Ratios, fractions, decimals and percentages

The use of number to quantify physical systems, processes and quantities runs across pretty much all of the three sciences and so the need to be confident with numbers and simple calculations is effectively a core skill in all subjects and most topics. However, it can be easy to make assumptions about the confidence and fluency students have with number and this can become a barrier to their understanding. If they struggle with the numbers this can impact their understanding of the real thing that the number is representing. Summarised below are some of the common challenges students may face and then there are some example activities and resources that you may wish to use.

It is important that students understand that ratios, fractions, decimals and percentages are all just ways of describing divisions of a whole number. Ratios are fractions which express how many times one number can be divided by another. The ratio p : q corresponds to the fraction ^p/_q

Pupils need to be confident and fluent with fractions as numbers on a number line as well as operators. So 1/2 can be interpreted as a number on the number line between 0 and 1 and also as an operator divide by 2.

Decimals can also be described as fractions where the denominator is a power of 10 . We write decimal fractions with a decimal point. Percentages are fractions where the denominator is 100.

In many cases students see these different representations as unrelated and the connections are not made. Encourage students to move between the different representations and discuss what makes a particular representation more appropriate in a given situation. For example when rearranging an equation student often make the mistake of wanting to convert fractions to decimals which makes the processes involved less transparent.

Suggest that students are flexible in the way they choose which representation to use and also in the way they work using mental, written or calculator strategies. The representation used should be accurate and convenient given the context. For example finding 25% of a quantity is equivalent to finding a quarter so divide by 4.

Students will have their own mental strategies, for example finding a quarter by halving and halving again. But, to make sure they understand what they are doing and can generalise it may be useful to check they understand that the line in a fraction stands for divide.

Students may think that it is impossible to have more than 100% or a fraction greater than 1. Emphasise scientific contexts where such proportions have real meaning.

Based on the fact that ¹/₁₀ is 0.1 pupils may wrongly generalise that, for example, ¹/₆ is 0.6.

Ratio is a way of comparing quantities, parts of a single quantity, e.g. mixing paint or one quantity with another quantity, e.g. real life distance and distance on a map.

When ratio is used to compare one quantity with another, some connected phrases are ‘to every’ or ‘for every’ or ‘as many as’, e.g. the scale drawing shows 1cm to every 50cm