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?
Explore how matrices can fix vectors and vector directions.
Can you make matrices which will fix one lucky vector and crush another to zero?
Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.
Where should runners start the 200m race so that they have all run the same distance by the finish?
See how enormously large quantities can cancel out to give a good approximation to the factorial function.
Here are several equations from real life. Can you work out which measurements are possible from each equation?
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
Which line graph, equations and physical processes go together?
Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?
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.
Get further into power series using the fascinating Bessel's equation.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
Looking at small values of functions. Motivating the existence of the Taylor expansion.
Which dilutions can you make using only 10ml pipettes?
Can Jo make a gym bag for her trainers from the piece of fabric she has?
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
Formulate and investigate a simple mathematical model for the design of a table mat.
This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.
Get some practice using big and small numbers in chemistry.
How efficiently can you pack together disks?
How much energy has gone into warming the planet?
Build up the concept of the Taylor series
Use simple trigonometry to calculate the distance along the flight path from London to Sydney.
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 possibilities for reaction rates versus concentrations with this non-linear differential equation
By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.
Look at the advanced way of viewing sin and cos through their power series.
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?
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
Work with numbers big and small to estimate and calculate various quantities in physical contexts.
Which units would you choose best to fit these situations?
When you change the units, do the numbers get bigger or smaller?
Are these statistical statements sometimes, always or never true? Or it is impossible to say?
Explore the relationship between resistance and temperature
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
A problem about genetics and the transmission of disease.
Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.
Match the descriptions of physical processes to these differential equations.
In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?
Invent scenarios which would give rise to these probability density functions.