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Slope: Intuitive Introduction and Exploration

Slope: Intuitive Introduction and Exploration
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2017-03-09 04:24
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This GeoGebra example dynamically illustrates the concept of slope for a line, allowing users to interactively modify the line and observe changes in slope. The core mathematical concepts involved include the definition of slope, linear equations, and the geometric interpretation of rate of change. The construction utilizes various GeoGebra commands such as GCD, If, OrthogonalLine, Intersect, Line, Polygon, Distance, Segment, Angle, Slope, Dilate, Vector, Midpoint, Mirror, along with objects like points A and B, deltaY, deltaX, and others, to create a modifiable diagram that calculates and displays slope in real-time. It is suitable for middle or high school mathematics education, providing an intuitive way to explore slope properties.

GeoGebra Commands & Parameters

  1. GCD(deltaY, deltaX) → GCDm
  2. GCD(numerYint, deltaX) → GCDy
  3. If(sgn(deltaY / deltaX) ≥ 0, "", "-") → textSignN
  4. If(sgn(numerYint / deltaX) > 0, "+", "-") → textSign
  5. If(deltaXR ≟ 1, textSignN + deltaYR, textSignN + "\, \frac{" + deltaYR + "}{" + deltaXR + "}") → textslopeN
  6. If(denomYintR ≟ 1, textSign + numerYintR, textSign + "\frac{" + numerYintR + "}{" + denomYintR + "}") → textYint
  7. If(denomYintR ≟ 1, If[numerYintR ≟ 1, textSign, textSign + numerYintR], textSign + "\frac{" + numerYintR + "}{" + denomYintR + "}") → textMID
  8. OrthogonalLine(B, xAxis) → a_2
  9. OrthogonalLine(A, yAxis) → c
  10. Intersect(a_2, c) → C
  11. Line(A, B) → d
  12. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  13. Distance(B, C) → distanceBC
  14. Distance(A, C) → distanceAC
  15. Intersect(a_2, c) → D
  16. Segment(A, D) → e
  17. Segment(C, B) → f
  18. GCD(deltaY_1, deltaX_1) → GCDm_1
  19. If(sgn(deltaY_1 / deltaX_1) ≥ 0, "", "-") → textSignN_1
  20. If(deltaXR_1 ≟ 1, textSignN_1 + deltaYR_1, textSignN_1 + "\, \frac{" + deltaYR_1 + "}{" + deltaXR_1 + "}") → textslopeN_1
  21. Line(A_1, B_1) → d_1
  22. Angle(B, D, A) → α
  23. Segment(A, B) → g
  24. GCD(deltaY_2, deltaX_2) → GCDm_2
  25. If(sgn(deltaY_2 / deltaX_2) ≥ 0, "", "-") → textSignN_2
  26. If(deltaXR_2 ≟ 1, textSignN_2 + deltaYR_2, textSignN_2 + "\, \frac{" + deltaYR_2 + "}{" + deltaXR_2 + "}") → textslopeN_2
  27. Line(A_2, B_2) → d_2
  28. Segment(A_2, B_2) → g_1
  29. GCD(deltaY_3, deltaX_3) → GCDm_3
  30. If(sgn(deltaY_3 / deltaX_3) ≥ 0, "", "-") → textSignN_3
  31. If(deltaXR_3 ≟ 1, textSignN_3 + deltaYR_3, textSignN_3 + "\, \frac{" + deltaYR_3 + "}{" + deltaXR_3 + "}") → textslopeN_3
  32. Line(A_3, B_3) → d_3
  33. Segment(A_3, B_3) → g_2
  34. Slope(g) → h
  35. Dilate(A, If[0 < i ≤ 1, 1 - i, 0], D) → A'
  36. Vector(A, A') → u
  37. Dilate(D, If[1 < i ≤ 2, 2 - i, 0], B) → D'
  38. Vector(D, D') → v
  39. Line(A, B) → j
  40. Midpoint(A, C) → E
  41. Mirror(B, k) → B'
  42. Distance(A, B') → distanceAB'
  43. Distance(B', B) → distanceB'B
  44. Polygon(A, B', B) → t1, b_5, a_6, b'
  45. Dilate(A, If[0 < i ≤ 1, 1 - i, 0], B') → A'_1
  46. Vector(A, A'_1) → w
  47. Dilate(B', If[1 < i ≤ 2, 2 - i, 0], B) → B''
  48. Vector(B', B'') → l
  49. Intersect(m, n) → F
  50. Dilate(A, If[0 < i ≤ 1, 1 - i, 0], F) → A'_2
  51. Vector(A, A'_2) → p
  52. Dilate(F, If[1 < i ≤ 2, 2 - i, 0], B) → F'
  53. Vector(F, F') → q
  54. Distance(A, F) → distanceAF
  55. Distance(F, B) → distanceFB
  56. Polygon(A, F, B) → t3, b_6, a_7, f_1