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Open Middle: Equal Ratios and Equation Construction via Dilation

Open Middle: Equal Ratios and Equation Construction via Dilation
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2021-03-24 02:53
PolygonSequenceTransformationGeometric Construction
This example demonstrates the “open middle” approach by constructing equal-ratio relationships inside a geometric configuration and turning those relationships into clear equation-style text. It focuses on key concepts such as equal ratios (proportional segments created by special points), linear-equation thinking for expressing relationships, and geometric similarity via dilation (scaling) that produces corresponding points. Construction-wise, it uses Polygon to build the base shapes, FractionText and Textfield to display ratio data as fractions, and If to control conditional display of information. Sequence and RandomElement help generate or iterate through point/ratio setups, making the theme reusable with varied parameters, while Dilate performs the geometric transformation establishing proportional correspondences. Mod and Text support systematic labeling and organization. The figure relies on multiple labeled points and polygon objects (including several poly* groups) together with auxiliary objects (f, g, h, i, etc.) for the interactive labeling and equation presentation. Suitable for middle to high school learners, especially for inquiry-based lessons connecting geometry reasoning with algebraic expression.

GeoGebra Commands & Parameters

  1. Polygon(A0, B, 4) → poly0, f, g, h, i, A, C
  2. Polygon(A3, H, 4) → poly3, j, k, l, m, E, G
  3. Polygon(A7, P, 4) → poly7, n, p, q, r, I, K
  4. Polygon(A1, D, 4) → poly1, s, t, a, b, M, O
  5. Polygon(A6, N, 4) → poly6, c, d, e, f_1, Q, S
  6. Polygon(A2, H2, 4) → poly2, k_1, l_1, m_1, n_1, W, Z
  7. Polygon(A8, A8 + (2, 0), 4) → poly8, g_1, h_1, i_1, j_1, U, V
  8. Polygon(A9, T, 4) → poly9, p_1, q_1, r_1, s_1, A_1, B_1
  9. Polygon(A4, J, 4) → poly4, t_1, a_1, b_1, c_1, C_1, D_1
  10. Polygon(A5, L, 4) → poly5, d_1, e_1, f_2, g_2, E_1, F_1
  11. Polygon(K_1, L_1, 4) → poly10, l_2, m_2, n_2, p_2, M_1, N_1
  12. Polygon(O_1, P_1, 4) → poly12, q_2, r_2, s_2, t_2, Q_1, R_1
  13. Polygon(S_1, T_1, 4) → poly13, a_2, b_2, c_2, d_2, U_1, V_1
  14. Polygon(C_2, D_2, 4) → poly15, i_3, j_3, k_3, l_3, E_2, F_2
  15. Polygon(G_2, H_2, 4) → poly16, m_3, n_3, p_3, q_3, I_2, J_2
  16. Polygon(G_{3}, H_{3}, 4) → poly17, m_{4}, n_{4}, p_{4}, q_{4}, I_{3}, J_{3}
  17. Polygon(C_{3}, D_{3}, 4) → poly14, i_{4}, j_{4}, k_{4}, l_{4}, E_{3}, F_{3}
  18. FractionText(u) → text4
  19. If(aleft > aright, "Left side is bigger.", aleft < aright, "Right side is bigger", "Both sides are equal!") → text9
  20. Textfield(w) → InputBox1
  21. If(o_1 < 0, "less than", o_1 > 0, "greater than", o_1 ≟ 0, "equal to") → text12
  22. Sequence(i, i, 1, 80, 1) → l1
  23. Sequence(i, i, 1, 80, 1) → l2
  24. RandomElement(l1) → o
  25. RandomElement(l2) → w_1
  26. RandomElement(Sequence[i, i, -9, 9, 1]) → e_2
  27. Dilate(pic11, 1 / 4, A_2) → pic11'
  28. Dilate(pic12, 1 / 5, O_2) → pic12'
  29. Mod(Numerator[(i * 2)], Denominator[(i * 2)]) → j_2
  30. Mod(21, 7) → k_2
  31. Mod((10 * p14) + p17 - p12, p16) → i_2
  32. If(oddtest, "odd", "even") → text20

AI Drawing Prompt Reference

Draw a geometric figure with four parallelograms and one rhombus, demonstrating equal ratios through point positions and scaling transformations. First, draw four parallelograms, then generate a rhombus by adjusting point coordinates and applying scaling transformations. Finally, add fraction texts and conditional texts to display equal ratios and size comparisons.
The prompt above was compiled from the original drawing commands and is for reference only.