Learn Hardcore Russian: Учитель хочет представить новый план.

If you said Учитель представляет новый план, that would mean The teacher is presenting / presents a new plan (a separate sentence with only one verb), not wants to present.

Why is it хочет and not something like хотит?

Хотеть is irregular in the present tense. Its main forms are:

я хочу – I want
ты хочешь – you want (singular, informal)
он / она / оно хочет – he / she / it wants
мы хотим – we want
вы хотите – you want (plural / formal)
они хотят – they want

So for учитель (he/she as “the teacher”), you must use хочет (3rd person singular), not хотит.

What case is новый план in, and why?

Новый план is in the accusative case as a direct object of the verb представить.

Кто? – Учитель (nominative subject)
Что хочет сделать? – представить (infinitive verb)
Что представить? – новый план (accusative object)

For inanimate masculine nouns, the accusative form is the same as the nominative:

nominative: новый план
accusative: новый план (no change)

Why isn’t it нового плана instead of новый план?

Нового плана is the genitive case, not accusative. You use genitive in other contexts, for example:

after some prepositions: у нового плана (by / at the new plan – context-dependent)
to show “of”: детали нового плана (details of the new plan)
often in negative sentences: Он не имеет нового плана. (He doesn’t have a new plan.)

Here we simply have a direct object of представить, so we use the accusative:

представить (что?) новый план – to present a new plan

Why is план masculine, and how do we know that новый must be masculine too?

План is grammatically masculine. Many Russian nouns ending in a consonant (like план) are masculine.

Adjectives must agree with the noun in:

gender (masculine / feminine / neuter),
number (singular / plural),
case.

So in singular accusative with a masculine inanimate noun план, we use the masculine form of новый:

masculine: новый план
feminine: новая идея (new idea)
neuter: новое правило (new rule)
plural: новые планы (new plans)

What is the difference between представить and представлять?

This is an aspect difference:

представить – perfective, focuses on a single completed action:
- Учитель хочет представить новый план.
  The teacher wants to (at some point) present a new plan (as one event).
представлять – imperfective, focuses on a process, repeated or habitual actions:
- Учитель часто представляет новые планы.
  The teacher often presents new plans.
- Он будет представлять свой проект завтра.
  He will be presenting his project tomorrow (process / ongoing).

With хотеть, using представить suggests wanting the result (to have it presented once).
Using представлять with хочет is possible in different contexts, e.g. Он хочет представлять интересы компании. (He wants to represent the company’s interests – an ongoing role).

Does the perfective infinitive представить here refer to the future?

Yes, a perfective infinitive usually refers to a future completed action when combined with a present-tense verb of desire, intention, obligation, etc.

Учитель хочет представить новый план.
Literally: The teacher wants (now) to present (once, in the future) a new plan.

The wanting is in the present, the presenting is expected to happen in the future as a completed action.

Do we need a word for “to” before представить, like in English “to present”?

No. In Russian, the infinitive itself corresponds to “to + verb” in English.

English: wants to present
Russian: хочет представить

You don’t add an extra word for “to”; the infinitive ending (-ть in представить) already encodes that.

Why is there no word for “a” or “the” in учитель or новый план?

Russian has no articles (a, an, the). Definiteness or indefiniteness is understood from context, word order, and sometimes stress or intonation.

So Учитель хочет представить новый план can mean:

The teacher wants to present a new plan, or
A teacher wants to present a new plan, depending on context.

Learners must get used to understanding “a/the” from the situation, not from a specific word.

Is учитель always male? How do you say female teacher?

Grammatically, учитель is masculine, and traditionally refers to a male teacher.

For a clearly female teacher, you can use:

учительница – a feminine form.

Examples:

Учитель хочет представить новый план. – The (male) teacher wants to present a new plan.
Учительница хочет представить новый план. – The (female) teacher wants to present a new plan.

In modern usage, some people may still say учитель about a female teacher (especially in official contexts), but the grammar (adjectives, past tense verbs) will remain masculine in that case.

Can we change the word order, like Учитель новый план хочет представить?

Yes, Russian allows more flexible word order than English. You can say:

Учитель хочет представить новый план. (neutral, most common)
Учитель новый план хочет представить.
Новый план хочет представить учитель.

The meaning stays essentially the same, but the emphasis changes:

Putting новый план earlier can emphasize the new plan.
Putting учитель at the end can highlight who specifically wants to present it.

For learners, the safest and most neutral pattern is:

Subject – verb – object: Учитель хочет представить новый план.

Why doesn’t представить take a preposition, like представить к новому плану or something similar?

Because представить in the sense of “to present / introduce something” is a transitive verb that normally takes a direct object in the accusative case without a preposition.

представить (что?) новый план – to present a new plan
представить (кого?) студента группе – to introduce a student to the group

You would use prepositions with представить in other meanings or structures (e.g., представить кого-то кому-то – to introduce someone to someone), but not for the simple “present a plan” pattern in this sentence.

Does представить only mean “to present,” or can it also mean “to imagine”?

Представить has several meanings, depending on context:

To present / introduce
- Учитель хочет представить новый план.
  The teacher wants to present a new plan.
To introduce (people)
- Разрешите представить моего друга.
  Allow me to introduce my friend.
To imagine – usually as представить себе (“to imagine to oneself”)
- Представь себе, что ты в Москве.
  Imagine that you are in Moscow.

In your sentence, from the object новый план, context clearly tells us it means to present / introduce (not “imagine”).