Changing areas, changing volumes
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:
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:
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?
If I know two rectangles have the same area, how can I decide, just by looking at their dimensions, which has the greater perimeter?
If I know two cuboids have the same volume, how can I decide, just by looking at their dimensions, which has the greater surface area?
Well done to Philip from Wilson's School who submitted this solution.
Why do this problem?
Working on this problem will give students a deeper understanding of the relationship between volume and surface area, and how they change as the dimensions of a cuboid are altered.
Possible approach
This problem follows on from Changing Areas, Changing Perimeters. We suggest students start with the rectangles task from that problem to introduce them to the structure of the grid they will be using here.
For those who finish quickly, ask them the question from the problem about extending the grid like this:
Towards the end of the lesson, bring the class together to share any efficient strategies they used to complete the task.
Pose the question "If I know two cuboids have the same volume, how can I decide, just by looking at their dimensions, which has the greater surface area?"
Draw out students' ideas about the properties of long and thin cuboids as opposed to those that are almost cubes.
(This is the three dimensional analogue of short and fat rectangles having a smaller perimeter than long thin ones, when their areas are equal.)
Finally, discuss the possible content of the four extra spaces in the extended grid and strategies they used to generate possibilities.
Possible support
Changing Areas, Changing Perimeters provides a twodimensional version of this threedimensional problem.
Possible extension
Challenge students to design their own set of nine cards that can be arranged in this way. If students are restricted to whole numbers it is quite challenging to create cuboids with equal surface areas.