wǒ zài jiā xiě zhōngwén zuòyè.

Mandarin doesn’t use a separate verb like English “am / is / are” in this kind of sentence.

In 我在家写中文作业:

• 我 = I
• 在家 = at home
• 写中文作业 = write / do Chinese homework

You can think of 在 here as a verb meaning “to be located at” (at/in). So the whole thing is more like “I am at home (and) write Chinese homework.” There’s no extra copula needed.

Here 在 is mainly a location marker:

• 在 + place = at / in + place

So 在家 = at home.

There is another use where 在 (or 正在) comes before a verb to show a progressive action:

• 我在写中文作业。 = I am writing Chinese homework (right now).

In your sentence, 在 is tied to 家 (place), not directly to 写. If you really want to emphasize the ongoing action, you can say:

• 我在家正在写中文作业。 = I am (currently) at home, writing Chinese homework.

All three are possible, but they feel different:

• 在家

  • Default, very natural.
  • Usually implies “at (my) home” if the subject is 我.
  • Most common in everyday speech.

• 在家里

  • Literally “inside the home/house.”
  • Slight extra emphasis on inside the home, but often interchangeable with 在家.

• 在我的家

  • Emphasizes “at my home (not someone else’s)”.
  • Used when you need to contrast locations, e.g.
    
    • 他在他家，我在我的家。 = He’s at his home; I’m at my home.
  • Sounds a bit heavy if there’s no need for that contrast.

For a simple statement like this, 我在家写中文作业 is the most natural.

Mandarin has a fairly stable word-order pattern:

Subject + (Time) + 在 + Place + Verb + Object

So:

• 我 (subject)
• 在家 (place phrase with 在)
• 写 (verb)
• 中文作业 (object)

You generally cannot put 在 + place between the verb and its object in this kind of sentence, so:

• ✗ 我写在家中文作业 — unnatural / wrong
• ✓ 我在家写中文作业 — correct

If you add a time word, it fits like this:

• 我今天在家写中文作业。
  (Subject + Time + 在 + Place + Verb + Object)

Mandarin verbs usually don’t change form for tense. 我在家写中文作业 can mean:

• “I am writing Chinese homework at home (now).”
• “I (usually) write my Chinese homework at home.”

Context decides which reading is intended. To make it clearer:

• For ongoing / right now:

  • 我在家正在写中文作业。
  • 我在家写中文作业呢。

• For habit / routine:

  • 我常常在家写中文作业。
  • 我一般在家写中文作业。

Yes, and they’re both common, but there’s a nuance:

• 做作业 = do homework (general; any type of tasks)
• 写作业 = write homework (emphasizes the writing part)

For 中文作业, which often involves writing characters, both are possible:

• 我在家做中文作业。 = I do my Chinese homework at home.
• 我在家写中文作业。 = I (specifically) write my Chinese homework at home.

If the homework is mainly writing characters, 写 feels very natural. If it also includes reading, listening, exercises, etc., 做中文作业 sounds more general.

Both 中文 and 汉语 relate to Chinese, but:

• 中文

  • Literally “Chinese writing / Chinese language (as a whole)”.
  • Often used for the subject “Chinese” in school, especially including reading and writing.
  • 中文作业 sounds very natural as “Chinese class homework.”

• 汉语

  • More specifically the Chinese language of the Han people, usually as a spoken / language system concept.
  • 汉语作业 is also understandable, but is heard less often in everyday school talk; textbooks and formal contexts might use it more.

In this context, 中文作业 is the most typical and natural way to say “Chinese homework.”

In Mandarin, you use measure words when you’re counting or specifying quantity:

• 一份中文作业 = one piece/set of Chinese homework
• 两项作业 = two assignments/tasks

But when you are just talking about homework in general as an activity, you don’t need a measure word:

• 写中文作业 = write/do Chinese homework (in general)

Saying 写一个中文作业 sounds odd, because 作业 is usually not treated as a single, countable item in that way. Use a more natural measure word if needed:

• 写一份中文作业。
• 完成这份中文作业。

Yes:

• 写作业 = write/do homework (no subject specified)
• 写中文作业 = write/do Chinese homework (subject specified)

So:

• 我在家写作业。 = I do homework at home.
• 我在家写中文作业。 = I do my Chinese homework at home.

If the context already makes it clear that you’re talking about Chinese homework, people will often just say 写作业.

Yes, if the subject is clear from context, Mandarin often drops it:

• (我) 在家写中文作业。

In a conversation where everyone knows you’re talking about your own schedule, 在家写中文作业 would typically be understood as “(I) do my Chinese homework at home.”

• 在家我写中文作业。

  • Possible, but sounds a bit marked/emphatic, like stressing “At home, I write Chinese homework.”
  • Often used in contrasts:
    
    • 在学校我上课，在家我写中文作业。

• 我写中文作业在家。

  • Generally unnatural.
  • With this kind of verb and object, the place phrase with 在 normally goes before the verb, not after the object.

The safest and most natural pattern is:

• 我在家写中文作业。
• 我今天在家写中文作业。
• 放学以后我在家写中文作业。

It’s neutral and everyday:

• Very natural in spoken Chinese.
• Fine in informal writing (messages, diaries, etc.).

In more formal or written contexts (like a report or essay), you might see slightly more formal verbs or structures:

• 我在家完成中文作业。 = I complete my Chinese homework at home.
• 我通常在家完成中文作业。

But for normal conversation, 我在家写中文作业 is perfectly appropriate and idiomatic.