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

Created on 21 January 2019 by ncetm_administrator
Updated on 07 February 2019 by ncetm_administrator

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:can also be expressed as $\inline \fn_jvn \small \frac{a}{b}$, 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 $\inline \fn_jvn \small \frac{1}{2}$ 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 $\inline \fn_jvn \small \frac{1}{3}$.

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.

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30 March 2019 22:05
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.
30 March 2019 00:33
When talking about the word “difference”, some people just confuse about its different uses and the preposition associated with it in mathematics.

Just like ‘sum’ is the result of addition of two numbers, ‘difference’ is the result of subtraction of two numbers.

Thus, in 3 – 5 = –2, the result –2 is the difference.

When we say ‘difference between’ two numbers, it should be the result of the larger number minus the smaller number.

Thus, the difference between 3 and 5 is 5 – 3 = 2.

However, in algebra, when two numbers are represented by a and b, we don’t know which number is larger. We will use

difference of a and b = a – b,

and difference between a and b = | a – b |.

(KS2 and KS3 students may not know the absolute value sign.)

For instance, when saying difference of two squares a^2 and b^2, we often write

a^2 – b^2 = (a + b)(a – b),

without bothering whether it is positive or negative,

Regarding the arithmetic sequence, the definition is when the (n + 1) term – the nth term is a constant, this constant is called the common difference.

e.g. 3, 7 , 11, 15, …

Since 7 – 3 = 11 – 7 = 15 – 11 = 4,

this sequence is an arithmetic sequence with common difference 4.

23, 20, 17, 14, …

Since 20 – 23 = 17 – 20 = 14 – 17 = –3,

this sequence is an arithmetic sequence with common difference –3.

Here, we see again that a difference can be a negative number.

3, 7, 3, 7, 3, 7, …

In this sequence, the difference between two adjacent terms = | 7 – 3 | = 4 is a constant.

But it is not an arithmetic sequence. This is because

The 2nd term – the 1st term ≠ the 3rd term – the 2nd term.

So in the definition of an arithmetic sequence, we cannot say a sequence is an arithmetic sequence if the difference between a term and its preceding term is a constant.
28 March 2019 15:20
EmilySo, I think there's some confusion: I'm not sure that anyone here has *ignored* any basic definitions. If we are talking about definitions, then a ratio a:b can be *evaluated* as a/b. In other words, a/b is the *value* of the ratio of a:b. Without such an evaluation, we cannot proceed with arithmetic on such ratios as sin(x).

Nor, indeed, would we be able to calculate the sum of a geometric sequence, for example.

There are a few examples in maths where terms which have a specific meaning and definition are to be taken in a different context when used in other places, or in general situations.

One of these circumstances is where we use "common *difference*" when considering a linear sequence (aka arithmetic sequence - I'm not even going to try to discuss why it's called an *arithmetic* sequence). The early Greek mathematicians had no symbol for zero, and no specific way of representing negative numbers. So a sequence such as 15,13,11,9, ... has a difference of 2. A sequence such as 2,4,6,8,... also has a difference of 2. To say that the former example has a difference of *negative* 2, is to misuse the word difference. We might legitimately (and in many ways *lazily*) state that it has a difference of -2, because we understand the vagiaries of language. But this assumes that our audience has the same understanding, and we should be aware that if the audience does NOT have such an understanding, then we might be causing some confusion.

Another example is y = mx + c (obviously you see the link here to linear sequences?!) as an *EQUATION* of a straight line. Elsewhere, however, we see that the definition of an equation involves only one variable, the value of which we might determine.

The definition of a formula, however, is where there are 2 or more variables, and given the values of all but one of those variables, we might form an equation in order to solve for the remaining variable.

I am yet to find a textbook, or exam question, which refers to y = mx + c as a formula.

When our audience consists of students who are still forming a complete understanding of ratio, values of ratios, fractions, decimals, etc., it pays to be mindful of such language/terminology issues, and we might want to address, or at least be aware of, our own use of specific language and terminology. I think, in most cases, it is not difficult to do so.
28 March 2019 10:06
Some people ignore the basic definition of ratio and say that the ratio a : b cannot be represented as a/b. Apart from pi and the golden ratio, the common ratio of a geometric sequence is often represented as an integer or a fraction.

