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Fraction Multiplication Demo by Auston B Cron

Fraction Multiplication Demo by Auston B Cron
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2020-03-02 10:23
PolygonVectorTrigonometryGeometric Construction
This example dynamically demonstrates the geometric meaning of fraction multiplication, showing that the product of two fractions equals the area of the resulting parallelogram. It involves core math concepts like fraction multiplication, similar triangles, parallelograms, equations, hypothesis testing, and congruence. Construction highlights include GeoGebra commands such as OrthogonalLine, Circle, Intersect, Polygon, Ray, Point, Vector, Translate, Segment, Line, Mirror, Midpoint, GCD, Sequence, Element, and Text. Suitable for middle school and above students learning fraction multiplication and its geometric applications, or for mathematical exploration activities.

GeoGebra Commands & Parameters

  1. OrthogonalLine(A, yAxis) → e
  2. OrthogonalLine(A, e) → f
  3. Circle(A, Taille) → h
  4. Intersect(h, e, 1) → B
  5. Intersect(h, f, 2) → C
  6. Intersect(h, e, 1) → D
  7. OrthogonalLine(B, e) → i
  8. Intersect(h, f, 2) → E
  9. OrthogonalLine(C, f) → j
  10. Intersect(i, j) → F
  11. Polygon(A, B, F, C) → P, a_1, b_1, f_1, c_1
  12. Circle(A, VarRayon) → k
  13. Intersect(k, a_1) → G,
  14. Circle(G, A) → p
  15. Intersect(p, a_1, 2) → H
  16. Circle(H, G) → q
  17. Intersect(q, a_1) → I, J
  18. Circle(J, H) → r
  19. Intersect(r, a_1) → K, L
  20. Circle(L, J) → s
  21. Intersect(s, a_1) → M, N
  22. Circle(N, L) → t
  23. Intersect(t, a_1) → O, Q
  24. Circle(Q, N) → d_1
  25. Intersect(d_1, a_1) → R, S
  26. Circle(S, Q) → e_1
  27. Intersect(e_1, a_1) → T, U
  28. Circle(U, S) → g_1
  29. Intersect(g_1, a_1) → V, W
  30. Circle(W, U) → h_1
  31. OrthogonalLine(G, a_1) → l
  32. OrthogonalLine(H, a_1) → m
  33. OrthogonalLine(J, a_1) → n
  34. OrthogonalLine(L, a_1) → i_1
  35. OrthogonalLine(N, a_1) → j_1
  36. OrthogonalLine(Q, a_1) → k_1
  37. OrthogonalLine(S, a_1) → l_1
  38. OrthogonalLine(U, a_1) → m_1
  39. OrthogonalLine(W, a_1) → n_1
  40. Intersect(f_1, l) → Y
  41. Intersect(f_1, m) → Z
  42. Intersect(f_1, n) → A_1
  43. Intersect(f_1, i_1) → B_1
  44. Intersect(f_1, j_1) → C_1
  45. Intersect(f_1, k_1) → D_1
  46. Intersect(l_1, f_1) → E_1
  47. Intersect(f_1, m_1) → F_1
  48. Intersect(f_1, n_1) → G_1
  49. Ray(A, B) → p_1
  50. Point(p_1) → H_1
  51. Vector(A, H_1) → u
  52. Translate(A, u) → A'
  53. Translate(B, u) → B'
  54. Translate(F, u) → F'
  55. Translate(C, u) → C'
  56. Polygon(A', B', F', C') → P', a', b', f', c'
  57. OrthogonalLine(V, a_1) → q_1
  58. Circle(V, U) → r_1
  59. Segment(G, Y) → s_1
  60. Segment(H, Z) → t_1
  61. Segment(J, A_1) → a_2
  62. Segment(L, B_1) → b_2
  63. Segment(N, C_1) → c_2
  64. Segment(Q, D_1) → d_2
  65. Segment(S, E_1) → e_2
