Po lekcji zapisuję w zeszycie trzynaście nowych słów.

Because Polish often omits subject pronouns when they are clear from the verb form.

Here, zapisuję means I write down / I am writing down, and the ending -ę already tells you the subject is I.

So:

• zapisuję = I write down
• ja zapisuję = I write down with extra emphasis, contrast, or clarification

A Polish speaker would usually add ja only if they want to stress I, for example:

• Ja zapisuję, a on nie. = I write it down, but he doesn’t.

Because po here means after, and after po Polish normally uses the locative case.

So:

• lekcja = lesson
• po lekcji = after the lesson

This is a very common pattern:

• po pracy = after work
• po obiedzie = after lunch
• po spotkaniu = after the meeting

So lekcji is the locative singular form of lekcja.

Because w here means in, and when it describes location, it takes the locative case.

So:

• zeszyt = notebook / exercise book
• w zeszycie = in the notebook

The sentence is talking about where the words are being written down, not movement toward something.

Compare:

• w zeszycie = in the notebook
• na stole = on the table
• w pokoju = in the room

So zeszycie is the locative singular of zeszyt.

Both are in the locative singular.

Why?

• po
  • locative → po lekcji
• w
  • locative → w zeszycie

Base forms:

• lekcja → lekcji
• zeszyt → zeszycie

The locative is used after certain prepositions, especially ones like w, na, o, po.

Zapisuję comes from the verb zapisywać, which means to write down, to note down, or to record in writing.

So in this sentence, it means something like:

• I write down
• I note down

It is more specific than just piszę (I write).
Zapisuję suggests that you are putting something down in writing so you remember it or keep a record of it.

For example:

• Piszę list. = I am writing a letter.
• Zapisuję słowa. = I am writing down words.

It can mean either, depending on context.

Because zapisuję is an imperfective present form, it can express:

1. something happening now

   • I am writing down...

2. a repeated or habitual action

   • I write down...
   • I usually write down...

In this sentence, Po lekcji zapisuję... often sounds like a routine/habit:

• After the lesson, I write down thirteen new words.

But in the right context, it could also describe what is happening now after a particular lesson.

This is a difference of aspect.

• zapisuję = imperfective
• zapiszę = perfective future

So:

• zapisuję focuses on the action as ongoing, repeated, or habitual
• zapiszę focuses on completing it once

Examples:

• Po lekcji zapisuję nowe słowa.
  After the lesson, I write down new words.
  This sounds like a routine or an ongoing activity.

• Po lekcji zapiszę nowe słowa.
  After the lesson, I will write down the new words.
  This refers to one completed future action.

Because in Polish, numbers 5 and higher usually require the noun to be in the genitive plural.

So:

• słowo = word
• słowa = words
• słów = of words / words after 5+

After trzynaście (thirteen), you need:

• trzynaście słów

And if there is an adjective, it also goes into the genitive plural:

• nowych słów = new words after a number like 13

So:

• trzynaście nowych słów = thirteen new words

Compare:

• dwa nowe słowa = two new words
• trzy nowe słowa = three new words
• cztery nowe słowa = four new words
• pięć nowych słów = five new words
• trzynaście nowych słów = thirteen new words

Because nowych has to match słów, and słów is genitive plural after trzynaście.

So the pattern is:

• nowe słowa = new words
• trzynaście nowych słów = thirteen new words

Both the adjective and the noun change:

• nowe → nowych
• słowa → słów

This is a very common pattern after numerals from 5 upward.

It is the genitive plural of słowo.

The main forms are:

• słowo = word
• słowa = words
• słów = of words / words after 5+

This form may look surprising at first, but it is a normal Polish pattern. Many nouns have a special genitive plural form that you simply need to learn.

So in this sentence, słów appears because trzynaście requires the genitive plural.

Yes. Polish word order is more flexible than English because grammatical roles are often shown by case endings, not just by position.

The original sentence is natural and neutral:

• Po lekcji zapisuję w zeszycie trzynaście nowych słów.

But other orders are also possible, with different emphasis:

• Po lekcji zapisuję trzynaście nowych słów w zeszycie.
• Trzynaście nowych słów zapisuję w zeszycie po lekcji.

These versions are still understandable, but the emphasis shifts.

A rough guide:

• putting something earlier often gives it more focus
• the original version sounds quite natural as a neutral statement

Usually zeszyt means a school notebook or exercise book—the kind with pages for class notes, homework, vocabulary, and so on.

So in this sentence, w zeszycie sounds very natural.

A few related words:

• zeszyt = notebook / exercise book
• notatnik = notepad / notebook for notes
• notes = small notebook, memo pad

In a school-learning context, zeszyt is exactly the word you would expect.