Pattern, shape and symmetry

This week is about different aspects of pattern, shape and symmetry. There are some tessellations to explore, shapes to make from other shapes, patterns you can make involving rotational symmetry, problems to solve with line symmetry and some statements to reason through. At the end there are a few other ideas and puzzles if you want to take these ideas a little further.


Take a look at this tessellation of regular hexagons and see what happens when you mess about with it:

Try creating your own tessellation based on the ideas you’ve just explored.

Then try to find examples of other tessellations around your home – if you’re not sure where to start, try the kitchen or bathroom.

Send in pictures or diagrams of your tessellations to Student Showcase.


For the second activity you will need to find some scrap paper and cut a square from it. If you are not sure whether it’s a perfect square, try putting the opposite corners together to check for the diagonal line of symmetry.

If you don’t have any scissors at home, you should be able to tear your scrap paper neatly either by using the edge of a ruler or by folding it really crisply first and then tearing it carefully.

Draw a line from one corner to the middle of a side.
Cut along the line.
Use the 2 pieces to make a TRIANGLE first.
Now try to use your two pieces to make each one of these QUADRILATERALS.

The last part of this activity is try to make as many different PENTAGONS as you can with your two pieces. Show someone else each of your different pentagons if you can (or draw the outline of each of them so you can show someone later).


Now there’s a chance to explore two different types symmetry – first rotational symmetry and then line (or mirror) symmetry.

Explore different patterns with rotational symmetry order 4
Hint: the order of rotational symmetry is the number of times the shape will fit inside itself exactly through a full turn.

Where would you place an additional square to make each of the shapes symmetrical?
Decide where the line of symmetry would be.
How about if you were to remove one square (from the original)?

Work on this problem interactively or if you are able to, there is a link to print out the pieces to cut out and try to fit together.

For each of these statements decide whether you think it is ALWAYS, SOMETIMES or NEVER true. Discuss it with someone else if you can and see whether you can convince one another – examples and diagrams will help with this.

Further ideas to explore

The final activities for this week are two optional longer activities – a GeoGebra book all about Islamic geometric patterns that links together the ideas of pattern, shape and symmetry, followed by a clip and downloadable resource on Kajitsu puzzles.

There are some really interesting geometric patterns that you can discover more about through this interactive book – it takes you through examples of different patterns from a number of sites around the world and links to Islamic culture, architecture and sorts of research.

islamic geometric patterns
Islamic geometric patterns

If you like doing puzzles, there are some really interesting puzzles called Kajitsu shown in this video clip and pdf booklet that involve using symmetry as well as logic to solve them.

There is a booklet with instructions and many sets of puzzles here

ANSWERS – click below