Translation:The students went to school at 10 yesterday.
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What's wrong with "The student started class at ten yesterday." There's nothing to indicate whether there was one student or many. And the difference between "went to school" and "started class" is trivial in English. There's no character in the sentence to indicate motion in the past tense, i.e. "went", just as there is no word to indicate beginning, i.e. "started". My usual translation of shang4 ke4 is "begin class", matching "xia4 ke4", class over. Literal translation would be "up class" and "down class", which of course is not English.
Start class is 上课，课 literally translates to class. There may be little difference in English, but there is a difference in Chinese. Technically, for the translation to be correct, it should be 学生们. Because the speaker said 昨天, we can safely assume the action is taking place in the past. Chinese is a language where the speaker is in the present tense looking forward. There are no past tense verbs in Chinese, so you will have to pay attention to the time and day to determine what tense it is.
Starting and finishing can't really be states; change of state is inherent in their meaning.
But also, your question assumes a lot of things that aren't true.
Talking about when something happened is optional in Chinese. If talking about it is a mandatory part of the verb, that's tense. Chinese does not have tense.
State verbs are not an exception to tense marking, since there is no tense marking. Different verbs can be more or less friendly to the option of expressing time. State verbs are less friendly to it. But they are not the only reason why you wouldn't do it.
Even if you do want to express time, there is no rule that you have to do it with 了. Expressing time isn't even the main purpose of 了, it's a side effect.
了 marks the completedness of an action when that's relevant. A completed action is, naturally, usually in the past, so 了 happens to usually also express past time.
Does this construction mean they started going to school at 10 and arrived some time later, or that they left some time earlier and then arrived at 10? In English the former would be "left for school" while the latter would be "got to school". In English the construction with "went" is potentially ambiguous.
I just got marked wrong for writing "The students started school at 10:00 yesterday." It insisted I said "went to school" and not "started school".
I am feeling very frustrated with this whole section. This is the second time through and second time getting this particular exercise marked wrong.
I'm a native English speaker and I think "Went off to school" has the connotation of going away to a boarding school or a college where you live on-campus, i.e. not just walking or taking the bus to a school in town. So when I hear "He went off to school." I picture someone going away for a few weeks or months and coming back only on breaks, whereas when I hear "Went to school" it's more general, it could mean that but it more likely just means the person went to school (in the same town) on that particular day.