regular octagon loops

in England this task was devised by a group called SEAC (a national assessment agency), many moons ago (1992) to assess students on an investigative task
green plastic octagon templates were distributed to all England schools
they might be lurking, dustily, in cupboards

be aware that octagons drawn on squared paper are not regular and lead to erroneous data

you can buy regular octagon paper plates for a group to work with
but accurate recording is still difficult...

the 'growing in a line' case can be explored using a semi-regular tessellation of regular octagons and squares

regular hexagon loops

this task could be initiated with the simpler, 'square loops'

there are various physical regular hexagons available (e.g. ATM mats and Pattern Block shapes) that can be helpful for some students

regular hexagon sheets ensure that recording is relatively easy

regular octagon loops are also interesting but much harder to record (square based octagons being slightly misleading)

a task sheet, for students

it's better for students to select this option themselves rather than be advised to take it...

some students might find some ready-prepared loops to be helpful
















a powerpoint is here

teacher's notes here

isometric shape areas

draw several isosceles trapeziums on an isometric grid
try to find relationships between the three variables
and the number of (unit) triangles in the shape

this task is made more interesting by an involvement of three variables

it can be hard (but hopefully beneficial) for some students to grasp the notion of measuring 'area' with a unit triangle rather than unit squares

using sides that follow the isometric grid lines

the three sides can be labelled so that algebraic rules can be recorded easily

students can identify that a + b = c and give an explanation for this

rules for the 'area' usually involved two of the three variables

some other rules used the half-way line (or two lines)



it is interesting to try to relate versions of the rules for the area to each other

for an equilateral triangle

on an isometric grid, the number of unit triangles is the square of the length, as can be seen by rearranging the shape

this can also be used for isosceles trapeziums since they can be viewed as a difference of two triangles



'straight on' parallelograms (with a horizontal base and a sloping side at 60 or 120 degrees to this base) are slightly easier to analyse:

so a triangle, with one horizontal side and another going off this at 60 or 120 degrees, will have an area of: ab (half a parallelogram)

the 'area' of hexagons with rotational symmetry order 2 can build upon the area of parallelograms and/or the area of equilateral triangles

reasons for this rule can be identified from the diagrams:

three parallelograms:

or: a triangle, take away three other triangles









students can also explore the area of  'skew' equilateral triangles

describing these by isometric 'vectors' these are:
(1 , 2) top left
(3 , 1) top right
(1 , 1) bottom









for a vector (a , b) the rule for the equilateral triangle 'area' is:

and this can be justified using an equilateral triangle surrounded by three half parallelograms

painted cube

the 'painted cube' exploration has plenty of potential for developing, testing and justifying general rules

the general problem: a cube (made up of n by n by n 'cubelets') is dipped into orange paint so that only it's outside faces are painted

when broken apart, how many 'cubelets' will have one face, two faces, three faces and no faces painted orange?

and the task is good for looking at equivalent task in different dimensions: 1D, 2D, 3D, 4D...

for example, the 2-D equivalent would be painting around the edges of an n by n square and seeing how many of the 'squarelets' have 0 , 1 and 2 edges painted

students can be invited to say what will happen in
4 - dimensions...

sacks

establish that there is just one solution

tower of hanoi

for four disks the15 moves are:

what patterns are there in the moves of each colour disk?

try to develop a code or procedure for the moves to complete the puzzle - how could you remember how to do the puzzle?

three disks
three spikes

try to string together a sequence of 7 states (plus the first) to complete the puzzle

parabola template and stretched graphs

tangram tasks

there are several tangram interactives available
e.g. this one (utilising geogebra)

the tangram dissection:

find how to fit all of the 7 tangram pieces into these outlines:

you could consider the perimeters
(there are 4)

[in terms of multiples of the square root of 2 and integers]

an interesting set of 3 shapes that look the same size but are not (a kind of paradox)

find outline shapes that have just one line of symmetry (using all 7 tangram pieces)
some examples (several more can be found):

find an outline shape with rotational symmetry, order 4, using all the tangram pieces
(slightly cheating, with a gap in the centre)
or the square