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.

Use vectors and matrices to explore the symmetries of crystals.

How would you go about estimating populations of dolphins?

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

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

Match the charts of these functions to the charts of their integrals.

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

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?

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

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

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

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?

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

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

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

Match the descriptions of physical processes to these differential equations.

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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.

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

This problem explores the biology behind Rudolph's glowing red nose.

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

Build up the concept of the Taylor series

Work out the numerical values for these physical quantities.

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

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?

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

Which line graph, equations and physical processes go together?

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

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?

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

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

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

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

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

Analyse these beautiful biological images and attempt to rank them in size order.

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

Was it possible that this dangerous driving penalty was issued in error?

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