### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Zios and Zepts

## Zios and Zepts

On the planet Vuv there are two sorts of creatures. The Zios have $3$ legs and the Zepts have $7$ legs.

The great planetary explorer Nico, who first discovered the planet, saw a crowd of Zios and Zepts. He managed to see that there was more than one of each kind of creature before they saw him. Suddenly they all rolled over onto their backs and put their legs in the air.

He counted $52$ legs. How many Zios and how many Zepts were there?
Do you think there are any different answers to this that woud work?

### Why do this problem?

This problem focuses on numbers that are multiples of $3$ and $7$ and offers a motivating context in which to develop instant recall of them. It also provides an opportunity for learners to discuss alternative strategies and to consider the merits of each.

### Possible approach

Children enjoy considering strange forms of life in imaginary planets and these weird creatures with odd numbers of legs should appeal to them. You could begin by telling the story in more detail and introducing a simpler version of the problem, for example, if Nico saw $6$ legs what creatures might he have seen? And how about $14$ legs? Or $13$ legs? Ask them to work in pairs, perhaps on mini-whiteboards, to think about these first challenges. As they work, observe different ways of recording and thinking about the problem which you can share with the whole group.

After this learners could work in pairs on the actual problem which you could present orally with the image.  Again, listening to learners as they work can be an interesting insight to their thinking. Some children may like to use practical resources to help them with their thinking such as modelling clay and headless matches.

At the end of the session learners could share not only their solutions but also their methods. Some may have used a trial and improvement approach, either with the materials provided or using pictures; some may have written lists of multiples of $3$ and $7$ in a very systematic way and then made totals. Children could consider whether some strategies were more effective than others, although of course a strategy that is effective for one child may be very confusing for another. Having the freedom to approach this problem in any way is key here, but in talking to others, some pupils might change the way they work, which is interesting in itself.

### Key questions

What have you tried so far?
What happens if you replace this seven-legged creature with some three-legged creatures?
What happens if you replace these three-legged creature with some seven-legged creatures?

### Possible extension

Learners could investigate other possible numbers of legs of a group of Zios and Zepts. Perhaps some can be done in more than one way. How many ways can be found for each target number? The investigations can go on and on!

### Possible support

Starting with lower totals and using practical equipment to support them will help some children. They may like to write down all the multiples of $3$ and $7$ as an aid. Some children might benefit from trying the Spiders and Flies problem first which involves lower totals.