Learn Hardcore Polish: Dzieci biegają bez butów po piasku, a rodzice opalają się na ręcznikach. - The children run without shoes on the sand, and the parents sunbathe on the towels.

biegać – biegają here (imperfective, frequent/indefinite running, running around)
biec – biegną (imperfective, one concrete movement in some direction, often with a goal)

In this sentence, the idea is that the children are running around on the sand (not just running once from point A to B).
That kind of scattered, repeated movement is usually expressed with biegać, so biegają is natural.

If you said Dzieci biegną po piasku, it would sound more like they are running across the sand in some specific direction (e.g. to the water).

What grammatical form is dzieci and can it be singular?

Dzieci is:

nominative plural of dziecko (child)
also the form for several other cases (genitive, dative, locative), depending on context

In this sentence it is nominative plural – it is the subject:

Dzieci (who?) biegają – Children run.

There is no singular dzieci. For one child you say:

dziecko = a child
dzieci = children

Why is it biegają and opalają (present tense) and not some continuous tense?

Polish does not have a separate continuous tense like English are running / are sunbathing.

Both:

biegają (they run / they are running)
opalają się (they sunbathe / they are sunbathing)

are present tense imperfective forms. Context decides whether you understand them as:

right now: The children are running… the parents are sunbathing…
regularly / habitually: Children run… parents sunbathe…

Here, with a concrete scene, it’s understood as right now.

Why is it bez butów and not bez buty?

The preposition bez (without) always takes the genitive case.

buty (NOM plural) = shoes
butów (GEN plural) = of shoes

So after bez, you must use butów:

bez butów = without shoes

Using bez buty would be incorrect grammar.

What case is butów, and how is its form built?

butów is genitive plural of but (shoe).

Basic forms:

but – nominative singular
buty – nominative plural
butu – genitive singular
butów – genitive plural

The -ów ending is common for masculine genitive plural nouns, especially concrete countable ones (dom – domów, pies – psów, etc.).

Why is it po piasku and not po piasek?

The preposition po can take:

locative (where? on/over/along something, after something)
accusative (sometimes for searching or distribution, in certain idioms)

Here, po piasku expresses moving on/over the surface of the sand, so locative is used:

piasek – nominative singular
piasku – genitive or locative singular
(in this sentence, it is locative after po)

Po piasek (accusative) would mean something like to go for sand (to fetch sand), not on the sand.

What is the difference between po piasku, na piasku, and w piasku?

All three are possible but with different nuances:

po piasku – movement on / across the sand
→ Dzieci biegają po piasku = The children are running on the sand (across its surface).
na piasku – location on the sand (often more static)
→ Dzieci siedzą na piasku = The children are sitting on the sand.
w piasku – in the sand, partly or fully inside it
→ Dzieci grzebią w piasku = The children are digging in the sand.

In your sentence, movement over the sand is in focus, so po piasku is natural.

Why is it rodzice and not a singular form?

Rodzice means parents (plural). Polish normally uses a plural noun for parents:

rodzic (singular) = a parent – grammatically exists, but is less common in everyday speech
rodzice (plural) = parents – standard word

Here, rodzice is nominative plural, the subject of opalają się:

rodzice (who?) opalają się – the parents are sunbathing

Why do we use a and not i between the two clauses?

Both a and i can translate as and, but:

i usually just adds things, no contrast
→ Mama i tata = Mum and dad
a often marks a contrast, comparison or mild shift between parts of a sentence
→ This is happening, *whereas that is happening.*

In Dzieci biegają…, a rodzice opalają się…, a suggests a natural contrast:

the children are doing one thing
the parents are doing something different

You could use i, but a sounds more idiomatic here.

Why do we need się in opalają się?

Się is a reflexive particle. It can mark:

a truly reflexive action (doing something to oneself)
a reflexive verb whose meaning is idiomatic

Opalać się literally is to sunbathe (to tan oneself). The verb is normally used with się:

opalają się – they are sunbathing

Without się (opalają) it would sound like they are tanning something/someone else, which is not what we mean here.

Why is it na ręcznikach and not na ręczniki?

The preposition na (on) can take:

locative – for location (where?)
accusative – for direction / movement (onto where?)

Here, na ręcznikach describes location:

rodzice opalają się na ręcznikach = the parents are sunbathing on (their) towels

So na + locative is used:

ręcznik – nominative singular
ręczniki – nominative plural
ręcznikach – locative plural

If you said położyli ręczniki na piasku (they put the towels on the sand), na piasku is also locative (where they end up).
But położyli się na ręczniki would use accusative to stress movement onto the towels (less usual than na ręcznikach here, but possible in certain contexts).

Is the word order fixed, or could I move words around?

Polish word order is flexible, especially compared to English. The basic, neutral order here is:

Dzieci biegają bez butów po piasku, a rodzice opalają się na ręcznikach.

You can move some elements for emphasis, for example:

Dzieci po piasku biegają bez butów.
Bez butów dzieci biegają po piasku.
Na ręcznikach opalają się rodzice.

All are grammatically correct but may sound more or less natural or emphatic. The original order is the most neutral and typical.

How would I explicitly say their shoes or on their towels?

Polish often omits possessives when the owner is obvious from context (like body parts or personal items). So:

bez butów – naturally understood as without their shoes
na ręcznikach – naturally understood as on their towels

If you do want to emphasize ownership:

bez swoich butów – without their own shoes
na swoich ręcznikach – on their own towels

Swoich is the reflexive possessive adjective, used when the owner is also the subject (here, dzieci / rodzice).

What aspect do biegają and opalają się have?

Both are imperfective verbs in the present tense:

biegać → biegają
opalać się → opalają się

Imperfective verbs in Polish:

describe ongoing, repeated, or habitual actions
can be used in present, past, and future (with auxiliary for future)

If you wanted to use perfective forms (to highlight a completed act), you’d use different verbs:

pobiegać (to run for a while)
opalić się (to get a suntan / to become tanned)

But those would not fit well in a simple present-time description of a scene.