# Growing Rectangles

## Problem

*Growing Rectangles printable sheet*

Imagine a rectangle with an area of $20$cm$^2$

What could its length and width be? *List at least five different combinations.*

If you enlarge each of your rectangles by a scale factor of 2, what would their new dimensions be?

What would their areas be?

What do you notice?

What happens when you enlarge rectangles with different areas by a scale factor of 2?

What if you enlarge them by a scale factor of 3? Or 4? Or 5 ...? Or $k$?

What if $k$ is a fraction?

Explain and justify any conclusions you come to.

Do your conclusions apply to plane shapes other than rectangles?

Now explore what happens to the surface area and volume of different cuboids when they are enlarged by different scale factors.

Do your conclusions apply to solids other than cuboids?

*This problem is based on an idea suggested by Tabitha Gould.*

## Getting Started

## Student Solutions

We received concise statements of general rules based on clearly presented evidence:

Matthew, Holly and Isobel from Staunton & Corse C of E School discovered the relationship between the enlargement scale factor and the effect on the area:

We started by writing 5 different dimensions with an area of $20$cm$^2$:

Length x Width:

Then we enlarged them by a scale factor of 2:

0.5 x 160, 1 x 80, 2 x 40, 4 x 20, 8 x 10

We then took the areas of both rectangles ($20$cm$^2$ and $80$cm$^2$) and figured out a way that 20 could be turned into 80.

Well, obviously you multiply it by 4.

But if you enlarge them by a scale factor of 3:

0.25 x 80, 0.5 x 40, 1 x 20, 2 x 10, 4 x 5

become:

0.75 x 240, 1.5 x 120, 3 x 60, 6 x 30, 12 x 15

This time the areas are $20$cm$^2$ and $180$cm$^2$ which means that the areas have been multiplied by 9.

So the question we asked ourselves was:

What does 2 have in common with 4 as 3 has in common with 9?

The answer is: you square the scale factor to find out how much bigger the areas have become.

Akintunde from Wilson's School generalised for two and three dimensions:

A rectangle with area $20$cm$^2$ could have the following dimensions:

Length by Width:

4 by 5, 2 by 10, 20 by 1, 40 by 0.5, 50 by 0.4

When enlarged by a scale factor of 2 the dimensions become:

8 by 10, 4 by 20, 40 by 2, 80 by 1, 100 by 0.8

All these rectangles have an area of $80$cm$^2$

The area is four times larger.

A rectangle with an area of $10$cm$^2$ could have the following dimensions:

5 by 2, 1 by 10, 20 by 0.5

When enlarged by a scale factor of 2 the dimensions become:

10 by 4, 2 by 20, 40 by 1

All these rectangles have an area of $40$cm$^2$

Again, the area is four times larger than the original.

I think that maybe whatever scale factor you enlarge the rectangle by, the area is enlarged by the square of the scale factor:

Starting with the rectangles with an area of $10$cm$^2$:

5 by 2, 1 by 10, 20 by 0.5

When they are enlarged by a scale factor of 3, the dimensions become:

15 by 6, 3 by 30, 60 by 1.5

All these rectangles have an area of $90$cm$^2$

The areas are nine times larger.

And when the rectangles with an area of $10$cm$^2$ are enlarged by a scale factor of 4 the dimensions become:

20 by 8, 4 by 40, 80 by 2

All these rectangles have an area of $160$cm$^2$

The areas are sixteen times larger.

And when the rectangles with an area of $10$cm$^2$ are enlarged by a scale factor of 0.5 the dimensions become:

2.5 by 1, 0.5 by 5, 10 by 0.25

All these rectangles have an area of $2.5$cm$^2$

The areas are four times smaller (you multiply the original area by 0.25).

I think that for any scale factor, you square it and then multiply that new number by the original area.

e.g. SF$2$ and an area of $20$cm$^2$:

2$^2$ is 4 and $20$cm$^2$ x 4 is $80$cm$^2$

I think this happens because the two dimensions of a rectangle are both multiplied by three when the rectangle is increased by a Scale Factor of 3 so overall the area of the rectangle is nine-times bigger (3 x 3).

If you increase a rectangle by a scale factor k, the area of the new rectangle will be k$^2$ times the old area of the rectangle.

Taking a triangle with a base of 2, a height of 3 and an area of 3:

When enlarged by a scale factor of 2 it becomes a triangle with a base of 4, a height of 6 and an area of 12.

The area is 4 TIMES LARGER.

When enlarged by a scale factor of 3 it becomes a triangle with a base of 6, a height of 9 and an area of 27.

The area is 9 TIMES LARGER.

When enlarged by a scale factor of 4 it becomes a triangle with a base of 8, a height of 12 and an area of 48.

The area is 16 TIMES LARGER.

I think this rule applies to all plane shapes because when different rectangles and triangles are increased by different scale factors, the increase in area is the same.

Taking a cuboid with a width of 2, a length of 2, a height of 3, a surface area of 32 and a volume of 12:

When enlarged by a scale factor of 2 it becomes a cuboid with a width of 4, a length of 4, a height of 6, a surface area of 128 and a volume of 96.

The surface area is 4 TIMES LARGER and the volume is 8 TIMES LARGER.

The surface area increased by 4 (2$^2$) and the volume increased by 8 (2$^3$).

The volume takes in 3 dimensions and the surface area takes in 2 dimensions.

I have concluded that with rectangles and cuboids, whatever the scale factor is:

you square the scale factor and then multiply that number by the old area to find the new area,

and you cube the scale factor and times that number by the old volume to find the new volume.

Based on the fact that the rule about scale factors seems to work for all plane shapes, I predict that 3D shapes will follow the same rule as for cuboids.

Nice work - well done to you all.

## Teachers' Resources

### 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 printable worksheet may be useful: Growing Rectangles*

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.

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 $2, 3, 4, ... k$

### 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.

### Possible extension

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