W moim piórniku są cztery kredki.

Because w meaning in usually requires the locative case when it refers to location.

So:

• basic form: piórnik = pencil case
• after w: w piórniku = in the pencil case

This is a very common pattern in Polish:

• w domu = in the house
• w sklepie = in the shop
• w piórniku = in the pencil case

So the ending changes because the noun is in the locative singular.

For the same reason: the adjective or possessive word has to match the noun’s case, number, and gender.

The dictionary form is:

• mój piórnik = my pencil case

But after w, the noun goes into the locative, and mój changes too:

• w moim piórniku = in my pencil case

So both words change together:

• mój → moim
• piórnik → piórniku

This kind of agreement is very important in Polish.

It is the locative singular.

More specifically:

• w
  • location usually takes the locative
• piórnik is singular
• moim agrees with piórniku

So the whole phrase w moim piórniku is a locative phrase meaning in my pencil case.

Because the thing that exists / is located is plural: cztery kredki = four crayons.

In Polish:

• jest = is for singular
• są = are for plural

Compare:

• W moim piórniku jest kredka. = There is a crayon in my pencil case.
• W moim piórniku są cztery kredki. = There are four crayons in my pencil case.

So są agrees with the plural noun phrase cztery kredki.

Polish usually just uses jest or są.

So English There are four crayons in my pencil case is expressed literally more like:

• In my pencil case are four crayons.

Polish does not need a special word like English there in this kind of sentence.

So:

• jest can mean is or there is
• są can mean are or there are

The meaning comes from context and word order.

Because after the numerals 2, 3, 4 (and numbers ending in those, with some exceptions), Polish normally uses the nominative plural of the noun.

So:

• jedna kredka = one crayon
• dwie kredki = two crayons
• trzy kredki = three crayons
• cztery kredki = four crayons

That is why you see kredki, not kredka.

Polish numerals can change depending on gender and case.

Here, kredka is a feminine noun, and the phrase is in the nominative plural, so the correct form is:

• cztery kredki

For comparison:

• czterej chłopcy is a masculine-personal type form
• cztery książki
• cztery kredki
• cztery okna

At a beginner level, the main thing to remember is that with a noun like kredka, four is cztery.

The basic dictionary form is kredka, meaning crayon or sometimes colored pencil, depending on context.

In this sentence:

• kredka = singular
• kredki = plural form used after cztery

So:

• To jest kredka. = This is a crayon.
• To są kredki. = These are crayons.

No, Polish word order is fairly flexible, although some orders sound more natural in certain contexts.

This sentence is a very natural way to say it because it starts with the location:

• W moim piórniku są cztery kredki.

You could also say:

• Cztery kredki są w moim piórniku.

But that version puts more focus on four crayons.

Very often, Polish puts known/background information first and new information later. Here, in my pencil case is the setting, and four crayons is the new information.

Yes. In normal writing, using the digit 4 is fine in many contexts.

So both are acceptable:

• W moim piórniku są cztery kredki.
• W moim piórniku są 4 kredki.

In more careful or formal language-learning contexts, the full written form cztery is often preferred.

In modern everyday Polish, piórnik usually means pencil case.

Historically, it is related to pióro meaning feather or pen, but for learners the practical meaning is simply:

• piórnik = pencil case

So w moim piórniku should be understood as in my pencil case.

Moim is pronounced roughly like MO-yim, but smoothly, not as two fully separate words.

A few useful hints:

• m is like English m
• o is a short o
• i softens the previous sound a bit and combines into the ending
• the stress is normally on the second-to-last syllable: MO-im

You do not need to pronounce it exactly like English spelling suggests; listening to native audio is the best way to get comfortable with it.

Usually no, not in this sentence by itself.

Swoim means one’s own, and it normally refers back to the subject of the sentence. But this sentence does not have a clear subject like I, you, or she owning the pencil case.

So:

• W moim piórniku są cztery kredki. = There are four crayons in my pencil case.

If the sentence were connected to a subject, then swój might work:

• Mam cztery kredki w swoim piórniku. = I have four crayons in my pencil case.

Here swoim clearly refers back to I.

Not always.

With w, if it means in as a location, it usually takes the locative:

• w domu
• w szkole
• w piórniku

But with motion into something, Polish often uses w with the accusative:

• Wkładam kredki w piórnik. = I am putting the crayons into the pencil case.

So the case depends on meaning:

• location → usually locative
• movement into → often accusative

In your sentence, it is clearly location, so locative is used.