Learn Hardcore Russian: Учитель объясняет правило другим студентам. - The teacher explains the rule to other students.

Russian present tense of an imperfective verb covers both simple present and present continuous meanings in English. Context decides which sounds more natural in translation.

Why is правило in this form? What case is it, and why does it look like the dictionary form?

Правило is in the accusative singular (direct object of the verb).

Dictionary form: правило (nominative singular, neuter).
Accusative singular of most inanimate neuter nouns is identical to the nominative singular.

So:

Nominative: правило (the rule – subject)
Accusative: правило (the rule – object)

In this sentence, правило is what is being explained, so it is the direct object → accusative case. It just happens to look the same as the nominative.

Why do we say другим студентам and not something with a preposition like к студентам for “to the other students”?

Russian often uses the dative case without a preposition to express the idea of “to someone”.

другим студентам is dative plural:
- другой → другим (dative plural)
- студент → студентам (dative plural)

So другим студентам literally means “to (the) other students”.

You could say объяснять что‑то студентам (to explain something to students) quite naturally. A preposition like к is usually used to show physical motion towards something (e.g. идти к студентам – to go to the students), not for indirect objects of verbs like объяснять.

Why is другим in that form? What is it agreeing with?

Другим is the dative plural form of the adjective/pronoun другой (other).

It must agree with the noun студентам in:

gender: (plural, so gender-neutral plural)
number: plural
case: dative

So you get:

Nominative plural: другие студенты – other students
Dative plural: другим студентам – to other students

The ‑им ending is typical for dative plural of adjectives and some pronouns in modern standard Russian.

What case is студентам, and how is that form built?

Студентам is dative plural of студент (student).

Declension:

Nominative plural: студенты (students)
Dative plural: студентам (to students)

For many masculine nouns ending in a consonant, the dative plural ending is ‑ам:

друг → друзьям
учитель → учителям
студент → студентам

Here студентам marks the indirect object: the recipients of the explanation.

Why is there no word for “the” or “a” in this sentence?

Russian has no articles (no direct equivalents of “a/an” or “the”).

So учитель, правило, and студентам can be translated, depending on context, as:

учитель – a teacher / the teacher
правило – a rule / the rule
другим студентам – to other students / to the other students

Russian relies on context, word order, and sometimes additional words (like этот – this, тот – that) to convey the same distinctions that English shows with articles.

Is учитель always masculine? How would you refer to a female teacher?

Учитель is grammatically masculine. It can refer to:

a male teacher
a teacher of unknown or irrelevant gender
occasionally, a female teacher (especially in formal, job-title style)

There is a distinctly feminine form:

учительница – a female teacher (especially in school contexts)

So you might say:

Учитель объясняет правило… – The teacher explains the rule… (gender-neutral or male)
Учительница объясняет правило… – The (female) teacher explains the rule…

What is the perfective partner of объясняет, and how would that change the meaning?

The imperfective verb is объяснять → объясняет (he/she explains / is explaining).

The common perfective partner is объяснить:

объяснит (3rd person singular future): he/she will explain / will have explained

Difference:

объяснять (imperfective): focuses on the process, repeated or ongoing action.
- Учитель объясняет правило. – The teacher is in the process of explaining / habitually explains the rule.
объяснить (perfective): focuses on the result, completion.
- Учитель объяснит правило. – The teacher will explain the rule (and finish explaining it).

You normally don’t use perfective in the present tense form to mean real present; объяснит is future.

Can I change the word order, like Учитель другим студентам объясняет правило? Does it change the meaning?

Yes, Russian word order is more flexible than English. All of these are grammatically correct:

Учитель объясняет правило другим студентам.
Учитель другим студентам объясняет правило.
Правило учитель объясняет другим студентам.
Другим студентам учитель объясняет правило.

The basic meaning stays the same (teacher – explains – rule – to other students), but the emphasis shifts:

Starting with учитель: focus on who is doing it.
Starting with правило: focus on what is being explained.
Starting with другим студентам: focus on to whom it is being explained.

The version you gave is the most “neutral” or textbook‑like order.

How do you pronounce объясняет? Is the б silent?

Объясняет is pronounced approximately like: [ab‑yis‑NYA‑yet].

Details:

о in unstressed position sounds more like a: [a]
б is not silent, but it blends with the following ъ / я and palatalization. To an English ear it can sound subtle.
объяс‑ is something like [ab‑yis‑] (with a quick b).
Stress is on ‑ня‑: объяс/ня́/ет → объясня́ет

So: ob‑yas‑NYA‑yet (with the stress on NYA).

What is the difference between студент and ученик? Could I say ученикам here?

Студент usually means university/college student (or similar higher‑education context).
Ученик usually means school pupil (primary/secondary school).

You could say:

Учитель объясняет правило другим ученикам. – The teacher explains the rule to the other pupils.

The grammar is the same (dative plural: ученикам), but the age/educational level implied is different.

Why is there no word like “to” before другим студентам?

In Russian, the dative case itself often carries the meaning “to” or “for”, so no separate word is needed.

English: to other students
Russian: другим студентам
(dative plural → “to other students”)

So the function of English “to” is built into the ‑им / ‑ам dative endings in другим and студентам.