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

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.

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?

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

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

Which line graph, equations and physical processes go together?

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

How would you go about estimating populations of dolphins?

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

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

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

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

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

Get some practice using big and small numbers in chemistry.

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

Explore the relationship between resistance and temperature

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

Build up the concept of the Taylor series

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

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

Match the descriptions of physical processes to these differential equations.

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

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

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

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

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

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

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

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

Use vectors and matrices to explore the symmetries of crystals.

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

How would you design the tiering of seats in a stadium so that all spectators have a good view?

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

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

Which dilutions can you make using only 10ml pipettes?

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

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?

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

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