Why is increasing entropy sometimes referred to as "the arrow of time"?

Here's my pick at this:
Entropy (or some form of disorder) will always increase in time.

Thus if you look at snapshots of a system and see that it's entropy is increasing you can tell the flow (or arrow) of time!

Search '2nd Law of thermodynamics' for more info! :)

You need to be more specific.  i really don't know what a "motion map" is.you can draw a graph of an object's position, distance, displacement, speed,velocity, or acceleration versus time, or you can draw a graph of the object's x-position versus its y-position and not give any information about time at all.if the graph presents position or displacement against time,  then constant velocity is a straight line . . i.e. a line that has  constant slope.if the graph shows speed against time, then constant velocityis a horizontal piece of line.if the graph shows acceleration against time, then constant  velocity is a piece of line that lies on the x-axis . . it's value is zero.