Learn Hardcore Norwegian: Til tross for høy arbeidsmengde prøver begge foreldrene å beskytte familietiden, slik at barna ikke blir for trøtte. - Despite the high workload both parents try to protect the family time, so that the children do not become too tired.

Til tross for literally means in spite of or despite. It is a prepositional expression and must be followed by a noun (or noun phrase):

Til tross for høy arbeidsmengde = In spite of a high workload

Selv om is a subordinating conjunction, like even though / although, and it must be followed by a clause (with subject + verb):

Selv om foreldrene har høy arbeidsmengde, prøver de å beskytte familietiden.
= Even though the parents have a high workload, they try to protect family time.

So:

til tross for + noun
selv om + clause

Why is it høy arbeidsmengde and not høye arbeidsmengde?

Arbeidsmengde is a feminine noun (also treated as common gender), and it is singular and indefinite here.

Adjective rules (for a regular adjective like høy – high, tall):

Singular indefinite: høy arbeidsmengde
Singular definite: den høye arbeidsmengden
Plural indefinite: høye arbeidsmengder
Plural definite: de høye arbeidsmengdene

Because we are talking about one, non-specific workload (not “the workload” or “workloads”), we use:

høy (singular, indefinite form of the adjective)
arbeidsmengde (singular, indefinite noun)

So høye would be wrong here because that is the plural (or definite) adjective form.

Could you also say stor arbeidsmengde or mye arbeid instead of høy arbeidsmengde? Do they mean the same?

Yes, you could, with slightly different nuances:

høy arbeidsmengde – literally high workload. Very normal and idiomatic, especially in more formal or written language.
stor arbeidsmengde – large workload. Also possible and understandable; sounds natural in many contexts.
mye arbeid – a lot of work. This is simpler and very common in everyday speech.

All three can often be used to express the same idea, but:

arbeidsmengde is a bit more technical/neutral, like “workload” as a measurable quantity.
mye arbeid is more general and colloquial.

In your sentence, høy arbeidsmengde is perfectly natural, slightly on the formal/neutral side.

Why is the verb order prøver begge foreldrene and not begge foreldrene prøver?

This is because of the verb-second (V2) rule in Norwegian main clauses.

The sentence starts with a fronted adverbial phrase:
Til tross for høy arbeidsmengde (Despite a high workload).
In a main clause, the finite verb must come second, no matter what is first.
So the order becomes:
- Fronted element: Til tross for høy arbeidsmengde
- Verb: prøver
- Subject: begge foreldrene

Full structure:

Til tross for høy arbeidsmengde | prøver | begge foreldrene | å beskytte familietiden ...

If the subject were first, with no fronted phrase, then yes, you would say:

Begge foreldrene prøver å beskytte familietiden.

Why is it begge foreldrene and not begge foreldre?

Begge foreldrene literally means both the parents.

begge = both
forelder = parent (singular)
foreldre = parents (plural, indefinite)
foreldrene = the parents (plural, definite)

When begge refers to a specific, known pair (here, the children’s own parents), Norwegian almost always uses the definite plural:

begge foreldrene = both (of) the parents
begge barna = both (of) the children
begge lærerne = both (of) the teachers

Begge foreldre is grammatically possible but sounds more like “both parents” in a general sense, not specifically these parents. In this context, the definite form begge foreldrene is natural and idiomatic.

Why do we say å beskytte familietiden with å and the definite form familietiden?

Two things are happening here:

The infinitive marker å
- prøver å beskytte = try to protect
  In Norwegian, many verbs that are followed by another verb in the infinitive require å:
- prøver å beskytte – try to protect
- liker å lese – like to read
- planlegger å reise – plan to travel
Definite form of the noun: familietiden
- familietid = family time (general concept, indefinite)
- familietiden = the family time (specific, definite)

Here, we are talking about their specific family time (the time they have together as a family), so the definite form is natural:

å beskytte familietiden = to protect the family time (they have).

Saying å beskytte familietid would sound more like protecting “family time” as a general abstract concept, which is not what is meant here.

What does slik at mean here, and how is it different from sånn at, så at, or for at?

In this sentence, slik at introduces a result clause and means so that or in such a way that:

... prøver begge foreldrene å beskytte familietiden, slik at barna ikke blir for trøtte.
= ... the parents try to protect family time, so that the children don’t get too tired.

Comparisons:

slik at – neutral, slightly more formal; often used for result or consequence.
sånn at – more informal/colloquial; often interchangeable with slik at in speech.
så at – also exists, but less common and can sound a bit old-fashioned or stylistically marked.
for at – usually expresses purpose (similar to in order that):
- De beskytter familietiden, for at barna ikke skal bli for trøtte.
  = They protect family time in order that the children will not get too tired.

In your sentence, slik at nicely expresses the consequence/result of protecting family time.

Why is there a comma before slik at?

Norwegian comma rules require a comma before most subordinate clauses introduced by subordinating conjunctions like at, fordi, når, hvis, selv om, slik at, etc., when they follow the main clause.

Here:

Main clause: Til tross for høy arbeidsmengde prøver begge foreldrene å beskytte familietiden
Subordinate clause: slik at barna ikke blir for trøtte

So we write:

..., slik at barna ikke blir for trøtte.

The comma marks the boundary between the main clause and the subordinate clause.

Why is it barna and not barnene?

Barn has an irregular definite plural:

Singular indefinite: et barn – a child
Singular definite: barnet – the child
Plural indefinite: barn – children
Plural definite: barna – the children

There is no form “barnene” in standard Norwegian; barna is the only correct definite plural.

So barna here means the children (specific: their children).

Why is it ikke blir and not ikke er? What is the difference between bli and være in this context?

bli means to become / to get (a change of state).
være means to be (a state).

ikke blir for trøtte = don’t become / don’t get too tired
If you said ikke er for trøtte, it would mean are not too tired (already in that state).

The idea in the sentence is to prevent the children from reaching a state of being too tired, so bli (become) is the natural verb:

slik at barna ikke blir for trøtte
= so that the children don’t get too tired.

Why is it for trøtte and not for trøtt?

Two points:

Adjective agreement
- barna = the children (plural)
- The adjective trøtt (tired) must agree in number (and definiteness) with the noun.
- Plural form of trøtt is trøtte.

So:

barnet er trøtt – the child is tired (singular)
barna er trøtte – the children are tired (plural)

The word for Here for means too (excessively), not for in the sense of “for someone”.

for trøtt – too tired (singular)
for trøtte – too tired (plural)

Since barna is plural, we must say:

barna ... blir for trøtte
= the children get too tired.