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.

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?

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?

Work out the numerical values for these physical quantities.

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

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

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

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

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

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

Have you ever wondered what it would be like to race against Usain Bolt?

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

Which line graph, equations and physical processes go together?

How would you go about estimating populations of dolphins?

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

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.

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

Which dilutions can you make using only 10ml pipettes?

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

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?

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

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.

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

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

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

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

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?

Practice your skills of measurement and estimation using this interactive measurement tool based around fascinating images from biology.

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

Which units would you choose best to fit these situations?

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

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

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

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.

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

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

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

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