Девочка подошла к доске и начала писать заголовок синим маркером.

Because девочка is the subject of the sentence — the person doing the actions.

In Russian, the subject is normally in the nominative case. So:

• девочка = the girl as subject
• девочку would be accusative, used when the girl is the object of an action

Here, the girl is the one who approached and began writing, so nominative is correct.

Подошла is the past tense feminine singular form of подойти.

It shows three things at once:

• past tense: the action happened in the past
• feminine: because девочка is feminine
• singular: because there is one girl

Compare:

• он подошёл = he approached
• она подошла = she approached
• они подошли = they approached

So подошла agrees with девочка.

Подойти к specifically means to come up to, to approach, or to walk up to something.

So:

• подошла к доске = she came up to the board

This is more specific than:

• шла = was walking
• пошла = set off / started going
• пришла = arrived / came

If the idea is that she moved from her place and went right up to the board, подошла к доске is the natural choice.

After the preposition к, Russian uses the dative case.

So:

• доска = board (dictionary form, nominative)
• к доске = to/toward the board (dative)

This is just a rule of the preposition:

• к столу = to the table
• к окну = to the window
• к доске = to the board

So доске is the dative singular form of доска.

Because подойти к means to come up to something, not to go onto it.

• к доске = up to the board
• на доску would literally mean onto the board

In a classroom, a student goes up to the board, not onto the board, so к доске is correct.

Russian commonly expresses beginning an action with:

• начать / начать́(ся)
  • infinitive

So:

• начала писать = began to write / started writing

This is completely normal Russian structure.

You could think of it as:

• начала = started
• писать = to write

Together they mean began writing.

Because after начать (to begin/start), Russian usually uses the imperfective infinitive to show the action as a process.

So:

• начала писать = began writing

Using написать here would sound less natural in this basic context, because написать is perfective and focuses on completing the writing.

Very often:

• начал читать
• начала говорить
• начали писать

all use the imperfective verb.

Заголовок is the direct object of писать — it is the thing she began to write.

The verb писать takes the accusative case for a direct object. But with a masculine inanimate noun like заголовок, the accusative singular looks the same as the nominative.

So:

• nominative: заголовок
• accusative: заголовок

That is why the form does not visibly change.

Синим маркером is in the instrumental case.

The instrumental is often used to show the tool or means by which something is done:

• писать ручкой = write with a pen
• резать ножом = cut with a knife
• синим маркером = with a blue marker

So here:

• маркером = instrumental singular of маркер
• синим = instrumental singular of синий, matching маркером

Together, синим маркером means with a blue marker.

Because adjectives must agree with the nouns they describe in case, number, and gender.

Here, маркером is:

• masculine
• singular
• instrumental

So the adjective must also be masculine singular instrumental:

• синий маркер = blue marker
• синим маркером = with a blue marker

The ending changes because the noun’s case changes.

Russian does not have articles like the and a/an.

So a noun like девочка can mean:

• a girl
• the girl

and the exact meaning depends on context.

The same is true for:

• доска = a board / the board
• заголовок = a title / the title
• маркер = a marker / the marker

Russian relies on context instead of articles.

Yes. Russian word order is more flexible than English because the case endings show grammatical roles.

The sentence as given is neutral and natural:

• Девочка подошла к доске и начала писать заголовок синим маркером.

But other orders are possible for emphasis, for example:

• К доске подошла девочка...
  This emphasizes to the board
• Синим маркером девочка начала писать заголовок...
  This emphasizes with a blue marker

Even though word order can change, the original version is the most straightforward and neutral.

Yes. И means and, and here it links two past actions done by the same subject:

• подошла к доске = approached the board
• начала писать заголовок = began to write the title

So the structure is:

• [She approached the board] and [began writing the title].

This is very common Russian sentence structure.

Because Russian often omits subject pronouns when the subject is already clearly stated.

Here, девочка already tells us who did the actions, so adding она would be unnecessary.

Russian often prefers:

• Девочка подошла... rather than
• Девочка, она подошла...

The noun itself is enough.