One root holds the number itself, the act of multiplying by three, and the fraction you get when you cut by three. The cardinal שלושה (three), the ordinal שלישי (third), the verb לשלש (to triple) and the noun שליש (a third) are not four vocabulary items: they are the same three letters bent into number, action, rank and portion. Learn the root and the whole arithmetic family opens at once. 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.
Only the pi'el is lexicalized here, and that fits its job: pi'el is the intensive / factitive voice, so it naturally means "make threefold". The number-words below are nouns and adjectives, not verbs, which is why this root lives mostly outside the binyan system.
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.