Learn Hardcore German: Unsere Mannschaft gewann den ersten kleinen Wettbewerb. - Our team won the first small competition.

gewann is the Präteritum form of gewinnen.
It’s often used in written narratives, reports or formal contexts.
In everyday spoken German you would more commonly hear:
Unsere Mannschaft hat den ersten kleinen Wettbewerb gewonnen.

What case is den ersten kleinen Wettbewerb, and how do I know?

den ersten kleinen Wettbewerb is in the accusative case (direct object). You can tell because:

The verb gewinnen takes a direct object (Wen oder was? → den Wettbewerb).
The masculine definite article der changes to den in the accusative.

Why do the adjectives ersten and kleinen both end in -en here?

When an adjective follows a definite article or a determiner with a clear case/number/gender marker (like den in Akusativ Maskulinum), it takes the weak (schwache) ending -en in all genders:
den (acc. masc.) + ersten (weak) + kleinen (weak) + Wettbewerb

What makes ersten an ordinal number and how is it formed?

ersten is the ordinal form of eins (one), meaning “first.”
Ordinals in German get adjective endings just like descriptive adjectives.
Here, because it follows den (acc. masc.), it takes -en: den ersten …

Why is kleinen translated as “small” and not “little”?

Both “small” and “little” are valid translations of klein. Choice depends on context:

“small” emphasizes size or scale.
“little” can feel more colloquial or affectionate.
In this sentence “small competition” simply indicates it wasn’t a major event.

Why do we say Unsere Mannschaft instead of Die Mannschaft von uns?

unsere is a possessive pronoun meaning “our,” directly modifying Mannschaft.
It’s more natural than using a postposed genitive construction for everyday speech.
Mannschaft is feminine, so in nominative singular the possessive ends in -e: unsere Mannschaft.

What gender and number is Mannschaft, and how does that affect the sentence?

Mannschaft is a feminine noun, singular.
In the nominative case it’s die Mannschaft, so with a possessive: unsere Mannschaft.
The verb gewann agrees with a singular subject: Unsere Mannschaft gewann.

Could I swap word order and say Gewann unsere Mannschaft den ersten kleinen Wettbewerb?

Not in a standard declarative sentence. German clause structure requires:

Finite verb in second position.
Subject usually first (unless you intentionally front something else).
So the correct order is Unsere Mannschaft (S) gewann (V) ….
If you front gewann, it becomes a question: Gewann unsere Mannschaft …?

Is it possible to say Unsere Mannschaft hat den ersten kleinen Wettbewerb gewonnen?

Yes. That uses the Perfekt tense:

Auxiliary hat
- past participle gewonnen.
It’s more common in spoken German for past events.
Meaning remains the same.

Why not omit the article and say Unsere Mannschaft gewann ersten kleinen Wettbewerb?

In German, countable singular nouns generally require an article or determiner. Omitting it would sound unnatural. You need den to mark the accusative masculine singular.