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.

Build up the concept of the Taylor series

Which of these infinitely deep vessels will eventually full up?

Use vectors and matrices to explore the symmetries of crystals.

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.

How would you go about estimating populations of dolphins?

Get further into power series using the fascinating Bessel's equation.

Match the charts of these functions to the charts of their integrals.

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

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.

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

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

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

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

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

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

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.

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

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.

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

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

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

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

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

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?

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

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

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

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.

Get some practice using big and small numbers in chemistry.

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

Which line graph, equations and physical processes go together?

Explore the relationship between resistance and temperature

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

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