Learn Hardcore German: Der Bauer und die Bäuerin zeigen uns stolz ihren neuen Traktor. - The farmer and the female farmer proudly show us their new tractor.

uns is the dative form of wir and means “to us”.
zeigen (to show) normally takes:
- a dative person (to whom?) → uns
- and an accusative thing (what?) → ihren neuen Traktor

So you get: Der Bauer und die Bäuerin (subject) zeigen uns (to us, dative) ihren neuen Traktor (accusative object).

Which cases are used with zeigen in this sentence?

zeigen often takes two objects:

Dative for the person: uns (to us)
Accusative for the thing: ihren neuen Traktor (their new tractor)

So the structure is:

Subjekt: Der Bauer und die Bäuerin
Dativobjekt: uns
Akkusativobjekt: ihren neuen Traktor

What does ihren mean here – is it “her”, “their”, or “your (formal)”?

The basic possessive stem ihr- can mean:

her (3rd person singular feminine)
their (3rd person plural)
your (formal: Sie)

In this sentence, the owners are Der Bauer und die Bäuerin = they, so ihr- here means “their”.
So ihren neuen Traktor = their new tractor.

Grammatically it could also be “her new tractor” or “your new tractor”, but the context (a plural subject “they”) makes “their” the only reasonable reading.

Why is it ihren and not just ihr?

ihr is the dictionary form of the possessive, but it has to be declined (get an ending) to match the noun’s case, gender, and number.

Noun: Traktor → masculine, singular
Case: accusative (direct object)
Possessive determiner before a masculine accusative singular noun takes -en.

So:

masculine accusative: ihren Traktor (not ihr Traktor)

The same happens with the adjective after it, hence neuen Traktor (also masculine accusative → -en).

What case is ihren neuen Traktor, and how do I see that from the endings?

ihren neuen Traktor is in the accusative singular masculine.

Clues:

It’s the direct object of zeigen → usually accusative.
Noun Traktor is masculine.
The endings -en on both ihren and neuen are typical for:
- masculine accusative singular after a possessive or other determiner.

A mini pattern (with a possessive/article before it):

Nominative: ihr neuer Traktor (their new tractor – subject)
Accusative: ihren neuen Traktor (their new tractor – object)

Why is stolz here and not stolze or stolzen?

In this sentence, stolz functions as an adverb (“proudly”), describing how they show the tractor.

Adverbs in German:

usually do not change their form
do not agree in gender, number, or case

So you simply use stolz:

Sie zeigen uns stolz ihren neuen Traktor. = They proudly show us their new tractor.

Forms like stolze or stolzen are adjectival and used before nouns, e.g.:

der stolze Bauer (the proud farmer)
dem stolzen Bauern (to the proud farmer)

Where can stolz go in the sentence? Can I move it around?

Yes, stolz is fairly flexible, but some positions sound more natural than others.

Neutral, common option (as given):

Der Bauer und die Bäuerin zeigen uns stolz ihren neuen Traktor.

Other possible placements:

Der Bauer und die Bäuerin zeigen uns ihren neuen Traktor stolz.
→ grammatically okay, but sounds a bit marked / unusual; emphasis is a bit odd.
Der Bauer und die Bäuerin zeigen stolz uns ihren neuen Traktor.
→ unusual; pronouns like uns normally come earlier in the “middle field”.

General tendency in the middle field:

Pronouns (e.g. uns)
Short adverbs (e.g. stolz)
Longer noun phrases (e.g. ihren neuen Traktor)

So zeigen uns stolz ihren neuen Traktor is the most natural version.

What is the difference between Bauer and Bäuerin, and why does the vowel change?

Bauer and Bäuerin are the masculine and feminine forms of the same profession:

der Bauer = male farmer
die Bäuerin = female farmer

To make many feminine forms in German, you add -in and often add an Umlaut to the vowel:

der Arzt → die Ärztin
der Lehrer → die Lehrerin
der Bauer → die Bäuerin (here au → äu)

So the vowel change (Umlaut) is a regular pattern in many masculine–feminine pairs.

Why is it der Bauer but die Bäuerin?

Because of grammatical gender and the feminine suffix -in:

Bauer is a masculine noun → it takes the article der in the nominative singular: der Bauer.
Adding -in usually makes a feminine noun for a female person → Bäuerin, which takes die: die Bäuerin.

So:

der Bauer (he, the male farmer)
die Bäuerin (she, the female farmer)

Why are Bauer, Bäuerin, and Traktor capitalized?

In German, all nouns are capitalized, regardless of where they appear in the sentence.

Here:

Bauer (farmer) – noun → capitalized
Bäuerin (female farmer) – noun → capitalized
Traktor (tractor) – noun → capitalized

Adjectives and verbs, like neuen, zeigen, stolz, are not capitalized (unless they start the sentence or are part of a proper name).

Can I say zu uns instead of uns, like “show the tractor to us”?

No, not with zeigen in this structure.

In German, zeigen normally uses a dative object for the person, without a preposition:

jemandem etwas zeigen = to show someone something

Thus:

Der Bauer und die Bäuerin zeigen uns ihren neuen Traktor. ✅
… zeigen zu uns ihren neuen Traktor. ❌ (ungrammatical)

uns already expresses “to us” here; no extra zu is needed.