A Physics Problem in Calculus. In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)$ as if it positive or negative. 

Now, various references stated that if we let the upward direction to be positive then $a$ is negative and if we let downward to be the positive direction, then $a$ is positive. The problem in their claim is that they did not explain well how they arrived with that conclusion. 

My question now is that, why is the acceleration $a$ negative if we choose the upward direction to be positive. Note: I need a simple but comprehensive answer. Thanks in advance.
The gravity force is downward and so is the acceleration (by $F=ma$).

So if you choose a downward axis, the acceleration is positive.

And if you choose an upward axis, the acceleration is negative.
You can model this on a linear one dimensional space, the "heigth" $x$ of the particle. Different forces are acting on the ball, on the one hand the gravitation, which is directed to the "ground", on the other hand in the beginning a force is applied towards the "top" as the ball is thrown in that direction. You can derive the equations of motion through Newtons formular $F(t)=m\ddot{x}(t)$ and the superposition of forces, the resulting force is the sum of each force.