"The children were with my mother."
Translation:Die Kinder waren bei meiner Mutter.
"bei meiner Mutter" doesn't necessarily refer to the mother's place. It just means that the children were together with your mother. If the place they visited is specified like in your museum example, the preposition changes to 'mit': 'Die Kinder waren mit meiner Mutter im Museum'. If they were at the museum, but you don't specify the place, 'Die Kinder waren bei meiner Mutter' is fine to use.
Good to know--thanks! That's something I wouldn't have guessed, as a non-native speaker. The next question is whether "Die Kinder waren mit meiner Mutter." is a possible native utterance, or whether it sounds wrong ("non-native"), if there's no additional "im Park/im Museum/einkaufen"?
Cool! That makes sense, and "rings a bell", based on the German I've heard read.
This is interesting stuff, because languages never map perfectly one to another (in other words, we can't just learn equivalent words phrases, we also need to learn where the "meaning boundaries" are different, or function differently.) What "doesn't work" is often unexpected (E.g. "ich bin heiss" "The rain stops soon", or in this case "mit meiner Mutter" ohne "zusammen"). The amount of exposure/practice necessary to know, say, that "bei meiner Mutter" works is much smaller than the amount it would take (me) to learn that a native speaker wouldn't just say "mit meiner Mutter" in this case. (My judgement is clouded by the fact that the equivalent English phrase is completely natural.) And a German speaker who would say, conversely, in English: "The children were by my mother" or "the children were with my mother together" would be making the opposite (incorrect) inference. If Duolingo wanted to help learners acquire some "deep knowledge", they could include explanatory notes, about "bei meiner Mutter" or "mit meiner Mutter zusammen." being "native" options. (It's also interesting because, of course, "Die Kinder waren mit meiner Mutter" totally "works" in the sense that it successfully conveys meaning. It just fails the "sounds like a German" test! :7 )