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

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.

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

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?

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

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.

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

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

Use vectors and matrices to explore the symmetries of crystals.

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

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

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

Which of these infinitely deep vessels will eventually full up?

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

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

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

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

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.

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

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.

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

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

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

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

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

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

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

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

Which dilutions can you make using only 10ml pipettes?

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

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.

The design technology curriculum requires students to be able to represent 3-dimensional objects on paper. This article introduces some of the mathematical ideas which underlie such methods.

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

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

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

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?

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.

Get some practice using big and small numbers in chemistry.

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