median
don steward
mathematics teaching 10 ~ 16

Sunday 31 October 2010

pythagoras pitch







pythagoras plank



















find the length of the plank (or long stick) in this right angled triangle
that just touches a 3m square, in the corner (as the diagram shows)

Friday 29 October 2010

pythagoras justification















when shapes are similar their areas are in proportion to the squares of corresponding sides

drop a perpendicular from the apex at the right angle to form two similar triangles (relatively straightforward to justify) which are similar to the original triangle

k(5^2) = k(4^2) + k(3^2)
and this generalises...

not that helpful as a means of justifying the theorem maybe, but a neat enough way of tying in the area property of similar shapes

[ from the website betterexplained ]

Wednesday 20 October 2010

number plus reciprocal

what is the smallest positive value that a number plus its reciprocal can have?

test this with decimals and fractions

trying to prove the result might involve rearranging an inequality




a number added to its reciprocal is 2.9, what is it?

a number added to its reciprocal is 2.05, what is it?

a number added to its reciprocal is 2.625, what is it?

Saturday 9 October 2010

Penrose, kites and darts



a kite and a dart (arrowhead) fit together to make a rhombus
find relationships between the angles in the kite and in the arrowhead

actually in Penrose's tiles the smallest angle in the kite is 36 degrees and the dart has three equal angles

what are the other angles?

Sunday 3 October 2010

trapezium property

construct any trapezium (a trapezoid in the US) and draw the two diagonals

at the point where the diagonals meet, construct a line parallel to the two parallel sides of the trapezium

what do you notice?
can you prove it?




this is an old problem, utilised in Michael de Villiers' admirable work on proof and geometry



use similar triangles to show that the length is the harmonic mean of 'a' and 'b'