A sequence is called a geometric sequence when the ratio of the (n + 1) term to the nth term is a constant. The constant is called the common ratio of the sequence.

e.g. For the sequence 2, 6, 18, 54, …,

as 6 : 2 = 18 : 6 = 54 : 18 = 3 : 1 is a constant,

this sequence is a geometric sequence whose common ratio = 3 : 1 = 3.

For the sequence 192, 96, 48, 24, … ,

as 96 : 192 = 48 : 96 = 24 : 48 = 1 : 2 is a constant,

this sequence is a geometric sequence whose common ratio = 1 : 2 = ½.

The common ratio is usually denoted by the letter r and applies to:

the general term = ar^(n – 1),

the sum to infinity = a/(1 – r).

In applying these formula, r is usually represented as an integer or a fraction, rather than the colon form.
07 March 2019 05:38
Euclidean concepts of ratio perfectly explains pi and phi as ratios, in terms of comparative magnitudes of 'like' objects. I totally agree with your definitions of pi and phi as ratios. My point, albeit with far too much digression (sorry!), is the irony that in order to find accurate values for these ratios, then the ratios by themselves are inadequate. We rely on the further definitions from number theory and/or calculus.

The Greeks understood this problem, (ask Hippasus! - or Pythagoras who apparently/allegedly refused to admit the issue), such as when trying to evaluate a ratio in a square of diagonal:side - since sqrt(2) is irrational.

When using sin(angle) to solve a problem to find the length of a side of a triangle, I nearly always want the value of sin(angle) as part of my calculation. Hence, when referring to such answers collectively as "trig values", I see this term as perfectly adequate and sufficient.
07 March 2019 02:03
Whatever names and definitions you are using for pi and the golden ratio, you cannot deny the nature that they are ratios. I wonder what definition about pi a teacher would use when teaching a KS3 class.

We should teach students proper mathematics terms. Sine, cosine and tangent of an angle can be called trigonometric ratios or trigonometric functions. A value of these ratios or functions, such as sin 60°, can be called a “value of the trigonometric ratio” or “trigonometric function value”, but not “trigonometric value”. This is the reason why you cannot find the term “trigonometric value” in a textbook.
06 March 2019 04:09
Just to finish off ...

Gwen explicitly quotes Vivian in the final paragraph of her article, and continues to make a striking and telling comment, "...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."

I note with a little disappointment, that many texts continue to discuss trigonometry using the term "trig ratios" as opposed to a preferable and more precise "trig values". These include Pearson/Edexcel current GCSE (9-1) Higher, 700+ page textbook; the iconic Bostock & Chandler 1990, 500+page "Core Maths for Advanced Level", and most (all?) Edexcel mark schemes and even Examiners' Reports. [Other exam boards are available ... and possibly use "trig ratios" ... I'll leave it for others to check].

Refreshingly (?) CGP current GCSE guides make no mention of either "trig ratios", nor "trig values", but stick to naming/using only "trig formulae" and "trig functions" [I checked them out in a local Waterstones earlier].

Returning to the classroom, briefly, I'd like to consider a time late September/early October when you've just got to grips with your new Y8 class and your prescribed sequence of lessons on fractions. With trepidation, you enter the class, armed with a fab lesson plan based on equivalent fractions leading to perhaps a few students being able to add & subtract different denominator fractions (eg 5/9 - 2/7) by the end of the lesson. But, you've a fairly hopeful starter to 'test the water' which involves maybe a dozen questions with sums, differences, products and quotients of various pairs of fractions.

How do you feel when all of the class responds with "never ever add the bottoms, but make sure before adding/subtracting that the bottoms are the same by finding lowest common denominator, then just add/take the tops" and "when multiplying, make sure you cancel, first, any top with any bottom, then simply times the tops & times the bottoms" ??!

