Doug's Tablecloth

A Special History

This problem began with a real cloth that would not fit back into the real drawer it had come from after first being used for a dinner party. Sally Farr, who had come from Queensland to investigate the use of tasks with Indigenous students was present as Doug tried - unsuccessfully - to refold the cloth to fit. She suggested there could be a task in this problem ... and she was right. Later, towards the end of the 1990s, Per Berggren, a well-known and respected teacher in Sweden purchased Doug's Tablecloth to add to his school's collection of tasks (mattegömmor). He was so puzzled by it that he eventually flew to Australia to try it with the actual drawer and actual cloth which had first generated the problem at Doug's house. As you can see, it was no less puzzling 'in the flesh', but perseverance won out in the end. See below for students struggling (and succeeding!) with this task.

Iceberg A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The strategy of working backwards plays a big part in this task: You can solve the problem by trial and improve and then look back and ask What could I have learnt from the numbers in the problem?. You can work backwards from the given numbers to solve the problem in the first place. As in the second part of the task, you can work backwards from the solution to decide the exact measurement of the other side of the cloth which would allow it to fit into both the depth and width of the drawer. Once the problem has been solved the problem can be seen as a whole piece of cloth folded into parts and this opens the door to exploring fractions. Move to paper and fold to make sections. The rows represent one fraction of the whole piece, the columns another fraction and the individual cells another. Again working backwards is appropriate, eg: How will you fold a piece of paper so that the rows represent halves and the columns represent thirds?

Whole Class Investigation Tasks 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.

To convert this task to a whole class investigation each pair (or student) will need a cloth the correct size. Perhaps a parent, or the craft faculty, could cut sufficient pieces. This only has to be done once if the pieces are stored after each use. Some schools use filing boxes made from recycled cardboard to store class sets of task materials. This is an efficient use of preparation time - one preparation, multiple uses. In the future, when teachers reach the part of the curriculum document that requires a whole class investigation of a task, they simply select the appropriate box from the shelf and everything is there. Some teachers have thought of using paper instead of cloth for Doug's Tablecloth, but there is a lot of experimental folding done before the problem is solved, and paper holds its creases. Consequently, paper becomes confusing. In the long run material also retains creases, but it can easily be ironed flat again after a few uses. At this stage, Doug's Tablecloth does not have a matching lesson on Maths300.

Is it in Maths With Attitude? Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Doug's Tablecloth task is an integral part of: MWA Number & Computation Years 5 & 6 MWA Number & Computation Years 7 & 8

Students At Work Helen's Year 7 class at Simonds Catholic College, West Melbourne.

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