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## 'Changing Areas, Changing Volumes' printed from http://nrich.maths.org/

### 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.

Once students are familiar with the grid structure, hand out

these cards and
invite students to work in pairs to arrange the cards in a grid
like this:

Students could use multilink cubes or draw each cuboid on

isometric paper to support them as they work on the task.

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 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.

### Possible support

Changing Areas,
Changing Perimeters provides a two-dimensional version of this
three-dimensional problem.