There are a few grammatical differences between British English and American English:
UNIT 7A-B and 13A
BRITISH
The present perfect is used for an action in the past with a result now:
* I've lost my key. Have you seen it?
* Sally isn't here. She's gone out.
The present perfect is used with just, already and yet:
* A: What time is he leaving?
B: He has already left.
* Have you finished your work yet?
AMERICAN
The present perfect OR past simple can be used:
* I've lost my key. Have you seen it? or I lost my key. Did you see it?
* Sally isn't here. She's gone out. She went out.
The present perfect OR past simple can be used:
* I'm not hungry. I've just had lunch. I've just had lunch.
* A: What time is he leaving?
B: He has already left. He already left.
* Have you finished your work yet? or Did you finish your work yet?
UNIT 17B
BRITISH: have a bath/have a shower
AMERICAN: take a bath/take a shower
UNIT 22D
BRITISH
Will or shall can be used with I/we:
* I will/shall be late this evening. The questions shall I...? and shall we...? are used to ask for advice etc.:
* Which way shall we go?
AMERICAN
Shall is unusual:
* I will be late this evening. Should I...? and should we...? are used to ask for advice etc.:
* Which way should we go?
UNIT 32B
BRITISH
You can use needn't (do) or don't need to (do):
* We needn't hurry. or We don't need to hurry.
AMERICAN
Needn't is unusual. The usual form is don't need to:
* We don't need to hurry.
UNIT 34 A-B
BRITISH
After demand, insist etc. you can use should:
* I demanded that he should apologize.
* We insisted that something should be done about the problem.
AMERICAN:
The subjunctive is normally used. Should is unusual after demand, insist etc.:
* I demanded that he apologize.
* We insisted that something be done about the problem.
Many verbs ending in ~ise in British English (apologise/organise/specialise etc.) are spelt with ~ize (apologize/organize/specialize etc.) in American English.
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