Law of Trichotomy and Solution Sets

Share Cognitive Science Implications for Teaching Mathematics
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I (and other teachers) have learned that sometimes examples we present do not accomplish their intended purpose.  Students frequently focus on something other than the principle illustration.  That may well have happened with the illustrations of The Law of Trichotomy presented from November 1, through December 27.  Surely everyone who reads these illustrations recognizes that The Law of Trichotomy is important in each of the examples.

However, the unifying properties may have eluded some readers.  These are revealed when all the examples are viewed in total.  They are not obvious from any single one of the examples.  The collection of all these examples reveal:

  • The graph of an equation or inequality is a picture of its solution set.
  • An equation and its two inequality siblings are closely related by The Law of Trichotomy.
  • When considering an equation or inequality, simultaneously consider its two siblings.
  • The union of solutions sets of an equation and its two inequality siblings is the Real numbers.
  • The three solution sets (equation and two inequalities) have no elements in common.
  • The graph of an equation is the boundary between the graphs of the two inequality siblings.
  • When the three solution sets are graphed on the same coordinate system, the entire system is used.
  • Solving and one of the three provides the solutions of the other two.
  • To solve an equation or an inequality, solve the easy one and deduce the desired solution set.

We want our students to use this as the basis for solving any equation or inequality.  Such a view of equations and inequalities will facilitate the student’s ability to solve new problems.

Teachers must realize that this process will be tweaked to accommodate future complications (variables in the denominator, square roots, etc.).  When appropriate that tweaking process should be presented as a normal mathematics process.

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