Multiply by 6 with Dice: Interactive Random Polygons and Dilation

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2020-02-09 13:56

① This example demonstrates how rolling a dice generates random points and corresponding polygons (e.g., hexagons) on a plane, illustrating the relationship between multiplying by 6 and geometric dilation (scaling). ② Core math concepts include multiplication, plane figures (polygons), dilation (similarity transformation), randomness, and conditional probability, with quadratic equations as an extended context. ③ Construction highlights: GeoGebra commands such as Shuffle, Random, Point, Circle, Polygon, Rotate, Angle, Midpoint, Textfield, and Intersect are used to build dice face dot arrays (AP to JP), position lists (positionlist, pl), dice list (dlist), and center point A, with random rotation and scaling for dynamic visuals and a text input for interaction. ④ Suitable for upper elementary to middle school students, this helps intuitively grasp the link between multiplication and geometric transformations, and serves as an introduction to probability and random simulation.

GeoGebra Commands & Parameters

Shuffle(positionlist) → pl, {(9, 21), (14, 16), (2, 16), (5, 10), (21, 21), (21, 15), (8, 15), (19.234824979926557, 9.296710623986465), (3, 22), (15, 22)}
Element(pl, 1) → A
Element(pl, 2) → B
Element(pl, 3) → C
Element(pl, 4) → D
Element(pl, 5) → E
Element(pl, 6) → F
Element(pl, 7) → G
Element(pl, 8) → H
Element(pl, 9) → I
Element(pl, 10) → J
Circle(A, sd) → c
Circle(B, sd) → d
Circle(C, sd) → e
Circle(D, sd) → f
Circle(E, sd) → g
Circle(J, sd) → h
Circle(I, sd) → k
Circle(H, sd) → p
Circle(G, sd) → q
Random(90, 270) → a1
Circle(F, sd) → r
Point(c) → K
Point(d) → L
Point(e) → M
Point(f) → N
Point(g) → O
Rotate(K, a1°, A) → K'
Angle(K, A, K') → α
Polygon(A, K', 4) → poly1, i, j, l, m, P, Q
Random(0, 360) → a2
Random(0, 360) → a3
Random(0, 360) → a4
Random(0, 360) → a5
Random(0, 360) → a6
Random(0, 360) → a7
Random(0, 360) → a8
Random(0, 360) → a9
Random(0, 360) → a10
Point(r) → R
Point(q) → S
Point(p) → T
Point(k) → U
Point(h) → V
Rotate(L, a2°, B) → L'
Angle(L, B, L') → β
Rotate(M, a3°, C) → M'
Angle(M, C, M') → γ
Rotate(N, a4°, D) → N'
Angle(N, D, N') → δ
Rotate(O, a5°, E) → O'
Angle(O, E, O') → ε
Rotate(R, a6°, F) → R'
Angle(R, F, R') → ζ
Rotate(S, a7°, G) → S'
Angle(S, G, S') → η
Rotate(T, a8°, H) → T'
Angle(T, H, T') → θ
Rotate(U, a9°, I) → U'
Angle(U, I, U') → ι
Rotate(V, a10°, J) → V'
Angle(V, J, V') → κ
Polygon(B, L', 4) → poly2, n, s, t, a, W, Z
Polygon(C, M', 4) → poly3, b, f_1, g_1, h_1, A_1, B_1
Polygon(D, N', 4) → poly4, i_1, j_1, k_1, l_1, C_1, D_1
Polygon(E, O', 4) → poly5, m_1, n_1, p_1, q_1, E_1, F_1
Polygon(F, R', 4) → poly6, r_1, s_1, t_1, a_1, G_1, H_1
Polygon(G, S', 4) → poly7, b_1, c_1, d_1, e_1, I_1, J_1
Polygon(H, T', 4) → poly8, f_2, g_2, h_2, i_2, K_1, L_1
Polygon(I, U', 4) → poly9, j_2, k_2, l_2, m_2, M_1, N_1
Polygon(J, V', 4) → poly10, n_2, p_2, q_2, r_2, O_1, P_1
