Time For TilingTask 140 ... Years 4  8SummarySue is planning to build a square courtyard with dark and light tiles so that the main diagonals are dark. She realises that the patterns will be subtly different depending on whether she uses an even or odd side square. The odd ones always have one central tile. Her challenge becomes:

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IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
The answers to the questions on the card can be found by making all or part of the model courtyard and are: 7 x 7 needs 13 dark tiles and 36 light tiles.From the number patterns it could be predicted that if n is odd: n x n needs [n + (n  1)] dark tiles and (n  1)^{2} light tiles.This prediction would only be correct if [n + (n  1) + (n  1)^{2}] simplifies to n^{2}. Looking at even squares in a similar way, ie: by construction, gives: 6 x 6 needs 12 dark tiles and 24 light tiles.From the number patterns it could be predicted that if n is even: n x n needs (n + n) dark tiles and [(n  1)^{2}  1] light tiles (or n[n  2]).This prediction would only be correct if [n + n + (n  1)^{2}  1] simplifies to n^{2}. ExtensionsSo far this is only playing with numbers and patterns. Can we reconcile these generalisations with a visual explanation? The hint is in the dark tiles in each case.
The diagonal always has the same number of tiles as the side of the square  it's not the same length, but it does have the same number of tiles.
But the mathematics asks: Can I check it another way?. In this context, can we see the problem another way? The answer is always yes and one way to see this problem if from the point of view of the light tiles. They are always in four clusters.
The number of tiles in the outside line of a cluster (vertical line in the case of the yellow clusters) is always two less than n because the end ones are part of the dark diagonal. Each row as you move toward the centre is then 2 less than the previous one, also because of the diagonals.
Note: If you need to explore ways of summing series such as those above, then Task 51, Staircase, explores that challenge by making towers of blocks to represent each number and placing them side by side to make a staircase. Further extensions to this problem include:

Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 
In preparation for this investigation you will need to make up sandwich bags of around 20 tiles or blocks in each of two colours. Alternatively, Poly Plug can be used, as indicated in the link below. You will also need some graph paper. Establish the initial story shell and encourage exploration. As students construct and count particular squares, gather the data on the whiteboard. Follow through the investigation guided by the information above. At this stage, Time For Tiling does not have a matching lesson on Maths300. Visit Time For Tiling on Poly Plug & Tasks. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Time For Tiling task is an integral part of:
