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

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?

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

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

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

Which of these infinitely deep vessels will eventually full up?

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

How would you go about estimating populations of dolphins?

Use vectors and matrices to explore the symmetries of crystals.

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

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

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

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

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

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.

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

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.

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

Can you construct a cubic equation with a certain distance between its turning points?

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

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

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?

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.

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.

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

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

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

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

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

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.

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.

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

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

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

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

Which dilutions can you make using only 10ml pipettes?

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