Adding Fractions: Why Avoiding Common Denominators Works

I wrote this post showing that adding fractions can be done using the algebraic definition of addition of rationals:

MathHeadInc (via twitter) has requested a video showing why this works.

I aim to please.

Here is the video showing why adding fractions using a common denominator is the same as the definition of addition of rationals:

What do you think? Will this help convince your kids that “the trick” is okay to use? Share your experiences in the comments.