Учительница кивнула, и по её лёгкому кивку мы сразу поняли, что можно начинать.

Учительница means female teacher, while учитель usually means male teacher (though in some contexts it can also be used more generally for teacher).

Since the verb later is кивнула, which is feminine past tense, the sentence is clearly talking about a woman, so учительница fits naturally.

Кивнула is the past tense, feminine singular form of кивнуть (to nod).

Russian past tense agrees with gender and number:

• кивнул = he nodded
• кивнула = she nodded
• кивнуло = it nodded
• кивнули = they nodded

So because учительница is feminine, the verb becomes кивнула.

This is an aspect question.

• кивнула is perfective
• кивала is imperfective

Here, кивнула shows a single completed action: she gave one nod.

If you said кивала, it would suggest something more like:

• she was nodding
• she kept nodding
• she nodded repeatedly

In this sentence, the teacher gives one quick nod, and that signal lets everyone understand they can begin, so кивнула is the natural choice.

It means something like:

• from her slight nod
• by her slight nod
• judging from her slight nod
• at her slight nod / signal

The preposition по here is used in a sense like based on or from the sign of.

So the idea is:

We immediately understood from her slight nod that we could start.

It is not literally along her nod or anything like that. This is a very normal Russian way to express understanding something from a sign, gesture, expression, signal, etc.

Because the preposition по here requires the dative case.

The noun is:

• nominative: кивок
• dative: кивку

So:

• по кивку = from/by the nod

This is a very useful pattern to remember:

• по голосу = by the voice
• по лицу = from the face/expression
• по взгляду = by the look/glance
• по жесту = by the gesture

Because её here is a possessive word meaning her:

• её кивок = her nod

A very important point: Russian possessive его, её, их do not change form.

So even though кивку is dative, the possessive stays её:

• её кивок
• по её кивку
• с её кивком

not ей кивку

By contrast, ей is usually the dative form meaning to her.

So:

• я дал ей книгу = I gave her a book
• по её кивку = from her nod

Different functions, different forms.

Because лёгкому has to agree with кивку, which is:

• masculine
• singular
• dative

So the adjective лёгкий changes to лёгкому.

Agreement here is:

• по её лёгкому кивку

All the pieces fit together around кивку.

Also, лёгкий кивок here does not mean light in weight. It means something more like:

• slight
• small
• subtle
• gentle

So по её лёгкому кивку = from her slight/subtle nod.

Again, this is about aspect.

• поняли is perfective
• понимали is imperfective

Поняли means we understood / realized as a completed moment.

That fits the sentence well: the teacher nodded, and at that moment we immediately understood.

If you used понимали, it would sound more like an ongoing state of understanding, which is less natural here.

So:

• мы сразу поняли = we immediately understood
• good for a sudden realization

Because и is joining two full clauses, each with its own predicate:

1. Учительница кивнула
2. мы сразу поняли, что можно начинать

Since these are two separate clauses in a compound sentence, Russian uses a comma before и.

So the structure is:

• [She nodded], and [we immediately understood ...]

Because что introduces a subordinate clause:

• мы сразу поняли
• что можно начинать

Literally:

• we immediately understood
• that it was possible to begin

Russian normally puts a comma before что when it introduces this kind of clause.

This is an impersonal construction.

• можно = it is possible / one may / it is allowed
• начинать = to begin / to start

So можно начинать means:

• we can start
• it’s okay to begin
• we may begin
• it’s time to start (depending on context)

Russian often uses можно + infinitive without naming the subject directly. English often prefers an explicit subject like we.

So even though Russian does not say мы можем начинать, the meaning is understood from context.

Both can be possible in some contexts, but they are not exactly the same.

• начинать = imperfective
• начать = perfective

With можно, the imperfective infinitive is very common when the meaning is simply it is now possible/allowed to start doing something.

So можно начинать sounds natural as:

• now we may begin
• we can get started

If you said можно начать, that would focus more on the single act of beginning. It is also possible in Russian, but можно начинать often sounds smoother in a situation like a teacher giving the signal to begin an activity.

Сразу means:

• immediately
• at once
• right away

It shows that the listeners did not hesitate or need any explanation. The nod was enough, and they understood instantly.

So:

• мы сразу поняли = we understood immediately

The word order is natural and expressive, but not absolutely fixed.

The sentence puts the signal first and then the interpretation:

• Учительница кивнула = first, the action happens
• и по её лёгкому кивку мы сразу поняли = then, based on that signal, we understood
• что можно начинать = what we understood

This order feels very natural in Russian storytelling.

In particular, по её лёгкому кивку appears early because it explains how or from what sign we understood. Russian often places this kind of phrase before the main verb for emphasis or smooth flow.

Literally yes, but in natural English that usually sounds awkward.

Here лёгкий means something more like:

• slight
• small
• subtle
• gentle

So better translations would be:

• her slight nod
• her subtle nod
• a small nod from her

That is a good example of how Russian adjectives do not always match the most literal English adjective in actual usage.