15 February 2020

The Rule of Three

Multiplication is vexation;
Division is as bad;
The Rule of Three doth puzzle me,
And Practice drives me mad.

You may know this classic nursery rhyme, which was apparently written by John Napier in 1570.

Most modern English maths teachers don't know much about the Rule of Three. I hadn't come across it myself until I started reading old textbooks, where it is ubiquitous. In Hodder's Artihmetick (1702) the Rule of Three is described as the 'Golden Rule' (for as Gold transcends all other metals, so doth this Rule all others in Arithmetick').
Some of you will not know what the Rule of Three is, so allow me to explain. The dictionary definition is as follows:

noun: rule of three; plural noun: rules of three
  1. a method of finding a number in the same ratio to a given number as exists between two other given numbers.

When I explain it in conference presentations, I usually use this amusing example which I found on a maths website explaining how to solve proportion problems. The problem given is:

"Today we are going to go on a school excursion and we need to make sandwiches for the whole class. If we need 2 loaves of bread to make sandwiches for my 4 siblings, how many loaves of bread will we need in order to make sandwiches for all 24 students in the class?"

Now I think we can all agree that this question is silly. Two loaves of bread to make sandwiches for four siblings? Those sandwich are immense.
Anyway, the website determines that this is a direct proportion problem, so tells us to use the Direct Rule of Three (as opposed to the Inverse Rule of Three, which is different). It provides the following method:

Now I know that many of us will be confused as why they felt the need for a formula here. What they've essentially done is cross-multiply: 4 times x is equal to 2 times 24, then we solve for x. But this isn't how I would approach this question. I'd just say something like this:

"He needed 2 loaves for 4 sandwiches, and now he needs six times the number of sandwiches so he needs six times the number of loaves".

I don't use an equation to solve problems like this. I use multiplicative reasoning. It's logic. 

In England, teachers tend to solve proportion problems using a unitary method and/or scale factors. A popular layout is seen in this lovely resource from Don Steward
The Rule of Three is explained very clearly on its Wikipedia page. Here it says that:

"The Rule of Three was an historical shorthand version for a particular form of cross-multiplication that could be taught to students by rote. It was considered the height of Colonial math education and still figures in the French national curriculum for secondary education.".

From my conversations with teachers I get the impression it is not just taught in France, but is taught all over the world. In fact, I get the impression they think it's weird that we don't use it.

Yesterday I attended the Harris Federation Maths Conference and, coincidentally, saw teachers from other parts of Europe use the Rule of Three on two separate occasions. They described it as cross-multiplication rather than the Rule of Three.

The first was a teacher from Poland who spoke to me after I delivered my session on Unit Conversions (I didn't catch his name - but if he's reading this then thank you for speaking to me - I love talking about methods!).

Converting metric units is done shockingly badly at GCSE in this country so I ran a workshop on how to teach it in more depth. I talked about various methods for doing unit conversions and showed them (for interest, not because I recommend teaching it) the zany factor-label method. At the end he asked me whether I'd considered using cross-multiplication to convert metric units.

Say we want to convert 300 milliliters to litres - this is the sort of question that is often answered incorrectly at GCSE. He said that he'd lay out his workings like this:

Then he'd cross-multiply.

Cross-multiply is a term I tend to avoid using but I know it is widely used around the world. I once worked with an American teacher who used it all the time, for lots of different problems, assuming that our students knew what she meant (I don't think they did). The dictionary definition of cross-multiply is "to clear an equation of fractions when each side consists of a fraction with a single denominator by multiplying the numerator of each side by the denominator of the other side and equating the two products obtained". Apparently the term has only been used since the 1950s (if that's true then it's a relatively new term in mathematics). 

In this case we have no fractions, but we have quantities that are in proportion which means they can be treated (or indeed, written) as fractions. We cross multiply like this:
This gives the equation 1000? = 300, so to find the value of ? (or whatever symbol we use here), we solve the equation by dividing 300 by 1000.

For students who know how to solve simple equations, it is a very reliable, efficient and straightforward approach for unit conversions. And indeed, some teachers may use it in this country, but I believe it is far more popular in other parts of the world. 

I think that the Rule of Three fell out of common use in England quite a long time ago. Perhaps this is because it was considered mechanical and done without understanding...?  I don't know for sure but I'd be interested to hear from teachers who were educated in England in the 1950s and 60s who can tell me whether they were taught the Rule of Three.

