Next Size Up
2 out of 3
The challenge for you is to make a string of six (or more!) graded cubes.
Problem
Long ago, I made a string of ten graded cubes.
Each edge of the smallest cube was one centimetre long. Each edge of the largest cube was ten centimetres long.
I labelled each one with its volume in cubic centimetres.
The challenge for you is to make a string of graded cubes. You might do better working in a team, so encourage your friends to help!
We would love to hear how you made your own cubes and to see photos of them.
Getting Started
Using squared card or paper will help.
You could download this squared sheet and print it out on thin card. You will need several sheets to make the larger cubes.
How can you draw a net of a cube? You could look at this problem.
Student Solutions
Class 5AM from St Peter's School, Barcelona, worked on this problem in groups. Here is their report.
Yusuf from Columbia Primary School told us:
We made our cubes using modular origami. We had to measure each piece of paper to make sure it was the correct size and that it would make the right sized cube. We worked out the volume of each cube and wrote it on the side.
Kathy and Pete from Barley Hill Primary School decided to see what patterns they could find by making cuboids.
First we drew the nets of our cuboids.
We noticed that the volume of our first set where we just added one square to the length of the rectangle went up by $1$ each time. The volume of our second set went up by $4$ each time. This is different to the set of cubes in the problem which went up different amounts each time because we only changed one length of our shape.
Here is a picture of their graded cuboids.
Well done for sending in your pictures.
There are lots more shapes you could make - you could see what happens to the volume if you change more than one side of a cuboid.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem is fundamentally a practical one which involves careful measuring and drawing. But by making a string of graded cubes for themselves, children's understanding of length and volume will be deepened far more than through just calculating volumes of hypothetical cubes. The problem could be used as a group or team activity where learners work together for a common purpose.
Possible approach
You could begin by showing the group the string of cubes pictured in the problem and invite them to talk in pairs about what they see. Open up the discussion so that the whole group participates and eventually comes to a consensus about what is pictured. You can then introduce the challenge to make a set of cubes for themselves.
This problem could be a good way to introduce the idea of a net of a cube, or you could choose to set the challenge having already worked on nets. The group's prior experience of nets will influence the way you facilitate the activity and the frequency with which you bring them all together to share progress.
Invite groups of learners to work together on this task and try not to be too prescriptive in terms of the way they approach it and the materials they use. It might be worth ensuring that sharp pencils, rulers, thin card (preferably squared), glue and sticky tape are available, should a group require them. (This squared sheet for printing onto thin card might be useful.) Some groups might want to use the computer to create their net. Thin thread and a needle will also be needed to hang up the shapes.
It might be that the whole class decides to distribute the work amongst the groups so that one or two sets are produced in total.
Key questions
How will you make a cube from paper/card?
What can you tell me about a cube?
How do you know how big each cube will be?
How will you share out the work?
Possible extension
Learners could make a string of other more ambitious shapes such as tetrahedra or octahedra although calculating the exact volume of these might be too tricky.
Possible support
Some children will find this net a useful starting point.