Learn Hardcore Italian: È lo scoiattolo a cui diamo spesso una nocciola quando studiamo matematica all’aperto. - It is the squirrel to which we often give a hazelnut when we study mathematics outdoors.

C’è (short for ci è) means “there is” or “there are” and introduces existence.
È simply means “it is” or “he/she/it is.”
In our sentence, È lo scoiattolo… is a cleft construction used to put focus on lo scoiattolo (“It’s the squirrel…”), not to state mere existence.

Why do we use lo before scoiattolo instead of il?

Italian has special rules for the definite article:

Use lo before masculine nouns starting with s + consonant, z, ps, pn, x, y, or gn.
Scoiattolo begins with s + c, so its article is lo, not il.

What does a cui mean, and why do we use cui here instead of che or quale?

Cui is a relative pronoun that must be paired with a preposition when linking back to its antecedent.
Here the verb dare (“to give”) requires an indirect object introduced by a (“to”), so we get a cui (“to whom”).
You can’t use che after a preposition, and while al quale is possible, a cui is more common in everyday speech.

Could we replace a cui with al quale? What’s the difference?

Yes:

Al quale = “to which” (more formal, often written style).
A cui = “to whom/which” (colloquial, very common in spoken Italian).
Both mean the same grammatically; choice depends on register and style.

Why is spesso placed after diamo? Could we also say spesso diamo?

Adverbs of frequency in Italian are quite flexible:

You can put spesso before the verb (spesso diamo) or after it (diamo spesso).
Putting it after the verb is very normal in spoken Italian; before the verb can sound a bit more emphatic or formal.

Why is there no article before matematica?

When talking about subjects of study (or activities) in a general sense, Italian often drops the article:

Studiamo matematica = “We study mathematics” (in general).
If you were referring to a specific math course, you might say la matematica.

What does all’aperto mean, and why the apostrophe?

All’aperto literally means “in the open (air),” i.e. “outdoors.”
It’s a contraction of a
- l’
  - aperto (where aperto has been nominalized as “the open-air space”).
The apostrophe marks the elision of l’ before the vowel a in aperto.

Why is una nocciola singular? Could we say delle nocciole instead?

Una nocciola emphasizes giving one single hazelnut each time.
Delle nocciole would mean “some hazelnuts” (plural), implying multiple nuts per occasion.
Choose singular or plural depending on how many nuts you want to highlight.