As I posted elsewhere, I believe that the "formal" teaching of English Grammar is a waste of valuable class time. Informal "gentle" correction of mistakes within the meaningful context developed from what students are reading, speaking about or listening to is when grammar will begin to be absorbed and be truly learned. The reason that I give the student is always, "When you say "mistake" I am not sure what you mean. Do you mean "mistake" or "your guess to their intent/meaning"?
It is crucial that they understand the "why" of your correction and that the correction comes within the context of "communication is breaking down".
If you feel that direct instruction in English Grammar is a necessary thing, you must tie this to a "true and good" purpose. Because their are about 4 billion people on earth speaking and typing and listening to English, incorrect and garbled English is actually becoming more and more understandable and comprehensible to more and more people. Most students have struggled through situations with poor English Grammar and reached understanding and communication.
The best way to create a perception, on the part of the student who understands that "perfect" English Grammar really isn't essential, that Grammar should be focused on is to set it into a "test preparation mode".
I would take them into my "confidence" with "We know perfect grammar isn't necessary for communication but in the context of preparing for the TOEIC or TOEFL exam it is necessary."
Then, I would introduce the Longman Preparation Series for the appropriate exam/level. TOEIC is all that is really necessary for most learners. TOEFL is much more intense because it is for those intending to go to universities in native speaking countries.
Longman has a very long standing and close relationship to the creators of the tests. Longman listening materials often use the same actors that the test prep people use. They were the first to have the new versions of the tests back in the last revision. They are the best, in my humble opinion.
Even having said this, I still believe that reading, speaking, listening and writing (particularly writing) are the most important and best ways of leading students to better grammar. In every language, published written materials are checked for grammar and contain a high level of correctness. Grammar is best learned and learned most quickly in the context of assimilation with gentle correction, just as we all learned our grammar, than in direct study as some kind of "knowable subject matter" like Algebra or Biology.