clapping patterns

Steve Reich's clapping music is quite an interesting application of structure to a simple rhythm

parallelogram law

in any parallelogram,

this can be proved using the cosine rule (and cos(180 - A) = - cosA)
but can also be established as an application of pythagoras' rule
and coordinates:

Eudoxus' ladder

Eudoxus of Cnidus (now the tip of SW Turkey) was a renowned mathematician (amongst other things) who lived around 400 BC

one of the methods he (reportedly) developed or adopted when studying irrationals was to use a number series (called his 'ladder') in order to approximate to the square root of 2:

how is the 'ladder' formed?
how can it be used to approximate to the square root of 2?









'Reaching the Core of AS Mathematics', available from the ATM, interestingly links this blended recursion 'ladder' to the expansion of:

work out and simplify this expression for n = 2, 3, 4 etc

what has it got to do with Edoxus' ladder and why?

in the limit,

how does this provide an approximation to the square root of 2?

lichenometry

people use the diameter or the rate of growth of lichen as a means of dating historical events

formulas have been devised to enable dates to be estimated

simpler formulas assume a circular growth pattern

a Pisa test item (international test, M047) provided a formula used in studies of glaciers











d is the diameter of the 'circle' in mm

t is the number of years after ice has disappeared

the test question asked students to use the formula to find an estimate for a diameter after 16 years

another question might have been:
for what two values of 't' does this formula predict the same numerical value for 'd' as for 't'?

harmonic mean

if one piece of data is an extreme outlier it is recommended to use the harmonic mean to more appropriately represent an 'average' for the data set (but there are problems if one of the items in the data set is zero...)

try this for some simple data sets: compare the arithmetical mean with the harmonic mean where one of the numbers is large or small compared with the rest

for just two numbers, the arithmetical, geometric and harmonic means (along with the root mean square) can be represented by the following lengths:

establish that these lengths are the various means
















the harmonic mean of two lengths occurs in the crossed ladders problem - for the height at which two crossed ladders 'meet' (i.e. 'h' from 'A' and 'B')

powers of 3

how many ellipses at each stage?

thirds and fifths

[based on an idea from the AQA GCSE problem solving questions 2008]

two strips of card are 30cm long
one is divided into thirds
the other is divided into fifths

what are the total overall lengths for these three arrangements?

what other lengths can be made?
why must the overall lengths all be even?

establish that all but four even lengths between 30cm and 60 cm (inclusive) can be made in this way (by lining up two of the fraction marks)

trees, powers of 2

powers were invented for fractals

repeated addition = multiplication
repeated multiplication = powering (or maybe indicing?)

how many branches at the ends, as the tree grows?

powers of 5

 how many pentagons are there altogether?


















these look a bit weird:

sculpture by Tony Cragg

















how many squares at each stage:

square diamond

quadratic growing rules (ii)

a ppt is here