Year 8 Common Task: Tessellation
Draw dots beforehand on your board and on
a piece of flip chart paper.
'I would like someone to come to the board
and draw me a four sided shape, whose corners are on the dots.'
Or else you could decide the shape and have it drawn already eg
'OK, now I would like someone to
come up and perform an action on the shape so that they can draw
an exact copy of that shape, touching it somehow, in a way that
us cover the whole board with copies of that shape, with no gaps
Allow pretty much any offer
at this stage - so long as it is a copy of the original shape and
is touching, eg:
'OK, James has defined an action
when he placed the second shape. I'd now like someone to come up
and perform the same action on the second shape, to create a third
At this point only one shape
is correct (shown below), do not allow any others, if there is disagreement
from the class ask them to describe what action James performed.
You may want to continue this
pattern for a while, but at some point bring the class' attention
back to the fact that you are trying to cover the board with no
gaps or overlaps. When agreement has been reached that this cannot
be done with this system, rub off all shapes except the first one
and repeat the first question.
At some point you will get
an action that works, eg:
Keep getting students to come
to the board to draw the next shape, and the one that is the inverse
action, until you judge the class are convinced the pattern could
continue in both directions for ever.
'From what we have done so far, we
can continue in this direction and this direction forever. Will
that cover the board with copies of the shape?'
'OK, so can someone come and place another copy
of the original shape, defining another action, that might help
us cover all of this (pointing to the rest of the board) area?'
After students have come up and the same
process you went through to define the first direction has been
repeated, you should get a board looking something like this:
At this point students may
be convinced they have shown enough to cover the whole board, if
not get more shapes following other lines. Emphasise that you have
defined actions in TWO directions. Around this time, invite students
to do a complete tessellation of this shape on dotty paper.
As the first people are finishing, put their
sheets on a board, and issue the challenge for this topic:
'These sheets show that we have a four sided shape
that tessellates - ie we can fill the page with copies of the shape,
with no overlaps and no gaps. Your challenge for this project is
to find a four sided shape that will not tessellate. (Write this
on the board.) Do your drawings on dotty paper. Remember you need
to define actions in two directions to get a tessellation to work.
Any tessellations that work will go on this board, and shapes that
you think cannot be tessellated can be drawn on this sheet here'.
Have a piece of flip chart paper prepared.
Where this can go
Common boards are important in this
topic - one for completed tessellations and one for shapes that
students think do not work. The latter provides a good source of
shapes for others to try.
Encourage students to be organised in trying types
of shapes and come up with conjectures eg 'All parallelograms will
Encourage students to write instructions for how
to tessellate their chosen shape, and ultimately instructions for
how to tessellate any shape, (eg parallelograms tessellate by translating
copies to the left/right and up/down).
A challenge to issue eg in the second lesson is:
what is the minimum number of shapes you have to show for everyone
to be able to see how the tessellation will continue. This can be
tested by inviting someone to come and draw eg three copies of their
shape, getting the rest of the class to complete the tessellation
and seeing if everyone has completed it in the same way.
If students are struggling with a particular shape
they can cut four copies out of card and see if they can fit the
shapes together first, before drawing. (This can also be done as
a whole class activity at the beginning of a lesson with one shape
that no one can get to tessellate - you may need to copy the outline
on to card.)
Keep emphasising that the shapes cannot be placed
randomly - students need to define two directions.
Another lesson start to do near the end of the
topic is to draw a shape similar to the one in 'A first lesson'
and label the angles a,b,c,d. Get students to tessellate the shape
and in each copy of the shape write in the angle labels. They should
find that one copy of each angle meets at each point - it is possible
to derive from this that the angles in that quadrilateral add up
to 360 degrees - how general is this?
This topic leads very naturally into either finding
angles in parallel lines (which is just tessellations of parallelograms)
and also into exploring which of the regular polygons tessellate
(either on their own or in combination). ATM MATs can be used for
students to draw around if working on regular polygons.
Guidelines for notes at the end of the topic
Tessellation means ...
We have found that all (fill
in what you did find ... parallelograms / quadrilaterals
/ triangles) tessellate, by (fill this
is if anyone discovered it ... eg rotating the shape 180
degrees about the mid-point of each side).
The interior angles of polygons are as follows
(have a table of values) ...
The following regular polygons tessellate, and
no others; triangles, squares, hexagons.
(Also give notes on angles in parallel lines,
etc, if you deal with this here.)
Types of quadrilateral; square, rectangle, parallelogram, trapezium,
kite, arrow-head (delta)
Transformation words; reflection, rotation, translation
Typical mathematical content
Angles (eg in quadrilaterals, parallel lines, polygons)
A homework to be done by everyon
Matt's conjecture: All triangles tessellate. Explore Matt's
conjecture, write down all your ideas and diagrams.