Copyright © University of Cambridge. All rights reserved.
The notes below describe a method of
engagement based around a technique called complex instruction. You
can see many examples of tasks which work well in groups on
the May 2010 NRICH publication.
Although this problem is group-worthy, it
can, of course, be attempted individually if you wish
Why do this problem?
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.
The problem involves working out sensible approximations to
physical quantities. Making good headway into the task will be
difficult and requires good use of teamwork.
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
You may want to make calculators, internet, squared or graph
paper, and coloured pens available for the Resource Manager in each
group to collect. As teacher, you (or the internet) will be a
resource containing knowledge of physical data, constants and
While groups are working divide the board up with the groups names
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. This is a good way of highlighting the
mathematical behaviours you want to promote, particularly with a
challenging task such as this.
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? Are there any facts or data
that you need but don't know?
Alternatively, your focus for feedback might be mathematical:
Group A - I like the way you set
out your assumptions clearly.
Group B - Can you provide
some error bounds on that calculation?
Group C - Good to see that
someone's checking the numerical calculations.
Make sure that while groups are working they are reminded of the
need to be ready to present some of their approximations at the
end, and that all are aware of how long they have left.
We assume that each group will record their reasoning, assumptions
and calculations in preparation for reporting back. There are many
ways that groups can report back. Here are just a few
- Every group is given a couple of minutes to report back to the
whole class, perhaps focussing on explaining two of their
approximations. Those listening can seek clarification and ask
questions. After each presentation, those listening are invited to
offer positive feedback. Finally, those presenting can suggest how
the group could have improved their work on the task.
- Everyone makes a poster to 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 some of their calculations and reasoning
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
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 assumptions have you made?
What other information do you need?
Are there any questions which give an exact answer?
Can you say anything about the accuracy of those answers which
this is an approximations question, it is possible to give rigorous
bounds on the quantities. Students could be encouraged to give a
known upper and lower bound for each quantity, with the focus on
providing as tight a bound as possible
Students could also compare their approximations with those of
others. Is there a sense in which one of best?
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.