I have a little book called 'How to Teach the Method of Unity' from 1883 which appears to argue against using The Rule of Three for solving proportion problems, in favour of using what we now call the unitary method.
If solving a problem such as "5 pencils cost £1.25, how much do 3 pencils cost?", it tells us to first work out how much one pencil costs, instead of using a Rule of Three formula or 'statement'. 

In its preface we are told that the New Code of 1882 says that teaching of the Rule of Three is to be by the Method of Unity, and then goes on to explain the justification for this. It tells us that the great aim of teaching arithmetic is to improve reasoning powers.
The second occasion in which I saw a cross-multiplication method yesterday was in a workshop about ICCAMS

At the start of the workshop we were asked a question: 

Three teachers went to the board to show their methods. The first use a scale factor method, noting that we could multiply by 1.5. He set his workings out in what I'd call the box style, similar to the Don Steward resource shown above. The second teacher use a visual approach (essentially a bar model).  The third teacher (I didn't catch where she was from unfortunately - somewhere in Europe) showed a cross multiplication method:

It was interesting to see this approach used twice in one day. I was already aware that these methods are widely used around the world, but what I am interested in is when and why we (for the most part) moved away from them in this country. As I've said above, I assume there was a move towards approaches that involved reasoning as opposed to procedure, and I am interested in how this move came about.

The approach of using cross-multiplication to solve proportion problems is one that all maths teachers in this country should know, even though most of us don't teach it. We need to know it because:
a) some of our students will learn it from elsewhere (e.g. from tutors)
b) some of our colleagues may teach it
c) some of our students will have learnt it in other countries before moving to our schools.

If we have students who use it, we just need to check that they understand why it works and are able to use it correctly when solving different types of problems (e.g. those involving inverse proportion).

Thank you to all teachers who shared their approaches with me yesterday. It's so important to talk about methods.


  1. I use the Cross multiplication method but I explain why it works first whilst doing equivalent fractions in year 7. I also teach the reasoning approach as sometimes that is the easier option. I use this multiplicative reasoning in lots of topics at GCSE.

  2. I was recently, last week in fact, teaching my yr10 class. We are following the crossover scheme from JustMaths. In Unit 15 - Ratio, I came across a the objective “write a ratio as a linear function”. I suspect that others in my faculty have ignored this objective and wished I had spotted this before anyone should have taught it, (I will be looking into the the first week back). The cross multiplication method works well for this when students put one ratio above the other but I was not convinced that they knew why they were doing it. Is there another way you would have done this?

    1. The method I describe for this in my post about ratio is essentially the Rule of Three. I don't say 'cross-multiply' but it's the same thing: https://www.resourceaholic.com/2017/12/ratio.html

  3. I just got hold of your book and have started going through it, also read the intro which I think is important. I went to school in Kenya, Libya, India and eventually on to my 8th school at my local comp in London finishing off Year 11, 12 and 13. I learnt a few methods adapting to whatever new method was presented to me through those years. My London school only had one A level set run by the head of maths. He was kind enough to let me use whatever methods I had learnt before, including the Hindi equivalent of the term 'Soh Cah Toa'.

    Years later when I started tutoring I realised that on the whole my tutees needed consistency in methods. To be taught one method at school and then another one by the tutor was causing problems in many cases where procedural fluency was still being built. I learnt not to get too possessive about any particular methods I liked. So in most cases I go along with whatever method they are learning at school to give them a reminder of the method and more spaced practice. On a case by case basis, I might introduce them to another modality, typically things like bar modelling in ratios or algebra tiles. Or if the number sense needs work then I will use a range of manipulatives including Dienes, counters and a rekenrek. But these decisions are carefully made and at all points I want to make sure the tutee makes the best of their time at school, their own practice and with me.

    1. Thanks Atul, that's really interesting. Sounds like you're doing awesome stuff with your tutees.

  4. It all makes sense now! - I was teaching some percentage topics with a class including Spanish and Mexican students, and guess what they were doing - very proficiently, but they didn't have the understanding as to why. This always makes me somewhat nervous, but I was happy that they continued using their method as long as they were secure in knowing how to layout their calculation (some weren't); also they are only here for a year.