Одним жестом учитель попросил нас помолчать.

Because the instrumental case is often used to show the means or method by which something is done.

So одним жестом means with one gesture or by a single gesture.

• жест = gesture
• жестом = by/with a gesture
• одним agrees with жестом in case, number, and gender

This is very common in Russian:

• одним словом = with one word
• взмахом руки = with a wave of the hand

So the phrase tells you how the teacher asked: not by speaking, but by making a gesture.

Одним is the instrumental singular form of один.

Here it agrees with жестом, which is masculine singular instrumental.

The forms are:

• nominative masculine: один
• instrumental masculine/neuter: одним

So:

• один жест = one gesture
• одним жестом = with one gesture

Even though one is a number in English, in Russian один behaves much like an adjective, so it changes form to match the noun.

Учитель is in the nominative case because it is the subject of the sentence.

It is the person who performed the action:

• учитель попросил = the teacher asked

Also, попросил is masculine singular past tense, which matches учитель, so we know the subject is masculine.

If the subject were feminine, you would get:

• учительница попросила

This is a very common Russian structure:

попросить + someone in the accusative + infinitive

It means to ask someone to do something.

So here:

• попросил = asked
• нас = us
• помолчать = to be quiet / to keep silent for a while

Together:

• попросил нас помолчать = asked us to be quiet

Other examples:

• Он попросил меня подождать. = He asked me to wait.
• Она попросила нас уйти. = She asked us to leave.

Because with попросить in this structure, the person being asked usually goes in the accusative case.

So:

• попросить кого? = to ask whom?

That gives:

• меня
• тебя
• нас
• его

In your sentence:

• учитель попросил нас = the teacher asked us

This is about aspect.

• молчать = imperfective, to be silent / to keep silent
• помолчать = perfective, to be silent for a while

In requests like this, помолчать is very natural because it suggests be quiet for a bit rather than describing silence as an ongoing state in general.

So the teacher is not making a general statement about silence. He is asking for a limited action:

• Please be quiet for a moment / for some time.

That is why помолчать works well here.

Not exactly.

• помолчать = to remain silent for a while
• замолчать = to fall silent, to become silent

So the difference is:

• замолчать focuses on the moment of becoming quiet
• помолчать focuses on staying quiet for some time

In this sentence, помолчать fits well because the teacher wants the students to keep quiet for a bit.

Compare:

• Учитель попросил нас замолчать. = The teacher asked us to stop talking / fall silent.
• Учитель попросил нас помолчать. = The teacher asked us to be quiet for a while.

Both are possible, but the nuance is slightly different.

Yes.

You could also say:

• Учитель попросил, чтобы мы помолчали.

This also means The teacher asked us to be quiet.

But the infinitive pattern is often more compact and very common when the subject of the infinitive is the person in the accusative:

• попросил нас помолчать

So both are grammatical, but the version with the infinitive is especially natural and concise.

Because попросить is the perfective partner of просить.

• просить = imperfective
• попросить = perfective

In this sentence, the teacher made one completed request, so perfective is natural.

• учитель попросил = the teacher asked

If you used просил, it would usually sound more like:

• he was asking
• he kept asking
• he asked repeatedly
• or it would depend on a larger context

So попросил fits a single finished act very well.

Putting Одним жестом first gives it extra prominence.

The sentence begins by highlighting how the teacher communicated:

• Одним жестом ... = With a single gesture ...

This sounds natural because the gesture is an important part of the scene.

Russian word order is more flexible than English, so other orders are also possible, for example:

• Учитель одним жестом попросил нас помолчать.

That version is also correct. The difference is mainly one of emphasis and style, not basic meaning.

Usually yes, but in natural English it is often best understood as with a single gesture.

It emphasizes that the teacher did not need words. One gesture was enough.

So it can feel both:

• literally: with one gesture
• stylistically: with a single gesture

Because the instrumental case by itself can express means or instrument without a preposition.

So Russian can simply say:

• жестом = by means of a gesture / with a gesture

This is different from English, which usually needs with.

Other examples:

• карандашом = with a pencil
• автобусом = by bus
• улыбкой = with a smile

A native speaker would understand it as something like:

• With a single gesture, the teacher asked us to be quiet.
• The teacher signaled us with one gesture to keep quiet.

The important ideas are:

• the teacher did not need words
• the request was made nonverbally
• the students were expected to stay quiet for a while

So the sentence sounds natural and vivid, especially in narration.