Die Lehrerin sagt, wir brauchen die ganze Aufgabe nicht abzuschreiben.

Because Die Lehrerin sagt introduces another clause: wir brauchen die ganze Aufgabe nicht abzuschreiben.

In German, a comma is normally used to separate clauses, including reported statements like this one. So the comma shows that the teacher says is followed by what she says.

Because this is a reported statement without dass.

German often has two options after verbs like sagen, denken, glauben:

• Die Lehrerin sagt, wir brauchen die ganze Aufgabe nicht abzuschreiben.
• Die Lehrerin sagt, dass wir die ganze Aufgabe nicht abzuschreiben brauchen.

If dass is omitted, the clause keeps normal main-clause word order, so the conjugated verb stays in second position: wir brauchen.

If dass is included, the clause becomes a regular subordinate clause, and the conjugated verb moves to the end.

This is the pattern brauchen + zu-infinitive, which means to need to do something.

With nicht, it means not need to do something or don’t have to do something:

• Wir brauchen das nicht zu machen. = We don’t need to do that.
• Wir brauchen die ganze Aufgabe nicht abzuschreiben. = We don’t need to copy out the whole exercise.

A useful point: in standard German, this brauchen construction is most common with a negative or limiting word such as nicht, nur, or kaum.

Because abschreiben is a separable verb:

• prefix: ab-
• base verb: schreiben

When a separable verb takes zu, the zu goes between the prefix and the verb stem:

• abschreiben → abzuschreiben
• anfangen → anzufangen
• aufstehen → aufzustehen

So abzuschreiben is the correct zu-infinitive form of abschreiben.

Here nicht negates the idea of having to copy it out.

In German, nicht is often placed before the part of the sentence it most directly negates. With brauchen + zu-infinitive, it commonly appears before the infinitive phrase:

• Wir brauchen das nicht zu machen.
• Wir brauchen die ganze Aufgabe nicht abzuschreiben.

So the sentence structure is basically:

• wir = subject
• brauchen = finite verb
• die ganze Aufgabe = object
• nicht abzuschreiben = negated infinitive phrase

Because Aufgabe is a feminine noun, and here it is the direct object of abschreiben.

So:

• noun: die Aufgabe
• case here: accusative
• gender: feminine
• number: singular

For feminine singular, the definite article is die in both nominative and accusative, so the form stays die Aufgabe.

The adjective ganze also matches that noun phrase. After the definite article die, the adjective takes the weak ending -e:

• die ganze Aufgabe

So the full phrase means the whole exercise/task.

abschreiben usually means to copy something by writing it down from another source.

That is different from plain schreiben, which just means to write.

So in this sentence, die ganze Aufgabe nicht abzuschreiben means not needing to copy out the whole exercise/task, not simply not needing to write anything at all.

Yes. Lehrerin means female teacher.

German often forms feminine job titles with -in:

• der Lehrer = male teacher
• die Lehrerin = female teacher

So this sentence specifically says that the teacher is female.

Yes. A very similar sentence would be:

• Die Lehrerin sagt, wir müssen die ganze Aufgabe nicht abschreiben.

In everyday German, that often also means we don’t have to copy the whole exercise.

But brauchen ... nicht zu ... is especially clear for lack of necessity:

• nicht brauchen zu = do not need to
• nicht müssen = often do not have to, depending on context

So the original sentence is a very natural way to express that something is unnecessary.