You are having lunch with a coworker. You saw another coworker, Lisa, yesterday and earlier today. She usually looks busy and stressed out, but yesterday and today she looked quite happy and relaxed. You wonder if your coworker also saw this. You ask her:
Have you noticed that Lisa's been acting a little strange lately?
You ask "Have you ___?" when you want to know if someone did something that there is still the possiblity of doing now or in the future. If the action is something that had to be done in the past, use "Did you ___?":
Did you see "Titanic" when it was out in theaters?
For something that is still possible, use "Have you ___?":
Have you seen "Titanic"?
To "notice" something is to see details and then realize what they mean. Simply seeing something isn't enough - you have to think about it as well.
You use "that" to express a fact that someone has noticed. What follows "that" is a clause, which is just like a sentence inside of another sentence:
I noticed that he wasn't wearing a wedding ring.
Use "has been ___ing" to describe an ongoing or continuous situation that's still going on now:
Akane's been cooking all day.
Or it can be a situation that you're not sure whether it's finished or it's going to continue:
It's been cloudy all morning, but it looks like the sun might come out soon.
When a person is "acting strange" it means that they are acting differently than they normally do. People often say this about someone when they think that the person has a secret that they are hiding.
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