I was teaching someone that she can say either "I will walk" or "I will be walking" but not "I will be walk" - She asked "Why can't I say so?" - Can someone say why it is wrong?
Whether we describe words like "will," "can," and "may" as modal verbs or auxiliary verbs is a matter of definition. But to maintain clarity, let's reserve auxiliary verbs to forms of "be" and "have." Then the modal verbs to indicate futurity are "will" and "shall."
The form of the simple future tense is the appropriate modal of futurity plus the main verb." Example: "He will explain it at the next meeting."
The form of the future progressive tense is the simple future of "be" plus the main verb's present participle. Example: "He will be explaining it during the demonstration." Like all progressive tenses, the time is indicated by the appropriate simple tense of the auxiliary verb, and the progressive aspect is indicated by using the auxiliary verb "be" plus the main verb's present participle.
The form of the future perfect tense is the simple future of "have" plus the main verb's past participle. Example: "He will have explained it long before I arrive." Like all perfect tenses, the time is indicated by the appropriate simple tense of the auxiliary verb, and the perfected aspect is indicated by using the auxiliary verb "have" plus the main verb's past participle.
The form simple future of "be" plus the main verb is not a form of any known tense in English and so has no meaning.
"I will be walking"
Is using the present continuous tense to talk about a future action:
... is formed from the present tense of the verb be and the present participle (-ing form) of a verb. - britishcouncil.org
I will walk is in the future tense, and the future tense does not require an auxiliary verb.
See the conjugations of the verb walk:
I will walk - reverso.net
The auxiliary verb be is used in a variety of situations but not with the future tense. It is used as already mentioned with the present continuous tense: walking:
1 Used with a present participle to form continuous tenses. ‘they are coming’ ‘he had been reading’ ‘she will be waiting’ - OLD.
For a full list of the uses of the auxiliary verb be you can see the definition of the auxiliary verb above, where they are listed.