median
don steward
maths teaching 10 ~ 16

Wednesday, 23 April 2014

bouncing

this fine task is from the admirable collection of problems on the 'five triangles' blog

[it is problem 112, 'billiards']

what is the gradient of the line so that when a ball bounces (perfectly elastically of course) twice it then hits the mid-point of the lower side of the square?



















as the 5 triangle blog points out, some variety of techniques can be used and trigonometry isn't needed
it can be helpful to have some previous consideration about what happens to the gradient of a line when it is reflected
or some consideration of when triangles are similar
or you could solve this, very neatly, by working with the 'principle of reflections'

having solved this task, there are obvious extensions:



















students could also consider the length of the path(s)

impossible picture

it looks as if you can dissect a square into 5 pieces that will assemble to make 3 smaller squares...


use pythagoras and simple surd expressions to show that it doesn't work...

Friday, 11 April 2014

mistrusting diagrams

students probably need to know that diagrams aren't always trustworthy...

they can be asked to establish what is wrong with these diagrams that lead to missing or extra areas after a dissection and a rearrangement

youtube clip



what is wrong with these?

Sunday, 6 April 2014

net tasks







plans and elevations

























many of these resources are adapted from a very helpful powerpoint produced by Mike Crowley - thanks to him











isometric pictures


shapes to copy
complete the cuboids so that all the lines are drawn to show all the little cubes

from an idea in the South Nottinghamshire Project (Journey into Maths) Shell Centre, 1978








also from an idea in the Journey into Maths resources


















a true test of isometric drawing skills - draw it freehand








write your name or initials in 3D - maybe on three faces of an isometric cube

Saturday, 5 April 2014

Robert Webb's work

these are some of the puzzles posed on Robert Webb's 'Stella' website

on the website you can click on the net that you think doesn't match the solid

 octahedron
 cuboctahedron

 dodecahedron

visualising

these tasks are from Peter Grabarchuk's fine collection at unipuzzle (cubic puzzles)
or from
Robert Webb's 'Stella' work - free trial software for doing many fascinating things with polyhedra
look for the 'puzzles' section in the gallery tab of the website

or they are versions of their work




 (Robert Webb's)
(based on Peter Grabarchuk's)
 (Peter Grabarchuk's)
(Peter Grabarchuk's)


















(Peter  Grabarchuk's)