### Continued Fractions I

An article introducing continued fractions with some simple puzzles for the reader.

### LOGO Challenge - Circles as Bugs

Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.

Using logo to investigate spirals

# Gnomon Dimensions

### Why do this problem?

This problem continues to investigate the sequence of shapes introduced in Building Gnomons Learners will need to use their skills of representation to communicate their ideas and justify their findings. This problem builds on learners' prior knowledge of how the Fibonacci sequence grows.

### Possible approach

Ask learners to work in small groups to investigate areas and dimensions of gnomons.

After a short time, draw all the groups together to share ideas about how they might organise their approaches and record findings. Which groups are working systematically, which have used effieicent recording methods?

It is most desirable for learners to develop their own representations to justify any patterns they find. However, if they are struggling the Hint contains one way of recording edge lengths in terms of Fibonacci numbers that might be a useful stimulus for discussion.

Is it possible to predict the dimensions of the gnomons in the sequence?
Encourage the use of diagrams and notation to explain how the pattern will continue and why.

Sharing findings and justifications might be achieved by the use of posters which groups present to the rest of the group. Use the opportunity for other learners to feedback on the clarity of what is presented.

### Key questions

How does the approach in the hint work?
How does the way you put pairs of gnomons together result in new Fibonacci numbers?

### Possible extension

The article Whirling Fibonacci Squares explores some of the ideas of the Fibonacci numbers in more detail.

### Possible support

Try Building Gnomons first. Sheep Talk could be used as an introduction to Fibonacci numbers.