  For simple graphs $G$ and $H$, their size Ramsey number $\hat{r}(G,H)$ is the
smallest possible size of $F$ such that for any red-blue coloring of its edges,
$F$ contains either a red $G$ or a blue $H$. Similarly, we can define the
connected size Ramsey number ${\hat{r}}_c(G,H)$ by adding the prerequisite that
$F$ must be connected. In this paper, we explore the relationships between
these size Ramsey numbers and give some results on their values for certain
classes of graphs. We are mainly interested in the cases where $G$ is either a
$2K_2$ or a $3K_2$, and where $H$ is either a cycle $C_n$ or a union of paths
$nP_m$. Additionally, we improve an upper bound regarding the values of
$\hat{r}(tK_2,P_m)$ and ${\hat{r}}_c(tK_2,P_m)$ for certain $t$ and $m$.
