(6905,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','adv higher','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6906,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','[ks3/ks4]','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6907,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','gcse','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6908,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','a level','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6909,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','[i/ii]','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6910,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','[iii/iv]','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6911,'82.8.167.110','1280348144','http://www.scruffs.shetland.co.uk/geobck.htm','mathematics','[v/vi]','mathematics/geometry','Wednesday  21:18  28/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Geometry resources - all in very extreme detail.*1*ANGLES*2*Angles - how to draw them*4*Angles - how to name them*4*Angles - measuring angles*4*Angles - types of angle 1 ( acute, etc)*4*Angles - types of angle 2 ( complementary, etc)*4*Angles - vertically opposite and corresponding - click on the link for an interactive java play *4*Angles - alternate - click on the link for an interactive java play*1*BEARINGS*2*Bearings*4*Example*1*PERIMETER, AREA AND VOLUME*2*Perimeter*4*Area*4*Area of a square*4*Area of a rectangle*4*Area of a triangle*4*Area of a circle*4*Area of a kite*4*Volume*4*Volume of a prism*4*Volume of a cuboid*4*Surface area*4*Composite area*4*Composite volume*4*Surface area of a cylinder - an Excel example for finding min surface area of a cylinder*1*THE CIRCLE*2*Calculating the circumference*4*Calculating the area*4*Circle - arcs, sectors overview*4*Circle - calculating length of arcs*4*Circle - calculating the area of a sector*4*Circle - chords and things*4**4*Arcs and sectors*4*Circle - pythagoras in a circle*4*Circle - angle in a semi circle*4*Circle - area and circumference - click on the link for an Excel based quiz. (380 kb)*1*THE CIRCLE (ADVANCED)*2*The circle - higher course examples*4*equation of circle : x2 + y2 = r2*4*Equation of circle : given centre, radius*4*Equation of circle : line joining diameter*4*General equation of circle*4*Intersection of line and circle*4*Tangents to a circle*4*Equation of a circle from chords*1*CO-ORDINATES*2*Cartesian coordinates*4*Reading cartesian coordinates*4*Plotting cartesian coordinates*4*Polar coordinates*1*3D SHAPES*2*3-d shapes*4*3-d shapes : planes*4*3-d shapes : prisms*1*MATRICES*2*Matrix algebra*4*Matrix multiplication*4**5*Determinants*6**4*2 x 2 matrix*4*3 x 3 matrix*4**4*Cofactor of a 3 x 3 matrix*4*Adjoint of a 3 x 3 matrix*4*Inverse of a square matrix*4*Transformation matrices*1*PYTHAGORAS*apos* THEOREM*2*The theorem of pythagoras*4*Finding the length of the hypotenuse*4*Finding the length of a shorter side*4*The converse of the theorem of pythagoras*1*QUADRILATERALS*2*Quadrilaterals*4*Quadrilaterals - line symmetry - click on the link for an interactive java play*4*Quadrilaterals - rotational symmetry - click on the link for an interactive java play*1*REFLECTIONS*2*Reflections - horizontal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4*Reflections - diagonal - click on the link for an interactive java play*4*Reflections - vertical - click on the link for an interactive java play*4**4**5*Pages at Math Is Fun - Maths Resources*6**4*Reflections*4*Rotations*4*Transformations*1*SCALE DRAWINGS*2*Map scales*4*The rule of 3*4*Example*4*Powerpoint 2003 ( 200 kb)*1*SIMILARITY*2*Congruence*4*Enlargements and reductions*4*Scale factor*4*The rule of 3*4*Similar shapes*4*Similar triangles*4*Scaled area*4*Scaled volume*1*THE STRAIGHT LINE*2*The straight line*4*Gradients*4*Equations and graphs*4*Plotting straight line graphs*4*Finding equations from straight line graphs*4*Straight lines, gradients and the y-intercept - click on the link for an Excel refresher (677 kb)*4*Straight lines - quick quiz*1*THE STRAIGHT LINE (ADVANCED)*2*y - b = m(x-a)*4*gradient = tan theta*4*distance formula*4*midpoint formula*4*perpendicular lines*4*collinearity*4*general equation of a straight line*1*TRIANGLES*2*Types of triangles*1*TRIANGLES (ADVANCED)*2*altitudes*4*centroid*4*circumcentre*4*circumcircle*4*concurrency*4*median*4*orthocentre*4*perpendicular bisector*1*VECTORS (ADVANCED)*2*Scalar and vector quantities*4*Components*4*Magnitude*4*Position vectors*4*Unit vector*4*Zero vector*4*Equal vectors*4*Adding vectors*4*Subtracting vectors*4*Collinearity*4*Perpendicular vectors*4*Angle between vectors*4*Scalar product*4*Example*4*Section formula*4**4*Vector product*4*Scalar triple product*4*Planes*4*Vector equations*4*Equations of a line*4*Intersections*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_g_e_o_b_c_k_._h_t_m','_m_a_t_h_s','_p_6_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _G_e_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_1_*_A_N_G_L_E_S_*_2_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _d_r_a_w_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _h_o_w_ _t_o_ _n_a_m_e_ _t_h_e_m_*_4_*_A_n_g_l_e_s_ _-_ _m_e_a_s_u_r_i_n_g_ _a_n_g_l_e_s_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _1_ _(_ _a_c_u_t_e_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _t_y_p_e_s_ _o_f_ _a_n_g_l_e_ _2_ _(_ _c_o_m_p_l_e_m_e_n_t_a_r_y_,_ _e_t_c_)_*_4_*_A_n_g_l_e_s_ _-_ _v_e_r_t_i_c_a_l_l_y_ _o_p_p_o_s_i_t_e_ _a_n_d_ _c_o_r_r_e_s_p_o_n_d_i_n_g_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_ _*_4_*_A_n_g_l_e_s_ _-_ _a_l_t_e_r_n_a_t_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_B_E_A_R_I_N_G_S_*_2_*_B_e_a_r_i_n_g_s_*_4_*_E_x_a_m_p_l_e_*_1_*_P_E_R_I_M_E_T_E_R_,_ _A_R_E_A_ _A_N_D_ _V_O_L_U_M_E_*_2_*_P_e_r_i_m_e_t_e_r_*_4_*_A_r_e_a_*_4_*_A_r_e_a_ _o_f_ _a_ _s_q_u_a_r_e_*_4_*_A_r_e_a_ _o_f_ _a_ _r_e_c_t_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _c_i_r_c_l_e_*_4_*_A_r_e_a_ _o_f_ _a_ _k_i_t_e_*_4_*_V_o_l_u_m_e_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _p_r_i_s_m_*_4_*_V_o_l_u_m_e_ _o_f_ _a_ _c_u_b_o_i_d_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _a_r_e_a_*_4_*_C_o_m_p_o_s_i_t_e_ _v_o_l_u_m_e_*_4_*_S_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_ _-_ _a_n_ _E_x_c_e_l_ _e_x_a_m_p_l_e_ _f_o_r_ _f_i_n_d_i_n_g_ _m_i_n_ _s_u_r_f_a_c_e_ _a_r_e_a_ _o_f_ _a_ _c_y_l_i_n_d_e_r_*_1_*_T_H_E_ _C_I_R_C_L_E_*_2_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _c_i_r_c_u_m_f_e_r_e_n_c_e_*_4_*_C_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_*_4_*_C_i_r_c_l_e_ _-_ _a_r_c_s_,_ _s_e_c_t_o_r_s_ _o_v_e_r_v_i_e_w_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _l_e_n_g_t_h_ _o_f_ _a_r_c_s_*_4_*_C_i_r_c_l_e_ _-_ _c_a_l_c_u_l_a_t_i_n_g_ _t_h_e_ _a_r_e_a_ _o_f_ _a_ _s_e_c_t_o_r_*_4_*_C_i_r_c_l_e_ _-_ _c_h_o_r_d_s_ _a_n_d_ _t_h_i_n_g_s_*_4_*_*_4_*_A_r_c_s_ _a_n_d_ _s_e_c_t_o_r_s_*_4_*_C_i_r_c_l_e_ _-_ _p_y_t_h_a_g_o_r_a_s_ _i_n_ _a_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_n_g_l_e_ _i_n_ _a_ _s_e_m_i_ _c_i_r_c_l_e_*_4_*_C_i_r_c_l_e_ _-_ _a_r_e_a_ _a_n_d_ _c_i_r_c_u_m_f_e_r_e_n_c_e_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _b_a_s_e_d_ _q_u_i_z_._ _(_3_8_0_ _k_b_)_*_1_*_T_H_E_ _C_I_R_C_L_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_T_h_e_ _c_i_r_c_l_e_ _-_ _h_i_g_h_e_r_ _c_o_u_r_s_e_ _e_x_a_m_p_l_e_s_*_4_*_e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _x_2_ _+_ _y_2_ _=_ _r_2_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _g_i_v_e_n_ _c_e_n_t_r_e_,_ _r_a_d_i_u_s_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_ _:_ _l_i_n_e_ _j_o_i_n_i_n_g_ _d_i_a_m_e_t_e_r_*_4_*_G_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _c_i_r_c_l_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_ _o_f_ _l_i_n_e_ _a_n_d_ _c_i_r_c_l_e_*_4_*_T_a_n_g_e_n_t_s_ _t_o_ _a_ _c_i_r_c_l_e_*_4_*_E_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _f_r_o_m_ _c_h_o_r_d_s_*_1_*_C_O_-_O_R_D_I_N_A_T_E_S_*_2_*_C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_R_e_a_d_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_l_o_t_t_i_n_g_ _c_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_s_*_4_*_P_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_*_1_*_3_D_ _S_H_A_P_E_S_*_2_*_3_-_d_ _s_h_a_p_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_l_a_n_e_s_*_4_*_3_-_d_ _s_h_a_p_e_s_ _:_ _p_r_i_s_m_s_*_1_*_M_A_T_R_I_C_E_S_*_2_*_M_a_t_r_i_x_ _a_l_g_e_b_r_a_*_4_*_M_a_t_r_i_x_ _m_u_l_t_i_p_l_i_c_a_t_i_o_n_*_4_*_*_5_*_D_e_t_e_r_m_i_n_a_n_t_s_*_6_*_*_4_*_2_ _x_ _2_ _m_a_t_r_i_x_*_4_*_3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_*_4_*_C_o_f_a_c_t_o_r_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_A_d_j_o_i_n_t_ _o_f_ _a_ _3_ _x_ _3_ _m_a_t_r_i_x_*_4_*_I_n_v_e_r_s_e_ _o_f_ _a_ _s_q_u_a_r_e_ _m_a_t_r_i_x_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_ _m_a_t_r_i_c_e_s_*_1_*_P_Y_T_H_A_G_O_R_A_S_*_a_p_o_s_*_ _T_H_E_O_R_E_M_*_2_*_T_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _l_e_n_g_t_h_ _o_f_ _a_ _s_h_o_r_t_e_r_ _s_i_d_e_*_4_*_T_h_e_ _c_o_n_v_e_r_s_e_ _o_f_ _t_h_e_ _t_h_e_o_r_e_m_ _o_f_ _p_y_t_h_a_g_o_r_a_s_*_1_*_Q_U_A_D_R_I_L_A_T_E_R_A_L_S_*_2_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _l_i_n_e_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_Q_u_a_d_r_i_l_a_t_e_r_a_l_s_ _-_ _r_o_t_a_t_i_o_n_a_l_ _s_y_m_m_e_t_r_y_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_1_*_R_E_F_L_E_C_T_I_O_N_S_*_2_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _h_o_r_i_z_o_n_t_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _d_i_a_g_o_n_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_R_e_f_l_e_c_t_i_o_n_s_ _-_ _v_e_r_t_i_c_a_l_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _i_n_t_e_r_a_c_t_i_v_e_ _j_a_v_a_ _p_l_a_y_*_4_*_*_4_*_*_5_*_P_a_g_e_s_ _a_t_ _M_a_t_h_ _I_s_ _F_u_n_ _-_ _M_a_t_h_s_ _R_e_s_o_u_r_c_e_s_*_6_*_*_4_*_R_e_f_l_e_c_t_i_o_n_s_*_4_*_R_o_t_a_t_i_o_n_s_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_*_1_*_S_C_A_L_E_ _D_R_A_W_I_N_G_S_*_2_*_M_a_p_ _s_c_a_l_e_s_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_E_x_a_m_p_l_e_*_4_*_P_o_w_e_r_p_o_i_n_t_ _2_0_0_3_ _(_ _2_0_0_ _k_b_)_*_1_*_S_I_M_I_L_A_R_I_T_Y_*_2_*_C_o_n_g_r_u_e_n_c_e_*_4_*_E_n_l_a_r_g_e_m_e_n_t_s_ _a_n_d_ _r_e_d_u_c_t_i_o_n_s_*_4_*_S_c_a_l_e_ _f_a_c_t_o_r_*_4_*_T_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_S_i_m_i_l_a_r_ _s_h_a_p_e_s_*_4_*_S_i_m_i_l_a_r_ _t_r_i_a_n_g_l_e_s_*_4_*_S_c_a_l_e_d_ _a_r_e_a_*_4_*_S_c_a_l_e_d_ _v_o_l_u_m_e_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_*_2_*_T_h_e_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_4_*_G_r_a_d_i_e_n_t_s_*_4_*_E_q_u_a_t_i_o_n_s_ _a_n_d_ _g_r_a_p_h_s_*_4_*_P_l_o_t_t_i_n_g_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_F_i_n_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _f_r_o_m_ _s_t_r_a_i_g_h_t_ _l_i_n_e_ _g_r_a_p_h_s_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_,_ _g_r_a_d_i_e_n_t_s_ _a_n_d_ _t_h_e_ _y_-_i_n_t_e_r_c_e_p_t_ _-_ _c_l_i_c_k_ _o_n_ _t_h_e_ _l_i_n_k_ _f_o_r_ _a_n_ _E_x_c_e_l_ _r_e_f_r_e_s_h_e_r_ _(_6_7_7_ _k_b_)_*_4_*_S_t_r_a_i_g_h_t_ _l_i_n_e_s_ _-_ _q_u_i_c_k_ _q_u_i_z_*_1_*_T_H_E_ _S_T_R_A_I_G_H_T_ _L_I_N_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_y_ _-_ _b_ _=_ _m_(_x_-_a_)_*_4_*_g_r_a_d_i_e_n_t_ _=_ _t_a_n_ _t_h_e_t_a_*_4_*_d_i_s_t_a_n_c_e_ _f_o_r_m_u_l_a_*_4_*_m_i_d_p_o_i_n_t_ _f_o_r_m_u_l_a_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _l_i_n_e_s_*_4_*_c_o_l_l_i_n_e_a_r_i_t_y_*_4_*_g_e_n_e_r_a_l_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _s_t_r_a_i_g_h_t_ _l_i_n_e_*_1_*_T_R_I_A_N_G_L_E_S_*_2_*_T_y_p_e_s_ _o_f_ _t_r_i_a_n_g_l_e_s_*_1_*_T_R_I_A_N_G_L_E_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_a_l_t_i_t_u_d_e_s_*_4_*_c_e_n_t_r_o_i_d_*_4_*_c_i_r_c_u_m_c_e_n_t_r_e_*_4_*_c_i_r_c_u_m_c_i_r_c_l_e_*_4_*_c_o_n_c_u_r_r_e_n_c_y_*_4_*_m_e_d_i_a_n_*_4_*_o_r_t_h_o_c_e_n_t_r_e_*_4_*_p_e_r_p_e_n_d_i_c_u_l_a_r_ _b_i_s_e_c_t_o_r_*_1_*_V_E_C_T_O_R_S_ _(_A_D_V_A_N_C_E_D_)_*_2_*_S_c_a_l_a_r_ _a_n_d_ _v_e_c_t_o_r_ _q_u_a_n_t_i_t_i_e_s_*_4_*_C_o_m_p_o_n_e_n_t_s_*_4_*_M_a_g_n_i_t_u_d_e_*_4_*_P_o_s_i_t_i_o_n_ _v_e_c_t_o_r_s_*_4_*_U_n_i_t_ _v_e_c_t_o_r_*_4_*_Z_e_r_o_ _v_e_c_t_o_r_*_4_*_E_q_u_a_l_ _v_e_c_t_o_r_s_*_4_*_A_d_d_i_n_g_ _v_e_c_t_o_r_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _v_e_c_t_o_r_s_*_4_*_C_o_l_l_i_n_e_a_r_i_t_y_*_4_*_P_e_r_p_e_n_d_i_c_u_l_a_r_ _v_e_c_t_o_r_s_*_4_*_A_n_g_l_e_ _b_e_t_w_e_e_n_ _v_e_c_t_o_r_s_*_4_*_S_c_a_l_a_r_ _p_r_o_d_u_c_t_*_4_*_E_x_a_m_p_l_e_*_4_*_S_e_c_t_i_o_n_ _f_o_r_m_u_l_a_*_4_*_*_4_*_V_e_c_t_o_r_ _p_r_o_d_u_c_t_*_4_*_S_c_a_l_a_r_ _t_r_i_p_l_e_ _p_r_o_d_u_c_t_*_4_*_P_l_a_n_e_s_*_4_*_V_e_c_t_o_r_ _e_q_u_a_t_i_o_n_s_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _a_ _l_i_n_e_*_4_*_I_n_t_e_r_s_e_c_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6920,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','higher','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6921,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','adv higher','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6922,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','[ks3/ks4]','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6923,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','gcse','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6924,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','a level','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6925,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','[i/ii]','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6926,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','[iii/iv]','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6927,'82.8.167.110','1280426833','http://www.scruffs.shetland.co.uk/trigbck.htm','mathematics','[v/vi]','mathematics/trigonometry','Thursday  19:08  29/07/10','1','0','1','1','1','0','0','0','0','0','6','1','anon','Massive site - lots of Trig/Trigonometry resources - all in very extreme detail.*4**4**1*BASIC TRIGONOMETRY*2*Definitions*4*Using Tan to find angle*4*Using Tan to find side*4*Using Sine to find angle*4*Using Sine to find side*4*Using Cos to find angle*4*Using Cos to find side*4*Which ratio*4*Finding the hypotenuse*1*FORMULAE*2*Trig refresher*4*sec, cosec and cot*4*Compound formulae*4*Double Angle*4*Half Angle*4*Product formulae*4*Multiple Angle*4*Algebra - Solving Trig equations*1*AREA OF A TRIANGLE, SINE RULE, COSINE RULE*2*Area of a triangle - derivation*4*Area of a triangle - example*4*Sine Rule - derivation*4*Sine Rule- Example*4*Cosine Rule - derivation*4*Cosine Rule- Examples*4*Sine and Cosine Rule- refresher*4*Which rule to use ? SSS, etc*1*COMPOUND FORMULAE (ADVANCED)*2*Deriving compound angle*4*Compound angle formulae - Examples*1*TRIG FAMILIES*2*Amplitude and Period definition*4*Click to see which quadrant an angle lies in - Excel (125 KB)*1*AMPLITUDE AND PHASE ANGLE*2*See how trig graphs are affected by a change in amplitude or phase angle - Excel (1737 KB)*1*RADIANS*2*Radians*4*Converting degrees to Radians*4*Exact values Radians and Degrees*4*Using exact values : Radians and Degrees*4*Conversion table*1*THE WAVE FUNCTION*2*Adding two waves*4*Subtracting two waves*4*Express in the form cos(x+a)*4*Express in the form sin(x-a)*4*Express in the form sin(x+a)*4*Multiple Angles*4*Maximum / Minimum Value*4*Sketching graphs*4*Solving equations*4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','3','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_t_r_i_g_b_c_k_._h_t_m','_m_a_t_h_s','_s_1_ _u_p_ _f_u_l_l','_1','','_1','_1','_1','0','0','0','0','0','_7','_1','','_M_a_s_s_i_v_e_ _s_i_t_e_ _-_ _l_o_t_s_ _o_f_ _T_r_i_g_/_T_r_i_g_o_n_o_m_e_t_r_y_ _r_e_s_o_u_r_c_e_s_ _-_ _a_l_l_ _i_n_ _v_e_r_y_ _e_x_t_r_e_m_e_ _d_e_t_a_i_l_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _T_R_I_G_O_N_O_M_E_T_R_Y_*_2_*_D_e_f_i_n_i_t_i_o_n_s_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _T_a_n_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _S_i_n_e_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _a_n_g_l_e_*_4_*_U_s_i_n_g_ _C_o_s_ _t_o_ _f_i_n_d_ _s_i_d_e_*_4_*_W_h_i_c_h_ _r_a_t_i_o_*_4_*_F_i_n_d_i_n_g_ _t_h_e_ _h_y_p_o_t_e_n_u_s_e_*_1_*_F_O_R_M_U_L_A_E_*_2_*_T_r_i_g_ _r_e_f_r_e_s_h_e_r_*_4_*_s_e_c_,_ _c_o_s_e_c_ _a_n_d_ _c_o_t_*_4_*_C_o_m_p_o_u_n_d_ _f_o_r_m_u_l_a_e_*_4_*_D_o_u_b_l_e_ _A_n_g_l_e_*_4_*_H_a_l_f_ _A_n_g_l_e_*_4_*_P_r_o_d_u_c_t_ _f_o_r_m_u_l_a_e_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_*_4_*_A_l_g_e_b_r_a_ _-_ _S_o_l_v_i_n_g_ _T_r_i_g_ _e_q_u_a_t_i_o_n_s_*_1_*_A_R_E_A_ _O_F_ _A_ _T_R_I_A_N_G_L_E_,_ _S_I_N_E_ _R_U_L_E_,_ _C_O_S_I_N_E_ _R_U_L_E_*_2_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_A_r_e_a_ _o_f_ _a_ _t_r_i_a_n_g_l_e_ _-_ _e_x_a_m_p_l_e_*_4_*_S_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_S_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_*_4_*_C_o_s_i_n_e_ _R_u_l_e_ _-_ _d_e_r_i_v_a_t_i_o_n_*_4_*_C_o_s_i_n_e_ _R_u_l_e_-_ _E_x_a_m_p_l_e_s_*_4_*_S_i_n_e_ _a_n_d_ _C_o_s_i_n_e_ _R_u_l_e_-_ _r_e_f_r_e_s_h_e_r_*_4_*_W_h_i_c_h_ _r_u_l_e_ _t_o_ _u_s_e_ _?_ _S_S_S_,_ _e_t_c_*_1_*_C_O_M_P_O_U_N_D_ _F_O_R_M_U_L_A_E_ _(_A_D_V_A_N_C_E_D_)_*_2_*_D_e_r_i_v_i_n_g_ _c_o_m_p_o_u_n_d_ _a_n_g_l_e_*_4_*_C_o_m_p_o_u_n_d_ _a_n_g_l_e_ _f_o_r_m_u_l_a_e_ _-_ _E_x_a_m_p_l_e_s_*_1_*_T_R_I_G_ _F_A_M_I_L_I_E_S_*_2_*_A_m_p_l_i_t_u_d_e_ _a_n_d_ _P_e_r_i_o_d_ _d_e_f_i_n_i_t_i_o_n_*_4_*_C_l_i_c_k_ _t_o_ _s_e_e_ _w_h_i_c_h_ _q_u_a_d_r_a_n_t_ _a_n_ _a_n_g_l_e_ _l_i_e_s_ _i_n_ _-_ _E_x_c_e_l_ _(_1_2_5_ _K_B_)_*_1_*_A_M_P_L_I_T_U_D_E_ _A_N_D_ _P_H_A_S_E_ _A_N_G_L_E_*_2_*_S_e_e_ _h_o_w_ _t_r_i_g_ _g_r_a_p_h_s_ _a_r_e_ _a_f_f_e_c_t_e_d_ _b_y_ _a_ _c_h_a_n_g_e_ _i_n_ _a_m_p_l_i_t_u_d_e_ _o_r_ _p_h_a_s_e_ _a_n_g_l_e_ _-_ _E_x_c_e_l_ _(_1_7_3_7_ _K_B_)_*_1_*_R_A_D_I_A_N_S_*_2_*_R_a_d_i_a_n_s_*_4_*_C_o_n_v_e_r_t_i_n_g_ _d_e_g_r_e_e_s_ _t_o_ _R_a_d_i_a_n_s_*_4_*_E_x_a_c_t_ _v_a_l_u_e_s_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_U_s_i_n_g_ _e_x_a_c_t_ _v_a_l_u_e_s_ _:_ _R_a_d_i_a_n_s_ _a_n_d_ _D_e_g_r_e_e_s_*_4_*_C_o_n_v_e_r_s_i_o_n_ _t_a_b_l_e_*_1_*_T_H_E_ _W_A_V_E_ _F_U_N_C_T_I_O_N_*_2_*_A_d_d_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_S_u_b_t_r_a_c_t_i_n_g_ _t_w_o_ _w_a_v_e_s_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _c_o_s_(_x_+_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_-_a_)_*_4_*_E_x_p_r_e_s_s_ _i_n_ _t_h_e_ _f_o_r_m_ _s_i_n_(_x_+_a_)_*_4_*_M_u_l_t_i_p_l_e_ _A_n_g_l_e_s_*_4_*_M_a_x_i_m_u_m_ _/_ _M_i_n_i_m_u_m_ _V_a_l_u_e_*_4_*_S_k_e_t_c_h_i_n_g_ _g_r_a_p_h_s_*_4_*_S_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_3','0','0'),(6928,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','[7th/8th]','mathematics/statistics','Thursday  19:31  29/07/10','0','0','0','1','1','0','0','0','0','0','3','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6929,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','10th grade','mathematics/statistics','Thursday  19:31  29/07/10','0','0','0','1','1','0','0','0','0','0','4','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6930,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','[s1/s2]','mathematics/statistics','Thursday  19:32  29/07/10','0','0','0','1','1','0','0','0','0','0','6','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6931,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','[s3/s4]','mathematics/statistics','Thursday  19:32  29/07/10','0','0','0','1','1','0','0','0','0','0','6','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6932,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','[ks3/ks4]','mathematics/statistics','Thursday  19:32  29/07/10','0','0','0','1','1','0','0','0','0','0','6','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6933,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','[i/ii]','mathematics/statistics','Thursday  19:32  29/07/10','0','0','0','1','1','0','0','0','0','0','6','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6934,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/statbck.htm','statistics','[iii/iv]','mathematics/statistics','Thursday  19:32  29/07/10','0','0','0','1','1','0','0','0','0','0','6','1','anon','Statistics/Stats resources.*4**4**1*BASIC STATISTICS*2*types of charts*4*mean, mode, median and range*4*frequency tables*4*stem and leaf diagram*4**4*pie charts*4*an Excel reminder. (17 kb)*4**4*5 figure summary and boxplots*4*box plot*4*dot plot*4*cumulative frequency (ogive)*4*an Excel reminder. (24 kb)*4**4*relative frequency*4*an Excel reminder. (23 kb)*4**4*line of best fit*4*an Excel reminder. (37 kb)*4**4*probability*4*scattergraphs*4*an Excel reminder. (37 kb)*4**4*standard deviation*4**4*standard deviation and box plot*4*an Excel reminder.