Learn Hardcore German: Die Kinder sind im Garten sehr aktiv. - The children are very active in the garden.

Nominative: der, die, das → die
Accusative: den, die, das → die
Dative: dem, der, dem → den (+ -n on most nouns: den Kindern)
Genitive: des, der, des → der

So:

die Kinder = nominative (or accusative) plural
den Kindern = dative plural
der Kinder = genitive plural

Because Kinder is the subject, die Kinder (nominative plural) is correct.

But die is also used for feminine singular nouns. How do I know that Kinder is plural here, not feminine singular?

You know from the noun form and the verb:

Form of the noun
- Singular: das Kind (neuter)
- Plural: die Kinder
  Since Kinder already looks like a plural form, die Kinder must be plural.
Verb agreement
- die Kinder sind → sind is 3rd person plural of sein.
- Feminine singular would take ist:
  - Die Frau ist im Garten.

So die Kinder = they (the children), not she.

What gender is Garten, and how do I know?

Garten is masculine: der Garten.

How to know:

You normally have to learn the gender with the noun, e.g. der Garten.
The form im Garten tells you it is in + DEM Garten (dative masculine), because im = in dem, and dem is dative masculine or neuter singular.
You cannot reliably guess the gender just from the ending -en; many masculine nouns have this ending, but so do some others.

So the dictionary form is der Garten (masculine).

Why is it im Garten and not in den Garten?

Because in can take dative or accusative, and the meaning here is location, not movement.

Dative = location (where?):
- Die Kinder sind im Garten.
  → The children are in the garden (they are already there, no movement).
Accusative = direction (where to?):
- Die Kinder gehen in den Garten.
  → The children go into the garden (movement towards it).

So:

im Garten = in dem Garten (dative, location)
in den Garten = accusative (movement, direction)

What exactly is im? Is it always interchangeable with in dem?

Im is a contraction of in dem:

im Garten = in dem Garten

It is used when in is followed by dem (dative masculine or neuter singular).

Other common contractions:

an dem → am (e.g. am Tisch)
in das → ins (e.g. ins Kino)
bei dem → beim
zu dem → zum
zu der → zur
von dem → vom

Is it always interchangeable?

Grammatically, yes: im Garten and in dem Garten mean the same.
In normal speech and writing, im is far more common; in dem Garten sounds more formal or emphatic.

Why does aktiv not have an ending, like aktive or aktiven?

Because aktiv is used as a predicate adjective after the verb sein, not directly before a noun.

In German:

After sein, werden, bleiben, adjectives stay in their basic form (no ending):
- Die Kinder sind aktiv.
- Der Mann ist müde.
- Die Blumen sind schön.
When an adjective stands directly before a noun, it takes an ending:
- die aktiven Kinder (the active children)
- ein aktives Kind (an active child)
- mit aktiven Kindern (with active children – dative plural)

In Die Kinder sind im Garten sehr aktiv, aktiv describes the state of the subject, so it doesn’t get an ending.

What role does sehr play in the sentence, and where can it go?

Sehr is an adverb that intensifies adjectives or other adverbs.
Here, it modifies aktiv:

sehr aktiv = very active

Position:

It normally comes directly before the word it modifies:
- Die Kinder sind im Garten sehr aktiv.

You can move the whole phrase sehr aktiv, but sehr itself stays before aktiv:

Die Kinder sind sehr aktiv im Garten. (also correct)
You cannot say Die Kinder sind aktiv sehr im Garten – that is wrong, because sehr does not belong with im Garten, it belongs with aktiv.

Is the word order im Garten sehr aktiv fixed, or can I move these parts around?

The verb position is relatively fixed; the rest is flexible.

General rule in main clauses:

The finite verb (sind) must be in second position.
The subject often comes first, but not always.

In your sentence:

Standard: Die Kinder sind im Garten sehr aktiv.
Also possible (and natural):
- Die Kinder sind sehr aktiv im Garten.
- Im Garten sind die Kinder sehr aktiv. (focus on in the garden)
- Sehr aktiv sind die Kinder im Garten. (more emphasis on very active, a bit marked)

What you cannot change:

You cannot move sind away from second position in a normal main clause.
You cannot split sehr and aktiv in an unnatural way.

So im Garten and sehr aktiv are movable blocks inside the sentence.

Why is it sind and not ist or seid?

Because die Kinder is 3rd person plural, and sind is the 3rd person plural form of sein.

Present tense of sein:

ich bin – I am
du bist – you are (singular, informal)
er/sie/es ist – he/she/it is
wir sind – we are
ihr seid – you are (plural, informal)
sie/Sie sind – they are / you are (formal)

Die Kinder = sie (they) → sie sind → Die Kinder sind …

Is im Garten a kind of adverb? What is its function?

Im Garten is a prepositional phrase (preposition in + article + noun) that functions as an adverbial of place.

It answers the question Wo? (Where?):
- Wo sind die Kinder sehr aktiv? – Im Garten.
In grammatical terms, it’s an adverbial phrase of location.

So even though it’s not a single-word adverb, it plays an adverbial role in the sentence.

Could I say Die Kinder haben im Garten viel Aktivität instead?

You could, but it sounds unusual and quite unnatural in everyday German.

Reasons:

German prefers adjectives with sein (or verbs) to describe how active someone is, rather than the abstract noun Aktivität.
viel Aktivität haben is more like saying have a lot of activity, which feels more technical or stylistically odd here.

More natural ways:

Die Kinder sind im Garten sehr aktiv. (your sentence)
Die Kinder sind im Garten besonders aktiv.
Die Kinder treiben im Garten viel Sport. (if you mean they do a lot of sports)

So for normal description of children being lively and energetic, sind … aktiv is the idiomatic choice.

How would the sentence change in the past tense?

Two common options:

Präteritum (simple past) – very common for sein:
- Die Kinder waren im Garten sehr aktiv.
  → The children were very active in the garden.
Perfekt (present perfect) – also possible:
- Die Kinder sind im Garten sehr aktiv gewesen.
  (literally: have been very active)

In everyday speech, for sein, Germans often prefer the simple past (waren) to the Perfekt in many regions, especially in the north.