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小数乘法演示

小数乘法演示
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2019-03-14 14:15
多边形向量解析几何几何作图
①本范例通过动态图形演示小数乘法的几何意义，将小数乘积与矩形面积关联；②涉及核心数学概念包括小数运算、矩形面积、向量运算、特殊点（如正交中心）；③构造要点运用了Segment（线段）、Polygon（多边形）、Sequence（数列）、If（条件判断）、Join（连接）、Element（元素）、Intersect（交点）、Point（点）、Vector（向量）、Translate（平移）等GeoGebra命令与对象；④适合初中及以上数学学习者理解乘法原理及几何应用。

GeoGebra 命令及参数

  1. Segment(A, D) → f
  2. Segment(D, C) → g
  3. Segment(C, B) → h
  4. Segment(B, A) → i
  5. Segment(E, F) → j
  6. Segment(F, G) → k
  7. Segment(G, H) → l
  8. Segment(H, E) → m
  9. Segment(I, J) → n
  10. Segment(J, K) → p
  11. Segment(K, L) → q
  12. Segment(I, L) → r
  13. Polygon((0, 20), (10, 20), (10, 21), (0, 21)) → q1, s, t, a, b
  14. Polygon((0, 22), (10, 22), (10, 23), (0, 23)) → q2, c, d, e, f_1
  15. Sequence((k, 24), k, 0, 7) → l1
  16. Sequence((k, 25), k, 0, 7) → l2
  17. Sequence(Polygon[Element[l1, k], Element[l1, k + 1], Element[l2, k + 1], Element[l2, k]], k, 1, 8, 2) → l3
  18. Sequence(Vector[Element[l4, k], Element[l5, k]], k, 1, 6) → l6
  19. If(0 < o < 1, Vector[Dilate[Element[l6, 1], o]], Vector[If[o ≥ 1, Vector[Element[l6, 1]]]]) → u
  20. If(1 ≤ o < 2, Vector[Dilate[Element[l6, 2], o - 1]], Vector[If[o ≥ 2, Vector[Element[l6, 2]]]]) → v
  21. If(2 ≤ o < 3, Vector[Dilate[Element[l6, 3], o - 2]], Vector[If[o ≥ 3, Vector[Element[l6, 3]]]]) → w
  22. If(3 ≤ o < 4, Vector[Dilate[Element[l6, 4], o - 3]], Vector[If[o ≥ 4, Vector[Element[l6, 4]]]]) → u_1
  23. If(4 ≤ o < 5, Vector[Dilate[Element[l6, 5], o - 4]], Vector[If[o ≥ 5, Vector[Element[l6, 5]]]]) → v_1
  24. If(5 ≤ o < 6, Vector[Dilate[Element[l6, 6], o - 5]], Vector[If[o ≥ 6, Vector[Element[l6, 6]]]]) → w_1
  25. Join({l8, l3}) → l9
  26. Sequence((0, k), k, 0, 9) → l11
  27. Sequence((10, k), k, 0, 9) → l12
  28. Sequence((12, k), k, 0, 9) → l13
  29. Sequence((22, k), k, 0, 9) → l14
  30. Sequence((24, k), k, 0, 9) → l15
  31. Sequence((25, k), k, 0, 9) → l16
  32. Sequence((26, k), k, 0, 9) → l17
  33. Sequence((27, k), k, 0, 9) → l18
  34. Sequence((28, k), k, 0, 9) → l19
  35. Sequence(Polygon[Element[l11, 11 - k], Element[l11, 10 - k], Element[l12, 10 - k], Element[l12, 11 - k]], k, 1, o - 5) → l20
  36. Sequence(Polygon[Element[l13, 11 - k], Element[l13, 10 - k], Element[l14, 10 - k], Element[l14, 11 - k]], k, 1, o - 5) → l21
  37. Sequence(Polygon[Element[l15, 11 - k], Element[l15, 10 - k], Element[l16, 10 - k], Element[l16, 11 - k]], k, 1, o - 5) → l22
  38. Sequence(Polygon[Element[l16, 11 - k], Element[l16, 10 - k], Element[l17, 10 - k], Element[l17, 11 - k]], k, 1, o - 5) → l23
  39. Sequence(Polygon[Element[l17, 11 - k], Element[l17, 10 - k], Element[l18, 10 - k], Element[l18, 11 - k]], k, 1, o - 5) → l24
  40. Sequence(Polygon[Element[l18, 11 - k], Element[l18, 10 - k], Element[l19, 10 - k], Element[l19, 11 - k]], k, 1, o - 5) → l25
  41. Sequence(Translate[Element[l9, k], Vector[Element[l7, k]]], k, 1, 2) → l10
  42. Sequence(Translate[Element[l9, k], Vector[Element[l7, k]]], k, 3, 6) → l26
