Negative values for velocity were introduced early in my
11th-grade physics course (Chapter 5, "Motion along a
straight-line path"). They follow directly from the math
we learned in 7th grade.

If you want to measure positions and velocities of things,
you probably start doing so without even thinking about
the positive and negative directions. If I'm driving a
car, the forward direction naturally gets positive values,
and if I later need to drive backwards, it naturally gets
negative values. If my language is written left-to-right,
I will naturally count up from zero from left to right,
and will naturally assign higher positive values to greater
distances and speeds to the right. If I then want to find
distances or speeds to the left, they will naturally get
negative values. If I am launching rockets, I would
naturally count my progress in altitude or vertical speed
as larger positive values, and start running when I see
large negative values.

On the other hand, if I were drilling into the Earth or
diving under water, I might assign larger positive values
to greater depths and speeds of descent, and later, as the
need arises, assign negative values to heights above the
waterline and speeds of ascent.

Using a single continuum for positive and negative values
of any physical quantity is much easier to deal with
mathematically and numerically than using two separate
equations to deal with motion in opposite directions, and
seems obviously to make much more sense.

-- Jeff, in Minneapolis