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.

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.

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

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

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

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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

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

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

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

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

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

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

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.

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

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?

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?

Use vectors and matrices to explore the symmetries of crystals.

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

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.

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.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

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?

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

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.

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

Which dilutions can you make using only 10ml pipettes?

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

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

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?

Which line graph, equations and physical processes go together?

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

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.

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

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

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