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Why do this problem?
By starting with concrete examples of enlarged rectangles (and
cuboids), students can build up a picture of what happens to a
shape when it is enlarged, and discover the relationship between
length, area and volume scale factors of enlargement.
Working in groups, students can take responsibility for
choosing which questions to ask themselves, and produce numerical,
spatial and algebraic explanations for their findings.
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 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 (Word
) to each group,
and make it clear that you are expecting every group to suggest
possible lines of enquiry once they've worked on the initial task.
Once students have had a chance to formulate some ideas, bring the
class together and collect on the board each group's suggestions.
This can be added to over the course of the session(s). Students
can choose to carry on working on their own ideas, or on someone
Of course, if students find it hard to suggest new lines of
enquiry, teachers can prompt using the questions suggested in the
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. At the end of the
task, every group will be expected to present their findings.
You may want to make square and isometric
, 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
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
using diagrams to explain how the enlarged shape relates to the
Group B - You've shown some
numerical relationships - how can you convince yourselves they will
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 happens to the area of a rectangle when I enlarge it by a
scale factor of $2, 3, 4, ... k$
What happens to the area of other shapes when I enlarge
What happens to the surface area and volume of solids when I
enlarge them by a scale factor of $2, 3, 4, ... k$
Can you explain what's going on with a diagram?
Can you construct an enlargement of a rectangle whose area is twice
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.