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.