Reasoning, convincing and proving

What have Fibonacci numbers got to do with Pythagorean triples?

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which relates triangular numbers to square numbers?

Can you find a rule which connects consecutive triangular numbers?

Here are some rods that are different colours. How could I make a yellow rod using white and red rods?

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.