# Proportional reasoning

Proportional reasoning takes fractions, decimals, ratios or percentages and places them in a problem solving context. It can present a different way of approaching some problems rather than immediately leaping to or looking for a formula that numbers can be plugged into without a real consideration for the relationships between quantities.

Many students find proportional reasoning problems difficult. This is often because they have not been introduced early enough to the multiplicative nature of proportion reasoning and struggle to recognise this. Instead, they use addition methods, or informal methods.

A nice science example that is relevant across both physics and chemistry is density. Being a compound variable, students can often get confused about what might happen to one quantity if another changed (whilst another remained the same). For example “if I had the same amount of gas in half the volume, what would happen to the density?

Proportion problems can often be solved quite easily using these informal methods when the numbers involved are simple multiples of each other as with halving and doubling. However, students need to be able to deal with the general case using the operations of multiplication and division. So whenever possible encourage students to use a single step multiplicative calculation. For example a 20% increase means finding ‘120% of …’ which means multiplying by 1.20.

This will be particularly important when they come to finding reverse percentages, “the final cost after 20% increase was, what was the original cost?” This is reversed by dividing by 1.20 NOT by subtracting 20%. Students who are not fluent with using single step multiplies will struggle with this concept and will want try to use subtraction because it is the inverse of addition.

Remind students that a proportional increase/decrease can be represented by a decimal multiplier. For example to increase a quantity by 43%, multiply by 1.43, to decrease by 12%, multiply by 0.88.

Some students may think that a multiplier always has to be greater than 1.

The word ‘similar’ means something much more precise in this mathematical context than it may in their everyday experience and this can cause confusion.