### Why do this problem?

This problem investigates rational and irrational numbers. Along
the way, there is the chance for some reasoned arguments and the
search for counter-examples, while learners are challenged to apply
their understanding of fractions and decimals.

### Possible approach

Start by investigating a simpler case, such as a search for
rational numbers between $\sqrt{2}$ and $\sqrt{3}$. Learners could
start by working out approximately where these two irrational
numbers lie on the number line, and consider the implications for a
fraction where the denominator was $1$. This could stimulate
discussion about when two surds will have a whole number between
them.

Some learners may wish to investigate fractions between surds
by looking at the approximate decimal representation of the surds.
This can lead to a convincing argument about the gap between
$\frac{p}{q}$ and $\frac{p+1}{q}$ as $q$ gets larger.

Another line of enquiry is to represent the problem as an
inequality: $\sqrt{2} < \frac{p}{q} < \sqrt{3}$ and square
all terms, rearrange, and look at the implication of choosing
different values of $q$.

After successfully investigating simpler cases, learners could
use their approach to search for rationals between $\sqrt{56}$ and
$\sqrt{58}$, with the aim of finding both denominators that don't
yield a solution, and proving that they are the only ones. Groups
could work together on making a convincing argument that there are
no more, and present their ideas to the class.

### Key questions

What happens to the gap between $\frac{p}{q}$ and
$\frac{p+1}{q}$ as $q$ gets larger?

### Possible extension

Tackling this problem without using a calculator is quite a
challenge! An interesting extension question to ponder is whether
there are pairs of distinct irrational numbers without any
rationals between them.

### Possible support

Simply seeking the two denominators which don't work, and verifying
that other small denominators do, is a good exercise in working
with fractions and decimals.