This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.

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

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

We all know that smoking poses a long term health risk and has the potential to cause cancer. But what actually happens when you light up a cigarette, place it to your mouth, take a tidal breath. . . .

There has been a murder on the Stevenson estate. Use your analytical chemistry skills to assess the crime scene and identify the cause of death...

Ever wondered what it would be like to vaporise a diamond? Find out inside...

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

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.

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

Work out the numerical values for these physical quantities.

Unearth the beautiful mathematics of symmetry whilst investigating the properties of crystal lattices

Dip your toe into the fascinating topic of genetics. From Mendel's theories to some cutting edge experimental techniques, this article gives an insight into some of the processes underlying. . . .

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

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

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

Investigate constructible images which contain rational areas.

Get some practice using big and small numbers in chemistry.

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

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

Read about the mathematics behind the measuring devices used in quantitative chemistry

How fast would you have to throw a ball upwards so that it would never land?

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

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

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

How much peel does an apple have?

Where we follow twizzles to places that no number has been before.

Investigate x to the power n plus 1 over x to the power n when x plus 1 over x equals 1.

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

Explore the properties of this different sort of differential equation.

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

Find all the periodic cycles and fixed points in this number sequence using any whole number as a starting point.

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

Build up the concept of the Taylor series

Which parts of these framework bridges are in tension and which parts are in compression?

Given the equation for the path followed by the back wheel of a bike, can you solve to find the equation followed by the front wheel?

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