How to connect to common ground when using the TB6612FNG
You should connect the arduino to the black and red binding posts (Gnd, Vin respectively) and leave the earth (green) binding post unconnected.  This will let the Arduino GND float relative to earth ground which is generally bad for ESD protection but good for preventing oscilloscope probing problems for those new to electronics.  

If you connect Gnd to Earth and then try to connect an oscilloscope ground on a probe to anything except Gnd on the Arduino you will create a short through the ground on the instruments and possibly damage the Arduino or oscilloscope depending on the power supply configuration.