Za domem mamy małe boisko rodzinne dla dzieci.

Za domem literally means behind the house.

The noun dom (house) is here in the instrumental case: domem.
The preposition za can take:

• instrumental for a static location:
  
  • Za domem mamy… = We have (something) behind the house. (location)
• accusative for movement to a position behind something:
  
  • Idziemy za dom. = We are going behind the house. (movement to a place)

Because the sentence describes where the playground is (not where someone is moving), za needs the instrumental, so domem is correct.

Ask yourself whether the verb describes:

1. Location (where something is / happens) → use instrumental

   • Za domem jest ogród. = The garden is behind the house.
   • Siedzimy za stołem. = We are sitting behind the table.

2. Movement (where something goes / is put) → use accusative

   • Idziemy za dom. = We are going behind the house.
   • Połóż to za telewizor. = Put it behind the TV.

In your sentence, mamy describes what we have in a certain place (location), so you use za domem (instrumental).

• mamy = we have (1st person plural of mieć, to have)
• jest = there is / it is (3rd person singular of być, to be)
• są = there are / they are (3rd person plural of być, to be)

Za domem mamy małe boisko rodzinne dla dzieci.
= Behind the house we have a small family playing field for the children.

You could rephrase with jest:

• Za domem jest małe boisko rodzinne dla dzieci.
  = Behind the house there is a small family playing field for the children.

Both are correct, but:

• mamy emphasizes ownership / possession (we own it, it belongs to us).
• jest/są just states existence / presence in that place.

In Polish, the verb ending usually shows who the subject is, so subject pronouns (ja, ty, my, wy, on…) are often omitted.

• mamy already tells you it’s we (1st person plural).

• my mamy is also correct, but you normally use my only for emphasis or contrast:

  • To my mamy boisko, nie oni.
    It’s we who have the field, not them.

In a neutral sentence, Za domem mamy… is the natural form.

Boisko (playing field, court) is neuter.

Typical patterns:

• Nouns ending in -o or -e are often neuter:
  
  • boisko, okno, krzesło, morze

For neuter singular nouns in the nominative and accusative, adjectives usually end in -e:

• małe boisko – small field
• rodzinne boisko – family field

Both małe and rodzinne are:

• neuter
• singular
• accusative (object of mamy)

They must agree in gender, number, and case with boisko.
That’s why we say małe boisko rodzinne, not mały boisko rodzinne or rodzinna boisko.

The adjective must match the noun boisko:

• boisko = neuter singular
• neuter singular adjectives (in nominative/accusative) → typically -e

So:

• małe boisko (correct: neuter)
• mały boisko (incorrect: mały is masculine, doesn’t match boisko)

Some quick patterns for nominative/accusative singular:

• masculine inanimate: mały dom (small house)
• feminine: mała książka (small book)
• neuter: małe boisko (small field)

Boisko usually means playing field / sports field / court, depending on context:

• boisko piłkarskie – football (soccer) pitch
• boisko do koszykówki – basketball court

Rodzinne comes from rodzina (family) and means family as an adjective: family, family-type, for family use.

So małe boisko rodzinne suggests:

• a small field/court used by the family (not an official sports facility),
• maybe something like a little private pitch for the kids and family to play on.

If you meant a playground with swings, slides, etc., you’d more often say:

• plac zabaw – playground
  
  • Za domem mamy mały plac zabaw dla dzieci.

Polish word order is fairly flexible, and both of these are correct:

• Za domem mamy małe boisko rodzinne dla dzieci.
• Mamy małe boisko rodzinne za domem dla dzieci.

Differences are mostly in emphasis:

• Starting with Za domem emphasizes the location (where it is).
• Starting with Mamy emphasizes possession (what we have).

Other natural variants:

• Za domem mamy rodzinne małe boisko dla dzieci.
• Za domem mamy dla dzieci małe boisko rodzinne.

In normal, neutral speech, the original version is very natural. Polish allows reordering as long as grammar (cases, agreements) is correct, but some orders can sound clumsy if overcomplicated.

dla means for (in the sense of “intended for / on behalf of”).

dla always takes the genitive case.

• dziecko = a child (singular)
  
  • genitive singular: dziecka → dla dziecka (for a child / for the child)
• dzieci = children (plural)
  
  • genitive plural is also dzieci (same form as nominative/accusative)

So:

• dla dziecka – for (one) child
• dla dzieci – for (the) children

In your sentence, dla dzieci = for the children, and it is in genitive plural after dla.

dzieci is plural (children).

The base forms:

• nominative singular: dziecko – a child
• nominative plural: dzieci – children
• genitive plural: dzieci – of children / (for) children

In dla dzieci, dzieci is genitive plural, required by dla.
It just happens that the nominative plural and genitive plural look the same for this word.

You add a possessive adjective that agrees with dzieci (plural).

• dla naszych dzieci – for our children

Breakdown:

• nasz (our) → plural genitive: naszych
• dzieci (children) → genitive plural: dzieci

So you can say:

• Za domem mamy małe boisko rodzinne dla naszych dzieci.
  = Behind the house we have a small family playing field for our children.

Some key points:

• Stress is always on the second-to-last syllable in each word:
  • ZA DO-mem MA-my MA-łe bo-IS-ko ro-DZIN-ne DLA DZIE-ci
• z in za – like z in zoo.
• domem – do-mem, like “DO-mem”.
• ł in małe – like English w: MA-we.
• boisko – roughly bo-ees-ko (the oi is two sounds, not like English “boy”).
• dz in rodzinne – like the j in jeans.
• nn in rodzinne – double n, a bit longer n sound.
• dzieci – DZIE-ci:
  • dz like j in “jeans”
  • ci similar to soft tch (like ć), so JE-chee approximately.

Said smoothly, it sounds like one flowing phrase, with the main stress in each word on the penultimate syllable.