Module 4: Abstracting

 

 

 

Abstracting as a cognitive tool is the idea of stripping down a complicated idea to one facet or viewpoint in order to highlight that specific facet and present it in a simplified way.  The theme of graphing in algebra has many ideas that work together to communicate the entirety of the themes, so I chose to abstract the intersection point of lines (also know as the graphical solutions to systems of equations).  A line all by itself is made up of many individual properties.  The slope of the line, the location of the line, and the fact that there are an infinite number of points making up the line.  When we add another line to the system, this line also has these properties, but now the point where these two line cross becomes a very significant location.  Of the infinite points making up either line, this one specific point is the only solution for (x,y) that satisfies both equations of the lines.  This point must be highlighted and elevated above all the rest of the points.  The importance of this location on the graph cannot be understated, and this is without even giving a real world context to the lines and system of equations.

 

In my first abstraction, I chose to create an animated GIF.  I stripped the lines down to be represented by strings of dots rather than representing every single point by drawing the solid line.  The three colors are from the RGB scale red, green and blue.  When these three colors of light blend they create white light so the only location where we see white is when the three points intersect and blend together.  Thus the blinking white light at the point of intersection.  By animating the line  I also created an effect such that the lines move differently depending on how you allow your eyes to receive the movements of the points.  The lines can be seen moving in one direction, the opposite direction, moving toward the white point or moving away from the white point.  However, through this all, the point of intersection remains the focus of the GIF.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The second abstraction I did was a word picture.  To highlight the importance of where lines intersect I chose to hide the intersection points from where the line segments intersect while forming each letter in the phrase “INTERSECTING LINES”.  I left enough of the line segments visible that the phrase can still be seen with some effort, however the absence of the intersections highlights the importance of how they help communicate the phrase when they are present.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the theme of graphing it is important to understand that each point on a line represents a solution to the equation of the line.  So in a system of equations the most important of all these points is the one which shares a location on both lines.  This special point is when the two equations are the same. Assuming the equations represent a real situation, this would be the point when the two situations represented by the lines have the same outcome.  The idea of abstracting away the other properties of the lines and highlighting the intersection point further draws attention to this location and amplifies the importance of its existence.