Problem
Mum’s kitchen floor is square and is fitted by 64 square tiles in a 8 x 8 array. Mum has chosen black and white tiles.   She could have the tiles laid so that they looked like a chessboard but she was hoping for something a bit unusual. The tile man sketched something that had reflective and rotational symmetry. What did he suggest?

What is this problem about?
The problem is an exploration of symmetry. There are many ways to answer this question but it gives every child a chance to produce a correct answer.

The Extension is an old problem in disguise. It is probably a question that the children need to think about over a week or so.

Achievement Objectives
Geometry (Level 3)
- describe patterns in terms of reflective and rotational symmetry
- design and make a pattern which involves reflection and rotation

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (guess and check, make a drawing, use equipment)

Resources
Paper squares for tiling
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- create a pattern that involves reflection and rotation

Teaching sequence

1. Introduce the pattern using a chess board (64 squares of alternating colour).Discuss the symmetries of the board.
2. Pose the problem.
3. Brainstorm for ways to solve the problem – (use equipment, draw, guess and check).
4. As the children work on the problem in pairs ask questions that focus their thinking on the symmetries of the pattern.
   How have you used reflection?
   How have you used rotation?
5. After the children have completed the floor pattern ask them to record the symmetries that it contains.
6. Share patterns – display on wall and discuss.

Extension to the problem
After all that, Mum decided to have the floor tiles laid like a chessboard. Now while Mum was redecorating her kitchen she had some cupboards built. Two of these were placed in the opposite corners of the room and took up a whole tile each. (This meant she  needed to use 62 square tiles now.)

The tile man said that there was a special on. He had a combined tile that consisted of a black tile stuck to a white tile. Could Mum tile her floor with these combination tiles and so save herself some money?

 Solution to the problem
There are a large number of possible answers here. Each one can easily be checked to see that it has the right symmetries. Perhaps you could fax or e-mail us some of your classes answers.