Why It Works

The butterfly is not a trick — it's a visual shortcut for the algebraic process of finding a common denominator. Cross-multiplying a/b and c/d gives ad and bc over the common denominator bd. The butterfly just makes this process intuitive and hard to mess up.

Teachers sometimes discourage the butterfly because students use it without understanding why. But once you know the algebra behind it, there's no reason to avoid a method that is fast, reliable, and visually memorable. Understanding and efficiency are not enemies.