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.
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Use simple trigonometry to calculate the distance along the flight path from London to Sydney.
Explore the shape of a square after it is transformed by the action of a matrix.
How efficiently can you pack together disks?
Get further into power series using the fascinating Bessel's equation.
Look at the advanced way of viewing sin and cos through their power series.
See how enormously large quantities can cancel out to give a good approximation to the factorial function.
Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?
In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.
Build up the concept of the Taylor series
Looking at small values of functions. Motivating the existence of the Taylor expansion.
Where should runners start the 200m race so that they have all run the same distance by the finish?
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.
How would you design the tiering of seats in a stadium so that all spectators have a good view?
Can you sketch these difficult curves, which have uses in mathematical modelling?
Explore the properties of matrix transformations with these 10 stimulating questions.
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?
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?
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?
Use vectors and matrices to explore the symmetries of crystals.
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.
Is it really greener to go on the bus, or to buy local?
Have you ever wondered what it would be like to race against Usain Bolt?
Get some practice using big and small numbers in chemistry.
Explore how matrices can fix vectors and vector directions.
Formulate and investigate a simple mathematical model for the design of a table mat.
Explore the properties of perspective drawing.
Which dilutions can you make using only 10ml pipettes?
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?
Can you work out which processes are represented by the graphs?
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.
Can you work out what this procedure is doing?
Work out the numerical values for these physical quantities.
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
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?
Work with numbers big and small to estimate and calulate various quantities in biological contexts.
Which units would you choose best to fit these situations?