**ARTICLE**

## Why might the word ‘ratio’ cause confusion for students learning trigonometry?

**Gwen Tresidder watched Vivian Yuanyuan, a maths teacher from Shanghai, teach a lesson on ratio to a Y8 class in Solihull. It made Gwen reflect on how, and maybe why, the word ‘ratio’ often seems to puzzle students when they first encounter trigonometry. **

Ever been faced with a look of utter confusion when you mention trig ‘ratios’ to a Year 10 student, only to realise that it’s the word ‘ratio’ that is causing the confusion? Say the word ‘ratio’ and for most students the image that pops into their head is of two numbers separated by a colon.

As a teacher, once I recognised this barrier, my usual way of dealing with it was to explain to the class that (by the way) a ratio can also be expressed as a single number, obtained by dividing the first number by the second, and then showing how this applied to the lengths of triangle sides.

Did I ever wonder about the introduction of the concept of ratio earlier on in a student’s school career? Not really. I’m not sure it ever occurred to me to teach about ratio as a single number when covering the topic in Year 8. But that’s exactly what I saw a teacher, Vivian Yuanyuan from Shanghai, do with a Year 8 class in Solihull.

Right from the beginning, she showed a ratio in the *a:b* format and the equivalent division to accompany it:

There was a discussion with the students about what the 3 in this calculation represents (the number of attempts Dylan needs to score one basket) and the equivalent calculation for Emma’s score.

Shortly after, Vivian explained that the ratio *a*:*b *can also be expressed as , something else I have been guilty of avoiding or glossing over, because of the confusion about what the resulting fraction represents:

In this case there are the number of oranges than there are apples, but this requires the understanding that the ‘whole’ is the ‘number of apples’. Instinctively, looking at this picture, the ‘whole’ is ‘all the fruit’, of which the oranges are .

How much better that these difficult areas are identified and defined for students rather than glossed over. If I’m honest, I suspect that my understanding of the difficulty was not clear enough to express it as I have done now, until I had seen the lesson in Solihull.

Strikingly, Vivian named every term. For the ratio *a:b*, she called *a* the *preceding* *item* and *b* the *latter item* (English language maths dictionaries use the words *antecedent* and *consequent*) and the quotient of *a* ÷ *b* she called the *ratio value*. Perhaps just having the terminology would have helped me to see it explicitly and teach it explicitly. If I could describe the numbers used in trigonometry as ‘ratio values’ and expect students to be familiar with the concept and term from Year 8 (or primary school), how much easier would it be for helping students to get their heads round trigonometric concepts?

Gwen Tresidder, the NCETM Communications Manager and a former secondary maths teacher, was observing a lesson at Tudor Grange Academy in Solihull in January 2019. The school was one of eight secondary schools (and 35 primaries) playing host to teachers from Shanghai as part of the 2018/19 China–England Teacher Exchange, within the Maths Hubs Programme.

Interestingly, what you've posted about 'difference', although it makes sense & is erudite & eloquent, disagrees with most mathematicians' definition.

Here's one:

http://mathforum.org/library/drmath/view/69177.html

In terms of the word arithmetic, my question was 'why' we use that word when describing a certain type of progression. I wasn't asking for a definition of 'arithmetic progression'. And here we see it's purpose (look at the bottom of the page):

https://www.dictionary.com/browse/arithmetic

However, the post highlights the difficulties of language, especially the written word, when trying to discuss mathematics precisely. More importantly, I think it shows how a young student might easily be confused by our various uses of many of the words specific to maths, as well as many of the words used in maths which are also in common usage in other places.

So, back to ratio, and to trig.

Given that a ratio a:b can be evaluated as a/b so that we might carry out various arithmetical processes, I still think it's wrong to say that a:b is equivalent to a/b. We need to take more care.

If A = B, then logically, that's the same as A<=>B (using logical notation). Separately, a:b => a/b, then a/b => a:b

And although the former is always true, there are many situations where the latter is not true. Hence a:b is not EQUIVALENT to a/b.

Finally, the original article supposes, or at least implies, that students who understand ratio will therefore be more susceptible to understanding trig through trig ratios. The article also says, more obviously, that introducing fraction equivalents to ratios at the same time as discussing ratio is a good thing.

In my opinion, both of these claims need closer attention.

In my experience, students who have a good or better than good understanding of simple ratio, equivalent ratio and other ratio-only problems, together with good understanding of fraction, will more easily link the two. Here's a great resource to use on making those links:

https://variationtheory.com/2019/03/30/sharing-in-a-ratio-fill-in-the-gaps/

Note that this is headed "Variation Theory". In my day such resource would be found under 'good teaching'. It's good to see pedagogy training revert to using such material, rather than the prescribe the use of only a few questions on a topic.

In my experience, most students can master SOHCAHTOA quite easily, and without direct recourse to ideas of ratios. The most difficult is to solve for the denominator part of the formula, or to the angle (but only for the first time when they don't know of the existence of shift-sin, for example). Recall is a different proposition, but understanding ratio, or otherwise is not the key. Rather lack of algebra skills required to rearrange, and/or to lay out working in a sensible or consistent way. Sine/Cosine Rules and Area formula easily follow, but again, poor algebra/layout skills hinder medium/long-term retention.

It's at this point that ratio relationships MIGHT be useful. Not only to help with trig, but understanding trig from a ratio perspective might also reinforce/deepen ratio understanding.

However, it is my experience that even solid Grade 7+ students struggle with concepts/problems/questions which involve relating trig to ratio. I very much hope that this changes with subsequent year groups who have had more meta-cognitive skills development/training/practice. I believe they will be more open to trying to make links and finding ways to solve trig questions which in many cases can be handled more efficiently or elegantly through ratio due to similarity/enlargement, or via stretching transforms.