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

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

Which of these infinitely deep vessels will eventually full up?

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

How would you go about estimating populations of dolphins?

Use vectors and matrices to explore the symmetries of crystals.

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

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

Which dilutions can you make using only 10ml pipettes?

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

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?

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.

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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...

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

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

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.

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