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The Development of Algebra - 2

This is the second article in a two part series on the history of Algebra from about 2000 BCE to about 1000 CE.

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Finding 3D Stacks

Can you find a way of representing these arrangements of balls?

3D Stacks

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3


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.
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.