Min bror blev flov, da læreren opdagede, at hans historie om lektierne var en løgn.

Because bror is an en-word in Danish: en bror. With en-words, you use min.

• min bror = my brother
• mit hus = my house
• mine brødre = my brothers

So the choice depends on the gender and number of the noun.

Lærer means teacher, while læreren means the teacher.

In Danish, the definite article is often attached to the end of the noun:

• en lærer = a teacher
• læreren = the teacher

So læreren shows that this is a specific teacher, not just any teacher.

Blev is the past tense of blive, which often means become or get.

So:

• blev flov = became embarrassed / got embarrassed
• var flov = was embarrassed

Using blev focuses on the change of state: the brother was not embarrassed before, and then he became embarrassed when the teacher found out.

Not quite.

Flov describes how a person feels: embarrassed, awkward, sheepish, a bit ashamed.

Pinlig usually describes a thing or situation: embarrassing, awkward.

So you can think of it like this:

• Han blev flov = He became embarrassed
• Situationen var pinlig = The situation was embarrassing

A pinlig situation can make someone feel flov.

Here, da means when, but specifically for a single event in the past.

So:

• da læreren opdagede ... = when the teacher discovered ...

Danish usually uses når for repeated events, general truths, or often future meaning.

Compare:

• Da jeg kom hjem, ringede han. = When I got home, he called.
• Når jeg kommer hjem, ringer jeg. = When I get home, I call / I will call.

Since this sentence describes one completed event in the past, da is the natural choice.

Because this version of the sentence uses start comma, a common Danish comma style.

Both da and at introduce subordinate clauses, and with start comma you put a comma before them:

• Min bror blev flov, da ...
• ... opdagede, at ...

In modern Danish, many people also write without these two commas, depending on the comma system they use. So you may also see:

Min bror blev flov da læreren opdagede at hans historie om lektierne var en løgn.

Both are acceptable.

At here means that and introduces a content clause.

So:

• læreren opdagede, at ... = the teacher discovered that ...

This clause tells you what the teacher discovered.

English often leaves out that, but Danish very often keeps at.

Here, hans most naturally refers to the brother.

It is not sin because Danish sin/sit/sine normally refer back to the subject of the same clause. In the clause:

at hans historie om lektierne var en løgn

the grammatical subject is the whole noun phrase hans historie om lektierne. So sin would not work to refer back to min bror.

That is why Danish uses hans here.

A useful comparison:

• Min bror glemte sine lektier. = My brother forgot his own homework.

There, min bror is the subject of the same clause, so sine works.

It can be, but here it is closer to story, explanation, account, excuse.

In this sentence, hans historie om lektierne means the story or explanation he gave about the homework. Because the sentence later says it was a lie, historie here has the sense of excuse or made-up story.

Lektierne is the definite plural form: the homework.

So:

• lektier = homework / assignments in a general sense
• lektierne = the homework, meaning the specific homework being talked about

In this sentence, it refers to the particular homework situation that the brother was talking about. That is why the definite form sounds natural.

Because løgn is an en-word:

• en løgn = a lie
• løgnen = the lie

So after var, you say var en løgn = was a lie.

This is just noun gender: some nouns take en, others take et.

Because the whole sentence is told from a past viewpoint.

The verbs are all in the past:

• blev = became
• opdagede = discovered
• var = was

So var fits the time frame of the sentence. It does not necessarily mean the story stopped being a lie later; it just matches the past narration.

It has three parts:

1. Min bror blev flov
   Main clause: My brother became embarrassed

2. da læreren opdagede
   Subordinate time clause: when the teacher discovered

3. at hans historie om lektierne var en løgn
   Subordinate content clause: that his story about the homework was a lie

So the sentence is built like this:

main clause + da-clause + at-clause

This is a very common Danish pattern. The word order looks fairly close to English here, but if you added something like ikke, you would more clearly see Danish subordinate-clause word order:

• da læreren ikke opdagede ...
• at hans historie ikke var en løgn