This paper on grammatical terminology argued that most had been developed to analyse language rather than teach it. A correction is needed. Antonio de Nebrija, author of the first Spanish grammar, was asked by Queen Isabella why she needed a grammar, as she already spoke Spanish. The reply was that Spain was about to make huge conquests, and the people she conqured needed to learn Spainish. I still think the main point stands, and that teaching children has too often been based on the way adults understand the language, rathr than children who are in the process of learning it. The main sources of terminolgy were the grammatical studies of the renaissance and twentieth-century linguistics, both of which use language that is obscure, and sometimes unnecessarily compressed, the most celebrated example being the "fronted adverbial", which, expanded, means an adverb, adverbial phrase or adverbial clause, placed before the main clause of a sentence. Otherwise known as a starter.
The two most important terms when construcitng a sentence are subject and verb. Both are discussed in the paper, and this short posting add some points on verb groups. Chomsky's idea of the "verb phrase" conveys the idea that a verb can consist of more than one word. Verbs are important words, and like people, can be said to have a name, in English beginning with to - to be, to play, to have. Most of th time, when two verbs work together, the second says its name - I like to play foootball, to eat pizza, to go travelling..etc.
Sometimes, when a verb tells us how well or otherwise we do something - I can/can't play the piano - it is so closely joined to the next verb that the "to" is dropped. David Crystal in The Cambridge Encyclopaedia of English, lists just nine - can, could, may, might, will, would, shall, should, must. There is, fortunately, no interference between the way these verbs work and normal speech, but thee term attached to them - "modal" - is anything but clear. It is a term more often used in German grammar, and David Crystal does not even mention it in his latest book on Grammar, Making Sense. The OED notes it as descending from "mood" 1569, expresing some form of uncertainty, but this does not make sense. Cinderella shall go to the ball expresses something close to certainty rathr than mood, as does "I can't swim". However, the term modal is part of the glossary of the national curriculum, and is included in some SAT testing, so pupils need to know it, whether anyone really knows what it means or not. I would teach the group first, and add the term afterwards, whether or not we can see (modal construction) any point to it. If I had to call these verbs something, it would be "companion verbs". They can't operate on their own - I can, or can't, on its own means nothing, unless we know what it is I can or can't do. The companion may preced the verb, or refer to one in the previous sentence, in a form of tag sentence. You can't do that. Yes, I can. (Whatever it was.)