### Making choices to get a job done

A comfortable and accessible place to start introducing the idea of limits to possibilities, without worrying about the detail of mechanisms, is in getting simple lifting jobs done. Here choices are made, but there are hard limits to the options, and necessary trade-offs. A trade-off is a useful handle for a linked pair of quantities, say force and distance. Piaget got there first with compensated quantities, so I'm suggesting avoiding labelling them compensated quantities. (Not only is it probably unwise to step on his turf, but there are Piagetian studies of compensation for flasks of different cross-sectional areas, and of the relationship between mass, volume and density, which are not the focus here). I suggest that we can fruitfully use trade-off when choosing values of a pair of physical quantities whose product is either power or energy, to achieve some required value in joule or watt.

You might start with pulleys, used to support a load that is too heavy for you alone. You hold only one rope of the number that leave the load, so only have to support a fraction of the load, the remaining ropes account for the remainder. However, the trade-off is now that you have to pull through lots of rope.

You could extend this sharing idea to a hydraulic machine, where increasing the area under the load again allows you to choose to support less, but still find that you have chosen to push further.

You could represent the idea of a trade-off as the area of a box, one side represented by each of the quantities involved in the trade-off:

The orange bar, as ever, is the quantity of energy needed to perform the task. Not enough energy: the task is impossible. Choose how to perform the task—large force and small distance or small force and large distance.

This pattern of trade-offs applies to filling any store or calculating the power in any pathway, for example.

Three preparatory steps for a gentle path: a semi-quantitative representations that can be reasoned with; thinking with constraint relationships; choosing trade-offs.