The case letter: der becomes den

The article changes again when the noun's job changes

Knowing that Mann is masculine is not the end of it. The article changes its last letter with the noun's job, and that letter is what the case colours mark:

123

1. 1der Junge
2. 2das Buch
3. 3dem Mädchen

Der Junge gibt dem Mädchen das Buch: the boy gives the girl the book. The giver is der, the thing given das, and the girl who receives it dem.

That one letter is why German word order can move. Watch it decide who sees whom:

The second sentence puts the child first for emphasis, and der still flags the man as the one seeing. In the third, the order matches the first, the meaning is the opposite, and the only signal is den. English cannot do this: its word order is spent on grammar.

Mistakes English speakers make

What to actually do about it

1. Never write a noun without its article. In notes and on flashcards: der Tag, not Tag.
2. Learn the masculine first. der, den, dem, des change the most, so they carry the most information.
3. Read the endings before the word order. Find the article, find the subject, and a scrambled sentence resolves.

The four jobs have names, nominative, accusative, dative and genitive: they are the German case system, the next thing worth reading. A second verb waiting at the end of the clause belongs to the modal verbs.