Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . .

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

Find the equation from which to calculate the resistance of an infinite network of resistances.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.