Your teacher is either confused, over-applying a rule, or using a grammar book that's a hundred years old! In other words, he or she is wrong. In fact, the response "I too" is never used in modern English. You do need to use "I"- the subject pronoun - when you have a verb in the phrase. But if there is no verb, as with " xx too" , we use what's called a disjunctive pronoun - and the first person disjunctive pronoun is ME. It's the same as in French, when you would say 'Moi aussi'. We can also sometimes use the disjunctive pronoun with the verb 'to be', which is why we answer the question "Who's there?" by saying "It's me!" (or 'C'est moi!' in French) There are three ways of responding to the comment 'I'm tired' with short answers. You can say: "So am I." "I am, too." or simply "Me too" Or to respond to a statement such as "I like travelling", the short-answer responses are these: "So do I" "I do, too" or simply "Me too". And don't forget the negative responses - Neither do I, Neither am I, Neither can I etc etc ...or simply "Me neither".

You could also use a "So + verb + I" structure. A: I love traveling. B: So do I. A: I'm tired. B: So am I.

Speaking as an American, I would always say "me too" even though what your teacher says is true, grammatically speaking. Also, I would never say "I'm too" in that manner. Here's an example of idiomatic English: I'm too tired to continue. Me too.

Hi, I give this answer only by the point of view of a general syntax. "Me too" and "I too" are both correct but they are based on different structures besides they don't have a different using. -- "I too" is built upon a Personal Subject Pronoun therefore is intended directly to carry a Verb: --- "I too sing America/ I am the darkest brother...", or ---"I like tea" >> "I too (like tea)" -- "Me too" is built using a Personal OBJECT Pronoun and it is the result of an Argument Complement: "About me, I too ...": eg --- "I like tea" >> "Me too, (I like tea)". To make a comparison with Japanese, this structure is such sistematically overused that it is usually confused with the subject( while it is really not), mainly through the using of the "WA" particle: "Anata wa, kami no ge ka, kuroi desu" ---> "For what concerns you, your hair are black" = "You" is not the real subject. As you might have therefore noticed, the using of one or the other looks to be a personal preference.

'Me too' sounds natural to me but I'm no grammar expert. If I were to use 'I', it would be, 'I as well' not 'I too' but I have no idea why, other than it sounds 'better' to me.