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Escher Tessellations

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Escher Tesselations


I've been seeing tessellations in lots of different places and realise that they are being used in so many situations such as tiling on both walls and floors.
At visits to various garages and big buildings I've seen one particular style of tile that's good to create and tessellate. Here's how it goes.
A - You start with a rectangle that is twice as long as it is wide.
B - Then you make a roof shape at one end - this one happens to be as high as a quarter of the base of the triangle it has made.
C - You then put the same triangle on the other end.
D - On one side you take a bite out just the same size as the roof triangle.
E - Then do the same at the other end of the opposite side.
5tiles


Now you have the tile shape for tessellating like the floors I've seen.
One way I've seen is like this:

garage Tiles1

At another garage I saw a tile like this:
2ndTile
These were put together like this [I've coloured them to see the pattern more easily]:

easy2ndTile
or

2ndTilesB


Now it's time for you to try and find some ways of putting shapes - like these two examples - together!

Let us know how you made your shape and do send us some pictures of your tessellations.

 

Why do this problem?

This very practical activity creates an opportunity for investigating properties of shapes in an informal way. Asking them to tackle this in pairs or small groups will mean that talking to each other will be necessary, and this discussion can reveal a great deal to you about their knowledge and mathematical skills.

Possible approach

It is important to consider what part the recording of this work plays in the development of children's concepts relating to tessellation.

Key questions


Possible extension

Those of you who are Escher fans will probably have resources that can extend this activity.
You may have carried out transformations (like the first one on the rectangle) on other shapes.
A very simple idea can be explored with equilateral triangles - an Escher favourite:

1stOdd Tri2ndOdd Tri
These can be put together in different ways.
easyOdd
easy2nd

Again there's lots of opportunity for discussion with certain pupils. For example, "Is there always a need for two different kinds of the transformed triangles?"

Possible support