Symmetry · Usually met around P3
Rotational symmetry: turn it and it looks the same
Turn it once round: count the times it looks the same.
How to do it
Method
Turn it once round: count the times it looks the same
4 blades alike: 1 + 1 + 1 + 1 = 4 times
Another way
Trace and turn
Pin the centre, turn, and note each match
In full
Turn the shape slowly about its centre through one whole turn and count how many times on the way it looks exactly as it did at the start (counting the return to the start). A square looks the same after every quarter turn, 4 times in a whole turn; the letters N, S and Z look the same after half a turn, 2 times; a shape that only looks the same when it is back where it started matches just once in a whole turn and has no rotational symmetry. A windmill with n blades all alike matches n times in a whole turn. The colours must match too: blades in two colours by turns need to move two blades along. Another way: trace the shape on thin paper, pin its centre, turn it, and note each match; or find the part that repeats (a blade, say) and count how many times it appears. Turning and folding are different: N has no line of symmetry, yet half a turn makes it look the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Turn this windmill one whole turn about its centre. How many times on the way (counting the return to the start) does it look exactly as it did at the start?
Show the answer and working
Answer: 5 times
It has 5 blades alike, and it looks the same each time a blade reaches the next one's place: 5 times in a whole turn.
Example 2
Turn this letter half round (upside down): it looks exactly as it did before.
True or false?
Show the answer and working
Answer: False
Imagine turning the paper half round: the top row ends up at the bottom and left and right swap too. Compare square by square.
Example 3
This windmill has 4 blades of the same shape, some in another colour. Turn it one whole turn about its centre. How many times does it look exactly as at the start (colours too)?
Show the answer and working
Answer: 1 times
The colours must land where they were too: just 1 times in a whole turn.
Practise it in the game →Symmetry · Free to start, unlimited questions, no download