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

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.

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

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

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.

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?

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

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?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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

Use vectors and matrices to explore the symmetries of crystals.

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

Which of these infinitely deep vessels will eventually full up?

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

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

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

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

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

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.

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

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

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

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?

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

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

Was it possible that this dangerous driving penalty was issued in error?

Work with numbers big and small to estimate and calculate various quantities in physical contexts.

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.

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

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

Look at the advanced way of viewing sin and cos through their power series.

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

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

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.

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.