### Why do this problem?

Most students will be familiar with the $A$ paper sizes but be
unaware of their special properties. This problem explores ratio
and scale factors, with the possibility of working with surds and
indices if students choose to tackle the extension questions.

### Possible approach

For students who have not yet met the relationship between length
and area scale factors,

Growing Rectangles
offers a good introduction.

Take a sheet of $A4$ paper, hold it up to show the class.

"Here is a sheet of $A4$ paper."

Fold it in half so that it is $A5$ size.

"Does anyone know what size this paper is?"

"$A5$"
"What special relationships can you think of between $A4$ and $A5$
paper?"

"The area is
half"
"The long side on the $A5$ is the
same as the short side on the $A4$"
"They are similar
shapes"
If similarity doesn't emerge as a suggestion from the class, use
the example of a photocopier - something printed on $A4$ paper can
be enlarged to fit onto $A3$ without distortion, so they must be
similar rectangles.

Set students the challenge of working out the scale factor of
(length) enlargement needed to get from any size of paper to any
other. You may wish to scaffold the task for your students by first
asking them to work from $A4$ to $A2$ and then to $A3$, $A1$ and
$A0$, before working out the scale factors necessary to go from
larger to smaller sizes.

Finally, you may wish to spend some time with the students
expressing the relationships they have discovered in a general
form.

The possible extension below is suitable for students who have met
negative and fractional indices.

### Key questions

When I enlarge from $A4$ to $A2$ how does the area change? How
do the lengths change?

What if I went from $A(n)$ to $A(n-2)$?

When I enlarge from $A4$ to $A3$ how does the area change? How
do the lengths change?

What if I went from $A(n)$ to $A(n-1)$?

What if I went from $A(n)$ to $A(n-x)$?

What happens when I go from a larger sheet to a smaller
one?

### Possible extension

Starting with the definitions of the $A$ paper sizes given in the
problem, challenge students to work out the dimensions first of
$A0$ paper and then of $A4$ paper, which they can then check by
measuring.

### Possible support

Start by looking at the relationship between the even numbered
paper sizes ($A4$ to $A2$, $A0$ and so on).