Nach der Stunde stecke ich das Handout und das Arbeitsblatt in dieselbe Tasche und gehe zufrieden nach Hause.

Because nach takes the dative case in this meaning.

• die Stunde → nominative
• der Stunde → dative

So after nach, you say nach der Stunde.

Also, here Stunde usually means class / lesson rather than just hour, depending on context.

German is a verb-second (V2) language in main clauses. That means the conjugated verb has to come in the second position.

In this sentence, Nach der Stunde is placed first, so the verb stecke must come next:

• Nach der Stunde | stecke | ich ...

If you started with the subject instead, you could also say:

• Ich stecke nach der Stunde ...

Both are possible, but the original sentence emphasizes after class by putting it first.

Here it most likely means lesson, class, or period.

German Stunde can mean:

• hour in a time sense
• lesson/class period in a school or course context

So nach der Stunde often means after class or after the lesson.

Those are the grammatical genders of the nouns, and both are neuter, so they take das in the nominative and accusative singular.

• das Handout
• das Arbeitsblatt

A few notes:

• Handout is a loanword from English, but in German it is treated as a regular noun and is capitalized.
• Arbeitsblatt is a compound noun:
  • Arbeit = work
  • Blatt = sheet/page
    The final part, Blatt, determines the gender, so the whole word is neuter: das Arbeitsblatt.

Because in can take either the dative or the accusative, depending on the meaning.

Here, the sentence describes movement into the bag, so German uses the accusative:

• in dieselbe Tasche = into the same bag

Compare:

• Ich stecke das Handout in dieselbe Tasche.
  Movement into the bag → accusative
• Das Handout ist in derselben Tasche.
  Location in the bag → dative

So:

• dieselbe Tasche = accusative feminine
• derselben Tasche = dative feminine

This is a very common question.

In careful German:

• dieselbe Tasche = the very same bag, one identical object
• die gleiche Tasche = a bag of the same kind/model, but not necessarily the exact same physical bag

So in your sentence, dieselbe Tasche means both papers go into one and the same bag.

In everyday speech, many people use dieselbe and die gleiche less strictly, but the traditional distinction is still useful and often taught.

Because it is one determiner/pronoun form in German, not two separate words.

So the standard spelling is:

• dieselbe Tasche
• dasselbe Buch
• derselbe Lehrer

Not:

• die selbe Tasche

The declined forms change with gender, case, and number, but they stay one word.

Stecken can mean different things depending on how it is used.

Here, ich stecke ... in ... means:

• I put
• I stick
• I tuck

So Ich stecke das Handout in die Tasche means I put/tuck the handout into the bag.

This verb often suggests putting something into something else, especially in a practical, physical way.

Compare:

• Ich lege das Buch auf den Tisch. = I lay/put the book on the table.
• Ich stecke das Papier in die Tasche. = I stick/tuck the paper into the bag.

Because the same subject applies to both verbs.

German, like English, can join two verbs with und when they have the same subject:

• Ich stecke ... und gehe ...
• I put ... and go ...

You do not need to repeat ich.

You could repeat it for emphasis or clarity in some contexts, but normally you would not.

Because here zufrieden is being used adverbially, not as an attributive adjective before a noun.

German adjectives change endings when they come before nouns:

• ein zufriedener Student

But when they describe how someone feels or acts, they usually stay unchanged:

• Ich bin zufrieden.
• Ich gehe zufrieden nach Hause.

So in this sentence, zufrieden means something like contentedly / satisfied.

Because German distinguishes between motion toward home and location at home.

• nach Hause = to home / homeward → movement
• zu Hause = at home → location

So:

• Ich gehe nach Hause. = I’m going home.
• Ich bin zu Hause. = I’m at home.

That is why the sentence says gehe zufrieden nach Hause.

Because nach Hause is a fixed expression in German.

You usually just say:

• nach Hause gehen
• nach Hause kommen
• nach Hause fahren

without an article.

This is one of those common idiomatic phrases that learners simply need to get used to.

This is normal German main-clause word order.

The first clause is:

• Nach der Stunde stecke ich das Handout und das Arbeitsblatt in dieselbe Tasche

Then it is joined with a second verb phrase:

• und gehe zufrieden nach Hause

Because und connects two main-clause elements here, the verb gehe comes early, just as you would expect in a main clause.

Also, zufrieden comes before nach Hause quite naturally because it describes the subject’s state while going home.

Yes, often you could.

• nach der Stunde = after the lesson / after the class period
• nach dem Unterricht = after class / after the teaching session / after instruction

The best choice depends on context:

• Stunde often refers to one specific class period
• Unterricht can sound a bit broader or more general

So both may work, but they are not always perfectly identical in nuance.