A big part of math is about *transformations* and *structure*.  For a wide audience in a short time, you can give an inkling about this in a short time along the following lines:

- "Numbers'' are really abstract things, not just  "quantities''. They can correspond to transformations --- for example, dilations and translations of the line.  Show how this corresponds to multiplication and addition. Negative numbers are flips.
This explains "negative times negative is positive," and shows $x^2=1$ has two solutions.

- Solving $x^2=-1$ corresponds to answering the question "What can you do twice to get a flip?"  Likely as not someone will think of rotating 90 degrees.  Dilations, translations and rotations of the *plane* are *complex numbers*.

- "What about rotations in 3D?" Demonstrate they don't always commute.  A lot of math is about understanding the rules and concepts that govern much more general transformations of complicated kinds of data.