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?

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.

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

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.

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

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

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

Which line graph, equations and physical processes go together?

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

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.

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

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

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.

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

Which dilutions can you make using only 10ml pipettes?

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

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

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

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

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

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

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

Get some practice using big and small numbers in chemistry.

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

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

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.

Which units would you choose best to fit these situations?

Build up the concept of the Taylor series

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

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

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

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.

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

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

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.

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

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