don steward
mathematics teaching 10 ~ 16

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

Monday, 11 December 2017

shape quiz

20 questions
[needs downloading for the animations to work]

Friday, 8 December 2017

equable trapeziums and right angled triangles

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
not involving pythagoras
 the areas of the three triangles sum to the area of the large (right angled) triangle
showing the radius = 2 another way

involving pythagoras
d = 8 so
4 + 2n = 8

equable orthogonal octagons

an equable shape has the same value for the area as for the perimeter
set up an equation
from A = P
and solve it

different ways to chop up the octagons

equable orthogonal hexagons

an 'equable' shape has the same value for the perimeter as the value for the area
this work can involve setting up and solving linear equations and forming linear relationships
and generalisations that can be proved

an orthogonal polygon just has right angles and 270 degree angles (that I call left angles) inside it
[sometimes these are called 'rectilinear polygons'
but 'rectilinear' seems to mean 'bounded by straight lines' rather than involving multiples of 90 angles]

Thursday, 30 November 2017

product of three consecutive numbers

On one of Ed Southall's tweets
[ September 16th 2017 ]

interesting to think why it works: diagrammatically, as well as algebraically
[ the algebra is probably simpler with (n - 1) n (n + 1) + n ]

how is a general rule adapted for three numbers that are in (a constant difference) sequence rather than being consecutive?

fraction generalising

this is based on a problem featured by five triangles [tweeted on March 9th 2017]

Monday, 27 November 2017

Saturday, 25 November 2017

quadratic expression versions

this task was devised by Martin Wilson, of Harrogate

Friday, 24 November 2017

concentric circular rings

the powerpoint is here

thanks to David Wells for the first question

Thursday, 23 November 2017

multiplication find the gaps

Tony Gardiner has produced many such questions

these problems involve consecutive digits

 with a clue
 with another clue
using the FACT key on a calculator to help