Learn Hardcore German: Der Lehrer prüft heute die Hausaufgaben sorgfältig. - The teacher checks the homework carefully today.

Because Lehrer is a masculine noun in the nominative singular case, serving as the subject. In German you must match the definite article to the noun’s gender, number, and case—here der is the correct masculine nominative singular article.

Why is the verb prüft written that way?

“prüft” is the third person singular present tense of the regular verb prüfen (“to check/examine”). You remove the infinitive ending -en to get the stem prüf- and add -t for er/sie/es, giving er prüft.

What case is die Hausaufgaben in, and how do you know?

It’s in the accusative plural because it’s the direct object of prüft (it answers “what is being checked?”). The accusative plural article is die, which coincides with the nominative plural form—but you identify it as the object by its role after the verb.

Why is Hausaufgaben plural when English uses “homework” as an uncountable noun?

In German, Hausaufgabe is a countable noun meaning “an assignment” or “exercise.” You normally talk about a collection of tasks—die Hausaufgaben—whereas English treats “homework” as a mass noun without a plural form.

What kind of word is heute, and why does it follow immediately after prüft?

“heute” is an adverb of time. In German main clauses the finite verb must occupy the second position. After the verb you can place various elements; time adverbials often come right after the verb, before objects and manner or place adverbials.

Could you start the sentence with heute, and how would that affect the word order?

Yes. If you front heute, it becomes the first element and the verb stays in second position:
Heute prüft der Lehrer die Hausaufgaben sorgfältig.
The subject der Lehrer then follows the verb.

What kind of word is sorgfältig, and why doesn’t it change form here?

“sorgfältig” is an adjective used adverbially to mean “carefully.” When adjectives function as adverbs in German, they remain in their base form without any case or number endings.

Why is sorgfältig placed at the end of the sentence?

It’s an adverb of manner. In the default German main-clause sequence, you have time adverbials first, then objects, then manner adverbials, and finally place adverbials—so sorgfältig naturally comes at the end.

How do you pronounce the pf sound in prüft, and the ü vowel?

The “pf” is a voiceless labial affricate with closed lips for [p] then release into [f]. The “ü” is a front rounded vowel [y], roughly like the English “ee” sound with your lips rounded.