# 16 Inverse proportionality and equilibrium

Inverse proportional reasoning relies on a sound concept of ratios and proportionality.

The mathematical form for inverse proportions is yx=m. As y goes up so x must come down proportionally to keep m constant. At the concrete operational level this understanding is qualitative, a child can see that if a block of plasticene is flattened it still contains the same amount of plasticene because the loss in height has been compensated for by the increase in width. The mathematical expression of inverse proportions requires formal operational thinking.

Equilibrium is more complicated because it involves thinking about two ratios describing two inverse proportionalities at the same time. The mathematical formula is ab=cd. This involves holding in one’s mind four independent variables and their relationships to each other. For example, a child using a beam balance can calculate that a 25 N force must be three times further from the fulcrum if it is to balance a 75 N force, without moving the masses along by ‘trial and error’. This is formal operatonal thinking. At an intermediate stage a child will realise that a heavy mass must be nearer the fulcrum than a lighter mass to keep the balance. He may have a semi-quantitative idea of just how far out to put the mass but will still resort to ‘trying it out’ to formulate the ratios.

Science abounds with examples of equilibrium models, for example chemical equilibrium (where the entities being balanced are themselves concentrations i.e. ratios); osmotic and diffusion gradients; ecological equilibrium. Predicting what happens when these systems become unstable is an important scientific skill. We also talk about ‘balance of power’ politically and globally and use such models to describe economic stability or the strategic influence of countries.