Make Those Bracelets

This activity has been particularly created for the higher attaining pupils so these notes are shorter than usual for our activities.  It was used in the preliminary rounds of the Young Mathematicans' Award 2012.

Make Those Bracelets

Making bracelets.

Here are two pictures of boys, girls and parents in Africa making bracelets. They have lots of colourful beads to choose from. 

 

Let's think about making a bracelet using two different colours of beads. The smallest bracelet will have two of each colour and the largest five of each colour. 



          
 

The challenge

Find all the ways of arranging 4 and 5 beads on the bracelets, each time using just the two colours. Be careful not to have any arrangements the same!
Can you convince others that you have found them all?

Try this next for 6, 7, 8, 9 and 10 beads.

Final Challenge

A new person joins your group and wants to make bracelets. Can you put on paper for them some guidance so that they have a system to make sure that they find them all and know how to avoid repeats?

Why do this problem?


This activity engages the pupils in both a spatial and numerical context. It challenges their ability to see symmetrical reflections. It also gives them the freedom to choose how they go about the task - visualising in their head, using pencil and paper, beads, cubes or other counters 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 might be better.

Possible approach


You could introduce the task as presented as on the problem page.
You may find it appropriate to make use of this interactive help
useful as part of  designing the bracelets.

Key questions


Use open questions such as "tell me about this . . . . "
How did you decide on this approach to finding all the possibilities?


Possible extension


Explore the differences between this challenge and when the bracelets are like a non-reversable necklace.