two numbers sum to a square number

graphical interpretation

thanks to the Shell centre
a couple of these tasks are based on WJEC maths GCSE past paper questions

[from ukbubble]

straight line graphs and nth terms

maths GCSE questions (mostly based on WJEC past papers)

right-angled trapeziums

students initially consider the value that 'c' should have for the area to be a maximum

they can then look at their three different area values

and possibly establish that the differences between the differences will always differ by 1

e.g. for 3 , 4 , 5

• smallest area = 13.5
• middle area = 16
• largest area = 17.5

17.5 - 16 = 1.5
16 - 13.5 = 2.5

the differences from the middle one are 1 apart

proving this relationship involves quadratic expansions

big sprouts and Frodo Baggins

larger than usual sprouts were available this year (milder winter)
I've changed the numbers reported by a newspaper slightly
[thanks to mathspad]

Elijah Wood played the film role Frodo Baggins

new year

decimal scales

an example of how reversing a question can be more interesting
leading, ideally, to generalising

what is the relationship between 'a' and 'b' if the marker is at 0.25 (half-way between the second and third mark from the left)?

areas of isosceles trapeziums

a search for all the possible isosceles trapeziums on a 5 by 5 grid
can involve being systematic
once upon a time I thought I'd found 19 of them...
but then on a later occasion I could only find 18...
(help would be much appreciated...)

calculating their areas could involve an introduction to the square root of 2
although the areas can be found by subtracting (triangles) from a surrounding rectangle (frame)

the area of the second shape could be obtained by cutting it up or subtraction from a 3 by 3 square

and this related to the area of a trapezium formula

some isosceles trapeziums are drawn
with their vertices on the dots

draw some more

find their areas










what are the 12 different areas for the 19 (or is it 18?) altogether different isosceles trapeziums on a 5 by 5 dot grid

angles in tessellations

regular pentagons with rhombuses

what are the angles in the rhombuses?

regular pentagons with isosceles triangles

what are the angles in the isosceles triangles?

regular octagons with star octagons

what are the angles in the star octagons?

regular heptagons with 'bow-tie' octagons

what are the angles in the bow-tie?

show that a + b = 240 degrees
and 2a + b = 360 degrees

hence deduce the sizes of angle 'a' and angle 'b'

what is the relationship between angle 'a'
and angle 'b' ?

what is the relationship between angle 'b'
and angle 'a' ?

what is the relationship between angle 'c'
and angle 'a' ?

give reasons why c = 135

find a relationship between 'a' and 'b'

find a relationship between 'a' and 'd'




hence work out all of the angles in the pentagons

tessellation of pentagons discovered by:
Casey Mann
Jennifer McLoud
David von Deran

(University of Washington Bothell)

August 2015

'Cairo' pentagon tilings

a tessellation of a single symmetric pentagon,
can have (several) equal sides,
two (opposite) right angles

these come in various forms (have degrees of freedom)

found in Cairo as paving tiles
(not that old)

what relationship is there between the angles if two are right angles and two of the other (obtuse) angles are equal?

what if the 3 obtuse angles are equal?

David Bailey (from Grimsby, England, with a keen interest in recreational mathematics) has undertaken a very thorough analysis of the variety and dates of 'in situ' tiles
so far the oldest dated version he has been able to establish is 1956

due to the two 90 degree angles in the pentagon
the tessellation(s) have a 'skeleton' (as David Wells calls them) of squares

so the variety of this type of tessellation is created by the different angles in the rhombus:

the pentagon tessellation can be viewed with other square 'skeletons'
















or another way
are these squares?

or with isosceles (at least) triangles


what relationship is there for the apex and two equal base angles?














or with trapeziums
what is the relationship here?

















a version of the tiling can be created from the 4, 3, 3, 4, 3 semi-regular tessellation, as a dual (corners of the original tessellation become centres of the related one - and vice versus)

what are the angles in the tiles?

are the tiles congruent?

a tessellation of Cairo-like tiles can be drawn on isometric paper

but...  there are two different types of (non-congruent) pentagon here
establish that their angles are the same

David Bailey quotes Robert H Macmillan's claim that collinearity is a feature of some of the 'Cairo' tessellations and explores this

four of the sides are the same length

what are the angles in each pentagon tile for this arrangement?

use trigonometry to establish the angles in each pentagon for this arrangement

what collinearity is there? 

it is possible to have
all the sides the same length


















what are the angles for this equilateral pentagon tile?















if the triangle formed by joining the apex of the pentagon to the two bottom corners is equilateral

there is a collinearity

the triangle shown is equilateral

find angle 'a'
and establish the (seeming) collinearity property