don steward
mathematics teaching 10 ~ 16

Tuesday, 3 September 2019

a Keith Richardson-Jones picture

this picture by Keith Richardson-Jones (1974) is in the Tate, London UK

there is an interesting structure to it, that seems not too difficult to discern

Saturday, 31 August 2019

bidmas differences

one of the reasons why some people do not like to use the mnemonic 'bidmas' (or 'pemdas') is that it can lead to some students misguidedly working out division prior to multiplication (or vice versus for pemdas) and an addition prior to a subtraction

it ought perhaps to be labelled:

but this doesn't ensure a lack of confusion

Matt Dunbar has suggested using this graphic:

it seems to be important that students have some notion of a more 'powerful' operation
as well as a need for a global convention, to avoid ambiguity

it can be interesting to look at the difference between a calculation worked out (a) using a correct priority of operations and (b) incorrectly

there are three tasks here, looking at these differences for:

  1.  3 consecutive numbers
  2.  where the first number is 10 times the second and the third is the second add 1
  3.  where one result (incorrect version) is double that of the other (correct version)

a powerpoint is here

diagrams of the different ways that this can be worked out

as can be seen in the diagram above

as can be seen in the diagram

examples where one (incorrect) result is double the correct result

Friday, 16 August 2019


a tessellation is a way of covering the plane with (usually) just one shape
repeated, regularly, without any gaps
forming a structure
'stripes' emerge

Geoff Giles (of DIME materials fame, see some of his work here) made reference to a way of creating shapes that tessellate
by modifying triangles
he called these 'trisides'

these ideas are not quite the same as Geoff's but have the same principle structure (my article was published on transforming tessellating shapes into 'trisides' in Mathematics in School, volume 14, issue 3, may 1985)

to form a 'triside' from a given (tessellating) shape:
  • the routes/sections of lines, sometimes extended, between two dots (which are corners of a triangle, the 'triside') must have rotational symmetry, order 2
  • all of the perimeter (boundary) length sections of the original shape must be included within the three 'triside' sections

to create a 'triside' for a pentomino (all of which tessellate) usually requires going beyond the perimeter of the shape to place a dot (as a corner of a 'triside')

possibly with these 'construction' lines, the whole of the perimeter (lengths) must to be encompassed within three routes/sections, which all have rotational symmetry

by way of examples, here are some ways to transform four of the pentominoes
into 'trisides' :

can you see that each section (between the red dots) has rotational symmetry?

start somewhere
try to use as much of the perimeter as you can
maybe with additional, extension, lines
to place a second dot so that this section has rotational symmetry
then try to place a third dot so that the next section has rotational symmetry

if the third section also has rotational symmetry then the resulting three dots form a triangle of area equal to the original shapes (5 squares in this case)

this is a 'triside' transformation for the original shape

trisides for a particular tessellating shape are not usually unique

for example, one pentomino:

this pentomino turned into a 'triside' in four different ways

can you see that each section has rotational symmetry?

another fairly tough-to-'triside' pentomino:

other pentomino 'triside' constructions (there are probably others) with a starting point:

where will the other two 'triside' corners/vertices be?

other pentominoes will also transform to 'trisides':
maybe attempt these:

by means of this transformation, tessellations of each pentomino can be seen to have a triangle tessellation lying 'behind' them

in trying to turn tessellating shapes into 'trisides', one technique is to start somewhere (A) and then use rotational symmetry to locate B from A and also C from A
then find C from B
adjusting the outcomes so that all three sections between the dots have rotational symmetry

if you end up with the third dot not in the same place as another but in line with it, that is good because going half way between them enables a 'triside' to be created:

furthermore, it seems that many straight line bounded shapes that tessellate can be turned into 'trisides'

sometimes with a little bit of perseverance...

where will the other two corners of a 'triside' be?

in this way, tessellations of straight line bounded 'tiles' can be viewed as a tessellation of triangles....

are probably easier to explore (initially) than the pentominoes:

other options can be found
four different constructions for 'trisides' of the same quadrilateral (above) ~ two pairs of congruent triangles

how can you (easily) see that the 'trisides' (above) all have the same area? [= 18]
how can you see that this is the area of the quadrilateral
[drop a perpendicular from the top left corner]

it seems that for a quadrilateral you can start at a mid-point of any of the sides to form a 'triside'

even if it is concave:

this is a general method to turn a quadrilateral (since they always tessellate) into a triangle of equal area

a chevron (hexagon) turned into a 'triside'

maybe also consider how the triangle can be turned into the chevron by cutting and fitting

a regular hexagon turned into a 'triside':

how can you see that the areas will be the same?

are all tessellations of straight line bounded shapes
actually just tessellations of triangles?

e.g. for one of the pentagons ('house') that will tessellate:

sometimes, for a shape that is difficult to 'triside'
it can be helpful to form a 'quadriside' as an intermediate step and then change this quadrilateral into a 'triside'

for example:

first form a 'quadriside'
then transform the quadrilateral into a 'triside'

another example of transforming a tessellating shape into a 'triside' via a 'quadriside':

you might also want to check that this shape does tessellate

sometimes the construction lines are difficult
where do the other two points of a 'triside' for this (tessellating) shape go?

tessellation of a 7-sided shape:

a 'triside' for this heptagon:

the arrow tessellation with the tessellation of 'trisides' as a basis for it:

one of the types of pentagon that will tessellate has two sides parallel
[the 'house' pentagon above is another example of this shape]

an associated 'triside' for these (generality of) pentagons:

some tessellating shapes are difficult (if not impossible) to find 'trisides' for
but where this is the case it can be possible to find a 'triside' for a pair of the shapes

two examples:

an example of a 'Cairo' pentagon tiling:

two of the pentagons form a 'unit' for a tessellation

you can find a 'triside' for this pair

for this 12 sided polygon:

that can tessellate like this
a pair form a new tessellating unit
and you can find a 'triside' for this pair

one of the semi-regular tessellations (at least) can be turned into a 'triside'

so, I'm growing in my conviction that every tessellation of polygons has a triangle tessellation as a basis for the tessellation