## Swimming Pool Tiles

Most swimming pools are surrounded by tiles of some sort, like the pool in the photo above.

Imagine that we number the tiles around the pool.

In the image below, the pool takes up a 5 by 5 square and the space around the pool is five tiles wide:

Can you describe how the tiles are numbered?

Think about a series of numbers that you know about, for example square numbers, prime numbers, triangle numbers...

Explore where those numbers occur in the path around the pool.

What do you notice?

What happens if you look at the path those numbers make around a different sized pool? (You may find it useful to print off

this sheet which includes pools of different sizes up to 7 by 7.)

Please let us know what you discover!

### Why do this problem?

This activity is designed to nurture children's curiosity by introducing mathematics into a familiar non-mathematical context. Children might end up pursuing different ideas from each other and this freedom to explore may well encourage learners to persevere more than they might usually. In this way, they will immerse themselves in the particular number sequence they have chosen to use,
which will help them gain a deep understanding of its structure. This activity lends itself to pupils posing their own questions in the form “I wonder what would happen if...?”. (The further note at the foot of this page offers more support on curiosity in the classroom generally.)

### Possible approach

You could display the picture of the numbered tiles around the 5 by 5 pool and simply say that it represents a swimming pool surrounded by square tiles. Ask learners to say what they see and use the ensuing discussion to draw out the key features of the picture.

Try not to say too much more at this stage and set learners off on the task of choosing a number sequence to explore on the numbered tiles. After some time, you could bring everyone together for a mini plenary to share some discoveries so far and to discuss ways of working and/or recording methods that are proving useful. As they are being driven by their own curiosity, encourage learners
to share anything at all they have found/noticed. In addition, use the mini plenary to remind everyone that this is not just about pattern spotting, but they should try to explain and justify their 'noticings'.

### Key questions

Tell me about what you're doing.

What do you notice?

Can you explain why that happens?

### Possible extension

Encourage learners to pose their own questions along the lines of "I wonder what would happen if I...?" Here are some possible ideas to explore further:

- Walking along a straight path to the pool, adding up the numbers on the five tiles and exploring the totals for different paths.
- Walking around the pool travelling from tile to tile without getting wet and exploring the number sequences that result.
- Changing the size of the pool (this sheet contains pictures of pools up to 7 by 7)
- Investigating where the times tables numbers appear

### Possible support

These Word documents might be helpful, which already mark square numbers/prime numbers/triangle numbers on different sized grids:

SquareNumbers,

PrimeNumbers,

TriangularNumbers.

### Further note

You may be interested in the following talks given by Professor Susan Engels, which focus on encouraging curiosity and are available on YouTube:

The Rise and Fall of Curiosity - the extract from 23.50 to 37.15 on adult encouragement and teacher behaviour is particularly worth viewing

The Hungry Mind: The Origins of Curiosity - the extract from 8.22 to 12.29 on children asking questions is especially useful.