  66. Segment(U, F_1) → f_2
  67. Segment(W, G_1) → g_2
  68. Circle(H_1, VarRayon_2) → h_2
  69. Intersect(h_2, c', 1) → J_1
  70. Circle(J_1, H_1) → k_2
  71. Intersect(k_2, c', 2) → K_1
  72. Circle(K_1, J_1) → p_2
  73. Intersect(p_2, c', 2) → L_1
  74. Circle(L_1, K_1) → q_2
  75. Intersect(q_2, c', 2) → M_1
  76. Circle(M_1, L_1) → r_2
  77. Intersect(r_2, c', 2) → N_1
  78. Circle(N_1, M_1) → s_2
  79. Intersect(s_2, c') → O_1, P_1
  80. Circle(P_1, N_1) → t_2
  81. Intersect(t_2, c', 2) → Q_1
  82. Circle(Q_1, P_1) → c_3
  83. Intersect(c_3, c', 2) → R_1
  84. Circle(R_1, Q_1) → d_3
  85. Intersect(d_3, c', 2) → S_1
  86. Circle(S_1, R_1) → e_3
  87. OrthogonalLine(J_1, c') → i_2
  88. OrthogonalLine(K_1, c') → j_2
  89. OrthogonalLine(L_1, c') → l_2
  90. OrthogonalLine(M_1, c') → m_2
  91. OrthogonalLine(N_1, c') → n_2
  92. OrthogonalLine(P_1, c') → a_3
  93. OrthogonalLine(Q_1, c') → b_3
  94. OrthogonalLine(R_1, c') → f_3
  95. OrthogonalLine(S_1, c') → g_3
  96. Intersect(b', i_2) → T_1
  97. Intersect(b', j_2) → U_1
  98. Intersect(b', l_2) → V_1
  99. Intersect(b', m_2) → W_1
  100. Intersect(b', n_2) → Y_1
  101. Intersect(b', a_3) → Z_1
  102. Intersect(b', b_3) → A_2
  103. Intersect(b', f_3) → B_2
  104. Intersect(b', g_3) → C_2
  105. Segment(J_1, T_1) → h_3
  106. Segment(U_1, K_1) → i_3
  107. Segment(L_1, V_1) → j_3
  108. Segment(W_1, M_1) → k_3
  109. Segment(N_1, Y_1) → l_3
  110. Segment(Z_1, P_1) → m_3
  111. Segment(Q_1, A_2) → n_3
  112. Segment(B_2, R_1) → p_3
  113. Segment(S_1, C_2) → q_3
  114. Circle(A, largeur_1) → r_3
  115. Intersect(r_3, a_1) → D_2, S_2
  116. OrthogonalLine(D_2, f_1) → s_3
  117. Intersect(s_3, f_1) → E_2
  118. OrthogonalLine(E_2, s_3) → t_3
  119. Intersect(t_3, c_1) → F_2
  120. Polygon(A, D_2, E_2, F_2) → P_2, a_4, d_4, e_4, f_4
  121. OrthogonalLine(O, a_1) → b_4
  122. Intersect(b_4, f_1) → G_2
  123. Circle(O, N) → g_4
  124. Circle(O_1, N_1) → q_4
  125. OrthogonalLine(O_1, c') → t_4
  126. Circle(H_1, Largeur_2) → h_5
  127. Intersect(h_5, c') → F_3, B1
  128. OrthogonalLine(F_3, c') → i_5
  129. Intersect(i_5, b') → G_3
  130. Polygon(H_1, F_3, G_3, B') → P_3, h_6, f_6, g_6, b'_1
  131. Line(A, H_1) → a
  132. Point(a) → A1
  133. Line(A1, yAxis) → b
  134. Mirror(A', b) → A''
  135. Mirror(B', b) → B''
  136. Mirror(F', b) → F''
  137. Mirror(C', b) → C''
  138. Polygon(A'', B'', F'', C'') → P'', a'', b'', f'', c''
  139. Mirror(H_1, b) → H_1'
  140. Mirror(F_3, b) → F_3'
  141. Mirror(G_3, b) → G_3'
  142. Mirror(B', b) → B''_1
  143. Polygon(H_1', F_3', G_3', B''_1) → P_3', h_1', f_3', g_3', b''_1
  144. Mirror(J_1, b) → J_1'
  145. Mirror(T_1, b) → T_1'
  146. Segment(J_1', T_1') → c
  147. Mirror(U_1, b) → U_1'
  148. Mirror(K_1, b) → K_1'
  149. Segment(U_1', K_1') → d
  150. Mirror(L_1, b) → L_1'
  151. Mirror(V_1, b) → V_1'
  152. Segment(L_1', V_1') → g
  153. Mirror(W_1, b) → W_1'
