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:

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

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:

SF 2, AREA increases by 4
SF 3, AREA increases by 9

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.