CONCEPT 1 – Directed Line Segments
We can reference the same partition of a line segment by using the different endpoints of the directed segment. The diagram below demonstrates how you can reference the same location using either endpoint of the line segment. Note how the wording changes for these two descriptions.
|Teacher Note: Directed Line Segments are awkward... I'm not sure why we aren't calling these vectors and working with those. Notation order is very important here because it determines the order of the ratio partitioning.
CONCEPT 2 – Dilations
Partitioning a directed line segment can be done using dilation. Let me do a quick review of some key concepts about dilation. The first thing that I want to review and emphasize is that dilation is directly connected to slope. When we use a scale factor of 2, we are actually performing 2 slopes starting from the center of dilation. In other words, we do two runs and two rises to determine the new location. When we use a scale factor of 3, we are actually performing 3 slopes – three runs and three rises to determine the image.
These diagrams demonstrate the relationship between the dilation scale factor and the number of slopes that we do to determine the image. This relationship will be very helpful in partitioning a line segment.
|Teacher Note: The slope is never reduced. It is measuring two things for us... the rise and the run as well as the exact distance to the other end point. While reducing the slope maintains the slope it definately alters the distance. DON'T REDUCE THE SLOPE.
VERY IMPORTANT -- In the first example you might notice that WE DID NOT REDUCE THE SLOPE!!! The slope should not be reduced or altered in any way because those values not only represent the slope they also the distance and the direction to the point. In the same context, in the second example we do not want to move the negative value around. Changing the negative would not affect the slope but it would definitely alter the direction.
CONCEPT 3 – Partitioning a Directed Line Segment
Partitioning a line segment means to divide it up into pieces. To relate this to a dilation it means that we will be doing a reduction (0 < k < 1) so that the point will be on the segment.