Snake Coils

This activity has been particularly created for the most able, and so pupils might be expected to work independently from an adult. So these notes are shorter than usual for our activities.

Snake Coils

Imagine a $600$ cm snake that coils into a circle, but he's not going to eat himself.
He is going to coil himself around to make a double circle, then a treble circle and so on until he is coiled round like this rope.

Challenge 

Once in a circle how much further will he have to slither to make a double circle of his body. What's this length as a fraction of his total length?

How much further will he have to slither to make three circles of his body.  
What's this length as a fraction of his total length?

 

How much further will he have to slither to make four circles of his body.
What's this length as a fraction of his total length?

How much further will he have to slither to make five circles of his body.
What's this further length as a fraction of his total length?
 

Hint 


This diagram for the five coils might help you to imagine what is happening.
Each set of circles should be the same size and on top of each other, but we could not show that in a diagram. 

 


Final challenge


When the snake has slithered a total of $550$ cms along his body length, making smaller and smaller circles of his body, how many circles will there be?

Why do this problem?


This activity engages the pupils in both a spatial and numerical context. It gives them also the freedom to choose how they go about it - visualising in their head, using paper, string etc. that they have requested and/or making use of a spreadsheet. They can learn a lot from adopting one method and then realising that an alternative method would be better.

Possible approach


Presented as on the problem page.

Key questions


Very open questions, like "tell me about this . . . . "
What have you found out so far?

Possible extension


Consider the sizes of the circles as they get smaller and smaller.