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.

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.

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

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

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

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

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

Build up the concept of the Taylor series

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

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

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

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

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?

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

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

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

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

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

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

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

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

Is it really greener to go on the bus, or to buy local?

Use vectors and matrices to explore the symmetries of crystals.

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

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

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

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?

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.

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?

Get some practice using big and small numbers in chemistry.

Work out the numerical values for these physical quantities.

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

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

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

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

Why MUST these statistical statements probably be at least a little bit wrong?

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

Explore the relationship between resistance and temperature