Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Look at the advanced way of viewing sin and cos through their power series.

Work with numbers big and small to estimate and calculate various quantities in physical contexts.

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

Get further into power series using the fascinating Bessel's equation.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Build up the concept of the Taylor series

Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

Use vectors and matrices to explore the symmetries of crystals.

Explore the relationship between resistance and temperature

Explore the meaning of the scalar and vector cross products and see how the two are related.

Can you make matrices which will fix one lucky vector and crush another to zero?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Which line graph, equations and physical processes go together?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Can you work out which processes are represented by the graphs?

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Why MUST these statistical statements probably be at least a little bit wrong?

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

Go on a vector walk and determine which points on the walk are closest to the origin.

This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

Get some practice using big and small numbers in chemistry.

Formulate and investigate a simple mathematical model for the design of a table mat.

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Invent scenarios which would give rise to these probability density functions.

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

Explore the properties of matrix transformations with these 10 stimulating questions.

Explore the shape of a square after it is transformed by the action of a matrix.

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

Can you sketch these difficult curves, which have uses in mathematical modelling?

How do you choose your planting levels to minimise the total loss at harvest time?

Can Jo make a gym bag for her trainers from the piece of fabric she has?

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

Work with numbers big and small to estimate and calulate various quantities in biological contexts.

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.