Im Unterricht schreiben wir zuerst einen Hauptsatz und danach einen Nebensatz mit weil.

Im is the normal contraction of in dem.

Here, in is used for a location or situation, so it takes the dative case.
Unterricht is masculine: der Unterricht.
So:

• in dem Unterricht → im Unterricht

This is very common in German. In fact, im Unterricht sounds much more natural than the full in dem Unterricht in most contexts.

Because German main clauses normally put the finite verb in second position.

In this sentence, the first position is already taken by Im Unterricht. That means the verb must come next:

• Im Unterricht | schreiben | wir ...

If the sentence started with the subject instead, you could say:

• Wir schreiben im Unterricht zuerst einen Hauptsatz ...

Both are correct. The version with Im Unterricht simply puts more focus on the setting.

Both phrases are direct objects of schreiben, so they are in the accusative case.

Both nouns are masculine:

• der Hauptsatz
• der Nebensatz

The masculine accusative singular article is einen, so:

• einen Hauptsatz
• einen Nebensatz

That is why you do not see ein Hauptsatz here.

In German, all nouns are capitalized.

So these are capitalized because they are nouns:

• Hauptsatz
• Nebensatz
• Unterricht

But weil is a conjunction, not a noun, so it stays lowercase.

They are adverbs of sequence. They show the order of actions.

• zuerst = first
• danach = afterwards / after that

So the sentence is structured as:

• first, write one thing
• then, write the next thing

They help organize the sentence clearly and are very common in instructions.

Because und is simply joining two parts of the same clause:

• einen Hauptsatz
• danach einen Nebensatz mit weil

These are coordinated elements, so German does not normally use a comma before und here.

A comma would be wrong in this sentence.

Here, mit means something like using or with in a grammatical sense.

So einen Nebensatz mit weil means:

• a subordinate clause with/using weil
• in other words, a subordinate clause introduced by weil

This is a common way to talk about language in German. For example:

• ein Satz mit weil
• eine Frage mit warum

So mit weil does not mean that weil is a noun. It just means that weil is the word used to form that clause.

Normally, mit is followed by a dative noun phrase, such as:

• mit dem Lehrer
• mit einer Freundin

But here, weil is not acting like a normal noun phrase. It is being mentioned as a word used in grammar.

In that kind of language-about-language usage, German often leaves the word bare:

• ein Satz mit weil
• ein Text ohne Punkt
• ein Verb mit Präfix

So there is no article before weil.

A Hauptsatz is an independent clause. It can stand on its own as a complete sentence.

Example:

• Ich bleibe zu Hause.

A Nebensatz depends on another clause and usually cannot stand alone in the same way.

Example:

• weil ich krank bin

A Nebensatz is often introduced by words like:

• weil
• dass
• wenn
• obwohl

So in this sentence, the learners first write a main clause, then a subordinate clause introduced by weil.

In a subordinate clause introduced by weil, the finite verb usually goes to the end.

Example:

• Ich bleibe zu Hause, weil ich krank bin.

Notice the order:

• weil ich krank bin
• not weil ich bin krank

That verb-final pattern is one of the most important things to remember with weil.

Yes. That is also correct.

German allows some flexibility, as long as the finite verb stays in second position in a main clause.

So both are correct:

• Im Unterricht schreiben wir zuerst ...
• Wir schreiben im Unterricht zuerst ...

The difference is mainly emphasis:

• Im Unterricht ... highlights the setting first
• Wir ... highlights the subject first