3D stacks
Can you find a way of representing these arrangements of balls?
Age
7 to 14
Challenge level
Problem
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.
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These two layers together can also be viewed as representing 6 cubes.
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In the top picture the centre of each cube is shown with a light/dark brown ball.
In Doplication you would have had;
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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.
Teachers' Resources
Why do this problem?
This acts as a further extension to Doplication. It's an activity that is intended to give opportunities for those pupils to explore deeply using their intuition and flair in the areas of both spatial awareness and number replationships and patterns.
Possible approach
As this is designed for the highest attaining, it might be presented as on the website or in a one-to-one situation, encouraging discussion between adult and pupil. The pupils may need access to a spreadsheet once many number results are being acquired.
Key questions
Tell me about what you have found.
Can you describe the ways that you arrived at these numbers?
How did you construct this on the spreadsheet you used?