Examples
Randomizing a Theme 

You make an app that draw by coding an element called a turtle. You can code movement forward, backward, left, and right. You can also position it using coordinate plane. It to leave a trail of different colors and widths to draw shapes and patterns.

Here's a quick tutorial: Getting started with App Lab: Turtle

More videos:
How to create artistic shapes
More fun with turtles (and functions) 

You can also create an app that lets a user draw and create patterns.  Here's a  video for making a simple version:
How to Make a Drawing App 

Single Die Roller 

How to Make a Calculator 

Ethnomathematics is an approach to math that connects math and culture. Math is, of course, already connected to culture, but sometimes the connection is invisible. Ethnomathematics points out connections to all kinds of cultures beside math class and tests. Here's a list of sites that explore the connections:

What is ethnomathematics, and how can it help children in schools 

Math and Music
An Index of Lessons
Graphing Pitch and Musical Instruments
Jazz and Math: Improvisation Permutations
Jazz and Math: Rhythmic Innovations
Patterns in Music and Math