Modal verbs and modal auxiliaries (such as 'can' and 'should') are verbs that can be characterised as follows:
1) They have no non-finite form 
2) They don’t take an ‘s' for the 3rd person singular. 
3) They cannot be used with other modals in a sentence 
4) They are inverted for questions e.g. - Would you like a cup of tea? -Yes, I would.

Students find it difficult to learn modals and modal auxiliaries, firstly, because they take a myriad of forms, some of which have subtle distinctions. For example; can doesn't take the auxiliary do in a question (NOT do you can swim?) whereas have to does (Do you have to go to work tomorrow?).

Crucially, modals are different to other auxiliary verbs because they have meaning. Students need to be able to distinguish a modal verb's grammatical as well as semantic properties. There are different categories of meaning and the same modal is often used to express different concepts (You really must call your Mum (strong advice) / You must do your homework (obligation)/ you must be Dave's brother (deduction).

Although we don't teach them to our students in this way, it maybe helpful to know about the three categories of modals:

1) Deontic Modality: Used to offer, request, command and grant permission etc. e.g. Would you pass me the salt? You must be home by nine o'clock.

2) Epistemic Modality: Used to express the speaker's opinion about the truth of a proposition. E.g. You can't have finished already – The speaker is saying that the proposition isn't true.

3) Dynamic Modality: Refers to the subject of the sentence, not the speaker e.g. Jim can speak five languages. This is talking about the subject's abilities. It doesn't give information about the speaker.

Another difficulty for students is that we often use adjectives or passive structures to express these ideas of modality. E.g. Students are obliged to return their books by the end of term. Students who use these structures may sound a little stilted at times simply because the English language has a preference for modal verbs rather than modal adjectives.

Students also find that sometimes the negatives for modals don't always follow the pattern of other verbs. Consider this:

When is don't have to the opposite of must rather than mustn't?

Don't have to means it is unnecessary whereas mustn't means 'don't do it'.

When is can't the opposite of must?

So, modals have different meanings and different opposites in different contexts.

Consider which modals and semi modals (e.g. have to, used to, need etc. 'ones which don't follow the basic rules of modals; have past forms, third person forms, etc.) can be used to express the following concepts:

Ability: can, could.

I can play the piano. I could play the piano when I was younger

Possibility and probability: may, might, could (but NOT can), should, be going to, can't, supposed to.

Paul may/might/could have got lost. I'm not sure he had a map. 
Paul should be on his way now. 
I think it is going to rain. 
You can't have seen Grandma on the bus- she died years ago. 
According to his directions, this is supposed to be the bar where they serve the crazy cobra-blood drink and have yaks grazing in the toilets.

Imaginary and conditional situations: would, might, could, can, should, may should. Consider the difference in meaning between the following: 

If I had the money, I would buy it. 
If I had the money, I could buy it. 
If you have the time, you should do it. 
If you have time, you may/can do it. 
If we have time, we might do it

Deduction, expectation and conclusion: must, might, should, could, can't (but NOT mustn't), supposed to.

He must be Russian, he's drunk by midday on a Saturday afternoon. 
He might/could be Russian, he's paying for her meal. 
He can't be Russian; he's smiling on the metro. 
We're supposed to leave now. What's the delay? 
We should be leaving now. What's the delay?

Willingness, spontaneous decisions and promises: will, have to, would

The car won't start. 
I really have to start going to the gym.
I will do it tomorrow - you have my word. 
I would do it but I haven't got enough grease.

Reporting: can, could, should, might, may, must, would.

He said he can do it. 
He said he could do it. 
He said he should do it. 
He said he might do it. 
He said he may do it. 
He said he must do it. 
He said he would do it.

Making requests: may, could, might, can, should (formal 1st conditional).

May I have a glass of water? 
Could I have a glass of water? 
Might I have a glass of water? 
Should I have a glass of water, would it inconvenience you?

Making suggestions: may, might, could, can.

You may want to rephrase that. 
You might want to rephrase that. 
You could rephrase that. 
You can rephrase that.

Past habits: used to, would.

I used to play hockey at school. My grandfather would play with us in the garden for hours on end.

Obligation: must, have to, should, ought to, had better, need (n't), supposed to.

You must complete your project. 
You have to complete your project. 
You had better complete your project. 
You should complete your project. 
You ought to complete your project. 
You are supposed to complete your project. 
You need to complete your project. 
You needn't complete your project.

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