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Equations (Level 3) – Integrated Geometry and Algebra Exploration

Equations (Level 3) – Integrated Geometry and Algebra Exploration
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2016-04-11 00:41
PolygonSequenceCoordinate GeometryGeometric Construction
①This example demonstrates a composite geometric figure involving ellipses, polygons, segments, intersection points, midpoints, and text fields, designed as a level-3 equation practice. ②Core mathematical concepts include solving equations, properties of ellipses and polygons, relationships among points and segments, combinatorics, standard deviation, indefinite integrals, and planes, making it a multi-topic integrated activity. ③Construction highlights: from the summary, it uses GeoGebra commands such as If, Ellipse, Segment, Intersect, Point, Polygon, Midpoint, Sequence, and Textfield, creating numerous objects (dr, u, ul, r, dl, pul, mul, fmul, UL, fmur, mur, pur, ur, fmr, mr, pr, l, fml, L, ml, pl, R, UR, B, c, d, a, A, b, C, A', C', e, f, g, D, E, poly2, i, j) to achieve complex geometric construction and algebraic linkage. ④It is suitable for high school or early college students to explore the interplay between equations and geometric figures, and also serves as a reference for teachers creating interactive math lessons that foster dynamic understanding of algebra-geometry connections.

GeoGebra Commands & Parameters

  1. If((u * UL) ≟ R, 0, If[(u * UL) ≥ R, -1, 1]) → B
  2. Ellipse((-3, B), (-7, B), 2.25) → c
  3. Ellipse((3, (-B)), (7, (-B)), 2.25) → d
  4. Segment((-5, B), (-5, -3)) → a
  5. Intersect(c, a) → A,
  6. Segment((5, (-B)), (5, -3)) → b
  7. Intersect(d, b) → C,
  8. Segment(A, A') → e
  9. Segment(C, C') → f
  10. Segment(A', C') → g
  11. Intersect(yAxis, g) → D
  12. Point(yAxis) → E
  13. Polygon((10.455803599971945, 4.428079038697364), (11.45154195318282, 4.426628459182624), (11.45154195318282, 4.926628459182623), (10.455803599971945, 4.928079038697364)) → poly2, i, j, k, m
  14. Segment(D, E) → h
  15. Polygon(F, G, H, I) → poly3, f_1, g_1, h_1, i_1
  16. Midpoint(J, M) → O
  17. Sequence(Polygon[(-4.5, B - 0.5 + t / 2), (-3.5, B - 0.5 + t / 2), (-3.5, B + t / 2), (-4.5, B + t / 2)], t, 1, L, 1) → list1
  18. Polygon((-11.445852088028492, 4.308916490437969), (-10.445852088028493, 4.308916490437969), (-10.445852088028493, 4.8089164904379675), (-11.445852088028492, 4.8089164904379675)) → poly4, n, p, q, s
  19. Polygon((-10.543680266275816, -2.544527077316755), (-10.543680266275816, -1.5445270773167552), 4) → poly1
  20. Polygon((11.557939744940192, -2.5558711836942214), (11.557939744940192, -1.5558711836942218), 4) → poly1_3
  21. Polygon((-10.806626137271166, 7.362992675821548), (-9.806626137271168, 7.362992675821548), (-9.806626137271168, 7.8629926758215465), (-10.806626137271166, 7.8629926758215465)) → poly4_1, n_1, p_1, q_1, s_1
  22. Polygon((-10.806626137271166, 7.362992675821548), (-9.806626137271168, 7.362992675821548), (-9.806626137271168, 7.8629926758215465), (-10.806626137271166, 7.8629926758215465)) → poly4_2, n_2, p_2, q_2, s_2
  23. Sequence(Polygon[(-6.5, B - 1 + t), (-5.5, B - 1 + t), 4], t, 1, UL, 1) → list3
  24. Sequence(Polygon[(3.5, (-B) - 1 + t), (4.5, (-B) - 1 + t), 4], t, 1, UR, 1) → list4
  25. Sequence(Polygon[(5.5, (-B) - 0.5 + t / 2), (6.5, (-B) - 0.5 + t / 2), (6.5, (-B) + t / 2), (5.5, (-B) + t / 2)], t, 1, R, 1) → list2
  26. Textfield(fml) → input_fml
  27. Textfield(fmr) → InputBox1
  28. Textfield(fmul) → InputBox2
  29. Textfield(fmur) → InputBox3
  30. Textfield(dl) → InputBox4
  31. Textfield(dr) → InputBox5