Challenge Level

Why do this problem?

Introduce the task and together with the children, generate a symmetrical necklace. Challenge them to work in pairs to find as many other symmetrical necklaces as possible and when they think they have found as many as possible, they record them on strips of paper (you could use this sheet Beads.pdf cut into strips). (Giving separate slips of paper as opposed to a single worksheet will mean the children will be able to arrange and rearrange them.)

Can they now arrange the strips to help them to see whether they have found all the necklaces?

Whilst they are doing this, copy the results they have found onto the large strips and then invite children to come and hold them up at the front. Ask 'Do we have them all? How do we know? How can we be sure?" Encourage individual children to come to the front and rearrange the strips by moving their classmates around.

The most usual reponse to this is that the necklaces fall into pairs, each the inverse of the other (RRBBBBRR and BBRRRRBB, for example). Point out that this helps us to know that there should be an even number of solutions, but it doesn't prove that we have found them all - there might well be another pair that we haven't found yet.

Give some time for the children to rearrange their own strips and if anyone thinks they have found a way to show that they have found them all, invite them to come to the front and rearrange the large strips. If no-one does, you mght want to leave the children for some time - or even take the problem home - before sharing the proof below.

Focus first on the strips that begin with a red bead.

Arrange them so that the second red bead is in position 2, then position 3 and finally position 4:

RRBB BBRR

RBRB BRBR

RBBR RBBR

This must be the full set of necklaces begining with a red bead since the second bead has been placed in each of the three other places.

If there are three beginning with red, there must also be three beginning with blue.

This exemplification of every possible solution is known as 'proof by exhaustion'.

How are these different?

How are these the same?

How do you know that is symmetrical?

Is your pattern the same as ... 's?

Do you think there are any more? Why/why not?

Can you show me why you think youve got them all?

Once they have convinced you, offer them nine beads, five of one colour and four of another. What difference does this make? What about ten, eleven, etc? Can they state any rules about the number of beads they need to make a symmetrical necklace with two colours?