  154. Mirror(M_1, b) → M_1'
  155. Segment(W_1', M_1') → a1
  156. Mirror(N_1, b) → N_1'
  157. Mirror(Y_1, b) → Y_1'
  158. Segment(N_1', Y_1') → b1
  159. Mirror(Z_1, b) → Z_1'
  160. Mirror(P_1, b) → P_1'
  161. Segment(Z_1', P_1') → c1
  162. Mirror(Q_1, b) → Q_1'
  163. Mirror(A_2, b) → A_2'
  164. Segment(Q_1', A_2') → d1
  165. Mirror(B_2, b) → B_2'
  166. Mirror(R_1, b) → R_1'
  167. Segment(B_2', R_1') → e1
  168. Mirror(S_1, b) → S_1'
  169. Mirror(C_2, b) → C_2'
  170. Segment(S_1', C_2') → f1
  171. Vector(A, B'') → v
  172. Translate(A, v) → A'_1
  173. Translate(D_2, v) → D_2'
  174. Translate(E_2, v) → E_2'
  175. Translate(F_2, v) → F_2'
  176. Polygon(A'_1, D_2', E_2', F_2') → P_2', a'_1, d_2', e_2', f_2'
  177. Translate(G, v) → G'
  178. Translate(Y, v) → Y'
  179. Segment(G', Y') → g1
  180. Translate(H, v) → H'
  181. Translate(Z, v) → Z'
  182. Segment(H', Z') → h1
  183. Translate(J, v) → J'
  184. Translate(A_1, v) → A_1'
  185. Segment(J', A_1') → i1
  186. Translate(L, v) → L'
  187. Translate(B_1, v) → B_1'
  188. Segment(L', B_1') → j1
  189. Translate(N, v) → N'
  190. Translate(C_1, v) → C_1'
  191. Segment(N', C_1') → k1
  192. Translate(Q, v) → Q'
  193. Translate(D_1, v) → D_1'
  194. Segment(Q', D_1') → l1
  195. Translate(S, v) → S'
  196. Translate(E_1, v) → E_1'
  197. Segment(S', E_1') → m1
  198. Translate(U, v) → U'
  199. Translate(F_1, v) → F_1'
  200. Segment(U', F_1') → n1
  201. Translate(W, v) → W'
  202. Translate(G_1, v) → G_1'
  203. Segment(W', G_1') → p1
  204. Midpoint(f_1) → C1
  205. Midpoint(g_6) → E1
  206. Midpoint(c') → D1
  207. Midpoint(f'') → F1
  208. Midpoint(c'') → G1
  209. Midpoint(a_4) → H1
  210. Midpoint(a'_1) → I1
  211. Midpoint(g_3') → J1
  212. Circle(D1, 1.5) → q1
  213. Circle(C1, 2) → r1
  214. Circle(F1, 2) → s1
  215. Circle(G1, 1.5) → t1
  216. Line(C1, b) → a2
  217. Intersect(r1, a2) → K1, L1
  218. Line(L1, f_1) → b2
  219. Intersect(s1, b2) → M1, N1
  220. Line(F_2', b) → c2
  221. Line(C'', b) → d2
  222. Line(F, b) → e2
  223. Line(C, b) → f2
  224. Intersect(e2, b2) → O1
  225. Intersect(e2, f2) → P1
  226. Intersect(b2, f2) → Q1
  227. Vector(L1, O1) → w
  228. Vector(L1, Q1) → z
  229. Line(D_2, b) → g2
  230. Circle(H1, 1.5) → h2
  231. Line(H1, b) → i2
  232. Intersect(h2, i2) → R1, S1
  233. Line(R1, v) → j2
  234. Intersect(g2, j2) → U1
  235. Intersect(f2, j2) → V1
  236. Circle(I1, 1.5) → k2
  237. Intersect(k2, j2) → W1, Z1
  238. Line(D_2', b) → l2
  239. Intersect(l2, j2) → A2
  240. Intersect(c2, j2) → B2
  241. Intersect(b2, c2) → C2
  242. Intersect(b2, d2) → D2
  243. Vector(R1, U1) → u1
  244. Vector(R1, V1) → v1
  245. Vector(W1, A2) → w1
  246. Vector(W1, B2) → z1
  247. Vector(M1, C2) → u2
  248. Vector(M1, D2) → v2
  249. Line(F, C') → m2
  250. Line(D1, j2) → n2
  251. Intersect(q1, n2) → E2, F2
  252. Line(E2, b) → p2
  253. Intersect(p2, m2) → G2
  254. Line(H_1, B) → q2
