Module 6: Modeling and Dimensional Thinking

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Modeling is the process of distilling down or abstracting the most important properties or functions of an object or idea in order to present the idea in a more concrete and understandable form.  My theme of graphing was a large focus of this chapter on modeling because graphing is a model of a situation.  The Root-Bernsteins even discussed how every equation we can make can be given a real context.  Through teaching graphing I have come to see that there is a huge difference or gap between studying problems just to learn algorithms and techniques versus studying real world problems.  Slope and y-intercept on a graph and in an equation are abstract ideas that when disconnected from the real world seem trivial and useless.  However, when we start to give meaning to the slope (a rate of change) and the y-intercept (a starting place or amount), the equations take on real connections to our lives.  Even the most complex of equations can be given a real context and vice versa, some argue everything in life can be modeled by mathematics.  I am not sure I would go as far as to say everything is mathematics because I’m not so sure about things like love and faith being described by math, however, nearly everything can be explained with equations

 

One of the interesting things about my topic of graphing is that it also lends itself to modeling in multiple dimensions.  The first dimension can be represented on the number line where x represents a location.  The second dimension can be represented by the coordinate plane (2 number lines) and locations represented by (x, y).  The third dimension can be modeled by adding a third number line and a third coordinate (x, y, z). 

 

For my graphical representation I decided to take a 3D object and distill one perspective of it into a graph.  I chose to take stairs and look at them just from their function of taking us vertically all while traveling horizontal as well.  With my graphical model and equation I compared the horizontal distance one travels on stairs (x) to the vertical distance one travels (y).  The result prompts discussion questions such as what slope for stairs would be too high or low?  What if we don’t have enough horizontal space to travel the vertical distance with an acceptable slope?  Are there other designs for staircases that may be more efficient (spiral staircase)?  I think this method of taking an everyday, seemingly trivial object and modeling it with mathematics helps people start to see mathematics more and more in their lives and gives more meaning to what they are studying.