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?
Can you make matrices which will fix one lucky vector and crush another to zero?
Go on a vector walk and determine which points on the walk are
closest to the origin.
Explore how matrices can fix vectors and vector directions.
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.
Work out the numerical values for these physical quantities.
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Work with numbers big and small to estimate and calculate various quantities in physical contexts.
Here are several equations from real life. Can you work out which measurements are possible from each equation?
Which line graph, equations and physical processes go together?
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
Get further into power series using the fascinating Bessel's equation.
How would you go about estimating populations of dolphins?
Are these statistical statements sometimes, always or never true?
Or it is impossible to say?
This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
Get some practice using big and small numbers in chemistry.
Formulate and investigate a simple mathematical model for the design of a table mat.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
Explore the shape of a square after it is transformed by the action
of a matrix.
Explore the properties of perspective drawing.
Explore the properties of matrix transformations with these 10 stimulating questions.
See how enormously large quantities can cancel out to give a good
approximation to the factorial function.
How much energy has gone into warming the planet?
Build up the concept of the Taylor series
Look at the advanced way of viewing sin and cos through their power series.
Explore the relationship between resistance and temperature
Explore the possibilities for reaction rates versus concentrations
with this non-linear differential equation
Match the descriptions of physical processes to these differential
By exploring the concept of scale invariance, find the probability
that a random piece of real data begins with a 1.
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
Explore the meaning behind the algebra and geometry of matrices
with these 10 individual problems.
The probability that a passenger books a flight and does not turn
up is 0.05. For an aeroplane with 400 seats how many tickets can be
sold so that only 1% of flights are over-booked?
Can you sketch these difficult curves, which have uses in
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
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.
When you change the units, do the numbers get bigger or smaller?
Invent scenarios which would give rise to these probability density functions.
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.
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
How would you design the tiering of seats in a stadium so that all spectators have a good view?
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
Looking at small values of functions. Motivating the existence of
the Taylor expansion.
Which dilutions can you make using only 10ml pipettes?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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?
A problem about genetics and the transmission of disease.
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?