Kennarinn sýnir okkur hvernig á að nota reglustiku í stærðfræði.

The -inn is the suffixed definite article, so kennari means teacher, while kennarinn means the teacher.

Icelandic usually attaches the to the end of the noun instead of putting a separate word in front of it.

• kennari = a teacher / teacher
• kennarinn = the teacher

Sýnir is the present tense, 3rd person singular form of the verb sýna (to show).

So:

• ég sýni = I show
• þú sýnir = you show
• hann/hún/það sýnir = he/she/it shows

Here, the subject is kennarinn, which is singular, so sýnir is the correct form.

Because okkur is the correct case form for us here.

The pronoun við means we and is used as a subject.
But in this sentence, us is not the subject; it is the person being shown something.

For the verb sýna, Icelandic commonly uses:

• the thing shown
• plus the person shown in the dative

So:

• við = we
• okkur = us

That is why Kennarinn sýnir okkur ... is correct.

The verb sýnir has:

• okkur as the indirect object: to us
• the clause hvernig á að nota reglustiku í stærðfræði as the thing being shown

So the teacher is not just showing a physical object. The teacher is showing how to use a ruler in math.

This is a very common Icelandic way to say how to do something.

• hvernig = how
• á að = is supposed to / should / one does
• nota = use

Together, hvernig á að nota means something like:

• how to use
• more literally, how one is to use

This structure is extremely common in Icelandic:

• hvernig á að elda fisk = how to cook fish
• hvenær á að fara = when to go / when one should go
• hvað á að gera = what to do

The að belongs to the expression á að + infinitive.

So the structure is:

• á að nota = should use / is supposed to use

It is not that hvernig itself requires að.
The key unit is á að nota.

So you should think of this as:

• hvernig
  • á að nota

not as:

• hvernig að nota

Because reglustika is the object of nota (to use), and that object is in the accusative singular.

The dictionary form is:

• reglustika = ruler

But after nota, you need the accusative:

• nota reglustiku = use a ruler

So this is a case change, not a different word.

Icelandic does not have an indefinite article like English a/an.

So:

• reglustika can mean a ruler or just ruler, depending on context
• reglustikan would mean the ruler

In this sentence, reglustiku is indefinite, so English naturally translates it as a ruler.

Here í means in, and it is used in a non-motion sense: in math / in mathematics.

With í, Icelandic often uses:

• accusative for motion into something
• dative for location or being within something

In this sentence, the meaning is not movement into math, but context/field: in mathematics, so this is the dative use.

The noun stærðfræði happens to look the same here, so you do not see an obvious ending change.

No, here it means in math / in mathematics as a subject area or context.

So the sentence means the teacher shows us how to use a ruler in mathematics, not literally inside math or necessarily in the math classroom.

This kind of í + subject is very common:

• í ensku = in English
• í sögu = in history
• í stærðfræði = in math

There is no explicit subject inside hvernig á að nota ..., and that is normal.

This construction usually has a general meaning, like:

• how one uses
• how you use
• how to use

So it is not specifically saying:

• how the teacher uses it
• or how we use it

Even though in context the teacher is probably explaining it to us, the Icelandic construction itself is more general.

Yes. That would also be a natural way to express the idea.

Compare:

• hvernig á að nota reglustiku = how to use a ruler / how one should use a ruler
• hvernig maður notar reglustiku = how one uses a ruler / how people use a ruler

The version with á að is very common in explanations and instructions, so it fits this sentence well.

It is the normal word order for a main clause:

• Kennarinn = subject
• sýnir = verb
• okkur = indirect object
• then the clause being shown

So the structure is basically:

The teacher + shows + us + how to use a ruler in math

This is a straightforward declarative sentence. Icelandic does allow other word orders for emphasis, but this version is the neutral one.

Because the meaning here is not shows us that X is true, but shows us how to do something.

So Icelandic uses a question-word clause with hvernig (how) rather than a that-clause.

Compare:

• Kennarinn sýnir okkur hvernig á að nota reglustiku.
  = The teacher shows us how to use a ruler.

• Kennarinn segir okkur að þetta sé rétt.
  = The teacher tells us that this is correct.

So hvernig is exactly what you want for how in this sentence.