J Lo and A-Rod's Kids Are Major Factor in Trying to Mend Relationship

Jennifer Lopez and Alex Rodriguez reaching a breaking point came as a total shock to their kids -- and that's one major reason the couple's together right now in the Dominican Republic trying to patch things up.

Sources close to the couple tell TMZ ... neither of them expected news to break on Friday about their split, and as a result, they didn't have time to prepare their 4 children ... who've all grown very close.

We're told J Lo's 13-year-old twins, Emme and Max, and A-Rod's teenage daughters, 16-year-old Natasha and 12-year-old Ella, were all devastated by the discussions the modern family had on Friday of a possible break up. As one source put it, "There were many tears."

As we first told you ... people close to the couple acknowledged things were "bad" between J Lo and A-Rod that day -- but on Saturday morning they told us "We are working through some things," and there was no third party involved in their issues.

We're told the kids weighed heavily in the long conversation Alex and Jen had Friday night.

The fact is, both families have blended as one, enjoying holidays and vacations together and all the other fun stuff kids do, so there's a lot at stake.

TMZ broke the story ... Alex fueled up the jet and flew to the D.R. to be with Jennifer, who's been down there shooting a movie. They'd been apart for about 2 weeks, but on Monday morning Alex posted video from their oceanfront pad, tagged J Lo and said he was moving "onward" and "upward" in the new week.