Concepts used:

Network theory

Geometry

Spatial reasoning

Deductive Reasoning

Inductive Reasoning

Problem solving

Grade level:

Grades 5-12

Subject area:

Math

Standards

NCTM

- Geometry

- Problem Solving

- Reasoning & Proof

- Connections

1. -Representation
   

2. -Technology
   

NETS*S

1b, 3a, 6a, 6b

 Network Theory  (Kmz file)

Objectives

1. •Draw consecutive paths without traversing any segment twice.
   

2. •Write rules for network solutions
   

Lesson Description

Network Theory investigations deal with how paths can be drawn in a continuous circuit.  It is a topic considered to be a part of another theory named Graph Theory (link).  Both concepts come from a branch of geometry known as topology (link).  Topology is the study of how objects relate to one another in space and their properties when stretched or bent.  None of these topics would fall under what most people would consider to be traditional mathematics.  Nevertheless, they contain important mathematical ideas that can be applied to a multitude of other pursuits, such as science, computer science, and economics to name a few.

The students will attempt to solve three types of network “puzzles,” such as the infamous Konigsburg Bridge problem, in Google Earth.  The first type of problems asks student to draw a path around a set of island obstacles, the second type are bridge problems, and the third type are tracing puzzles.  The object in each is to draw a path that connects each point (or vertex)  in a sequence without repeating any segment (or edge) of the path.  Each set of problems gets increasingly difficult and some are unsolvable.  Almost all of the problems need to solved in a certain way by carefully choosing the most efficient paths.

The overall goal of the lesson is for students to make generaliza-tions  or rules on how these problems can be solved.  It is intended to be a constructive learning exercise in that students make sense of the ideas on their own.  Rather than supply them with the correct terminology, it would be better to have the students try to describe these ideas in their own way. 

The key ideas involved are:

1. •Paths can be drawn only if the the number of edges is odd for 0, 1, or 2 vertices.
   

2. •If only one vertex has an odd number of edges, then you must start and end at that vertex.
   

3. •If two vertices has an odd number of edges, then you must start on one and end on the other.
   

4. •If three or more vertices has an odd number of edges, then the path cannot be drawn.
   

Additionally, the students are asked how these ideas can be applied.  The nature of some of the problems, routes for instance, suggests one use.  Any transfer or flow between two points, be it money, mail, computer circuits, or plumbing, should be able to find a use of Network or Graph theories.

Download:

Network Problems - Google Earth Kmz file

Note:  The grade level listed for this lesson goes all the way to 12th grade.  That is because Network and Graph Theory prob-lems can become advanced topics with further investigation.  High school students can try to express the problems using vertex and edge graphs, or explore matrices or algorithms related to graph theory.

Here are some links for advanced study:

Graph Theory Tutorials by Chris Caldwell (link)

MegaMath’s Graph and Graph Theory (link)

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