... if only ... ?

what if .... ? ... what if this was to happen? & what if there was a similar situation when first exploring basic ratio with your Y8 class?

For various reasons, I pay close attention to the iGCSE Higher exams from Edexcel. I note that there's a common response, year on year, to questions in these exams involving ratio. Examiners' Reports point to "a high level of success" with early questions in the exam, even when some of these involve quite tricky manipulations between fractions, decimals & ratio/proportion to solve a problem, mostly presented in what we refer to as " a wordy" format.

In later questions where there is an application of ratio/proportion, we see "In order to make any real progress with this question, students had to recognise the fact that the small and large cone were similar and use the volume scale factor to find the length scale factor." or comments along similar (?) lines. I read these as "this question was badly handled".

I'm now seeing Gwen's article as a plea for a consistent approach to 'ratio'. By direct consequence, if we're consistent in approach to 'ratio', then why not in 'fractions' too. And what about area, volume, probability. What about ALGEBRA? Consistency begins with terminology, and ends with students confidently using the terminology themselves, without doubts caused by misuse of terminolgy previously, or elsewhere. If terms are used consistently, and constantly, then methods shown & used will be consistent, too? Outcomes - consistently good?

Here are a few of the papers/questions that I mentioned earlier:

7 June 2018, 4MA1/2H, Qns: 2,3,8,16,18-22

7 June 2018, 4MA0/4H, Qns: 2,5,9,18-20

24 May 2018, 4MA1/1H, Qns: 2,16

NB (not iGCSE): 6 Nov 2018, 1MA1/1H Q21 *

Great article, Gwen. Many thanks.
06 March 2019 04:08
Hi, Emily, good to hear from you again.

Firstly, I was going to point out that the value of the ratio sqrt(8):sqrt(2) = 2 , until i revisited your edited paragraph.

Second, I note that Wikipedia lists pi under the headline "mathematical constant" rather than ratio, and lists several definitions which bear no reliance on ratio. These include : infinite series, Euler's identity, Lindeman's proof of Legendre's and Euler's conjectures, and so on.

https://en.wikipedia.org/wiki/Pi

Intriguingly, there are 2 quite important definitions which are based on (arguably) ratio-value considerations. One is Weierstrass' integral definition (which would require a trig substitution since d/dx(arcsinx) = 1/sqrt(1-x^2). However, we note that trig functions may be defined as infinite series, and therefore not bear reliance on ratio. The other (better) one is Torricelli's proof of the volume of Gabriel's Trumpet. At the time, devoid of any knowledge of calculus, Torricelli used both Cavlieri's Principle (based on ratio values) AND (separately, for back-up, and slightly showing off!) a Euclidian ratio-based proof. He relied later on Marin Mersenne (yes, that Mersenne) to spread his proof to the wider mathematical/scientific communities throughout Europe

http://fredrickey.info/hm/CalcNotes/torricelli.pdf

I say 'better', since I'm linking Euler and Mersenne with their work on primes : they both worked on what would become Euler's Totient Function (where "phi (φ) function is a very important number theoretic function having a deep relationship to prime numbers and the so-called order of integers")

https://www.doc.ic.ac.uk/~mrh/330tutor/ch05s02.html

and so this links quite nicely to φ :

Again, variously defined without reference to ratio values, but rather quite easily as a solution to the quadratic equation [x^{2}-x-1=0].

φ is also labelled, without reference to ratio, as : golden mean or golden section (Latin: sectio aurea). Other names include medial section, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.

I find it unfortunate that in many cases, for example we see written "pi, the ratio of C:d = 3.1415926...", rather than perhaps "C:d has a value, pi=3.1415926...". Similar happens when we see φ defined.

Thirdly, with reference to "...some people still are not convinced that a ratio x : y is defined as x/y." then I should hope that they are not convinced of this.

