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Counting Counters

Age 7 to 11 Challenge Level:

Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many counters do you need to do this?
Imagine surrounding this ring with more counters.
How many more counters are needed now?
How many counters will there be altogether?

Were you right?

What about a bigger ring? And another? And another? 

How many counters will there be in the $9$th ring?
How many do you think there will be in the $100$th ring?
How would you predict how many counters there will be in any ring?

Why do this problem?

This problem gives an opportunity for children to generalise from a practical situation. It is easy to replicate these patterns in the classroom with counters, bottle tops etc.

Possible approach

You could begin by displaying the image in the problem and invite children to talk to a partner about what they notice.

After a short time, bring everyone together and ask pairs to share their thoughts.  Use their observations to clarify how the images have been created and then lead into the first question.  Ensure learners have access to counters and other similar resources to use in order to tackle the task, should they wish.

This task could extend over a few lessons and might culminate in pairs or small groups presenting findings to the rest of the class.

Key questions

Can you describe the pattern?
How many counters will you need to make the next 'one'?
How many counters have you used already?
Can you predict the number of counters you'll need for the next 'one'?
Can you predict the number of counters you'd need for any 'layer'?

Possible extension

Ask children to create their own increasing pattern so that they could predict the number of counters needed in any 'layer'.

Possible support

Having counters available to make the pattern, or drawing it, will help some children. They may find it useful to record the number of counters in each 'layer' too.