Learn Hardcore Russian: Вчера я получил низкую оценку по математике и немного расстроился. - Yesterday I got a low grade in math and felt a bit upset.

Because in this sentence низкая оценка is the direct object of the verb получил (got/received), so it has to be in the accusative case.

The base form is:
- низкая оценка – low grade (nominative, feminine singular)
In accusative feminine singular, the endings change:
- низкая → низкую
- оценка → оценку

Pattern:

Many feminine nouns ending in -а change to -у in the accusative:
книга → книгу, работа → работу, оценка → оценку.
Their agreeing adjectives also change to -ую / -юю:
новая книга → новую книгу, низкая оценка → низкую оценку.

So получил (что?) низкую оценку is grammatically correct: verb + object in accusative.

What exactly does оценка mean here? Is it a “grade”, a “mark”, or something else?

In a school/university context, оценка is the standard word for a grade / mark.

Common uses:

получить оценку – to get a grade
высокая / хорошая оценка – a high / good grade
низкая / плохая оценка – a low / bad grade
итоговая оценка – final grade

Outside school, оценка can also mean assessment / evaluation, but in this sentence with по математике it clearly means a school grade in math.

Why is получил used here and not получал? What is the difference?

This is the difference between perfective and imperfective aspect.

получить (perfective) → получил
Focus: a single completed result.
- Вчера я получил низкую оценку…
  = Yesterday, at one specific moment, I got that grade (completed event).
получать (imperfective) → получал
Focus: process, repeated or habitual action, or incomplete action.
- Вчера я получал низкую оценку…
  Sounds odd in isolation. It would suggest:
  - you were in the process of getting it (but we don’t care about the final result), or
  - you were getting low grades repeatedly over some period yesterday.

For a one-time event with a clear outcome (“I got a low grade”), Russian normally uses the perfective получил.

Why is the verb получил in the masculine form if the subject я has no gender?

In Russian, past tense verbs agree in gender with the subject.

The past tense is formed from the verb stem + -л and then:

masculine singular: usually just the stem + л
получил, пришёл, прочитал
feminine singular: -ла
получила, пришла, прочитала
neuter singular: -ло
было, началось
plural: -ли
получили, пришли

Я itself is gender-neutral, but the speaker’s gender is reflected in the verb:

A man says: Я получил низкую оценку…
A woman says: Я получила низкую оценку…

So получил here tells us that the speaker is male.

What does по mean in по математике, and why is математике in that form?

In this context, по with a school subject means “in (a subject)” / “in the subject of”. It is the usual way to talk about grades in a particular subject:

оценка по математике – a grade in math
оценка по истории – a grade in history
экзамен по биологии – an exam in biology

The preposition по here requires the dative case, and математика in the dative singular becomes математике:

Nominative: математика
Dative: математике

So низкую оценку по математике literally is “a low grade in mathematics”, with по + dative.

Could you use в математике instead of по математике? Would it change the meaning?

For grades and exams, в математике is not the normal phrasing. Native speakers say:

оценка по математике
пятёрка по русскому языку
экзамен по физике

В математике usually means “in mathematics” in a more abstract sense (e.g. in the field of mathematics), not specifically “in my math class as a school subject”.

So:

Я получил низкую оценку по математике – natural, standard.
Я получил низкую оценку в математике – sounds strange or wrong to a native speaker in this context.

What is the function of немного here? Is it like “a little” or “a bit”?

Yes. Немного here is an adverb meaning “a little / a bit / slightly” and it modifies the verb расстроился:

немного расстроился – “got a bit upset”, “was a little upset”

It softens the emotion:

расстроился – he got upset
сильно расстроился – he got very upset
немного расстроился – he got a bit upset (not too much)

Note: немного can also be used with nouns to mean “a little (amount of) something”, followed by genitive:

немного воды – a little water
немного времени – a little time

In your sentence it is clearly an adverb, not a quantity of a noun.

Why is расстроился reflexive (with -ся)? What would расстроил mean?

The verb here is расстроиться (perfective, reflexive) – to get upset, literally “to upset oneself”:

я расстроился – I got upset
она расстроилась – she got upset
они расстроились – they got upset

The -ся ending is the reflexive marker.

Without -ся, расстроить is a different verb: to upset (someone/something), to spoil, to ruin:

Он расстроил меня. – He upset me.
Дождь расстроил наши планы. – The rain ruined our plans.

So:

я расстроился – I became upset (no direct object)
я расстроил его – I upset him (direct object present)

In your sentence we are talking about the speaker’s own emotional reaction, so расстроился (reflexive) is correct.

Can I leave out я and just say Вчера получил низкую оценку по математике и немного расстроился?

Yes, you can. Russian often drops subject pronouns when the subject is clear from context or from the verb form.

Вчера получил низкую оценку…
Still clearly means “I got a low grade yesterday”, especially if the context is you talking about yourself.

In the second part и немного расстроился, the subject я is also understood; you do not need to repeat it:

Вчера получил низкую оценку по математике и немного расстроился.

This sounds natural and conversational. Including я is also fine, just slightly more explicit:

Вчера я получил низкую оценку по математике и немного расстроился.

Can I change the word order, for example to Я вчера получил низкую оценку по математике?

Yes. Russian word order is relatively flexible, and all of these are possible:

Вчера я получил низкую оценку по математике…
Я вчера получил низкую оценку по математике…
Я получил вчера низкую оценку по математике…

The basic meaning stays the same. Differences are mainly in rhythm and emphasis:

Starting with Вчера slightly emphasizes when it happened.
Starting with Я slightly emphasizes the subject I.
Placing вчера right after получил is also fine and neutral in speech.

For a learner, Вчера я получил… or Я вчера получил… are the most typical “neutral” patterns.

Is низкую оценку the most natural way to say “a bad grade”, or are there more common phrases?

Низкую оценку is correct and understandable, but in everyday speech Russians often prefer slightly different wording, depending on the grading system.

Some common options:

плохую оценку – a bad grade (very common and simple)
неудовлетворительную оценку – an unsatisfactory grade (formal, school language)
If grades are numbers (e.g. 2–5 system):
- получил двойку по математике – got a “2” in math (very natural in Russia)
- получил тройку – got a “3” (barely passing / mediocre)

So you could also say:

Вчера я получил плохую оценку по математике и немного расстроился.
Вчера я получил двойку по математике и немного расстроился.

Низкую оценку is fine, just a bit more literal (“low grade”) than people often say in casual conversation.

How do you pronounce this sentence, and where is the stress in each word?

Here is the sentence with the stressed vowels marked by an accent:

Вчера́ я получи́л ни́зкую оце́нку по матема́тике и немно́го расстро́ился.

Word by word:

Вчера́ – vcherá (stress on -ра́)
я – ya (only one syllable, no stress mark needed)
получи́л – poluchíł (stress on -и́л)
ни́зкую – nízku-yu (stress on ни́-)
оце́нку – otsénku (stress on -це́н-)
по – po (unstressed)
матема́тике – matemátike (stress on -ма́-)
и – i (unstressed)
немно́го – nemnógo (stress on -но́-)
расстро́ился – rasstróilsya (stress on -стро́й-)

You can think of the main rhythmic peaks as:
вчера́ / получи́л / ни́зкую / оце́нку / по матема́тике / немно́го / расстро́ился.