To Show That a Certain Module is Not Flat 
Let $k$ be a field and $R=k[x]$. Let $M=\dfrac{k[x, y]}{(x)\cap(x, y)^2}$.
  How to show that $M$ is not a flat $R$-module?

I am sorry but I am really blank on this one.

If $M$ is flat, then it must be torsion-free (see here), but this is not the case: $xy=0$ and $y\ne 0$ in $M$.  
