$H$ is a normal subgroup if and only if $aH=Ha$. Create the cosets $G/H$. If $a\in H$ then $aH=Ha$ because $G/H$ partitions the set $G$ into two sets. If $a\in G-H$ then $aH\not = H$ and $Ha\not = H$ thus $aH=Ha$ again because $G/H$ partitions the set into two sets.