Nach der Prüfung sammelt die Lehrerin die Arbeitsblätter ein.

Because nach takes the dative case when it means after.

• die Prüfung = the exam (dictionary form, nominative)
• nach der Prüfung = after the exam (dative)

Since Prüfung is a feminine noun, its article changes like this:

• nominative: die Prüfung
• dative: der Prüfung

So nach der Prüfung is the correct form.

German often puts a time expression at the beginning of a sentence to set the scene.

So:

• Nach der Prüfung sammelt die Lehrerin die Arbeitsblätter ein.

means literally:

• After the exam, the teacher collects the worksheets.

This is very normal in German. Starting with the time phrase gives it a little emphasis, but it does not change the basic meaning.

Because German main clauses follow the verb-second rule.

That means the finite verb must be in the second position in the clause.

Here, the first position is already taken by Nach der Prüfung, so the verb must come next:

1. Nach der Prüfung
2. sammelt
3. die Lehrerin
4. die Arbeitsblätter
5. ein

If you started with the subject instead, that would also be correct:

• Die Lehrerin sammelt nach der Prüfung die Arbeitsblätter ein.

Both are grammatical.

The basic verb is einsammeln, which means to collect in, to gather up, or in this context to collect.

It is a separable verb:

• full infinitive: einsammeln
• in a main clause: sammelt ... ein

So:

• Die Lehrerin sammelt die Arbeitsblätter ein.

This is why ein appears at the end of the sentence.

Because einsammeln is a separable verb.

In a normal main clause:

• the conjugated part goes near the beginning: sammelt
• the prefix goes to the end: ein

So:

• einsammeln → sammelt ... ein

This is a very common pattern in German:

• anrufen → Ich rufe dich an.
• aufstehen → Er steht früh auf.
• einsammeln → Sie sammelt die Arbeitsblätter ein.

In German, the subject does not always have to come before the verb.

Here, the subject is die Lehrerin because:

• the verb sammelt is singular
• die Lehrerin is singular and fits as the doer of the action
• die Arbeitsblätter is the thing being collected, so it is the object

German word order is more flexible than English word order. The subject often comes after the verb when something else is placed first.

Here, die Arbeitsblätter is the accusative plural direct object.

The teacher is collecting the worksheets, so those worksheets receive the action.

The tricky part is that for many plural nouns, the article die is the same in both nominative and accusative:

• nominative plural: die Arbeitsblätter
• accusative plural: die Arbeitsblätter

So the form looks the same, but its role in the sentence is different. Here it is the object, so it is accusative.

Arbeitsblätter is the plural of Arbeitsblatt.

• das Arbeitsblatt = worksheet
• die Arbeitsblätter = worksheets

It is a neuter noun in the singular:

• nominative singular: das Arbeitsblatt
• plural: die Arbeitsblätter

So in this sentence, the teacher is collecting more than one worksheet.

Yes, the German present tense often covers both simple present and near-future meaning, depending on context.

So:

• Nach der Prüfung sammelt die Lehrerin die Arbeitsblätter ein.

can mean:

• After the exam, the teacher collects the worksheets.
• After the exam, the teacher will collect the worksheets.

Usually the context tells you which one is meant.

Yes, absolutely.

That version is also correct:

• Die Lehrerin sammelt nach der Prüfung die Arbeitsblätter ein.

The difference is mainly focus:

• Nach der Prüfung ... puts the time first and emphasizes when
• Die Lehrerin ... starts with the subject and sounds more neutral in some contexts

German allows this kind of variation much more freely than English.

Lehrerin is a feminine noun and means female teacher.

So:

• die Lehrerin = the female teacher

The masculine version is:

• der Lehrer = the male teacher

The ending -in is a very common way to form a feminine noun referring to a person:

• Lehrer → Lehrerin
• Schüler → Schülerin

When nach is used as a preposition, yes, it normally takes the dative.

Examples:

• nach der Prüfung = after the exam
• nach dem Unterricht = after class
• nach der Arbeit = after work

But remember that nach can also have other meanings, such as to with cities/countries without articles:

• nach Berlin
• nach Deutschland

In all these uses, nach still goes with the dative.

German can express this idea in different ways, but nach + dative noun is the normal way to say after + noun.

So:

• nach der Prüfung = after the exam

If you wanted to say afterwards, you would use a different word, such as danach, but that would change the structure:

• Danach sammelt die Lehrerin die Arbeitsblätter ein.
  = Afterwards, the teacher collects the worksheets.

So nach der Prüfung is used because the sentence specifically says after the exam, not just afterwards.

You can divide the sentence like this:

• Nach der Prüfung = time phrase
• sammelt = conjugated verb
• die Lehrerin = subject
• die Arbeitsblätter = direct object
• ein = separable prefix belonging to einsammeln

So the full structure is:

Time expression + finite verb + subject + object + separable prefix

That is a very common German sentence pattern.