This is right out of Kassel's Quantum Groups book which I am self-studying. It is on page 14.
The general set-up is this. Let $A$ be a filtered algebra with filtration $F_0(A) \subset F_1(A) \subset \cdots \subset A$, and suppose that $I$ is a two-sided ideal of $A$. The quotient algebra is then filtered with filtration $F_i(A/I) = F_i(A)/F_i(A) \cap I$. We know that for a filtered algebra, $A$, there exists an associated graded algebra, called $\mathrm{gr}(A) = \oplus S_i$ where $S_i = F_i(A)/F_{i-1}(A)$.
Define $M(2)$ as the polynomial algebra $k[a,b,c,d]$ and define $SL(2) = M(2)/(ad - bc -1)$.
His first claim is that $\mathrm{gr}(A/I) = \oplus_{i \in \mathbb{N}} F_i(A)/(F_{i-1}(A) + F_i(A) \cap I)$. Following this, he claims that $\mathrm{gr}(SL(2)) \cong k[a,b,c,d]/(ad-bc)$ (note that the ideal $(ad-bc-1)$ is not generated by homogeneous elements so that $SL(2)$ is not graded).
I am unsure how one gets to these results, and any feedback on this would be greatly appreciated!