median
don steward
mathematics teaching 10 ~ 16

Tuesday, 9 January 2018

products of primes

the powerpoint is here
[ it needs to be downloaded for the animations to work ]




what is the common form of these pairs of numbers?

what, in general, will the hcf be?

the lcm?

Monday, 8 January 2018

pythagoras questions

these questions are from CBSE (India) Y10 papers


the top left question leads to the cosine rule

cos(180 - B) =
 - cos B




Tuesday, 2 January 2018

straight line graphs with ratio

Martin Wilson, of Harrogate, UK developed this idea to intermingle straight line graphs with ratios

it might be interesting to compare student's approaches to these problems

later questions have non-integer solutions



quadratic formula

the powerpoint is here






biggest square inside a right angled triangle

a good, if demanding, application of similar triangles





 general cases
proving a general result

what are the steps?


















unsurprisingly, I subsequently found that this has been explored by (amongst others) Marion Walter in FLM 21 in 2001 and MT 53 in 1970

her general results agree with those above (although there is a small numerical error on p.29 of her article)

she points out that 1/y^2 = 1/x^2 + 1/c^2 where 'c' is the hypotenuse

Monday, 18 December 2017

equable triangles

the proof that there are only five equable triangles (integer sided) was done in 1904

this version of a proof is largely due to David Wells with small bits of help from me
it is complex but involves only GCSE tools















equable right angled triangles

not using pythagoras
working out the radius of the incircle

three little triangles sum to the large one
an alternative method, using pythagoras, to find the radius of the incircle
proving that there are just two right angled equable triangles with integer lengths








two of the possible values duplicate the other two


Tuesday, 12 December 2017

equable isosceles triangles

the area value = perimeter value
for isosceles triangles (including an equilateral triangle, in Q2) with integer heights

this work involves: surds,
rationalising (simple) denominators
and pythagoras



three triangles sum to the large (isosceles) triangle

2a + 2c = ah since P = A

Friday, 8 December 2017

equable trapeziums

beginning to appreciate that all tangential polygons with an incircle radius = 2 are equable

set A = P and solve the equation
ignore the heights from the previous task

work out the missing lengths using the tangents to circle property
the expressions need to be simplified
the areas of the four triangles
add up to the area of the trapezium