# Wire cages

We can make 'wire cages' by using rods as the edges of cuboids. Can you work out what the volume of each wire cage will be? How many rods will you use?

*here at a 2D task which could be explored before this 3D task. You may wish to look *

Wire cages are used to contain large rocks in some building projects.

**Here is a small version**

**of a 2 by 3 by 4 wire cage cuboid**

**, made with plastic straws.**

Image

We can see 14 red rods of length 2 units, 12 blue rods of length 3 units and 10 green rods of length 4 units.

#### For all the challenges below, no rod is allowed to be longer than 6 units and each cage is always a cuboid.

Challenge 1

The example above of dimensions 2 by 3 by 4 uses exactly 36 rods and has a volume of 24 cube units.

Find all the cages that have a volume of 20 to 25 cube units inclusive and use 40 to 50 rods inclusive.

### Challenge 2

The example above of dimensions 2 by 3 by 4 uses exactly 36 rods.

Can you find a cuboid that uses exactly 28 rods?

Can you find all the cuboids that use exactly 28 rods?

Can you find all the cuboids that use exactly 28 rods?

### Challenge 3

Find at least one pair of cuboids of different dimensions, that when constructed:

- Use exactly 100 rods

AND

- Have each dimension greater than 2 rods.

- Use exactly 100 rods

AND

- Have each dimension greater than 2 rods.

This sheet contains a copy of the task.

You can use the applet below to see 3D models of different wire cages.

Move the Length, Width and Height sliders to change the dimensions of the cage.

Use the Rotate and Tilt sliders to view the cage from different angles.

You can choose whether your cage is transparent or opaque using the checkbox.

Move the Length, Width and Height sliders to change the dimensions of the cage.

Use the Rotate and Tilt sliders to view the cage from different angles.

You can choose whether your cage is transparent or opaque using the checkbox.

Thank you to Keya, Brodie, Ayali and Alex from Strand on the Green Junior School who sent in these solutions:

Challenge 1:

Image

Challenge 2:

Image

Challenge 3:

Image

They said:

We noticed that the numbers are all multiples of 4 and if you add 1 to one of the dimensions the number of rods goes up by 4.

Well done for laying out these solutions so clearly!

There are some more possible solutions to Challenge 3. If anybody has found more solutions and would like to let us know, or if you have any different ideas about this activity that you'd like to share with us, please email us.

### Why do this problem?

This task was designed for the 2018 National Young Mathematicians' Award, so it might be a particularly useful activity for a small group of your highest-attaining pupils to work on. It encourages a systematic approach and provides an opportunity for pupils to explain how they know they have all the solutions.### Possible approach

Begin by showing a model 'wire cage' like the one in the picture. Discuss how many long rods are used to make it, and what the lengths of these rods are. How can we find the volume of a cuboid? Make sure that pupils understand that the volume is the number of small cubes (one unit long in each direction) that can fit inside their cuboid - this will help them visualise how the volume changes when the lengths change.There is a great interactive resource here (in the 'Getting Started' section) which can be used to explore different cuboids.

When pupils understand the key ideas about 'wire cages', give them time to work in small groups on the three challenges.

Key questions

Explain what you are doing to find the next solution.Tell me about the number of rods that you have.

How do you know what the volume of that cuboid is?

How are you trying to get a particular total?

### Possible extension

Pupils could be encouraged to consider what happens when you start with a small cuboid and increase one of the dimensions by 1 many times, e.g. 1x2x3, 2x2x3, 3x2x3, 4x2x3... Then they could think about how many rods are needed for each and what the relationship is between the numbers.### Possible support

Some pupils may need some model making materials or pictures of other cuboids to help them visualise their 'wire cages'.Sharing different ways of thinking about the problem and different ways of recording that thinking can be very useful in developing all learners' problem-solving skills.