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

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

Work out the numerical values for these physical quantities.

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

Get further into power series using the fascinating Bessel's 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?

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

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

Which line graph, equations and physical processes go together?

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

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

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

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.

Get some practice using big and small numbers in chemistry.

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

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?

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

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

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

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

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.

Build up the concept of the Taylor series

Explore the relationship between resistance and temperature

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

Match the descriptions of physical processes to these differential equations.

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.

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

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

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

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?

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

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?

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

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

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

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

Which dilutions can you make using only 10ml pipettes?

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