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3D Stacks

Age 7 to 14 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.
In this example of a 2 by 3 by 4 we see in the red balls, as two layers of 3 by 4.

 new6
These two layers together can also be viewed as representing 6 cubes.

 
 
In the top picture the centre of each cube is shown with a light/dark brown ball.
 

In Doplication you would have had;
 
 axitate version
which we called $4$ ? $3$.
 
In this activity it would be called something different involving $4, 3, 2$. Just for now I am going to use @, so the diagram above represents $4$@$3$@$2$.
 
I invite you to explore many such arrangements and find a way of recording your results.
You might like to just look at the arrangements that are like cubes - $3$@$3$@$3$ , $4$@$4$@$4$, $5$@$5$@$5$, etc.
 
As with most mathematics it's good to compare so maybe compare the results for squares in Doplication and cubes in 3D Stacks.