Tiling

'Tiling' printed from https://nrich.maths.org/

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Tiling

Let's imagine we are tiling a floor or patio area.
Suppose the area is square and is 3 by 3.

Tiles come in three sizes: 1 by 1, 2 by 2 and 3 by 3.
You are allowed to use any arrangement of these tiles to cover the space completely.
However, none of the tiles can be cut.

For this challenge, we are interested in the total number of tiles you use for any arrangement.
For example, for the 3 by 3 square you could do it by using:

nine 1 by 1 tiles,

one 2 by 2 and five 1 by 1 tiles,

one 3 by 3 tile.

Therefore the smallest number of tiles is one and the largest is nine and the only other value you can get between one and nine is six (using one 2 by 2, and five 1 by 1).

Now imagine you also have tiles which are 4 by 4 in size.
What total numbers of tiles can you use for a square patio that is 4 by 4?
If you could have tiles which are 5 by 5 as well, what total numbers of tiles can you use for a square patio that is 5 by 5?

The final part of this activity is to examine carefully the answers you get for each of the three sizes of floors - 3 by 3, 4 by 4 and 5 by 5.

Now talk about and record what you notice.
Make some statements about what you therefore think will happen if someone takes the long time needed to explore a 6 by 6 and 7 by 7 square.

Why do this problem?

This low threshold high ceiling problem is a good one to introduce pupils to working investigatively. It requires children to work systematically, look for patterns, conjecture and justify, and is an engaging context in which to apply knowledge of factors and multiples and area.

Possible approach

It would be good to introduce this problem on the board or interactive whiteboard by having several 3 by 3 squares ready drawn, and if using an interactive board, having copies of the different tiles ready to drag across. Explain the task and ask learners to talk to partners about how the 3 by 3 patio could be tiled. They could jot pictures on mini-whiteboards. Share their ideas on the board, reinforcing the constraints so that pupils become very familiar with the context.

Set them off in pairs to try the 4 by 4 and 5 by 5 sizes. You may wish to have a table on the board to collate their findings so that children work as a class to produce all the possible ways for each size. As soon as a combination is found, it can be entered in the table for all to see. Look out for those that have a system which ensures that they find all possibilities - this may be, for example, starting with all the smallest tiles and gradually introducing larger ones in order.

Bring the group together and focus on justifying why certain total numbers of tiles aren't possible, then lead them onto making conjectures about the 6 by 6 and 7 by 7. Allow time for learners to think carefully and talk to each other, followed by a whole-class discussion in which you should be able to sit back and let the children put forward their own arguments. If they are convincing enough there will be no need to physically find all the possibilities!

Key questions

How are you deciding on which tiles to use?

Are you checking that you've got a different number of tiles each time?

What will the maximum and minimum numbers of tiles for the 6 by 6 and 7 by 7? How do you know?

What do the 3 by 3, 4 by 4 and 5 by 5 combinations have in common? Can you apply this to a 6 by 6 and 7 by 7?

Possible extension

Encourage children to make generalisations that would hold for any size patio. The Tiles on a Patio problem would make a good follow-up problem to this one.

Possible support

It would be useful to have squared paper and coloured pencils available, and even multiple coloured squares cut out to represent the different sized tiles, so that pupils can physically make the arrangements.