### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

### Take Ten

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3 cube?

# Changing Areas, Changing Volumes

### 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.