Lookagain,
The extra step was added specifically with you in mind because if someone does not know that tt=t^2,
they may also not understand what x/t/t represents.
The student needs to know prior to this problem that tt = t^2. If not, then the student is in the wrong
class, or needs remedial work to catch up in that class.
You have the reasoning backwards.
Let's put aside that algebraically that "x/t/t"
** is in awkward form.
It is unreasonable for the student to understand that at the beginning, as you
started your first step uisng the x/t/t kind of idea without showing where it came from.
So then the rest of the steps after that wouldn't matter, because the understanding
would have stopped at that point.
That's akin to the "door in the face" technique. It shuts things down.
However, if you had 1) started with your tt use (and explained why you were using
it instead of t^2), and 2) then explained why you were coming up with the
form of x/t/t, then the motivation for all of the steps to be understood and flow
together would have been there. That route would have been akin to the
"foot in the door" technique. And the foot pushes the door open wider and wider
for continued understanding for the justification of all of the steps.
- - - - - - - - - - -- - - - - - - -
** I'll make the distinction that I am not discouraging this use in general.
A foot/sec/sec (foot per second per second) which is equivalent to \(\displaystyle foot/sec^2\),
has quicker meaning to me, than the one with the exponent of 2 on "sec."
