My New Patio

'My New Patio' printed from https://nrich.maths.org/

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My New Patio

In the problem Tiles on a Patio , we were looking at tiling a patio. Some of you probably did that and had a good time. If you did not you might like to look back at that challenge and see what you come up with.

But, you know, when you look at gardens or patios you tend to find that they're not lovely rectangles or squares. Someone, quite sensibly, has a bush or small tree growing in it so that there are some tiles missing. Other people have a small pond or a sand pit for smaller brothers and sisters to use. So, what shall we try in this problem?

I thought that we would look at squares, but keep in mind these trees, bushes, sandpits etc. , and so we leave a square tile out.

Let's suppose that we have a space to tile that is $3$ x $3$ and we remove just one tile. It'll look a bit like this:-

Now like in the original challenge you can tile the area using only square tiles. But how about using as few tiles as possible? Perhaps this is the way?

There's not a lot of choice there so let's look at a larger area, like this one:-

Now, after playing around for some time I come to this arrangement, using $1$s, $2$s, and a $3$.

That uses a total of seven tiles. Now that seven is the key number that we want for each size square.

The $3$ x $3$ area used five tiles.
The $5$ x $5$ area used seven tiles.

So my challenge ends up with asking you to explore the SMALLEST number of tiles needed to tile a square area when a $1$ x $1$ space is missed out. You do not have to have the one that is missed out in the top left hand corner like I have done. You can have that hole [bush, pond etc. ] wherever you choose.

Well after you have done a few, stop and look at your results and be a 'Numbers Detective' and see what comes up.

If you are doing this with others I suggest you check each other's work to make sure that you have got the smallest number of tiles each time!

You will find probably that the bigger and bigger the area gets, the more excitement that comes!!

So, I went up to $29$ x $29$ for my area, how about you? Of course I have to ask the question:-

I wonder what would happen if ... it were triangles instead?
I wonder what would happen if ... there were two small squares missing?
I wonder what would happen if ... it was a rectangle of a particular area that was not a square?

Good luck!

Why do this problem?

This activity can be a rich source of exploration both in calculation and shape and space.

Possible approach

It's good to make it as practical as possibe for many pupils and working together through a few simple examples before getting them to explore on their own is helpful.

Key questions

Which tiles have you used in this one?

How many tiles altogether have you used?

Tell me about the way you are deciding which tiles to use.

Is there a pattern in the numbers for the smallest number of tiles?

Can you explain any patterns you've found?

Possible extension

Children could:

go much further with bigger starting squares
allow square tiles to be cut into half
have the space left in a different position and discuss whether this makes any difference to the tiles used
investigate whether there is anything in the number of each size of square tiles that are used in each successive sized area [i.e. area 4 x 4 uses ? of 1 x 1's; ? of 2 x 2's; ? of 3 x 3's]. How does that compare with the tiles that are used in the 5 x 5 area?
investigate whether there are different tiles that can be used to produce the same smallest number of tiles for any given area
try to find a system for filling up the area
make predictions based on their findings

Some pupils will benefit by going up to areas that exceed 20 x 20. It is certainly worthwhile looking at the implications of where the bush is in relation to the patio. I found it interesting when exploring the odd numbered sided patios with the bush in the centre position: factors, and even/odd numbers, and fractions of the lengths of sides, and addition and subtraction. What else could you want?

For more extension work

These pupils could work towards comparing results with different sized patios and making statements about those arrangements that use the same number of tiles but of different sizes. There is also the idea of looking at more squares that are not tiled because of Garden trees, ornaments etc. even a stream.

Possible support

As much practical equipment as possible will help many children.