### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

# More Twisting and Turning

### Why do this problem?

This problem follows on from Twisting and Turning, in which students are introduced to an intriguing trick which provides a context for practising manipulation of fractions. The trick is a hook to engage students' curiosity, leading to some intriguing mathematics to explore and explain, and ultimately generalise and prove.

### Possible approach

After working on Twisting and Turning, students should have seen one sequence of fractions that leads to a tangling and untangling of the ropes, and also disentangled one more sequence. This problem offers an opportunity to explore and find a strategy for disentangling any tangle.

Begin by reminding students of the two operations:
Twisting has the effect of adding 1: $$x \mapsto x+1$$
Turning transforms any number into the negative of its reciprocal: $$x \mapsto -\frac1x$$

"I wonder if it's possible to get back to zero from any fraction...

Let's start by exploring negative fractions with a denominator of 2,
e.g. $-\frac{5}{2}$ or $-\frac{17}{2}$ or $-\frac{23}{2}$.
Using the two operations, can you find a way to get to zero?"

Give students some time to work on this challenge, and then bring the class together to discuss. The following strategy might emerge:
Keep adding $1$ until you get to $\frac12$. Then a Turn gives $-2$ and two more Twists gives $0$.

"So we have a strategy for getting to zero from fractions of the form $-\frac{n}2$. Can anyone explain how this helps us with fractions of the form $\frac2{n}$?"
Start with a twist to get $-\frac{n}2$. Then use our previous strategy.

Next, students could explore fractions of the form $-\frac{n}3$.
This time there are two cases to consider: repeated twists eventually leads to either $\frac13$ or $\frac23$. In the first case, a turn leads to $-3$. In the second case, $\frac23$ is a fraction of the form $\frac2{n}$ so the strategy above can be used.

These explorations should help students to work towards a general strategy for untangling back to zero from any fraction. To finish off the lesson, you could challenge students to untangle a complicated fraction - this could be done practically using ropes or string, or using the interactive tool in the Getting Started section.

### Key questions

If you keep adding 1 to a fraction of the form $-\frac{n}2$, what eventually happens?
If you keep adding 1 to a fraction of the form $-\frac{n}3$, what eventually happens?
If you keep adding 1 to a fraction of the form $-\frac{a}{b}$, what eventually happens?

### Possible support

Students who are less fluent with fractions might find it useful to use the interactive tool in the Getting Started section to check their answers.

### Possible extension

All Tangled Up invites students to explore how to get from zero to particular fractions.