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

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

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

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

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

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

Which of these infinitely deep vessels will eventually full up?

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

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

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

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

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

Which line graph, equations and physical processes go together?

Use vectors and matrices to explore the symmetries of crystals.

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

Match the descriptions of physical processes to these differential equations.

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

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.

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

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

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?

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

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.

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

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

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

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

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

Get some practice using big and small numbers in chemistry.

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

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.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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

Which dilutions can you make using only 10ml pipettes?

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

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

Simple models which help us to investigate how epidemics grow and die out.

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.