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Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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Cuisenaire Rods

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Finding 3D Stacks

Age 7 to 11 Challenge Level:

This activity may be used as a follow-on for those who have successfully worked at Doplication.

 
Let me help you visualise this representation of a 3D situation.
The picture below shows a 2 by 3 by 4 arrangement of red spheres, which you might see as two layers of 3 by 4:

 new6

These two layers together could also be viewed as representing six cubes:
 

 
In the top picture the centre of each cube is shown with a light or dark brown sphere. 
 
How many spheres are there altogether in this 2 by 3 by 4 arrangement?
How did you count them?

Use your system of counting to find the total number of spheres for other sizes, for example 3 by 3 by 3;  3 by 4 by 4; 4 by 4 by 5. etc.

How would you find the number of spheres for any sized arrangement?