На столе у учителя лежали линейка, маркер и коробка мела.

Because на столе answers the question where? and uses the prepositional case after на.

• на стол = onto the table → direction/motion
• на столе = on the table → location

In this sentence, the objects are already there, so Russian uses the location form:

• На столе = on the table

This is a very common contrast:

• Книга лежит на столе. = The book is lying on the table.
• Положи книгу на стол. = Put the book on the table.

Here у учителя means something like at the teacher’s place / by the teacher / at the teacher’s desk area.

The preposition у often means:

• by
• near
• at
• sometimes belonging to in context

So:

• на столе у учителя = on the table by the teacher / on the teacher’s desk

In a classroom context, this is most naturally understood as on the teacher’s desk.

Because у requires the genitive case.

The base form is:

• учитель = teacher

After у, it becomes:

• у учителя

This is a very common pattern:

• у брата = by/at the brother’s; the brother has
• у дома = by the house
• у окна = by the window

So у учителя is not random—it is the normal case form required by у.

Because the preposition у strongly points you toward the genitive.

The form учителя can look confusing, because it can also be a plural-related form in other contexts. But here the sentence has:

• у учителя

After у, learners expect a genitive form, so the natural reading is:

• учителя = of the teacher / by the teacher

Context helps too. In a classroom sentence about things lying on a desk, у учителя is naturally understood as at the teacher’s desk.

Because the subject is a list of several things:

• линейка
• маркер
• коробка мела

Together, they make a plural subject, so the past-tense verb is plural:

• лежали = were lying

Compare:

• Линейка лежала. = The ruler was lying...
• Линейка и маркер лежали. = The ruler and marker were lying...

Even though each noun is singular by itself, the whole group is plural.

Russian usually uses plural agreement when several nouns are listed as the subject.

So:

• линейка, маркер и коробка мела лежали = standard, natural agreement

You may sometimes see different agreement patterns in other sentence types, especially with word order effects or summary-style phrasing, but for a normal sentence like this, лежали is the expected choice.

Because they are the subjects of the sentence, so they are in the nominative case.

The sentence is basically:

• A ruler, a marker, and a box of chalk were lying...

In Russian, subjects normally appear in the nominative:

• линейка
• маркер
• коробка

Nothing is being acted upon here; these things are simply the items that were lying on the table.

Russian word order is more flexible than English word order.

This sentence starts with the location:

• На столе у учителя = On the teacher’s desk

Then it gives the verb:

• лежали = were lying

Then it names the items.

This order is very natural in Russian when you want to present a scene:

• In that place, there were these things.

A more English-like order is also possible in some contexts:

• Линейка, маркер и коробка мела лежали на столе у учителя.

But the original version sounds very natural and descriptive.

Russian often prefers specific verbs of position instead of a general to be.

• были = were
• лежали = were lying

With objects resting horizontally or simply placed somewhere, лежать is very common.

So Russian often says:

• На столе лежала книга. = A book was lying on the table.
• На полке стояли чашки. = Cups were standing on the shelf.

This gives a more physical, natural picture than just были.

In this sentence, лежали suggests the school supplies were lying on the desk.

Yes, but it changes the feel.

• На столе у учителя были линейка, маркер и коробка мела. = There were a ruler, a marker, and a box of chalk on the teacher’s desk.

This is grammatical, but less vivid.
Лежали is more natural if you want to describe the physical placement of the objects.

So:

• были = neutral existence
• лежали = specific position/location

Because after a container noun like коробка (box), Russian usually puts the thing inside in the genitive case.

So:

• коробка мела = a box of chalk
• literally: box of chalk

This is similar to:

• чашка чая = a cup of tea
• бутылка воды = a bottle of water
• кусок хлеба = a piece of bread

Here:

• мел = chalk
• мела = genitive form, used after коробка

In Russian, мел is often treated as a material/substance noun—basically chalk as a material, not necessarily individual chalk sticks.

That is why:

• коробка мела sounds natural = a box of chalk

If you wanted to be more explicit about individual pieces, Russian could use words like:

• мелки = chalks / crayons (depending on context)
• кусочки мела = pieces of chalk

But in a classroom sentence, коробка мела is a very normal way to say a box of chalk.

Russian has no articles like English a/an/the.

So words like:

• линейка can mean a ruler or the ruler
• маркер can mean a marker or the marker

You figure it out from context.

In this sentence, English will usually translate these as a ruler, a marker, and a box of chalk, unless the context makes them specific.

A helpful breakdown is:

• На столе = on the table
• у учителя = by the teacher / at the teacher’s desk
• лежали = were lying
• линейка, маркер и коробка мела = a ruler, a marker, and a box of chalk

So the structure is roughly:

[location] + [verb] + [things located there]

That pattern is extremely common in Russian when describing what is somewhere.