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

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

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.

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?

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.

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

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

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

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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

Is it really greener to go on the bus, or to buy local?

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?

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

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

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

Use vectors and matrices to explore the symmetries of crystals.

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

Work out the numerical values for these physical quantities.

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

Which line graph, equations and physical processes go together?

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

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

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

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.

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

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

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

Get some practice using big and small numbers in chemistry.

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

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

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.

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

Match the descriptions of physical processes to these differential equations.

Which of these infinitely deep vessels will eventually full up?

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.

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