## Do We Really Need Least Common Denominators?

Using least common denominators (LCD’s) to add and subtract rational expressions (RE’s) is tedious and in many cases unnecessary.  I am attempting to teach the entire algebra chapter on rational expressions without mentioning them.  Instead I use “criss-cross applesauce (CCA)“, the product of the means equals the product of the extremes and the distributive property (factoring) – all useful in other contexts –  to quickly simplify and solve.  This way students have to remember fewer special cases  and fewer specialized procedures.  Of course the explanation for CCA requires  application of common denominators (not necessarily least).  After that why not cut to the chase.

So, after learning about reducing to lowest terms (Factor then Cancel) and multiplying and dividing RE’s, we turn to adding and subtracting RE’s, complex RE’s and solving equations containing RE’s as follows.

Add/Subtract RE’s with the Same Denominator

Add/Subtract RE’s with Different Denominators (No Common Factors)

Add/Subtract RE’s with Different Denominators with Common Factors

Simplify Complex RE’s

Note that when using this method to simplify complex RE’s,  it is easy to see the disallowed values of $x$ , for instance in the first example, $x\neq 0 \mbox { and } x\neq -5$.

Solving Equations Containing RE’s

Solving equations like these without multiplying by the LCD is often more complicated but has the virtue of finding all possible solutions including those disallowed because they make a denominator zero.  One of my students came up with the shortcut of dividing the common factors out and listing their zeros.

Of course there are plenty of examples, particularly with solving equations, where the LCD approach is “easier” and certainly the LCD method is an effective method for approaching certain problems, but for the part of algebra which is skill-based and most of it is (Just look at the problem sets and sample tests in a typical textbook.).  Why not teach the faster, cleaner method.

Two Notes.

1.  Sometimes we seem stuck on methods because they resemble the proof of the method.  Since sooner or later we should be interested in efficiency, show why the shortcut is true and then use it.  Divide test questions into skills – Show me you can do it and fast – and concepts – Show me why the method works.

2.  I used, for the first time, the math tool that comes with OneNote 2010.  This worked fine for creating a finished product.  I lecture with OneNote as my “blackboard” though and I don’t think the math renderer is fast enough to keep up with my handwriting in class.