Imagine a rectangle with an area of $20$cm$^2$
What could its length and width be? List at least five different combinations.

Imagine enlarging each of your rectangles by a scale factor of $2$:

List the dimensions of your enlarged rectangles and work out their areas.
What do you notice?

Try starting with rectangles with a different area and enlarge them by a scale factor of $2$.
What happens now?
Can you explain what's going on?

What happens to the area of a rectangle if you enlarge it by a scale factor of 3? Or 4? Or 5 ...?

What happens to the area of a rectangle if you enlarge it by a fractional scale factor?

What happens to the area of a rectangle if you enlarge it by a scale factor of $k$?

Explain and justify any conclusions you come to.
Do they 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.
Explain and justify any conclusions you come to.

Do your conclusions apply to solids other than cuboids?

This problem is based on an idea suggested by Tabitha George from Swavesey Village College