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

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

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

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

How would you go about estimating populations of dolphins?

Use vectors and matrices to explore the symmetries of crystals.

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

Which of these infinitely deep vessels will eventually full up?

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

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

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

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

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

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.

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

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

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.

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

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

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

Can you construct a cubic equation with a certain distance between its turning points?

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

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?

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

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

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.

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?

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.

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

Which dilutions can you make using only 10ml pipettes?

Build up the concept of the Taylor series

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.

Explore the relationship between resistance and temperature

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

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.

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