The model structure for chain complexes

Let $\text{Ch}$ be the category of (possibly unbounded) chain complexes of abelian groups. In this note we construct the standard Quillen model structure on $\text{Ch}$, by a method that is somewhat different from the standard one. Essentially, we use a functorial two-stage projective resolution for abelian groups, and build everything directly from that. This has the advantage of being very concrete, explicit and functorial. It does not rely on the small object argument, or make any explicit use of transfinite induction. On the other hand, it is not so conceptual, and it does use the fact that subgroups of free abelian groups are free, so it does not generalise to many rings other than $\mathbb{Z}$. We do not claim any great technical benefit for this approach, but it seems like an interesting alternative, and may be pedagogically useful.