Slope of a Line

SLOPE


ΔY ÷ ΔX      Change in y over the change in x       y₂ – y₁ ÷ x₂ – x₁

As graphing becomes a major part of our early education, the understanding of
the concept of  the slope of a line may become part of that understanding.   Our
students already discuss the idea of a positive or negative correlation within a
data set.  As they observe these correlations it gives them clues into the ideas of
how a line should look if it has a positive or negative slope.

How do we tell the story of slope?  This concept, a basic concept of The Calculus,
is critical in basic Algebra.  How do we teach slope?  Where do we start?  I think
we should start with a walk that begins at the origin of a graph.  We are going to
walk to our right.

FIRST : As we stand at (0,0), the origin we are at a point.  Imagine we move
six paces to the right and stand on (6,0).  Now imagine these are the endpoints
of a line segment.  What is the slope of that line segment?  If we know this we
also know the slope of the X–axis, or the slope of any horizontal line.  From
the formula above:  y₂ – y₁ ÷ x₂ – x₁
                                 0 – 0  ÷  6 – 0
                                       0  ÷  6 = 0
   
            THE SLOPE OF A HORIZONTAL 
                         LINE IS 0!
        
SECOND : From the point (6,0) let's walk to the point (42,6).  I hope we are
walking single file,  I don't want one of us to fall of the line.  Be serious, you
know we can't really walk on a line.  Oh well, it was my attempt at being
humorous.  Back to the slope of the line.
                               y₂ – y₁ ÷ x₂ – x₁
                                  6 – 0 ÷ 42 – 6
                                        6 ÷ 36
                                        1 ÷ 6  or 1/6
We are looking at a positive slope, much the way we think of a positive
correlation in a set of data points.

           THE SLOPE OF A LINE THAT 
     SLOPES FROM THE LOWER LEFT 
    TO THE UPPER RIGHT HAS A POSITIVE 
    SLOPE!


THIRD : We are now standing on the point (42,6).  Next we will walk
to the point (62,2).  We are walking from the upper left to the lower right.
Let's find the slope.
                             y₂ – y₁ ÷ x₂ – x₁
                              6 – 2  ÷  42  – 62
                                   4   ÷  –20
                                    1  ÷  –5  or  – (1/5)
Here we are looking at a segment that has negative slope much like a
negative correlation in a set of data points.

        THE SLOPE OF A LINE THAT 
  SLOPES FROM THE UPPER LEFT TO 
 THE LOWER RIGHT HAS A NEGATIVE 
 SLOPE!


FOURTH : We now stand at (62,2).  From here we can not walk, but we
will slide down the segment to (62,0).  This is a vertical move so we are looking
at the slope of a vertical segment which could be extended to be a
vertical line.
                          y₂ – y₁ ÷ x₂ – x₁
                             2 – 0 ÷ 62 – 62
                                   2 ÷ 0  is undefined, we say "no slope"

        THE SLOPE OF A VERTICAL LINE 
      IS UNDEFINED OR HAS NO SLOPE!


We should now be able to observe a line on a graph and tell if it
has a positive slope, negative slope, a slope of 0, or no slope.  If the
slope is positive or negative then we need to be able to solve for 
that exact value.  It many cases it will take many examples.