Circle(J, O_1) → c_2
Circle(H, K_1) → d_2
Circle(G, I_1) → e_2
Circle(F, G_1) → s_2
Circle(E, E_1) → t_2
Circle(D, C_1) → c_3
Circle(C, A_1) → d_3
Circle(B, W) → e_3
Circle(I, M_1) → f_3
Circle(A, P) → g_3
Midpoint(K', Q) → Q_1
Midpoint(Q_1, Q) → R_1
Midpoint(K', Q_1) → S_1
Midpoint(B, W) → T_1
Midpoint(T_1, W) → U_1
Midpoint(B, T_1) → V_1
Midpoint(C, A_1) → W_1
Midpoint(C, W_1) → Z_1
Midpoint(W_1, A_1) → A_2
Midpoint(D, C_1) → B_2
Midpoint(D, B_2) → C_2
Midpoint(B_2, C_1) → D_2
Midpoint(E, E_1) → E_2
Midpoint(E_2, E_1) → F_2
Midpoint(E, E_2) → G_2
Midpoint(F, G_1) → H_2
Midpoint(H_2, G_1) → I_2
Midpoint(F, H_2) → J_2
Midpoint(G, I_1) → K_2
Midpoint(G, K_2) → L_2
Midpoint(I_1, K_2) → M_2
Midpoint(H, K_1) → N_2
Midpoint(H, N_2) → O_2
Midpoint(N_2, K_1) → P_2
Midpoint(J, O_1) → Q_2
Midpoint(J, Q_2) → R_2
Midpoint(Q_2, O_1) → S_2
Midpoint(M_1, I) → T_2
Midpoint(M_1, T_2) → U_2
Midpoint(T_2, I) → V_2
Element(dl, dsl) → dn
Textfield(kiddice) → InputBox1
Textfield(kiddots) → InputBox2
Element(dl, dsl) → dice
Intersect(j, l) → E_3
Intersect(t, a) → F_3
Intersect(g_1, h_1) → G_3
Intersect(k_1, l_1) → H_3
Intersect(p_1, q_1) → I_3
Intersect(t_1, a_1) → J_3
Intersect(d_1, e_1) → K_3
Intersect(h_2, i_2) → L_3
Intersect(l_2, m_2) → M_3
Intersect(n_2, p_2) → N_3
Intersect(q_2, r_2) → O_3
Midpoint(A, Q_1) → P_3
Midpoint(Q_1, E_3) → Q_3
Midpoint(F_3, T_1) → R_3
Midpoint(L', T_1) → S_3
Midpoint(M', W_1) → T_3
Midpoint(W_1, G_3) → U_3
Midpoint(H_3, B_2) → V_3
Midpoint(B_2, N') → W_3
Midpoint(E_2, I_3) → Z_3
Midpoint(E_2, O') → A_4
Midpoint(R', H_2) → B_4
Midpoint(H_2, J_3) → C_4
Midpoint(K_2, S') → D_4
Midpoint(K_3, K_2) → E_4
Midpoint(L_3, N_2) → F_4
Midpoint(N_2, T') → G_4
Midpoint(U', T_2) → H_4
Midpoint(T_2, M_3) → I_4
Midpoint(N_3, Q_2) → J_4
Midpoint(Q_2, O_3) → K_4
Midpoint(Q_3, R_1) → B_3
Midpoint(S_1, P_3) → C_3
Midpoint(S_3, U_1) → D_3
Midpoint(V_1, R_3) → L_4
Midpoint(V_3, C_2) → M_4
Midpoint(D_2, W_3) → N_4
Midpoint(A_2, U_3) → O_4
Midpoint(T_3, Z_1) → P_4
Midpoint(U_2, I_4) → Q_4
Midpoint(H_4, V_2) → R_4
Midpoint(C_4, J_2) → S_4
Midpoint(I_2, B_4) → T_4
Midpoint(P_2, G_4) → U_4
Midpoint(F_4, O_2) → V_4
Midpoint(E_4, M_2) → W_4
Midpoint(L_2, D_4) → Z_4
Midpoint(K_4, S_2) → A_5
Midpoint(F_2, A_4) → B_5
Midpoint(Z_3, G_2) → C_5
Midpoint(R_2, J_4) → D_5

AI Drawing Prompt Reference

Generate ten random points on a plane)Skip one as the center. Draw equal-sized circles centered at nine of the points. Place a random point on each circle. Connect the center to a point on the first circle, then rotate that point around the center by a random angle to form an angle. Construct a regular polygon (e.g., a square) with the center as a vertex, using that angle. Repeat similar rotations for points on other circles, each with its own random angle. All points, circles, angles, and polygons together illustrate randomness and geometric transformation.

The prompt above was compiled from the original drawing commands and is for reference only.