The secret steps in the 3-times table are very simple

3*1 = 3 ---------------- 3
3*2 = 6 ---------------- 6
3*3 = 9 ---------------- 9

3*4 = 12 --------- 1+2 = 3
3*5 = 15 --------- 1+5 = 6 
3*6 = 18 --------- 1+8 = 9

3*7 = 21 --------- 2+1 = 3
3*8 = 24 --------- 2+4 = 6 
3*9 = 27 --------- 2+7 = 9 

3*10 = 30 -------- 3+0 = 3
3*11 = 33 -------- 3+3 = 6 
3*12 = 36 -------- 3+6 = 9