Commutators and student mistakes pre-service teacher projects

For pre-service teachers, projects can be given to develop lesson plans to use commutators to teach students to understand common errors.

Distributive Law Error

a*(x+y) = ax

Define a commutator

a*(x+y) – ax = ay

We can call this a fallacy distributive commutator.
One definition might be

[a,x,y] = a*(x+y) – ax = ay

Using this definition, we have that

[a,y,x] = ax

[a,x,y] + [a,y,x] = ax + ay = a*(x+y)

We might write this as

[a,(x,y)] = a*(x+y) – ax = ay

Project: Identify common errors with fractions. Show how using the fraction commutator [p,q] = p/q – q/p can identify these errors. Develop a lesson plan for students to compute these commutators.

Project: In algebra, a student might think a/(b+x) = a/b or a/x. Develop a commutator for this error. What are its properties?


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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