(49 kb)*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','1','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_s_t_a_t_b_c_k_._h_t_m','_s_t_a_t_s','_s_1_ _-_ _s_4','0','','0','_1','_1','0','0','0','0','0','_7','_1','','_S_t_a_t_i_s_t_i_c_s_/_S_t_a_t_s_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_ _S_T_A_T_I_S_T_I_C_S_*_2_*_t_y_p_e_s_ _o_f_ _c_h_a_r_t_s_*_4_*_m_e_a_n_,_ _m_o_d_e_,_ _m_e_d_i_a_n_ _a_n_d_ _r_a_n_g_e_*_4_*_f_r_e_q_u_e_n_c_y_ _t_a_b_l_e_s_*_4_*_s_t_e_m_ _a_n_d_ _l_e_a_f_ _d_i_a_g_r_a_m_*_4_*_*_4_*_p_i_e_ _c_h_a_r_t_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_1_7_ _k_b_)_*_4_*_*_4_*_5_ _f_i_g_u_r_e_ _s_u_m_m_a_r_y_ _a_n_d_ _b_o_x_p_l_o_t_s_*_4_*_b_o_x_ _p_l_o_t_*_4_*_d_o_t_ _p_l_o_t_*_4_*_c_u_m_u_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_ _(_o_g_i_v_e_)_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_4_ _k_b_)_*_4_*_*_4_*_r_e_l_a_t_i_v_e_ _f_r_e_q_u_e_n_c_y_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_2_3_ _k_b_)_*_4_*_*_4_*_l_i_n_e_ _o_f_ _b_e_s_t_ _f_i_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_p_r_o_b_a_b_i_l_i_t_y_*_4_*_s_c_a_t_t_e_r_g_r_a_p_h_s_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._ _(_3_7_ _k_b_)_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_*_4_*_*_4_*_s_t_a_n_d_a_r_d_ _d_e_v_i_a_t_i_o_n_ _a_n_d_ _b_o_x_ _p_l_o_t_*_4_*_a_n_ _E_x_c_e_l_ _r_e_m_i_n_d_e_r_._(_4_9_ _k_b_)_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_1','1','_1'),(6935,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[3rd/4th]','mathematical skills/numeracy','Thursday  19:58  29/07/10','0','0','0','1','1','0','0','0','1','0','2','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6936,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','5th grade','mathematical skills/numeracy','Thursday  19:58  29/07/10','0','0','0','1','1','0','0','0','1','0','2','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6937,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','6th grade','mathematical skills/numeracy','Thursday  19:58  29/07/10','0','0','0','1','1','0','0','0','1','0','3','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6938,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[7th/8th]','mathematical skills/numeracy','Thursday  19:58  29/07/10','0','0','0','1','1','0','0','0','1','0','3','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6939,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[9th/10th]','mathematical skills/numeracy','Thursday  19:58  29/07/10','0','0','0','1','1','0','0','0','1','0','4','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6940,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','p4','mathematical skills/numeracy','Thursday  19:59  29/07/10','0','0','0','1','1','0','0','0','1','0','5','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6941,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[p5/p6]','mathematical skills/numeracy','Thursday  19:59  29/07/10','0','0','0','1','1','0','0','0','1','0','5','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6942,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','p7','mathematical skills/numeracy','Thursday  19:59  29/07/10','0','0','0','1','1','0','0','0','1','0','5','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6943,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','ks2','mathematical skills/numeracy','Thursday  19:59  29/07/10','0','0','0','1','1','0','0','0','1','0','5','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6944,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[s1/s2]','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6945,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[s3/s4]','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6946,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','access 3','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6947,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','intermediate 1','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6948,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[ks3/ks4]','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6949,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[i/ii]','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6950,'82.8.167.110','1280428245','http://www.scruffs.shetland.co.uk/miscbck.htm','mathematics','[iii/iv]','mathematical skills/numeracy','Thursday  20:00  29/07/10','0','0','0','1','1','0','0','0','1','0','6','1','anon','Arithmetic/Numeracy resources.*4**4**1*BASICS*2*associativity, etc.*4*base 10 - hundreds, tens, units*1*ARITHMETIC*2*addition*4*subtraction*4**4*multiplication table*4*multiplying decimals*4*multiplying large numbers*4*multiplying by 10, 100, 1000*4**4*division - short division*4*division - long division*4*division - dividing a decimal*4*division - dividing by a decimal*4*dividing by 10, 100, 1000*1*ORDER OF OPERATIONS*2*bodmas*1*FRACTIONS*2*fractions - the basics*4*wholes, halves, thirds fifths*4*fractions, decimals, piecharts - an Excel spreadsheet*4**4*simplifying fractions*4*simplifying fractions using prime factors*4*improper to mixed number*4*mixed number to improper*4**4*adding/subtracting fractions*4*multiplying fractions*4*dividing fractions*4*dividing fractions- alternative method*4**4*calculating fractions of amounts*4*algebraic fractions*4*converting fractions to percentages*4*converting fractions to decimals*4*fractions, percentages, decimals you must know*1*PERCENTAGES*2*percentages*4*wholes, halves, thirds fifths as percentages*4*converting percentages to decimals*4*converting percentages to fractions*4*calculating vat*1*DECIMALS*2*converting decimals to fractions*4*converting decimals to percentages*4*fractions, decimals and percentages - an Excel spreadsheet (274 kb)*1*INTEGERS*2*integers*4*adding negative numbers*4*subtracting negative numbers*4*multiplying integers*4*dividing integers*1*RATIO*2*Ratio*4*Equivalent ratios*4*Equivalent ratios - Vedic maths*4*Proportion*4*Sharing in Proportion*4*Map scales*4*ratio - direct proportion*4*ratio - indirect