I sine noter bruger hun en forkortelse, men i opgaven skriver hun hele ordet.

Because sine is the reflexive possessive form. It refers back to the subject of the clause, which is hun.

So I sine noter means in her own notes.

If you used hendes instead, it would usually mean the notes belong to some other woman, not the hun in the sentence.

Because noter is plural.

The reflexive possessive changes like this:

• sin for singular common gender nouns
• sit for singular neuter nouns
• sine for plural nouns

Examples:

• sin bog = her own book
• sit hus = her own house
• sine noter = her own notes

This is because of the Danish V2 rule: in a main clause, the finite verb normally comes in the second position.

Here, the sentence starts with I sine noter and later i opgaven. Since those elements come first, the verb must come second:

• I sine noter bruger hun ...
• i opgaven skriver hun ...

If the sentence started with hun, then you would get:

• Hun bruger en forkortelse i sine noter.

Here i means in.

• i sine noter = in her notes
• i opgaven = in the assignment / in the exercise / in the paper

In English, the most natural translation can vary a bit depending on context, but i is the normal Danish preposition here.

Because possessives normally replace the article in Danish.

So you say:

• sine noter
• not de sine noter

Just like in English, you say her notes, not the her notes.

Because forkortelse is a common gender noun, so its indefinite article is en.

• en forkortelse = an abbreviation

The sentence uses the indefinite form because it is mentioning an abbreviation as a type or instance, not naming it as a specific previously identified one.

Forkortelse means abbreviation.

It comes from the idea of making something shorter. In this sentence, it means she uses a shortened form in her notes instead of writing the full word.

Because opgaven is the definite form: the assignment or the exercise.

• en opgave = an assignment / an exercise
• opgaven = the assignment / the exercise

The sentence is talking about a specific one, so the definite form is used.

Opgave is a broad Danish word. Depending on context, it can mean:

• task
• assignment
• exercise
• paper

In this sentence, it most likely means something like the assignment or the written exercise.

It means the whole word or the full word.

• ord is a neuter noun: et ord
• the definite form is ordet = the word

Hele means whole or entire. So hele ordet means she writes out the complete word instead of the abbreviation.

Because the meaning is different.

• hele ordet = the whole word / the word in full
• et helt ord = a whole word / a complete word

In this sentence, we are talking about a specific word that was abbreviated earlier, so Danish uses the definite form ordet.

Also, hele is the form used here before a definite noun.

Yes. Both are present tense forms:

• bruger = uses / is using
• skriver = writes / is writing

Danish does not use a continuous form the way English does. So the present tense can cover both simple present and present continuous, depending on context.

Here it most naturally sounds like a general habit or regular practice.

Because men joins two main clauses:

• I sine noter bruger hun en forkortelse
• men i opgaven skriver hun hele ordet

In standard Danish, it is normal to put a comma before men when it links two full clauses like this.