pdf of sample variance?

Let $Y_{1},Y_{2},...,Y_{n}~iid~N(\mu,\sigma)$ .

Q: Find the probability density function of $S^{2}$ .

I am not sure how to approach this problem. I tried using the mgf method, but failed to find a recognizable mgf. My teacher recomened that we use the transformation method to derive the pdf, but I not sure how with this function.

Here is what he showed on the board.

Let $S^{2}=\frac{\sigma^{2}}{(n-1)}\frac{(n-1)S^{2}}{\sigma^{2}}$ . Then, treat the preceeding quantity like $U=aY$ and use the tranformation method to find $f_{U}(u)$ .

I really don't know where to go from here. Do I let $U=s^{2}, a=\frac{\sigma^{2}}{(n-1)},$ and $Y=\frac{(n-1)S^{2}}{\sigma^{2}}$ ?