proportion*4*the rule of 3*1*ROUNDING*2*rounding to nearest - an Excel spreadsheet*4*rounding decimals*4*rounding to decimal places  - an Excel spreadsheet*4*significant figures*1*ROUNDING*2*Squares and Square Roots*1*STANDARD FORM (SCIENTIFIC NOTATION)*2*standard form*4*tolerance*1*TIME*2*time units*4*12 and 24 hour clock*4*12 hour clock*4*24 hour clock*4*converting 12 hour clock to 24 hour clock*4*converting 24 hour clock to 12 hour clock*4*calculating time differences*1*MEASUREMENTS AND UNITS*2*measurements - length conversions*4*measurements - mass conversions*4*measurements - volume conversions*1*SOCIAL ARITHMETIC*2*by the rule of 3*4*profit and loss*4*percentage profit and loss*4*back / reverse percentage*4*vat*4*appreciation*4*depreciation*4*payslips*4*net pay*4*overtime*4*superannuation*4*income tax*4*national insurance*4*utility bills*4*loans*4*hire purchase*4*insurance*4*simple interest*4*compound interest*4*foreign exchange*4**4**4*UK site but relevant to any pupils covering mathematics.*4**4* Extremely useful for teachers of Maths/Mathematics.','1','0','2','_w_w_w_._s_c_r_u_f_f_s_._s_h_e_t_l_a_n_d_._c_o_._u_k_/_m_i_s_c_b_c_k_._h_t_m','_m_a_t_h_s','_g_r_ _3_ _-_ _s_4','0','','0','_1','_1','0','0','0','_1','0','_7','_1','','_A_r_i_t_h_m_e_t_i_c_/_N_u_m_e_r_a_c_y_ _r_e_s_o_u_r_c_e_s_._*_4_*_*_4_*_*_1_*_B_A_S_I_C_S_*_2_*_a_s_s_o_c_i_a_t_i_v_i_t_y_,_ _e_t_c_._*_4_*_b_a_s_e_ _1_0_ _-_ _h_u_n_d_r_e_d_s_,_ _t_e_n_s_,_ _u_n_i_t_s_*_1_*_A_R_I_T_H_M_E_T_I_C_*_2_*_a_d_d_i_t_i_o_n_*_4_*_s_u_b_t_r_a_c_t_i_o_n_*_4_*_*_4_*_m_u_l_t_i_p_l_i_c_a_t_i_o_n_ _t_a_b_l_e_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _l_a_r_g_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_4_*_*_4_*_d_i_v_i_s_i_o_n_ _-_ _s_h_o_r_t_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _l_o_n_g_ _d_i_v_i_s_i_o_n_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_s_i_o_n_ _-_ _d_i_v_i_d_i_n_g_ _b_y_ _a_ _d_e_c_i_m_a_l_*_4_*_d_i_v_i_d_i_n_g_ _b_y_ _1_0_,_ _1_0_0_,_ _1_0_0_0_*_1_*_O_R_D_E_R_ _O_F_ _O_P_E_R_A_T_I_O_N_S_*_2_*_b_o_d_m_a_s_*_1_*_F_R_A_C_T_I_O_N_S_*_2_*_f_r_a_c_t_i_o_n_s_ _-_ _t_h_e_ _b_a_s_i_c_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_,_ _p_i_e_c_h_a_r_t_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_s_i_m_p_l_i_f_y_i_n_g_ _f_r_a_c_t_i_o_n_s_ _u_s_i_n_g_ _p_r_i_m_e_ _f_a_c_t_o_r_s_*_4_*_i_m_p_r_o_p_e_r_ _t_o_ _m_i_x_e_d_ _n_u_m_b_e_r_*_4_*_m_i_x_e_d_ _n_u_m_b_e_r_ _t_o_ _i_m_p_r_o_p_e_r_*_4_*_*_4_*_a_d_d_i_n_g_/_s_u_b_t_r_a_c_t_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_*_4_*_d_i_v_i_d_i_n_g_ _f_r_a_c_t_i_o_n_s_-_ _a_l_t_e_r_n_a_t_i_v_e_ _m_e_t_h_o_d_*_4_*_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _o_f_ _a_m_o_u_n_t_s_*_4_*_a_l_g_e_b_r_a_i_c_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _f_r_a_c_t_i_o_n_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _p_e_r_c_e_n_t_a_g_e_s_,_ _d_e_c_i_m_a_l_s_ _y_o_u_ _m_u_s_t_ _k_n_o_w_*_1_*_P_E_R_C_E_N_T_A_G_E_S_*_2_*_p_e_r_c_e_n_t_a_g_e_s_*_4_*_w_h_o_l_e_s_,_ _h_a_l_v_e_s_,_ _t_h_i_r_d_s_ _f_i_f_t_h_s_ _a_s_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _d_e_c_i_m_a_l_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _p_e_r_c_e_n_t_a_g_e_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _v_a_t_*_1_*_D_E_C_I_M_A_L_S_*_2_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _f_r_a_c_t_i_o_n_s_*_4_*_c_o_n_v_e_r_t_i_n_g_ _d_e_c_i_m_a_l_s_ _t_o_ _p_e_r_c_e_n_t_a_g_e_s_*_4_*_f_r_a_c_t_i_o_n_s_,_ _d_e_c_i_m_a_l_s_ _a_n_d_ _p_e_r_c_e_n_t_a_g_e_s_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_ _(_2_7_4_ _k_b_)_*_1_*_I_N_T_E_G_E_R_S_*_2_*_i_n_t_e_g_e_r_s_*_4_*_a_d_d_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_s_u_b_t_r_a_c_t_i_n_g_ _n_e_g_a_t_i_v_e_ _n_u_m_b_e_r_s_*_4_*_m_u_l_t_i_p_l_y_i_n_g_ _i_n_t_e_g_e_r_s_*_4_*_d_i_v_i_d_i_n_g_ _i_n_t_e_g_e_r_s_*_1_*_R_A_T_I_O_*_2_*_R_a_t_i_o_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_*_4_*_E_q_u_i_v_a_l_e_n_t_ _r_a_t_i_o_s_ _-_ _V_e_d_i_c_ _m_a_t_h_s_*_4_*_P_r_o_p_o_r_t_i_o_n_*_4_*_S_h_a_r_i_n_g_ _i_n_ _P_r_o_p_o_r_t_i_o_n_*_4_*_M_a_p_ _s_c_a_l_e_s_*_4_*_r_a_t_i_o_ _-_ _d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_r_a_t_i_o_ _-_ _i_n_d_i_r_e_c_t_ _p_r_o_p_o_r_t_i_o_n_*_4_*_t_h_e_ _r_u_l_e_ _o_f_ _3_*_1_*_R_O_U_N_D_I_N_G_*_2_*_r_o_u_n_d_i_n_g_ _t_o_ _n_e_a_r_e_s_t_ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_r_o_u_n_d_i_n_g_ _d_e_c_i_m_a_l_s_*_4_*_r_o_u_n_d_i_n_g_ _t_o_ _d_e_c_i_m_a_l_ _p_l_a_c_e_s_ _ _-_ _a_n_ _E_x_c_e_l_ _s_p_r_e_a_d_s_h_e_e_t_*_4_*_s_i_g_n_i_f_i_c_a_n_t_ _f_i_g_u_r_e_s_*_1_*_R_O_U_N_D_I_N_G_*_2_*_S_q_u_a_r_e_s_ _a_n_d_ _S_q_u_a_r_e_ _R_o_o_t_s_*_1_*_S_T_A_N_D_A_R_D_ _F_O_R_M_ _(_S_C_I_E_N_T_I_F_I_C_ _N_O_T_A_T_I_O_N_)_*_2_*_s_t_a_n_d_a_r_d_ _f_o_r_m_*_4_*_t_o_l_e_r_a_n_c_e_*_1_*_T_I_M_E_*_2_*_t_i_m_e_ _u_n_i_t_s_*_4_*_1_2_ _a_n_d_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _1_2_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _2_4_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_o_n_v_e_r_t_i_n_g_ _2_4_ _h_o_u_r_ _c_l_o_c_k_ _t_o_ _1_2_ _h_o_u_r_ _c_l_o_c_k_*_4_*_c_a_l_c_u_l_a_t_i_n_g_ _t_i_m_e_ _d_i_f_f_e_r_e_n_c_e_s_*_1_*_M_E_A_S_U_R_E_M_E_N_T_S_ _A_N_D_ _U_N_I_T_S_*_2_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _l_e_n_g_t_h_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _m_a_s_s_ _c_o_n_v_e_r_s_i_o_n_s_*_4_*_m_e_a_s_u_r_e_m_e_n_t_s_ _-_ _v_o_l_u_m_e_ _c_o_n_v_e_r_s_i_o_n_s_*_1_*_S_O_C_I_A_L_ _A_R_I_T_H_M_E_T_I_C_*_2_*_b_y_ _t_h_e_ _r_u_l_e_ _o_f_ _3_*_4_*_p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_p_e_r_c_e_n_t_a_g_e_ _p_r_o_f_i_t_ _a_n_d_ _l_o_s_s_*_4_*_b_a_c_k_ _/_ _r_e_v_e_r_s_e_ _p_e_r_c_e_n_t_a_g_e_*_4_*_v_a_t_*_4_*_a_p_p_r_e_c_i_a_t_i_o_n_*_4_*_d_e_p_r_e_c_i_a_t_i_o_n_*_4_*_p_a_y_s_l_i_p_s_*_4_*_n_e_t_ _p_a_y_*_4_*_o_v_e_r_t_i_m_e_*_4_*_s_u_p_e_r_a_n_n_u_a_t_i_o_n_*_4_*_i_n_c_o_m_e_ _t_a_x_*_4_*_n_a_t_i_o_n_a_l_ _i_n_s_u_r_a_n_c_e_*_4_*_u_t_i_l_i_t_y_ _b_i_l_l_s_*_4_*_l_o_a_n_s_*_4_*_h_i_r_e_ _p_u_r_c_h_a_s_e_*_4_*_i_n_s_u_r_a_n_c_e_*_4_*_s_i_m_p_l_e_ _i_n_t_e_r_e_s_t_*_4_*_c_o_m_p_o_u_n_d_ _i_n_t_e_r_e_s_t_*_4_*_f_o_r_e_i_g_n_ _e_x_c_h_a_n_g_e_*_4_*_*_4_*_*_4_*_U_K_ _s_i_t_e_ _b_u_t_ _r_e_l_e_v_a_n_t_ _t_o_ _a_n_y_ _p_u_p_i_l_s_ _c_o_v_e_r_i_n_g_ _m_a_t_h_e_m_a_t_i_c_s_._*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_.','_1','0','_2','0','0'),(6952,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','[11th/12th]','mathematics/algebra','Friday  10:59  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6953,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','ap','mathematics/algebra','Friday  10:59  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6954,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','[s3/s4]','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6955,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','[s5/s6]','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6956,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','intermediate 2','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6957,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','higher','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6958,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','adv higher','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6959,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','ks4','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6960,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','gcse','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6961,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','a level','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _e_x_p_o_n_e_n_t_s_ _a_n_d_ _t_h_e_i_r_ _p_r_o_p_e_r_t_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _e_x_t_e_n_d_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _s_e_c_t_i_o_n_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._*_4_*_R_e_a_l_ _E_x_p_o_n_e_n_t_s_ _-_ _T_h_i_s_ _i_s_ _a_ _s_h_o_r_t_ _a_c_k_n_o_w_l_e_d_g_m_e_n_t_ _t_h_a_t_ _t_h_e_ _e_x_p_o_n_e_n_t_ _p_r_o_p_e_r_t_i_e_s_ _f_r_o_m_ _t_h_e_ _p_r_e_v_i_o_u_s_ _t_w_o_ _s_e_c_t_i_o_n_s_ _w_i_l_l_ _h_o_l_d_ _f_o_r_ _a_n_y_ _r_e_a_l_ _e_x_p_o_n_e_n_t_._*_4_*_R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_d_i_c_a_l_ _n_o_t_a_t_i_o_n_ _a_n_d_ _r_e_l_a_t_e_ _r_a_d_i_c_a_l_s_ _t_o_ _r_a_t_i_o_n_a_l_ _e_x_p_o_n_e_n_t_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _r_a_d_i_c_a_l_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _i_n_c_l_u_d_i_n_g_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_ _a_n_d_ _m_u_l_t_i_p_l_y_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_a_c_t_o_r_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _ _-_T_h_i_s_ _i_s_ _t_h_e_ _m_o_s_t_ _i_m_p_o_r_t_a_n_t_ _s_e_c_t_i_o_n_ _o_f_ _a_l_l_ _t_h_e_ _p_r_e_l_i_m_i_n_a_r_i_e_s_._*_4_*_R_a_t_i_o_n_a_l_ _E_x_p_r_e_s_s_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_ _a_n_d_ _d_i_s_c_u_s_s_ _a_d_d_i_n_g_,_ _s_u_b_t_r_a_c_t_i_n_g_,_ _m_u_l_t_i_p_l_y_i_n_g_ _a_n_d_ _d_i_v_i_d_i_n_g_ _t_h_e_m_._*_4_*_C_o_m_p_l_e_x_ _N_u_m_b_e_r_s_ _-_ _H_e_r_e_ _i_s_ _a_ _v_e_r_y_ _q_u_i_c_k_ _p_r_i_m_e_r_ _o_n_ _c_o_m_p_l_e_x_ _n_u_m_b_e_r_s_ _a_n_d_ _h_o_w_ _t_o_ _m_a_n_i_p_u_l_a_t_e_ _t_h_e_m_._*_1_*_S_O_L_V_I_N_G_ _E_Q_U_A_T_I_O_N_S_ _A_N_D_ _I_N_E_Q_U_A_L_I_T_I_E_S_*_2_*_S_o_l_u_t_i_o_n_s_ _a_n_d_ _S_o_l_u_t_i_o_n_ _S_e_t_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_ _n_o_t_a_t_i_o_n_ _a_n_d_ _i_d_e_a_s_ _i_n_v_o_l_v_e_d_ _i_n_ _s_o_l_v_i_n_g_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_,_ _i_n_c_l_u_d_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _L_i_n_e_a_r_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _l_i_n_e_a_r_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_q_u_a_t_i_o_n_s_ _W_i_t_h_ _M_o_r_e_ _T_h_a_n_ _O_n_e_ _V_a_r_i_a_b_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _m_o_r_e_ _t_h_a_n_ _o_n_e_ _v_a_r_i_a_b_l_e_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _f_a_c_t_o_r_i_n_g_ _a_n_d_ _t_h_e_ _s_q_u_a_r_e_ _r_o_o_t_ _p_r_o_p_e_r_t_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_,_ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _f_i_n_i_s_h_ _u_p_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _*_4_*_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_o_m_p_l_e_t_i_n_g_ _t_h_e_ _s_q_u_a_r_e_ _a_n_d_ _q_u_a_d_r_a_t_i_c_ _f_o_r_m_u_l_a_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _:_ _A_ _S_u_m_m_a_r_y_ _-_ _W_e_*_a_p_o_s_*_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_d_u_r_e_ _f_o_r_ _d_e_t_e_r_m_i_n_i_n_g_ _w_h_i_c_h_ _m_e_t_h_o_d_ _t_o_ _u_s_e_ _i_n_ _s_o_l_v_i_n_g_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _t_h_e_ _d_i_s_c_r_i_m_i_n_a_n_t_._*_4_*_*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _Q_u_a_d_r_a_t_i_c_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _s_o_m_e_ _o_f_ _t_h_e_ _a_p_p_l_i_c_a_t_i_o_n_s_ _w_e_ _s_a_w_ _i_n_ _t_h_e_ _l_i_n_e_a_r_ _a_p_p_l_i_c_a_t_i_o_n_ _s_e_c_t_i_o_n_*_4_*_o_n_l_y_ _t_h_i_s_ _t_i_m_e_ _t_h_e_y_ _w_i_l_l_ _i_n_v_o_l_v_e_ _s_o_l_v_i_n_g_ _a_ _q_u_a_d_r_a_t_i_c_ _e_q_u_a_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _R_e_d_u_c_i_b_l_e_ _t_o_ _Q_u_a_d_r_a_t_i_c_ _F_o_r_m_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_a_n_ _b_e_ _r_e_d_u_c_e_d_ _t_o_ _q_u_a_d_r_a_t_i_c_ _i_n_ _f_o_r_m_._*_4_*_E_q_u_a_t_i_o_n_s_ _w_i_t_h_ _R_a_d_i_c_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _s_q_u_a_r_e_ _r_o_o_t_s_ _i_n_ _t_h_e_m_._*_4_*_*_4_*_L_i_n_e_a_r_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_t_a_r_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _b_y_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_a_r_ _i_n_e_q_u_a_l_i_t_i_e_s_._*_4_*_P_o_l_y_n_o_m_i_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_R_a_t_i_o_n_a_l_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _i_n_v_o_l_v_i_n_g_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_s_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _o_f_f_i_c_i_a_l_l_y_ _d_e_f_i_n_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_n_d_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_._*_4_*_A_b_s_o_l_u_t_e_ _V_a_l_u_e_ _I_n_e_q_u_a_l_i_t_i_e_s_ _-_ _W_e_ _w_i_l_l_ _s_o_l_v_e_ _i_n_e_q_u_a_l_i_t_i_e_s_ _t_h_a_t_ _i_n_v_o_l_v_e_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_G_R_A_P_H_I_N_G_ _A_N_D_ _F_U_N_C_T_I_O_N_S_*_2_*_G_r_a_p_h_i_n_g_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _C_a_r_t_e_s_i_a_n_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _a_n_d_ _m_o_s_t_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _r_e_v_i_e_w_ _t_h_e_ _m_a_i_n_ _i_d_e_a_s_ _f_r_o_m_ _t_h_e_ _s_t_u_d_y_ _o_f_ _l_i_n_e_s_ _i_n_c_l_u_d_i_n_g_ _s_l_o_p_e_ _a_n_d_ _t_h_e_ _s_p_e_c_i_a_l_ _f_o_r_m_s_ _o_f_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _l_i_n_e_._*_4_*_C_i_r_c_l_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _c_i_r_c_l_e_ _a_n_d_ _g_r_a_p_h_i_n_g_ _c_i_r_c_l_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_T_h_e_ _D_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _F_u_n_c_t_i_o_n_ _-_ _W_e_ _w_i_l_l_ _d_i_s_c_u_s_s_ _t_h_e_ _d_e_f_i_n_i_t_i_o_n_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _f_u_n_c_t_i_o_n_ _e_v_a_l_u_a_t_i_o_n_._*_4_*_*_4_*_G_r_a_p_h_i_n_g_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _g_r_a_p_h_i_n_g_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _a_l_s_o_ _g_r_a_p_h_ _s_o_m_e_ _p_i_e_c_e_w_i_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_C_o_m_b_i_n_i_n_g_ _f_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _b_a_s_i_c_ _a_r_i_t_h_m_e_t_i_c_ _i_n_v_o_l_v_i_n_g_ _f_u_n_c_t_i_o_n_s_ _a_s_ _w_e_l_l_ _a_s_ _f_u_n_c_t_i_o_n_ _c_o_m_p_o_s_i_t_i_o_n_._*_4_*_I_n_v_e_r_s_e_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _a_n_d_ _f_i_n_d_ _i_n_v_e_r_s_e_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_1_*_C_O_M_M_O_N_ _G_R_A_P_H_S_*_2_*_L_i_n_e_s_,_ _C_i_r_c_l_e_s_ _a_n_d_ _P_i_e_c_e_w_i_s_e_ _F_u_n_c_t_i_o_n_s_._*_4_*_P_a_r_a_b_o_l_a_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _p_a_r_a_b_o_l_a_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_._*_4_*_*_4_*_S_y_m_m_e_t_r_y_ _-_ _W_e_ _w_i_l_l_ _b_r_i_e_f_l_y_ _d_i_s_c_u_s_s_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_y_m_m_e_t_r_y_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_R_a_t_i_o_n_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _s_o_m_e_ _r_a_t_i_o_n_a_l_ _f_u_n_c_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _v_e_r_t_i_c_a_l_ _a_n_d_ _h_o_r_i_z_o_n_t_a_l_ _a_s_y_m_p_t_o_t_e_s_._*_1_*_P_O_L_Y_N_O_M_I_A_L_ _F_U_N_C_T_I_O_N_S_*_2_*_D_i_v_i_d_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _r_e_v_i_e_w_ _s_o_m_e_ _o_f_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _d_i_v_i_d_i_n_g_ _p_o_l_y_n_o_m_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_Z_e_r_o_e_s_/_R_o_o_t_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _d_e_f_i_n_e_ _j_u_s_t_ _w_h_a_t_ _z_e_r_o_e_s_/_r_o_o_t_s_ _o_f_ _p_o_l_y_n_o_m_i_a_l_s_ _a_r_e_ _a_n_d_ _g_i_v_e_ _s_o_m_e_ _o_f_ _t_h_e_ _m_o_r_e_ _i_m_p_o_r_t_a_n_t_ _f_a_c_t_s_ _c_o_n_c_e_r_n_i_n_g_ _t_h_e_m_._*_4_*_G_r_a_p_h_i_n_g_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _g_i_v_e_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _g_e_t_ _a_ _r_o_u_g_h_ _s_k_e_t_c_h_ _o_f_ _s_o_m_e_ _p_o_l_y_n_o_m_i_a_l_s_._*_4_*_F_i_n_d_i_n_g_ _Z_e_r_o_e_s_ _o_f_ _P_o_l_y_n_o_m_i_a_l_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _l_o_o_k_ _a_t_ _a_ _p_r_o_c_e_s_s_ _t_h_a_t_ _w_i_l_l_ _a_l_l_o_w_ _u_s_ _t_o_ _f_i_n_d_ _s_o_m_e_ _o_f_ _t_h_e_ _z_e_r_o_e_s_ _o_f_ _a_ _p_o_l_y_n_o_m_i_a_l_ _a_n_d_ _i_n_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _a_l_l_ _o_f_ _t_h_e_ _z_e_r_o_e_s_._*_4_*_P_a_r_t_i_a_l_ _F_r_a_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _p_r_o_c_e_s_s_ _o_f_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_s_ _a_n_d_ _f_i_n_d_i_n_g_ _t_h_e_ _p_a_r_t_i_a_l_ _f_r_a_c_t_i_o_n_ _d_e_c_o_m_p_o_s_i_t_i_o_n_ _o_f_ _a_ _r_a_t_i_o_n_a_l_ _e_x_p_r_e_s_s_i_o_n_._*_1_*_E_X_P_O_N_E_N_T_I_A_L_ _A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),