median
don steward
maths teaching 10 ~ 16

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




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:



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




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