Year 8 Common Task: Tessellation

A first lesson

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 might let
us cover the whole board with copies of that shape, with no gaps or overlaps.'

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 shape.'

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 tessellate.'

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.)

Vocabulary
Tessellation
Quadrilateral
Types of quadrilateral; square, rectangle, parallelogram, trapezium, kite, arrow-head (delta)
Transformation words; reflection, rotation, translation

Typical mathematical content
Tessellation
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.