### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

# Growing Rectangles

### 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. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.

### Possible approach

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 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 else's questions.
Of course, if students find it hard to suggest new lines of enquiry, teachers can prompt using the questions suggested in the problem.

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 dotty 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 using diagrams to explain how the enlarged shape relates to the original.
Group B - You've shown some numerical relationships - how can you convince yourselves they will always work?
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:
• 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 differently.
• 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.

### Key questions

If your focus is effective group work, this list of skills may be helpful (Word, pdf). 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 them?
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? Algebraically?

### Possible extension

Can you construct an enlargement of a rectangle whose area is twice the original?

### Possible support

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.