.

GeoGebra Commands & Parameters

  1. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  2. Vector(B, B') → u
  3. Dilate(A, If[2 < a ≤ 3, 3 - a, 0], (0, s)) → A'
  4. Vector(A, A') → v
  5. Dilate(A', If[3 < a ≤ 4, 4 - a, 0], D) → C'
  6. Dilate(D, If[5 < a ≤ 6, 6 - a, 0], (x(D), y(D) + s)) → D'
  7. Vector(D, D') → c
  8. Dilate(D', If[6 < a ≤ 7, 7 - a, 0], E) → E'
  9. Dilate(A, If[8 < a ≤ 9, 9 - a, 0], D) → A'_{1}
  10. Segment(A, A'_{1}) → g
  11. Dilate(D, If[9 < a ≤ 10, 10 - a, 0], E) → D'_{1}
  12. Segment(D, D'_{1}) → h
  13. Random(-10, 10) → b_1
  14. Random(-10, 10) → n_1
  15. Random(1, 10) → d_1
  16. If(s ≥ 0, "earn", "lose") → text7
  17. If(d > 1, "minutes", "minute") → text8
  18. Textfield(p) → InputBox3
  19. Textfield(o) → InputBox1
  20. If(o ≟ n_1 / d_1, "Yes!", "no") → text9
  21. If(p ≟ q, "Yes!", "no") → text10
  22. Vector(A', C') → w
  23. Vector(D', E') → e

AI Drawing Prompt Reference

Create a geometric diagram illustrating proportional relationships through dilation transformations, with vectors showing corresponding segment lengths and directions, connected to linear function graphs. The construction should highlight geometric correspondences under different scaling ratios, with interactive labels updating numeric values, and include randomly generated points to test proportional patterns for middle and early high school learners.
The prompt above was compiled from the original drawing commands and is for reference only.