median
don steward
mathematics teaching 10 ~ 16

## Saturday, 19 May 2012

### isometric shape areas

students can be asked to draw several isosceles trapeziums on an isometric grid and 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 good) for some students to grasp the notion of measuring 'area' with unit triangles rather than unit squares

(using only sides that follow the isometric grid lines)

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

many students identified that a + b = c and were able to offer some 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 the 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 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 unit triangle 'area' is:

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