Kennarinn hvatti hana til að halda áfram, og sú hvatning hjálpaði henni í prófinu.

Because -inn is the suffixed definite article in Icelandic.

• kennari = teacher
• kennarinn = the teacher

So Kennarinn means the teacher. In this sentence it is also the subject, so it appears in the nominative singular.

Hvatti is the 3rd person singular past tense of the verb hvetja, which means to encourage, to urge, or to exhort.

So:

• að hvetja = to encourage
• hann/hún hvatti = he/she encouraged

Because the subject is Kennarinn (singular), the verb is also singular: hvatti.

Because different Icelandic verbs require different cases for their objects.

Here:

• hvetja einhvern takes the accusative
• hjálpa einhverjum takes the dative

So the pronoun hún changes form depending on its grammatical role:

• hún = nominative, she
• hana = accusative, her
• henni = dative, her
• hennar = genitive, her

That is why the sentence has:

• hvatti hana = encouraged her
• hjálpaði henni = helped her

This is a very common Icelandic pattern.

• til að = to, in order to
• halda áfram = continue, keep going

After hvetja einhvern, Icelandic often uses til að + infinitive to say what someone is being encouraged to do:

• hvetja einhvern til að gera eitthvað = encourage someone to do something

So:

• hvatti hana til að halda áfram = encouraged her to keep going

You may also see the fuller form til þess að, but til að is very common.

Yes, it is best to learn halda áfram as a fixed expression.

Literally:

• halda = hold, keep
• áfram = forward, onward

But together halda áfram usually means:

• continue
• keep going
• carry on

So even though the literal pieces are helpful, the whole phrase should be treated as one common idiomatic unit.

Because sú is the feminine singular nominative form of the demonstrative sá / sú / það, meaning that.

The noun hvatning is feminine, so the demonstrative has to match it:

• masculine: sá
• feminine: sú
• neuter: það

So:

• sú hvatning = that encouragement

It refers back to the encouragement mentioned in the first clause. Icelandic often uses a demonstrative like this to point clearly to something just mentioned.

Yes. They come from the same root.

• hvetja = to encourage
• hvatti = encouraged
• hvatning = encouragement

So the sentence is built very neatly:

• first the action: the teacher encouraged her
• then the result: that encouragement helped her

This kind of verb–noun relationship is very common in Icelandic and useful to notice when building vocabulary.

Yes, hjálpa normally takes a dative object.

So:

• hjálpa mér = help me
• hjálpa þér = help you
• hjálpa henni = help her

This is one of those verbs whose case pattern you simply need to learn. For English speakers, this can feel unusual because English no longer marks these object-case differences very much, but in Icelandic they are essential.

Í prófinu means something like in the exam or on the exam, depending on how natural English would express it.

Breakdown:

• próf = exam, test
• prófinu = the exam in the dative singular
• í = in

The preposition í takes:

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

Here there is no motion, so we use the dative: í prófinu.

English often says on the exam, but Icelandic commonly uses í in expressions like this.

Yes: the sentence follows normal Icelandic main-clause word order, which is often described as verb-second.

In both clauses, the subject comes first, so the structure looks very similar to English:

• Kennarinn hvatti hana...
• sú hvatning hjálpaði henni...

That gives:

• subject
• finite verb
• rest of the clause

Icelandic can move other elements to the front, but the finite verb still tends to stay in the second position. So this sentence is a nice, straightforward example of standard word order.