One idea runs under all these words: a whole cut into parts. Divide a cake and each person gets a chelek (part); divide a number and you do chiluk (division); divide opinions and you reach a machloket (disagreement). The same root even gives smooth (חָלָק): a surface with no parts sticking out, nothing to catch on. Splitting and smoothness are the two faces of the same cut. Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Notice the split inside the verbs themselves: pa'al and pi'el both mean to divide (pi'el is the active, hands-on distributing), while hif'il jumps to the other sense of the root, smoothness, hence to slip or glide. The passive partners (nif'al, pu'al, huf'al) and the reflexive hitpa'el are not lexicalized here.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.