  255. Intersect(q2, p2) → H2
  256. Vector(E2, H2) → w2
  257. Vector(E2, G2) → z2
  258. Line(G_3, j2) → r2
  259. Circle(E1, 1.5) → s2
  260. Circle(J1, 1.5) → t2
  261. Intersect(t1, n2) → I2, J2
  262. Line(J2, b) → a3
  263. Intersect(q2, a3) → K2
  264. Intersect(m2, a3) → L2
  265. Vector(J2, K2) → u3
  266. Vector(J2, L2) → v3
  267. Line(J1, j2) → b3
  268. Intersect(t2, b3) → M2, N2
  269. Line(M2, b) → c3
  270. Intersect(c3, q2) → O2
  271. Intersect(r2, c3) → P2
  272. Vector(M2, O2) → w3
  273. Vector(M2, P2) → z3
  274. Intersect(s2, b3) → Q2, R2
  275. Line(R2, b) → d3
  276. Intersect(d3, q2) → S2
  277. Intersect(r2, d3) → U2
  278. Vector(R2, S2) → e3
  279. Vector(R2, U2) → f3
  280. Line(V2, xAxis) → g3
  281. Point(g3) → Z2
  282. Point(g3) → A3
  283. Point(g3) → B3
  284. Point(g3) → C3
  285. Point(g3) → D3
  286. Point(g3) → E3
  287. Segment(Z2, A3) → h3
  288. Segment(B3, C3) → i3
  289. Line(W2, h_6) → k3
  290. Line(F3, k3) → l3
  291. Point(g3) → G3
  292. Point(g3) → H3
  293. Point(g3) → I3
  294. Point(g3) → J3
  295. Segment(G3, H3) → j3
  296. Segment(I3, J3) → m3
  297. Intersect(c2, e) → M3
  298. Intersect(e, l2) → L3
  299. Line(P2, j2) → n3
  300. Intersect(c2, n3) → K3
  301. Midpoint(K3, L3) → N3
  302. Circle(N3, L3) → p3
  303. Intersect(p3, l2) → O3, P3
  304. Polygon(M3, L3, P3, K3) → poly1, m3_1, l3_1, p3_1, k3_1
  305. GCD((Num1 * Num2), (Den1 * Den2)) → qq
  306. Vector(F'', I_1) → u_1
  307. Translate(A'', u_1) → A'''
  308. Translate(B'', u_1) → B'''
  309. Translate(F'', u_1) → F'''
  310. Translate(C'', u_1) → C'''
  311. Polygon(A''', B''', F''', C''') → P''', a''', b''', f''', c'''
  312. Sequence(Translate[I_1, Vector[(i * Vector[I_1, C'''])]], i, 0, 1, 1 / ((Den1 * Den2 / qq))) → liste1
  313. Sequence(Translate[B''', Vector[(i * Vector[I_1, C'''])]], i, 0, 1, 1 / ((Den1 * Den2 / qq))) → liste2
  314. Sequence(Segment[Element[liste1, i], Element[liste2, i]], i, 1, Length[liste1]) → liste3
  315. Element(liste1, 1 + (Num1 * Num2 / qq)) → RR
  316. Element(liste2, 1 + (Num1 * Num2 / qq)) → H_2
  317. Polygon(B''', H_2, RR, I_1) → poly2, b'''_1, h_4, rr, i_4
  318. Line(K_2, L_2) → c_4
  319. Line(M_2, N_2) → j_4
  320. Line(O_2, Q_2) → k_4
  321. Text(If[qq ≠ 1, "Fraction is simplified by " + ((qq * "")), "impossible to simplified"]) → texte6

AI Drawing Prompt Reference

Draw a dynamic geometric demonstration of fraction multiplication showing that the product of two fractions equals the area of the resulting parallelogram. Include a point, two perpendicular lines, a circle, and multiple intersection points to form a parallelogram and similar triangles, utilizing vector, polygon, and reflection concepts to create a clear geometric model that visually demonstrates the geometric meaning of fraction multiplication.
The prompt above was compiled from the original drawing commands and is for reference only.