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This activity investigates how you might make squares and pentominoes from Polydron.

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3 Rings

Age 7 to 11
Challenge Level

3 Rings

I found a number of small bracelets and rings on a table some time ago and I noticed how some were on their own, others were touching at the edges, others were overlapping each other and some small ones had found themselves inside larger ones.
I took two of these, one ring and one bracelet, and explored what possibilities there were.

I thought that this would be the next challenge for you all. To look at the situation when you have three rings, circles, bracelets . . . . it doesn't matter what they are really or what size they are. They could even expand and get bigger or get smaller if you liked. But, thinking of the four things I noticed at the start:-





I wonder what would be the number of ways in which 3 such circles could be?

Here are some ways, remember I said they could be different sizes each time, but I've coloured them so that it is easy to know which one we are talking about.

Well I feel you could carry on at this point, just a few points to remember:-

When writing you must say something about each of the three circle/rings/bracelets.

Three separate ones could be anywhere yet separate and they would all count as one arrangement, and the same kind of things goes for any other arrangement, if the words are the same then, for this challenge the arrangement is the same.

You could now ask "I wonder what would happen if.....?"

Why do this problem?

This activity is a very simple investigation which allows for many levels of participation. There are those youngsters who can use three different sized rings and just "play around" with them. This provides a wonderful basis for most useful discussion as to the positioning of each of the three rings and their physical relationship to each other. Children who go about it more systematically seem to have great enjoyment in the systems that they use. These two groups have to deal with the fact that some arrangements may 'look' very different but in the context of this challenge are equivalent. This again leads to very worthwhile discussion.

Possible approach

I usually provide the children with just two different sized rings and have the third one as an imaginary one that they can alter the size of for their own use, each time.

Key questions

How would you talk about this ring?
What is this ring doing?

Possible extension

The more advanced youngsters who have no problems with the "look" and can "see" equivalent situations are usually able to go through the activity fairly quickly and are soon asking "I wonder what would happen if ... they were squares . . . the rings were all the same size. . . ?"


Possible support

Having an adult working with a small group so as to help the language to flow can sometimes be very effective.