  43. Sequence(Segment[Element[l13, 11 - k], Element[l14, 11 - k]], k, 1, o - 4) → l27
  44. Sequence(Segment[Element[l11, 11 - k], Element[l12, 11 - k]], k, 1, o - 4) → l28
  45. Sequence((k, 10), k, 1, 9) → l29
  46. Sequence((k, 0), k, 1, 9) → l30
  47. Sequence((24, k), k, 0, 10) → l34
  48. Sequence((28, k), k, 0, 10) → l35
  49. Element(l9, 3) → q3
  50. Element(l9, 4) → q4
  51. Element(l9, 5) → q5
  52. Element(l9, 6) → q6
  53. Intersect(f, i) → D_1
  54. Intersect(f, g) → E_1
  55. Intersect(g, h) → F_1
  56. Intersect(h, i) → G_1
  57. Polygon(D_1, E_1, F_1, G_1) → q7, d_1, e_1, g_1, h_1
  58. Intersect(j, k) → H_1
  59. Intersect(k, l) → I_1
  60. Intersect(l, m) → J_1
  61. Intersect(j, m) → K_1
  62. Polygon(H_1, I_1, J_1, K_1) → q8, i_1, j_1, k_1, l_1
  63. Point(r) → L_1
  64. Point(r) → O_1
  65. Sequence(Segment[Element[l35, 11 - k], Element[l34, 11 - k]], k, 1, o - 4) → l36
  66. Sequence((k, 0), k, 25, 28) → l32
  67. Sequence((k, 10), k, 25, 28) → l31
  68. Sequence(Segment[Element[l31, k], Element[l32, k]], k, 1, 4) → l33
  69. Sequence(k, k, 0.1, 1, 0.1) → l37
  70. Sequence(Dilate[Element[l33, 1], Element[l37, k], Element[l31, 1]], k, 1, o - 4) → l38
  71. Sequence(Dilate[Element[l33, 2], Element[l37, k], Element[l31, 2]], k, 1, o - 4) → l39
  72. Sequence(Dilate[Element[l33, 3], Element[l37, k], Element[l31, 3]], k, 1, o - 4) → l40
  73. Sequence(Dilate[Element[l33, 4], Element[l37, k], Element[l31, 4]], k, 1, o - 4) → l41
  74. Intersect(n, p) → M_1
  75. Point(l38) → N_1
  76. Point(r) → P_1
  77. Point(r) → Q_1
  78. Polygon(M_1, N_1, P_1, Q_1) → q9, m_1, n_1, p_1, r_1
  79. Point(l39) → R_1
  80. Point(r) → S_1
  81. Polygon(N_1, R_1, S_1, P_1) → q10, s_1, t_1, f_2, g_2
  82. Point(l40) → T_1
  83. Point(r) → U_1
  84. Polygon(R_1, T_1, U_1, S_1) → q11, h_2, i_2, j_2, k_2
  85. Point(l41) → V_1
  86. Point(r) → W_1
  87. Polygon(T_1, V_1, W_1, U_1) → q12, l_2, m_2, n_2, p_2
  88. Segment(Z_1, A_2) → q_2
  89. Segment(A_2, B_2) → r_2
  90. Segment(B_2, C_2) → s_2
  91. Segment(C_2, Z_1) → t_2
  92. Segment(D_2, E_2) → a_2
  93. Segment(E_2, F_2) → b_2
  94. Segment(F_2, G_2) → c_2
  95. Segment(G_2, D_2) → d_2
  96. Segment(H_2, I_2) → e_2
  97. Segment(I_2, J_2) → f_3
  98. Segment(J_2, K_2) → g_3
  99. Segment(K_2, H_2) → h_3
  100. Segment(L_2, M_2) → i_3
  101. Segment(M_2, N_2) → j_3
  102. Segment(N_2, O_2) → k_3
  103. Segment(O_2, L_2) → l_3
  104. Point(n) → P_2
  105. Point(p) → S_2
  106. Point(p) → V_2
  107. Point(p) → Z_2
  108. Intersect(s_2, t_2) → A_3
  109. Intersect(q_2, r_2) → B_3
  110. Intersect(a_2, d_2) → C_3
  111. Intersect(b_2, c_2) → D_3
  112. Point(e_2) → E_3
  113. Intersect(f_3, g_3) → F_3
  114. Intersect(i_3, l_3) → G_3
  115. Intersect(j_3, k_3) → H_3
  116. Intersect(s, b) → I_3
  117. Intersect(t, a) → J_3
  118. Intersect(c, f_1) → K_3
  119. Intersect(d, e) → L_3
  120. Point(f) → M_3
  121. Intersect(g, h) → N_3
  122. Point(j) → O_3
  123. Intersect(k, l) → P_3
  124. Vector(I_3, M_3) → u_2
  125. Vector(J_3, N_3) → v_2
  126. If(0 < o < 1, Vector[Dilate[u_2, o]]) → w_2
