Learn Hardcore Esperanto: Mi havas grandan familion kun tri infanoj.

Because familio is the direct object, it becomes familion.
The adjective granda must agree with the noun in case and number, so it also takes -n, becoming grandan.

Why doesn’t infanoj end in -n as well?

Infanoj is not a direct object here; it comes after the preposition kun (with), in the phrase kun tri infanoj (with three children).

In Esperanto:

The direct object of the verb takes -n.
A noun after a preposition (like kun, en, sur) normally does not take -n, unless the preposition specifically uses -n to show direction (which kun does not).

So:

familion takes -n because it is the direct object of havas.
infanoj stays in the basic plural form (no -n) because it is the object of the preposition kun, not of the verb directly.

Why is there no word for “a” in a big family?

Esperanto has only one article: la, which means “the”.
There is no separate word for “a/an”. Indefinite nouns are simply used without an article.

So:

Mi havas grandan familion = I have a big family.
Mi havas la grandan familion = I have the big family (some specific one that both speaker and listener know about).

The presence or absence of la tells you roughly what English would show with “the” versus “a/an”.

Why is familio singular even though it represents many people?

In Esperanto, as in English, familio (family) is treated as one unit, a single group:

familio = one family (a group)
familioj = several families

The many people inside the family are shown separately with infanoj (children).
So the sentence has:

familion – one family (singular, accusative)
tri infanoj – three children (plural)

This is the same pattern as English: I have a big family with three children (family = singular, children = plural).

Why is granda changed to grandan? How do adjective endings work?

Adjectives in Esperanto must agree with the noun they describe in:

Number: singular (-a) or plural (-aj)
Case: nominative (no -n) or accusative (-n)

The basic pattern is:

singular, no -n: granda familio
singular, with -n: grandan familion
plural, no -n: grandaj familioj
plural, with -n: grandajn familiojn

In Mi havas grandan familion, familion is singular accusative, so granda becomes grandan to match it.

Could the word order be Mi havas familion grandan kun tri infanoj or something similar?

Esperanto word order is fairly flexible, but there are natural preferences.

The most usual and neutral order is:

adjective before the noun: grandan familion

You can say familion grandan, especially for emphasis or in a poetic/literary style, but it sounds marked rather than neutral in everyday speech.

Reordering larger chunks like:

Mi havas familion kun tri infanoj grandan

starts to sound awkward or confusing. The safest and clearest structure is the original:

Mi havas grandan familion kun tri infanoj.

Why is it kun tri infanoj and not something like de tri infanoj?

Both kun and de can appear with familio, but they have slightly different nuances:

kun tri infanoj = with three children
Focuses on the family having / being accompanied by three children.
de tri infanoj = of three children
Often feels more like the family consists of three children (and may suggest you’re counting only children, not parents), though in practice people do sometimes use it to mean “a family of three children”.

In your sentence, kun tri infanoj is the most natural and clear way to say that your big family includes three children.

Could I just say Mi havas tri infanojn instead? Does that mean the same thing?

Mi havas tri infanojn means “I have three children.”
This communicates the number of children clearly, but it doesn’t mention the family as such.

Mi havas grandan familion kun tri infanoj.
Emphasizes both the existence of a family and the fact that it contains three children.
Mi havas tri infanojn.
Just states the number of children you have.

In many real conversations, if your main point is the number of kids, Mi havas tri infanojn is perfectly natural.

Why do we use havas (“have”) instead of a form of “to be”, like estas?

Esperanto separates “have” and “be” in a way very similar to English:

havi = to have
esti = to be

To express I have a big family, you use havi:

Mi havas grandan familion.

If you said:

Mia familio estas granda. = My family is big.

you’re describing the family’s size or characteristic, not stating possession. Both sentences are correct, but they say slightly different things, just as in English.

Why don’t we say Mia familio or mian familion? In English we say “my family.”

You can say:

Mi havas mian familion.

but this usually means something like I have my family with me or I still have my family (they are alive / safe). It emphasizes keeping/retaining that particular family.

In Mi havas grandan familion, you are not contrasting this family with some other one; you are just stating that you possess a big family. In Esperanto, it is normal not to use mia here, because the subject Mi plus havas already makes it clear whose family it is.

If you want to talk about your family as a subject, you would naturally say:

Mia familio estas granda. = My family is big.