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Why do this problem?
There are many ways of tackling this problem, so there is an
opportunity for students to share insights and consider the
benefits of the different approaches.
Students have an opportunity to play with algebra they might
not normally meet, forming cubic expressions that can be related
back to the physical properties of the cuboids they
This problem allows students to model the way mathematicians
work, by taking a starting point and then asking their own
questions and making conjectures that may lead to
This problem lends itself to collaborative working, both for
students who are inexperienced at working in a group and students
who are used to working in this way.
Many NRICH tasks have been designed with group work in mind.
gathered together a collection of short articles that outline the
merits of collaborative work, together with examples of teachers'
This is an ideal problem for students to tackle in groups of
four. Allocating these clear roles (Word
) can help the group to
work in a purposeful way - success on this task should be measured
by how effectively the members of 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
Hand out this task sheet (
) 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
You may want to make isometric paper, cubes, poster paper, and
coloured pens available for the Resource Manager in each group to
When groups have had a chance to work on the first two questions
(this may take more than a lesson!) allow some whole-class time for
groups to share their thoughts, questions, and ideas for possible
lines of enquiry. There are some suggestions of possible questions
to consider in the problem which can be used as prompts for classes
who are struggling to come up with conjectures of their own.
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
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
the different combinations of faces that can be painted - what
could you do with this information?
Group C - Good to see that
someone's checking that the answers are in line with your
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:
- Every group is given a couple of minutes to report back to the
whole class. Students can seek clarification and ask questions.
After each presentation, students are invited to offer positive
feedback. Finally, students can suggest how the group could have
improved their work on the task.
- Everyone's posters are put on display at the front of the room,
but only a couple of groups are selected to report back to the
whole class. Feedback and suggestions can be given in the same way
as above. Additionally, students from the groups which don't
present can be invited to share at the end anything they did
- Two people from each group move to join an adjacent group. The
two "hosts" explain their findings to the two "visitors". The
"visitors" act as critical friends, requiring clear mathematical
explanations and justifications. The "visitors" then comment on
anything they did differently in their own group.
If your focus is effective
, this list of skills may be helpful (Word
). Ask learners to
identify which skills they demonstrated, and which skills they need
to develop further.
If your focus is
mathematical, these prompts might be useful:
What solid are you left with when you remove the painted cubes? How
can you express this algebraically (for 1, 2, 3... faces painted)?
What does this tell you about the factors of the number of
There is only one way to end up with 45 unpainted cubes. Are
there any numbers of cubes you could end up with in more than one
How can you convince yourself that it is impossible to end up
with 50 unpainted cubes?
Devise a method for determining whether any given number of
unpainted cubes can arise from painting whole faces of a large
cube, and if so, how it can be done.
By working in groups with clearly assigned roles we are
encouraging students to take responsibility for ensuring that
everyone understands before the group moves on.
provides a suitable introduction to some of the ideas met in this