  127. If(0 < o < 1, Vector[Dilate[v_2, o]]) → u_3
  128. Translate(s, w_2) → s'
  129. Translate(b, w_2) → b'
  130. Translate(a, u_3) → a'
  131. Translate(t, u_3) → t'
  132. Point(h) → Q_3
  133. Segment(M_3, Q_3) → s_3
  134. Vector(K_3, O_3) → v_3
  135. Vector(L_3, P_3) → w_3
  136. If(1 < o < 2, Vector[Dilate[v_3, o - 1]]) → a_3
  137. If(1 < o < 2, Vector[Dilate[w_3, o - 1]]) → b_3
  138. Translate(f_1, a_3) → t_3
  139. Translate(c, a_3) → c_3
  140. Translate(e, b_3) → e'
  141. Translate(d, b_3) → d'
  142. Point(l) → R_3
  143. Segment(O_3, R_3) → d_3
  144. Vector(A_3, P_2) → e_3
  145. Vector(B_3, Z_2) → u_4
  146. If(2 < o < 3, Vector[Dilate[e_3, o - 2]]) → v_4
  147. If(2 < o < 3, Vector[Dilate[u_4, o - 2]]) → w_4
  148. Translate(C_2, v_4) → S_3
  149. Translate(Z_1, v_4) → T_3
  150. Segment(S_3, T_3) → f_4
  151. Translate(B_2, v_4) → U_3
  152. Translate(C_2, v_4) → V_3
  153. Segment(U_3, V_3) → g_4
  154. Translate(Z_1, w_4) → W_3
  155. Translate(A_2, w_4) → Z_3
  156. Segment(W_3, Z_3) → h_4
  157. Translate(A_2, w_4) → A_4
  158. Translate(B_2, w_4) → B_4
  159. Segment(A_4, B_4) → i_4
  160. Point(n) → T_2
  161. Point(l26) → U_2
  162. Segment(T_2, U_2) → m_3
  163. Point(p) → W_2
  164. Segment(U_2, W_2) → n_3
  165. Intersect(m_3, n_3) → C_4
  166. Segment(C_4, D_4) → p_3
  167. Segment(D_4, V_2) → q_3
  168. Segment(D_4, E_4) → r_3
  169. Segment(E_4, S_2) → j_4
  170. Vector(C_3, C_4) → a_4
  171. Vector(D_3, V_2) → b_4
  172. If(3 < o < 4, Vector[Dilate[a_4, o - 3]]) → c_4
  173. If(3 < o < 4, Vector[Dilate[b_4, o - 3]]) → d_4
  174. Translate(D_2, c_4) → F_4
  175. Translate(E_2, c_4) → G_4
  176. Segment(F_4, G_4) → k_4
  177. Translate(G_2, c_4) → H_4
  178. Translate(D_2, c_4) → I_4
  179. Segment(H_4, I_4) → l_4
  180. Translate(E_2, d_4) → J_4
  181. Translate(F_2, d_4) → K_4
  182. Segment(J_4, K_4) → m_4
  183. Translate(F_2, d_4) → L_4
  184. Translate(G_2, d_4) → M_4
  185. Segment(L_4, M_4) → n_4
  186. Vector(E_3, D_4) → e_4
  187. Vector(F_3, S_2) → p_4
  188. If(4 < o < 5, Vector[Dilate[e_4, o - 4]]) → q_4
  189. If(4 < o < 5, Vector[Dilate[p_4, o - 4]]) → r_4
  190. Translate(H_2, q_4) → N_4
  191. Translate(I_2, q_4) → O_4
  192. Segment(N_4, O_4) → s_4
  193. Translate(K_2, q_4) → P_4
  194. Translate(H_2, q_4) → Q_4
  195. Segment(P_4, Q_4) → t_4
  196. Translate(I_2, r_4) → R_4
  197. Translate(J_2, r_4) → S_4
  198. Segment(R_4, S_4) → f_5
  199. Translate(J_2, r_4) → T_4
  200. Translate(K_2, r_4) → U_4
  201. Segment(T_4, U_4) → g_5
  202. Vector(G_3, E_4) → u_5
  203. Vector(H_3, R_2) → v_5
  204. If(5 < o < 6, Vector[Dilate[u_5, o - 5]]) → w_5
  205. If(5 < o < 6, Vector[Dilate[v_5, o - 5]]) → a_5
  206. Translate(L_2, w_5) → V_4
  207. Translate(M_2, w_5) → W_4
  208. Segment(V_4, W_4) → h_5
  209. Translate(O_2, w_5) → Z_4
  210. Translate(L_2, w_5) → A_5
  211. Segment(Z_4, A_5) → i_5
  212. Translate(M_2, a_5) → B_5
  213. Translate(N_2, a_5) → C_5
  214. Segment(B_5, C_5) → j_5
  215. Translate(N_2, a_5) → D_5
  216. Translate(O_2, a_5) → E_5
  217. Segment(D_5, E_5) → k_5