Learn Arabic (Modern Standard): أخذ الماء يغلي في السخان.

In Arabic, أخذ is one of the "verbs of beginning" (أفعال الشروع). When followed by a subject and a present tense (imperfect) verb, its meaning shifts from "to take" to "to begin doing" the action of that second verb.

If the English translation is "began to boil", why is there no word for "to" (like أن) before يغلي?

Verbs of beginning like أخذ, بدأ, and جعل must be followed directly by a present tense verb without the particle أن (an). Using أن here would actually be grammatically incorrect in Standard Arabic, even though English requires "to".

What case ending does الماء take here, and why?

It takes the nominative case ending (a damma, making it الماءُ). This is because verbs of beginning behave syntactically like كان (was). Therefore, الماء is the formal subject (اسم) of أخذ, and the verbal sentence يغلي acts as its predicate in the accusative position.

What is the pattern of the word السخان (the heater) and how does it help with vocabulary?

It comes from the root س-خ-ن (s-kh-n, related to heat) and uses the pattern فَعّال (fa''aal). This specific pattern is highly productive in Modern Arabic for naming machines, appliances, and tools, such as ثلاجة (fridge) or غسالة (washing machine).