Here you will find ideas and stimuli for your exploration

An example of  how to create an experiment to model : Mentos & Diet Coke 

Doodling in maths class, videos by Vi Hart: 

Numberphile

Lots of ideas and inspiration

Newton's method to approximate solutions to equations

Conic Sections Hide and seek

List of possible topics to explore by Cornell Department of Mathematics

Graph Theory

More or Less

A BBC series where Tim Harford explains the numbers and statistics used in political debate, news and everyday life

Meet Cryptography

A site that explains different methods of encryption.

The Maths of Magic Squares

SquareCircle

many ideas to explore 

Roots of unity

the scientific american blog related to mathematical ideas.

Stewart Math: list of 18 projects to explore

Patterns in polynomials

Maths Research: Connect the Dots

Mathematical Art: Virtual Maths Museum

Origami: Maths Craft

Fractal Machine

Scientific American: Simulating the Spread of Disease

Las matemáticas del aparcamiento