Fractions as Operators and Transition to Proof

Fractions as operators are a good way to teach transition to proof.  First, because fractions are a key part of math education. Students stumble on fractions.  A big part of this is likely teachers don’t understand the abstraction or path of abstraction from fractions as operators to fractions as a number class in their own right.

Teaching proofs involving fractions as operators leads to recognizing that an infinite number of natural numbers are not in the domain of fractions, unless the fraction can be reduced to a natural number.

This can be proven as a theorem.  Those who want to use self discovery in a transition to proof class can set this proof as a problem.   Fractions as operators lead to simple sounding propositions that require grasping the concept of proof.

Fractions as operators also lead to recognizing structure in math.  Structure is definitions, theorems, and proofs.  The requirement to prove seemingly obvious statements, forces the student to think about how such proofs are done.   Each new proof  uses the hooks in the definition and prior theorems or axioms to justify each step.  Learning this type of reasoning is part of transition to proof.

Fractions as operators requires understanding fractions as functions.  Functions are a key concept to learn in transition to proof. Functions as sets of ordered pairs are key to understanding proofs about functions.  The recursion theorem of Dedekind illustrates that well.


About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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