Changing areas, changing volumes

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

This problem follows on from Changing Areas, Changing Perimeters.

 

Changing Areas, Changing Volumes printable sheet

 

Here are the dimensions of nine cuboids. You can download a set of cards here.

 

 
1 by 2 by 28
cuboid  
 
 
4 by 4 by 4
cube
 
 
2 by 4 by 7
cuboid  
 
 
1 by 2 by 26
cuboid  
 
 
2 by 4 by 6
cuboid  
 
 
4 by 5 by 6
cuboid  
 
 
4 by 5 by 7
cuboid  
 
 
1 by 2 by 24
cuboid  
 
 
1 by 4 by 14
cuboid  
 



The challenge is to arrange them in a 3 by 3 grid like the one below:

 

Image
Changing areas, changing volumes


 

 As you go from left to right, the surface area of the shapes must increase.

As you go from top to bottom, the volume of the shapes must increase.

All the cuboids in the middle column must have the same surface area.

All the cuboids on the middle row must have the same volume.

What reasoning can you use to help you to decide where each cuboid must go?

 

Once you've placed the nine cards, take a look at the extended grid below:

 
Image
Changing areas, changing volumes


The ticks represent the nine cards you've already placed. Can you create cards with dimensions for cuboids that could go in the four blank spaces that satisfy the same criteria? 

 

 

Can you design a set of cards of your own with a different cuboid in the centre?