Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface of the water make around the cube?
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
This approach could be used if you want the focus to be on the mathematical structure of the problem:
"Here is a 3 by 3 by 3 cube made up of 27 smaller cubes. Imagine I dipped it in a pot of yellow paint so that each face of the large cube was covered. Then after the paint has dried, imagine I split it into the 27 original small cubes. Can you work out how many cubes will have no paint on them? How many will have just one face painted? Or two faces painted, and so on."
Give students time to discuss with their partners and work out their answers. While they are working, circulate and observe the different approaches that students are using, and challenge them to explain any dubious reasoning.
Bring the group together to share their responses. Collect the answers for the number of cubes with 0, 1, 2, 3 ? faces painted, and note that they add up to 27. Invite students to explain how they worked it out.
"I'd like you to work on some cubes of different sizes until you are confident that you can always work out how many cubes will have 0, 1, 2 and 3 faces painted. In a while, I'll be choosing a much larger cube at random, and you?ll need to have an efficient method of working it out."
Here are some prompts that could be used if students get stuck:
"Where are the cubes with no faces painted?"
"Where are the cubes with 1, 2, 3 faces painted?"
"How many of each type of cube would you have in a 4 by 4 by 4? in a 7 by 7 by 7? in an n by n by n cube?"
Bring the class together and challenge them to explain how they can work out the number of cubes of each type in a 10 by 10 by 10 painted cube. Depending on the students? experience of working with algebra, you could work together on creating formulas for the number of cubes of each type in an n by n by n cube. The Solution contains a table that shows
the results very clearly.
This approach could be used if you want the focus to be on developing groupwork:
This is an ideal problem for students to tackle in groups of four. Allocating these clear roles (Word, pdf) can help the group to work in a purposeful way - success on this task should be measured by how effectively the group work together as well as by the solutions they reach.
Introduce the four group roles to the class. It may be appropriate, if this is the first time the class have worked in this way, to allocate particular roles to particular students. If the class work in roles over a series of lessons, it is desirable to make sure everyone experiences each role over time. For suggestions of team-building maths tasks for use with classes unfamiliar with group
work, take a look at this article and the accompanying resources.
Hand out this task sheet (Word, pdf) to each group, and make it clear that everyone needs to be ready to share their findings with the rest of the class at the end of the sessions. Exploring the full potential of this task is likely to take more than one
lesson, with time in each lesson for students to feed back ideas and share their thoughts and questions.
You may want to make isometric paper, cubes, poster paper, and coloured pens available for the Resource Manager in each group to collect.
While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together.
You may choose to focus on the way the students are co-operating:
Group A - Good to see you sharing different ways of thinking about the problem.
Group B - I like the way you are keeping a record of people's ideas and results.
Group C - Resource manager - is there anything your team needs?
Alternatively, your focus for feedback might be mathematical:
Group A - I like the way you are considering the structure of the cube.
Group B - You've identified which cubes end up with one face painted - can you think of a way of quickly counting them?
Group C - Good to see that someone's checking that the answers are in line with your predictions.
Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.
We assume that each group will record their diagrams, reasoning and generalisations on a large flipchart sheet in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:
Where are the cubes with no faces painted?
How do your algebraic expressions relate to the geometry of the situation?
Partially Painted Cube provides a suitable follow-up activity.