Po kartkówce nauczycielka powiedziała, że nowy semestr zaczynamy naprawdę dobrze.

It means after the quiz / after the short test.

• po = after
• kartkówce = quiz in the form required after po

So Po kartkówce sets the time: something happened after that quiz.

Because the preposition po usually requires the locative case when it means after.

The dictionary form is:

• kartkówka = a short quiz

After po, it changes:

• po kartkówce = after the quiz

So this is a normal case change:

• nominative: kartkówka
• locative singular: kartkówce

A kartkówka is a very common school word in Polish. It usually means a short classroom quiz, often:

• based on recent material
• shorter than a full test
• sometimes unannounced

It is more specific than a general English word like test.

Because nauczycielka means female teacher, and the past tense verb agrees with the subject’s gender.

• powiedziała = she said
• powiedział = he said

So:

• nauczycielka powiedziała = the female teacher said

If it were a male teacher, it would be:

• nauczyciel powiedział

Powiedziała comes from powiedzieć, which is usually perfective. It presents the action as a completed act of saying: she said it, and that’s the whole event.

• powiedziała = she said
• mówiła = she was saying / she used to say / she said (with a less bounded feel, depending on context)

In this sentence, the point is a single completed comment from the teacher, so powiedziała is the natural choice.

Że means that and introduces a subordinate clause.

• powiedziała, że... = she said that...

Just like English, Polish uses this word to report what someone said, thought, knew, etc.

Also notice the comma:

• powiedziała, że...

In Polish, a comma before że is normally required.

Because Polish does not follow English-style sequence of tenses in the same way.

So after a past verb like powiedziała, Polish can still use the present tense if the reported statement is true at that time or still current:

• powiedziała, że zaczynamy... = she said that we are starting...

English often shifts tense backward:

• she said that we were starting...

Polish usually keeps the tense that matches the situation more directly.

Because the speaker is talking about a group that includes the students, and possibly the teacher in a broader school/class sense.

• zaczynamy = we are starting

This sounds natural in Polish when talking about a shared school experience:

• zaczynamy nowy semestr = we are starting a new semester

It gives a collective feeling: the class is beginning the semester together.

Here nowy semestr is the object of zaczynamy.

The verb zaczynać can mean to start something, so:

• zaczynamy nowy semestr = we are starting a new semester

It is in the accusative case, but for a masculine inanimate singular noun like semestr, the accusative looks the same as the nominative:

• nominative: nowy semestr
• accusative: nowy semestr

So even though the form looks like the basic dictionary form, its function here is accusative.

Because this word describes how they are starting the semester, so Polish needs an adverb, not an adjective.

• dobry = good (adjective)
• dobrze = well (adverb)

So:

• zaczynamy dobrze = we are starting well

If you used dobry, it would need to describe a noun, not a verb.

It means really well.

• naprawdę = really / truly
• dobrze = well

So naprawdę dobrze strengthens the positive evaluation:

• not just well
• but really well

It sounds natural and conversational.

Because Polish has no articles like English a and the.

So Polish simply says:

• nowy semestr

and context tells you whether English should use:

• a new semester
• the new semester

This is very normal in Polish, and learners have to get used to relying on context instead of articles.

Yes. Polish word order is more flexible than English, because case endings help show grammatical roles.

This version is natural:

• Po kartkówce nauczycielka powiedziała, że nowy semestr zaczynamy naprawdę dobrze.

But you could also hear:

• Nauczycielka powiedziała po kartkówce, że...
• Po kartkówce powiedziała nauczycielka, że...
  (more marked, often for emphasis or style)

The original order is smooth and neutral: first the time phrase, then the subject and verb, then the reported statement.