# 10 things to say about ratio?

I was surprised to hear that in the pre-2013 Key Stage 3 national curriculum there was only one statement about ratio, whereas in the current version there are ten statements. I was surprised because I did not remember, from my work on the development of the current curriculum, that there were so many, so I had another look to see why many statements might have been needed to replace an earlier ‘one’.

The earlier statement was:

“Ratio and proportion: This includes percentages and applying concepts of ratio and proportion to contexts such as value for money, scales, plans and maps, cooking and statistical information (e.g. 9 out of 10 people prefer…)”

The current ten statements are:

“Ratio, proportion and rates of change

1. change freely between related standard units [for example time, length, area, volume/capacity, mass]
2. use scale factors, scale diagrams and maps
3. express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
4. use ratio notation, including reduction to simplest form
5. divide a given quantity into two parts in a given part : part or part : whole ratio; express the division of a quantity into two parts as a ratio
6. understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
7. relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
8. solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics
9. solve problems involving direct and inverse proportion, including graphical and algebraic representations
10. use compound units such as speed, unit pricing and density to solve problems”

Some of the content that is now listed under ratio, proportion and rates of change (RPR) appeared elsewhere in the previous curriculum, and there is more detail in the current curriculum than before, but there is nothing new, except for an emphasis on the relationship between ratio and fractions and an emphasis on expressions.  The change was to consolidate under one heading a network of ideas referring to multiplicative relationships between numbers and quantities so that teachers could think about RPR as an embracing network of language, notation and experience for learners, rather than as separate topics in the curriculum.

This approach starts in Year 6, for which the non-statutory guidance of the national curriculum includes:

“Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).

Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.

Pupils solve problems involving unequal quantities, for example, ‘for every egg you need three spoonfuls of flour’, ‘3/5 of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.”

Of course, it is hoped that the idea of one quantity as a multiple of another would be introduced earlier than Year 6, for example, when fractions are introduced by thinking of fractions as operators on quantities rather than only as parts of one whole shape. ‘Quantities’ could be discrete, such as a number of apples, or continuous, such as a volume of water.

The old curriculum concentrated on the uses of ratio, recognising that this was where the mathematics curriculum would be most useful across the whole school curriculum and in outside life. The current curriculum adds structural and abstract elements to this. Expressing relationships algebraically is the most obvious aspect and this has led to GCSE boards becoming inventive about the kinds of questions they ask. I have watched this development with interest and slight concern as people begin to discuss methods rather than meanings.

It is tempting to offer ‘rules of thumb’ to deal with ratio problems, but these tend to be misapplied.  For example, to answer “Orange paint is made from red and yellow paint in the ratio 3:5.  To make 40 litres how much would I need of each colour?” some people who had learnt methods rather than meaning multiplied 3 and 5 by 40 and give answers as 120 and 200 litres respectively. The teacher realised that they did not think about ratio as expressing parts of a whole as well as comparison of parts.

Random multiplying has also been seen with questions such as: “given a:b and b:c what is a:c?”  It matters a lot whether a, b, and c are numbers or quantities in the same units, or represent three different materials, or different measures of the same material.  A rule that ignores their meanings and works in all cases does not exist.

Let’s give the question some meaning by treating a, b and c as numbers: suppose one paint mixture contains white to red in ratio 3:5, and red to blue in ratio 5:7. Then any portion of it contains white to blue in ratio 3:7 and white:red:blue in ratio 3:5:7.  Just think about putting scoops of paint into the mixing pot to model this: I put in 3 scoops of white followed by 5 scoops of red; now I have 5 scoops of red so I add 7 scoops of blue to complete the second ratio.  There are 15 parts in the mixture (3+5+7). But is this the only meaning of the question?

Suppose now that ‘b’ stands for ‘red stuff’ rather than quantity or portion. For example: “If a tin of paint has been constructed by using a ratio of white to red as 2:3 and red to blue as a ratio of 4:5, what is the ratio of white to blue?”  It is tempting here to talk about cross-multiplication: multiply the numbers in the first ratio by 4 and those in the second by 3, but why? This process describes the red content as 12 parts of the whole and hence brings all quantities into measures that can be compared. ‘Cross multiplying’ achieves a common denominator, a common ‘measure’, that allows us to compare white and blue amounts to the red amount, and hence to each other.

A rod diagram might help if learners are familiar with using rods, but can be seen as an extra thing to do if they are not familiar: Having a ‘rule’ to deal with ‘what is a:c?’ questions by cross-multiplying is not enough to tackle all mixture questions. Let’s go back to the beginning and suppose that our two initial ratios a:b and b:c are actually paint mixtures in separate pots. Remember the first mixture a:b contains white to red in the ratio 3:5, and the second mixture, b:c contains red to blue in the ratio 5:7. If we want to combine them, then it matters whether we combine them in equal or unequal quantities.  For simplicity we choose equality to start with, say one litre of each. For red paint, the final mixture will contain 5/8 of a litre plus 5/12 of a litre, which is a fraction addition 5/8 + 5/12 of a litre.  Although both red amounts are given as ‘5’ they are not the same size of part; the whole number of parts matters. It will also contain white, 3/8 of a litre, and blue, 7/12 of a litre.  To compare all these we need a common denominator, 24.  Finally we have white 9/24; red 25/24; blue 14/24, so a ratio of 9:25:14 for white:red:blue (with an imagined total of 48 parts in the 2 litres). Thus a ratio problem turns into a fraction problem when the number of parts in each whole is taken into account.

Suppose now we combine unequal quantities of the two mixtures: Because the numbers used for textbook examples are usually quite small it is easy to become confused with 2s, 3s, 5s and 7s floating around so I am going to use big numbers to emphasise structure.  Let’s mix 37 litres of the 3:5 mixture with 41 litres of the 5:7 mixture.  I am using primes to avoid simplification which might obscure the structure.  This means I now have 37 lots of 5/8 of a litre of red plus 41 lots of 5/12 of a litre of red, i.e. (37×15 + 41×10)/24 of red.  Again it has turned into a fractions problem. Structurally, the use of primes can be replaced by letters to indicate that to turn a ratio into a quantity we can think of fractions of the whole – whatever the whole might be.

There is much more that can be said about mixture problems, but I hope this is enough to indicate the connections between ratio and fractions that were intended in the curriculum. Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.