Paving ViewsTask 157 ... Years 4  8SummaryWithin the story shell of an architect's work, students are asked to create a tiled courtyard with a rotationally symmetric pattern. There are many possibilities  just putting a tile on every square for example  but the challenge is to use the minimum number of tiles. Students don't have to know what rotational symmetry means to start with (although it is explained in part on the card), because the challenge makes sense in the context of the story. Once created, the design can be used to create more designs using transformation such as translation and reflections. 
Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
Rotational symmetry links this task to Knight Swap and Wallpaper Patterns. Perhaps this is an underrated aspect of mathematics learning, which is a pity since the tool is used in:
Getting back to the task, the solution to Architect Sanja's puzzle is:
The second image shows the original tiles using white dots. Eighteen tiles have been added. This is the minimum solution. The courtyard can be viewed from all four sides and will look the same. But what clues could the students use to develop this solution?
Extensions

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. 
Most schools have multicoloured collections of tiles and the tiles are usually about 2.5cm square. Using the table facility of a word processor, it is easy to create an 8x8 grid for these tiles, although your printer would need to operate with 0.5cm left and right margins. Two centimetre wooden blocks are an alternative and a grid to take those would fit better on the width of a page. Each pair of students would need about 40 tiles of the one colour. Hand out the grid sheets to each pair and display the equivalent on your electronic white board. Tell the story of Sanja the architect and fill in the starting point tiles on your display. Students match this on their grid. Now set the challenge and share ideas as groups work. End the lesson by reviewing the key steps in the process that led to the solution and highlighting the mathematics of rotational symmetry. Leave time for students to record the key steps and the answer in their journals. Use the lesson as a starting point for further investigations of rotation and other transformations. As suggested above there is room within this lesson to use drawing software or the drawing tools of a word processor. They all easily draw and colour grids which can be grouped as a unit and then rotated, reflected and translated. At this stage, Paving Views does not have a matching lesson on Maths300. 
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 Paving Views task is an integral part of:
