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Turning Granny

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

Turning Granny

Watch this short video clip:
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Make a granny knot and a reef knot for yourself.

Now investigate their symmetries. Start with the granny knot.
Does it look the same after a half turn about a vertical axis?
How about after a half turn about a horizontal axis from right to left?
Now try a half turn about a horizontal axis front to back. Does it look the same this time?
(Don't forget you can play the video again to remind you of these three axes.)

Try the same thing with a reef knot.

What is the same about the symmetries of the two knots?

What is different?

The idea of this problem is to use a mathematical property to distinguish between knots. This activity allows pupils to become familiar with the idea of rotational symmetry in a very practical context.You might like to try Reef and Granny before going on to this problem.

It would be a good idea for pupils to work in pairs and begin by making each type of knot together. You could ask them to suggest ways of distinguishing between the two differnent types - they may well have some novel ideas - and then introduce rotational symmetry. At first, it would be good simply to leave them to get on with the task but after a few minutes, bring the group together to invite them to share good ways of comparing a knot before and after a turn. Some might have drawn pictures, some might want to make another knot of the same type. Some may be able to hold an image of the knot in their heads and compare this mental image with the result after a turn. However they choose to tackle the problem, listen for pairs justifying their decisions to each other in clear ways using appropriate language.

There is no reason why this activity should stop at just the reef knot and the granny knot. Those who are keen might want to investigate the symmetries of other types of knots too.