Dots Intersected By Sloping Lines

Here is another game of mine played on a grid drawn on paper.
It seems like it might be a fun way to teach slopes or how to reduce
fractions.
Start by lightly drawing an n-by-n grid on paper, or better yet, use
graph paper.
(I suggest an n of about 8 for starters.)
Game is for any number of players >= 2.
First, player 1 draws a dot at any intersection of grid-lines (at a
lattice-point).
He then connects one of the grid's corners to this dot with a straight
line-segment.
Players alternate moving. On each move, a player first draws a dot at a
lattice-point of the grid and then connects the last dot drawn by the
previous player to the newly drawn dot by a
*straight* line-segment (where the segment terminates at these 2 dots).
The new dot must be drawn so that no line-segments coincide. (The
segments may cross, but they cannot be on top of each other with the
same slope.)
Also, the dots cannot be drawn on previously drawn segments or dots.
A player drawing a line-segment gets a point for every previously drawn
dot his/her line-segment intersects, considering the *exact* slope of
the segment and *exact* positions of the intersected dots
*in theory*, as if the dots and line-segments and grid had been drawn
perfectly.
So, for instance, if we have a dot at (2,2), and the last drawn dot is
at (0,1), and the player makes a new dot at (6,4), no matter how poorly
the lines and dots and grid are drawn, the player still gets a point
for the "intersected" point (2,2) (which was theoretically intersected
by the line-segment of slope -1/2).
The game continues until either a predetermined score is reached by one
player, or until every lattice-point is intersected by a lines-segment
or has a dot drawn on it.

thanks,
Leroy Quet