I do understand that the parenthesis can be dropped and ultimately it leads to the same end results. What I don’t understand is that when teachers teach, they often miss out this step: Like in the case of 3(x+4)-6(3+x)...They miss out (3x+12)+(-18-6x), they jump straight to 3x+12-18-6x. Is that middle step non-existent? Is that something made up by my mind?
I don't think I can tell you
why (or even
whether) the majority of teachers would skip that particular step. That's up to them. Most likely it would be because they are talking to students whom they expect to have passed that hurdle, so that they focus their attention on other issues. I suppose I probably would skip it for most students, though of course not the first time I did an example like this.
I hope I made it clear in my response to you that the step with parentheses
is entirely valid. It is just such a small step to the next form that we often don't bother to write it.
Actually, in your new example (which is considerably trickier than the first), there are many other steps that might be shown, if one wanted to show every detail:
3(x + 4) - 6(3 + x)
3(x + 4) + -[6(3 + x]
3(x + 4) + -1[6(3 + x)]
3(x + 4) + (-1*6)(3 + x)
3(x + 4) + (-6)(3 + x)
[3*x + 3*4] + [(-6)3 + (-6)x]
[3x + 12] + [-18 + -6x]
3x + 12 + -18 + -6x
3x + -6x + 12 + -18
(3 + -6)x + (12 + -18)
-3x + -6
-3x - 6
I would never write out all those steps; but that is what I am actually doing in simplifying the expression. We
must omit details at
some point in order to communicate clearly; and that requires assuming something about the reader or student.
The trouble is that any teacher of a group of students (and especially any author writing to many students they will never even see) can't know what each of them does or does not already know, and therefore will make some choices that are not suitable for all of them.
That's part of the reason I prefer tutoring individually. But even then, I have to find out what you are ready for by trying things out and seeing what results I get.
But back to your question, no, it is not your imagination. The step exists; it is just one that teachers may easily assume students can figure out for themselves.