Explaining, convincing and proving

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

Relate these algebraic expressions to geometrical diagrams.

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

Work in groups to try to create the best approximations to these physical quantities.

Have a go at being mathematically negative, by negating these statements.

Can you match the charts of these functions to the charts of their integrals?