Die Lehrerin schreibt mit weißer Kreide an die Tafel, während der Beamer schon das erste Bild auf die Leinwand wirft.

Because mit always takes the dative case.

• Kreide is feminine: die Kreide
• In the dative singular, that becomes der Kreide
• Since there is no article here, the adjective has to show the case ending itself, so you get weißer

So:

• mit weißer Kreide = with white chalk

German often leaves out the article when talking about a material or tool in a general way.

So mit weißer Kreide sounds natural for with white chalk.

If you said mit einer weißen Kreide, it would mean something more like with a white piece of chalk / with a certain white chalk stick, which is more specific.

Because an and auf are two-way prepositions. They can take either:

• accusative for direction / movement toward a target
• dative for location / position

Here, the writing goes onto the board, and the image is projected onto the screen, so German uses the accusative:

• an die Tafel
• auf die Leinwand

Compare:

• Die Lehrerin steht an der Tafel. = The teacher is standing at the board.
• Das Bild ist auf der Leinwand. = The image is on the screen.

Because während introduces a subordinate clause.

In German subordinate clauses, the conjugated verb goes to the end:

• ..., während der Beamer schon das erste Bild auf die Leinwand wirft.

That is one of the most important word-order rules in German.

Yes. In German, a subordinate clause introduced by a conjunction like während, weil, dass, or wenn is normally separated by a comma.

So the comma in this sentence is required:

• ..., während der Beamer ... wirft.

Here schon means already.

It suggests that the projector has begun showing something before you might expect or while something else is still happening.

So the feeling is:

• The teacher is still writing on the board,
• and meanwhile the projector is already showing the first image.

Yes. In everyday German, der Beamer usually means a video projector / digital projector.

This is a good thing to notice because it is a pseudo-English word: it sounds English, but native English speakers do not normally use beamer to mean projector.

Not literally. German often uses werfen in a figurative way for light, shadows, or projected images.

So:

• ein Bild auf die Leinwand werfen
• einen Schatten an die Wand werfen

This is idiomatic German. It works a bit like English saying that a projector casts an image.

Because after the definite article das, the adjective takes the weak ending -e in the accusative neuter singular.

• das Bild = the image/picture
• das erste Bild = the first image/picture

So erste is correct here.

In this context:

• die Tafel = the classroom board, especially a chalkboard here
• die Leinwand = the projection screen

So they are two different surfaces:

• the teacher writes on the Tafel
• the projector shows an image on the Leinwand

German usually uses an article with singular countable nouns much more often than English does.

So Die Lehrerin is the normal way to say the teacher.

Using Lehrerin with no article would sound unusual here unless it were part of a special style, headline, or predicate structure.

Yes.

In this sentence, während is a conjunction meaning while, because it introduces a full clause:

• während der Beamer ... wirft

But während can also be a preposition meaning during:

• während des Unterrichts = during the lesson

So the word can do two different jobs, depending on what comes after it.