Translation:I will go to work at a quarter past ten tomorrow morning.
There is a difference between how people phrase things informally or colloquially versus in writing or formal speech.
Bob: What time is it?
Joe: Quarter past ten.
The English phrase used here is a complete sentence and not an informal or short spoken response.
Google can search written material between 1800 and 2008 and compare the prevalence of a phrase.
According to the Google results I see, "it's a quarter past" is still more popular in British English as of 2008.
If you look at the results for all variants of English, the phrase has been used with "a" more often than without it.
Duo Lingo sentences are communication-oriented. Who's to say dropping the 'a' is not 'proper' grammar' if it is dropped by so many speakers? You don't determine 'proper' by frequency (Actually, in any case, you might check out British National Corpus because I believe it is more common to drop the 'a' now)., nor by arbitrary adherence to tradition. English evolves and gets rid of unnecessary function words. In my opinion, there are far too many uses of articles in English when many languages have none. Do you say it's a half past ten?" Technically, you could argue the 'a' should be there, but we don't use it because we often drop articles when they can be assumed.
I see several people have commented on this already. In English the definite article is optional, "a quarter past.." and "quarter past.." are both frequently used, and both should be accepted as a correct answer.
Some of the other comments seem to be a little harsh, perhaps even rude. We might say 10:15am, or 10:15 in the morning, or quarter past ten in the morning, or a quarter past ten in the morning. All are correct, just as multiple ways of saying this in Chinese will be correct (four according to my teacher), but a closer actual translation of this is "tomorrow morning at quarter (or a quarter) past ten", than tomorrow at 10:15am. It certainly not wrong, but it is not as accurate a translation.
Thanks to all the people working very hard to make this program helpful!
Nothing wrong with it; keep reporting. There are so many possible variations that they don't all get put into the database at once. It gets frustrating sometimes, but I often get messages from the DL Chinese team saying my suggestions are now accepted, so be patient (as I often have to remind myself) and know that the course is improving over time.
早上 Zǎoshang is used when referring to 'early' morning (from dawn to 8/9am), then from 9am to 12 you say 早上 Shàngwǔ.
会 can also be used for future tense: https://www.writtenchinese.com/past-present-future-tenses-mandarin-chinese/