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

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

Can you match the charts of these functions to the charts of their integrals?

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

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

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

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

Build up the concept of the Taylor series

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.

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

Use vectors and matrices to explore the symmetries of crystals.

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

Which of these infinitely deep vessels will eventually full up?

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

Here are several equations from real life. Can you work out which measurements are possible from each equation?

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

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

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?

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.

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

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

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.

Which line graph, equations and physical processes go together?

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

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

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

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

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.

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?

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

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

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

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

Get some practice using big and small numbers in chemistry.

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

Formulate and investigate a simple mathematical model for the design of a table mat.

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?

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

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.

Which dilutions can you make using only 10ml pipettes?