Suppose I am giving a math talk and I am going to write on the board Let A be a variable. What do I say while I write? Can I say Let's let A be a variable or should I write We let A be a variable and say the same thing?
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You should simply write and say
and the audience will understand what you are conveying.
are fine but unnecessarily long.
I'd use "Let A be a variable" in speaking, too.
"Let's let A..." is correct grammatically, but sounds unnatural.
I wouldn't use "Let us A...", it isn't correct.
"We let A be..." is correct, but I insist that the first option is the best one.
You're a teacher. Communication is possibly more important than correctness. For this reason, I would use:
Also, this is posed as a command to your students, which is applicable in your situation.
I would say
Another useful phrase would be