Learn Hardcore German: Die Lehrerin schreibt einen Beispielsatz an die Tafel und erklärt ihre persönliche Lernstrategie. - The teacher writes an example sentence on the board and explains her personal learning strategy.

In German, Lehrer is the general word for “teacher” and is grammatically masculine (der Lehrer).
The form Lehrerin is the feminine version: die Lehrerin = “female teacher”.

The ending -in usually marks a female person:
- der Student → die Studentin
- der Lehrer → die Lehrerin

The article changes with the noun’s grammatical gender:

der Lehrer (masculine)
die Lehrerin (feminine)

Here, the sentence is talking about a female teacher, so we use die Lehrerin.

Why do we say einen Beispielsatz and not ein Beispielsatz?

Because Beispielsatz is masculine and used as a direct object (accusative case).

The base noun is der Satz (“sentence”) → masculine.
In the accusative singular, the indefinite article for masculine changes from ein to einen:

Indefinite article (singular):

Nominative: ein Satz (subject)
Accusative: einen Satz (direct object)

In the sentence:

Was schreibt die Lehrerin? – einen Beispielsatz.
(direct object → accusative → einen)

What exactly does Beispielsatz mean, and why is it one word?

Beispielsatz is a compound noun made from:

das Beispiel = example
der Satz = sentence

Put together: der Beispielsatz = “example sentence”.

In German, it’s very common to join nouns into one long word:

Hausaufgabe (Haus + Aufgabe) = homework
Autotür (Auto + Tür) = car door
Beispielsatz (Beispiel + Satz) = example sentence

Because German likes compound nouns, it’s written as one word and capitalized as a noun.

Why is it an die Tafel and not auf die Tafel or zu der Tafel?

All three prepositions (an, auf, zu) can relate to places, but they are used differently and “an die Tafel schreiben” is the standard idiom.

an die Tafel schreiben = “to write on the board (chalk/marker on the surface)”
- an here means something like “onto / on(to) the vertical surface”.
- With a direction (movement), an takes the accusative: an die Tafel.
auf die Tafel schreiben would literally mean “to write onto the top of the board” (as if on a horizontal surface). It sounds odd in this context.
zu der Tafel gehen = “to go to the board” (towards it, not onto its surface).
- zu focuses on movement towards a place, not onto a surface.

So in German, the fixed expression for writing on the blackboard/whiteboard is an die Tafel schreiben.

Is an in schreibt ... an die Tafel a separable prefix like in other German verbs?

No. In this sentence, an is a preposition, not a separable verb prefix.

The verb here is simply schreiben (“to write”).
an die Tafel is a prepositional phrase (where/to where she writes).

A separable verb with an- would look like:

anschreiben: Die Lehrerin *schreibt die Eltern an*. (“writes to the parents”)

In schreibt einen Beispielsatz an die Tafel, we do not have the verb anschreiben; we have the basic verb schreiben + the prepositional phrase an die Tafel.

Why is it ihre persönliche Lernstrategie and what does ihre refer to?

ihre here means “her” and refers back to die Lehrerin.

In German, ihr-/ihre can mean:

her (for a feminine singular owner)
their (for a plural owner)
your (formal: Ihr/Ihre with capital I)

Here:

Subject: Die Lehrerin (she)
Something that belongs to her: ihre persönliche Lernstrategie = “her personal learning strategy”

Grammatically:

Lernstrategie is feminine: die Lernstrategie.
In accusative singular (direct object), feminine looks the same as nominative: die / eine / ihre Lernstrategie.
So ihre agrees with the feminine noun Lernstrategie and expresses possession by the teacher (= her).

Why is it persönliche Lernstrategie and not persönlichen Lernstrategie?

Because Lernstrategie is:

singular
feminine
in the accusative case
with a possessive determiner (ihre)

For feminine singular nouns, with possessive determiners (mein/dein/sein/ihr/unser/euer/Ihr), the strong adjective ending is -e in both nominative and accusative:

ihre persönliche Lernstrategie (nominative or accusative, feminine singular)

We would only see -en in other situations, for example:

mit ihrer persönlichen Lernstrategie (dative feminine)
ihre persönlichen Lernstrategien (plural)

What does Lernstrategie mean, and why is it one word?

Lernstrategie is another compound noun:

lernen / das Lernen = to learn / learning
die Strategie = strategy

Combined: die Lernstrategie = “learning strategy” / “study strategy”.

German usually writes such combinations as one word:

Lernziel (learning goal)
Lernmaterial (learning material)
Lernstrategie (learning strategy)

As a noun, it is capitalized: Lernstrategie.

Why are Lehrerin, Beispielsatz, Tafel, and Lernstrategie all capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence.

In this sentence:

Lehrerin (teacher) → noun
Beispielsatz (example sentence) → noun
Tafel (board) → noun
Lernstrategie (learning strategy) → noun

So they are all written with an initial capital letter.
Adjectives and verbs are normally not capitalized unless used as nouns (e.g., das Lernen).

Why is the verb form schreibt / erklärt and not something like “is writing / is explaining”?

German does not have a special progressive tense like English “is writing / is explaining”.

The simple present in German (schreibt, erklärt) can express:

present habitual: Sie schreibt jeden Tag. – “She writes every day.”
present ongoing: Sie schreibt gerade. – “She is writing (right now).”

So:

Die Lehrerin schreibt einen Beispielsatz an die Tafel und erklärt ihre persönliche Lernstrategie.

can naturally be translated as:

“The teacher is writing an example sentence on the board and is explaining her personal learning strategy.”

Why don’t we repeat the subject before erklärt, like Die Lehrerin schreibt … und sie erklärt …?

The conjunction und (“and”) can connect two main verbs that share the same subject.

Full version (possible, but longer):
Die Lehrerin schreibt einen Beispielsatz an die Tafel, und sie erklärt ihre persönliche Lernstrategie.
Normal, more natural version:
Die Lehrerin schreibt einen Beispielsatz an die Tafel und erklärt ihre persönliche Lernstrategie.

In German (and English), if the subject of both verbs is the same, you can say it once at the beginning and then just use: Subject + Verb1 … und Verb2 …

The subject Die Lehrerin is understood for both schreibt and erklärt.