~다잖다 vs. ~다니까 - -다잖아 = -다고 하잖아. I'm saying [They're saying] ... (repeat or confirm what someone else or you just said). - -다니까 = -다고 하니까 (그렇게 알아). I'm saying .... (repeats what you just said in response to a question or dubious response) A revised version of your example. - 어디 가? - 응, 학교 (가). (가고 있어 emphasizes "on one's way", which sounds a little unnatural in this context) - 어디? - 학교 간다니까. I said I'm going to school (-다잖아 is not as good as -다니까 when repeating what you just said yourself) A 3-way conversation example of -다잖아. 영숙: 영희야, 우리 같이 공원에 가자. 영희: 안 돼. 나 집에 가야 돼. 영숙: 야, 가자, 영희야. 영철: 집에 가야 된다잖아. ("She's saying she has to go home. Why insist?". Can't use -다니까 when repeating what someone else said)

다니까 is an abbreviation of 다고 하니까 다잖아 is an abbreviation of 다고 하지 않아 다잖아 is used for reminding the previous speaking. 다니까 is used for reminding the previous speaking and also expressiong the annoyance. 바쁘다니까요! = I told you I’m busy! 모른다니까! = I told you I don’t know 먹는다니까! = I told you I’m eating! 바쁘다잖아요! = I told you I’m busy, didn't I? 모른다잖아! = I told you I don’t know! Right? 먹는다잖아! = I told you I’m eating! Right?

가고 있다니까: 가고 있다 + 니까 가고 있다잖아: 가고 있다 + 잖아. These are a kind of quoted speech. Instead of saying "가고 있으니까, 가고 있잖아", you may choose to quote what you have already said. It gives effect of emphasizing the fact that you have already said that and thus expressing nuisance of "being annoyed".