### Equal Equilateral Triangles

Using the interactivity, can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

### The Square Hole

If the yellow equilateral triangle is taken as the unit for area, what size is the hole ?

### An Introduction to Irrational Numbers

Tim Rowland introduces irrational numbers

# Rationals Between

### 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.