powers of phi

equations involving algebraic fractions

the idea for these tasks is from a question in Tony Gardiner's 'mathematical puzzling' book (1987)













the questions can be solved by trial and improvement but are intended to encourage an application of how to proceed with fractions and other techniques

'wigwam' is a visual cue for the procedure of adding/subtracting fractions
(sometimes called other things but 'wigwam' is heaps better)

golden ratio 1.618033989...

watch the youtube clip
Golden Section IV by Jo Niemeyer

establish that the lengths indicated are in a 'golden' ratio















if you fit three identical circles inside a semicircle:

it is not too difficult to show that the ratio of the bigger radius to the smaller one is twice the golden ratio











the golden ratio occurs in various ratios of sides in a pentagon

the golden ratio is 2 cos 36

but the ratio can be deduced algebraically:

 

it is probably helpful to set a = 1

then, using similar triangles

a quadratic equation can be formed that has the golden ratio as the positive root








students might be interested in checking whether or not the following claim is  statistically reasonable: