## Thursday, 14 August 2014

## Tuesday, 12 August 2014

### equations involving algebraic fractions

the initial 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, hopefully, might encourage an application of fraction and other techniques

the questions can be solved by trial and improvement but, hopefully, might encourage an application of fraction and other techniques

## Friday, 8 August 2014

### 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:

## Wednesday, 16 July 2014

### enlargement scale factor root 2

a shape is shown

enlarge it by a scale factor that is the square root of 2

one of the shorter lengths or the shortest length is shown

then enlarge by this scale factor again

again a/the shorter/shortest length is shown

the shapes should not overlap with the original or each other

this can be done or confirmed by dissecting the shapes into simpler shapes (e.g. right angled triangles)

or you can use Pythagoras and multiply surds

students can compare the areas of the original and enlarged shapes

a previous version of this task seemed difficult:

enlarge it by a scale factor that is the square root of 2

one of the shorter lengths or the shortest length is shown

then enlarge by this scale factor again

again a/the shorter/shortest length is shown

the shapes should not overlap with the original or each other

this can be done or confirmed by dissecting the shapes into simpler shapes (e.g. right angled triangles)

or you can use Pythagoras and multiply surds

students can compare the areas of the original and enlarged shapes

a previous version of this task seemed difficult:

## Tuesday, 15 July 2014

## Sunday, 13 July 2014

### river crossing

some puzzles with a long history, variations appearing in many cultures

there are phone aps for several of these

a flash version of this at onlybestflash

see wikipedia article

NRICH 5916 goes full screen

there are phone aps for several of these

a flash version of this at onlybestflash

see wikipedia article

NRICH 5916 goes full screen

## Friday, 27 June 2014

## Thursday, 26 June 2014

### angle proofs

thanks to gogeometry

and the UKMT junior challenge

and Doug French, when he was at Hull University

for several of these ideas

Subscribe to:
Posts (Atom)