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

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

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.

Which line graph, equations and physical processes go together?

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Work out the numerical values for these physical quantities.

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

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

Here are several equations from real life. Can you work out which measurements are possible from each equation?

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

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

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

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

Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

Get some practice using big and small numbers in chemistry.

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

Where should runners start the 200m race so that they have all run the same distance by the finish?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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

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

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

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

Build up the concept of the Taylor series

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

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

Match the descriptions of physical processes to these differential equations.

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

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.

Explore the relationship between resistance and temperature

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

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

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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

Use trigonometry to determine whether solar eclipses on earth can be perfect.

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

When you change the units, do the numbers get bigger or smaller?

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

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

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

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

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?

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?

Which dilutions can you make using only 10ml pipettes?

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

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