Why do this problem?
Do this activity
to encourage pupils to look at patterns around them and find the many patterns that are usually there to discover. The Great Tiling Count can be used as a gentle introduction to investigations that bridge the spatial and the numerical aspects of mathematics. Younger children may be encouraged to
explore the situations very practically with real "tiles" of some kind, even if they are coloured cubes. Computer graphics packages will also be of use for those inclined towards using that technology.
If possible use this activity to actually get out and observe the patterns that are in your locality.
Using the problems as set out, there will be a need for most children to have access to squared paper and possibly triangular paper.
Some of the arithmetic that may come out of the exploration will be concerned with counting and addition as well as multiplication and the use of halves [as in the edge tiles]. Particular additions, as in the case of the first picture counting the triangular pieces, could result in discussions about adding up:-
$1 + 3 + 5 + 7 + 9$
and then when the smaller versions are considered, adding up:-
$1 + 3 + 5$
The significance of there always being four quarter pieces may be thought about.
There should be many opportunities for looking at the occurrence of even numbers ... and why?!
What shapes can you see?
What would you like to count?
What would a larger one look like?
Explore mosaic and tiling patterns that can be found online. Learners can create their own mosaic/tiling patterns and be able to describe the tiles that will be necessary.
Some pupils will need a nudge to get started on the mathematics that is associated with a pattern, also help in counting accurately where appropriate.