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

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.

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.

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?

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?

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

Can you draw the height-time chart as this complicated vessel fills with water?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

Can you make matrices which will fix one lucky vector and crush another to zero?

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

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.

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

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

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

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.

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?

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

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

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.

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

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

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

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

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

Build up the concept of the Taylor series

Get some practice using big and small numbers in chemistry.

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

Match the descriptions of physical processes to these differential equations.

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

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

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

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

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

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?

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

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.

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

Simple models which help us to investigate how epidemics grow and die out.

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

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

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

Investigate circuits and record your findings in this simple introduction to truth tables and logic.