Mum's Kitchen Floor

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?

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 students 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 students 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.

For the extension, colour the squares like a chessboard. When you remove two opposite squares you remove two squares of the same colour. So you have left 30 squares of one colour and 32 of the other. You can’t cover these with the combination tiles as each combination covers one square of each colour.