I would hope, however, that we more closely follow the definition thus:

given a ratio a:b, then the *ratio value* is defined as a/b.
05 March 2019 18:45
Sorry, my last paragraph should be:

Hence, when x and y are integers, x : y is equal to the fraction x/y. When x or y or both are irrational numbers, x : y may be equal to an irrational number. For example, sin 45 degree = 1 : sqrt(2) is an irrational number.
05 March 2019 18:30
I see that some people still are not convinced that a ratio x : y is defined as x/y. They say that 1 : 2 is not equivalent to ½.

Let us see the definitions of pi and the golden ratio.

We all know that pi is defined as the ratio of the circumference of a circle to its diameter.

See https://en.oxforddictionaries.com/definition/pi

and https://en.wikipedia.org/wiki/Pi

It is denoted by π. It is an irrational number equal to 3.14159 …. We simply write

π = ratio of the circumference of a circle to its diameter,

≈ 3.14159

without any colon notation.

The golden ratio is defined as the ratio of the length of a longer line segment to the length of a shorter line segment such that their ratio is equal to the sum of their lengths to the length of the longer line segment.

See https://www.britannica.com/science/golden-ratio

and https://en.wikipedia.org/wiki/Golden_ratio

The golden ratio is denoted by ϕ. By the definition,

ϕ = (a + b)/a = a/b,

where a is the length of the longer line segment and b is the length of the shorter line segment.

It is known that

ϕ = (1 + sqrt(5))/2 = 1.618 … is an irrational number.

Again, the golden ratio is not expressed in colon notation.

From the pages of the book The Elements about the definition of ratio (the link indicated by stefn), it mentions that ratios can be compared. If the ratios 5 : 6 and 7 : 8 cannot be written as the fractions 5/6 and 7/8, how can we compare their magnitudes?

Hence, when x and y are integers, x : y is equal to the fraction x/y. When x or y or both are irrational numbers, x : y is equal to another irrational number.
05 March 2019 10:51
@EmilySo

Yes, thanks for prompt reply. I was not confused by your post. There was a slight glitch in my connection and/or with the site when I posted (maybe why my last post appeared twice?!) and it was 'sent' while I was editing ... the post was meant to have included the following:

We can use a similar method when solving *any* variation problem.

eg:

d is proportional to the square of s, so that d=20 when s=2

Find d, when s=5 :

20:2^2 = d:5^2

giving

20:4 = d:5

so, d = 125 when s=5.
05 March 2019 10:20
@ stefn

I said that we can use proportion to solve the recipe problem. It may be my solution is too brief that causes confusion to you. The full solution should be:

ratio of eggs in a dish to eggs in the recipe = ratio of sugar in the dish to sugars in the recipe

(These two ratios are valid since each of them are ratio of the same kind.)

i.e. 12 : 3 = n : 2

12/3 = n/2

n = 12/3 × 2

n = 4

(Since the two ratios are equal and thus 12, 3, n and 2 are in proportion.)
05 March 2019 00:29
@EmilySo

You say that we use proportion to solve direct variation problems, but we can do the same by using similar ratios.

In order to represent eggs : teaspoons of sugar : flour, you say that we should not use ratio, since we are not comparing like units. Later, you use a representation of ratio in order to show a solution to a given problem.

Going back to the original article, I was not (and am still not) happy with the statement that 1:2 is equivalent to 1/2. It seems that other contributors agree with this original claim. My issue here is the use of the word 'equivalent'. That would suggest that 1/2 is then equivalent to 1:2, since that's how the word 'equivalent' is intended to be used.

However, students presented with the following question:

Find P(rolling an even number using a regular, unbiased die)

would achieve full marks for 1/2, or 50%, or 0.5 , but zero marks for 1:2 (or for 1:1).

As for trig values being referred to as ratios, Euclid had a real issue with various types of arithmetic being carried out on ratios. Ratios compare relative sizes, so yes, I can see the reasoning behind labelling them as trig ratios. But in doing so, I think we are inviting students to be rather confused by this. Especially as we never, ever see written, for example, sin(30) = 1:2

In order to create the graphs of the trig functions, we use a unit circle, where the (x,y) coordinate of a point on the circle can be replaced by (cos(theta), sin(theta)) ... [ where theta is, of course, the angle made between 'East' , or positive x-axis, and the radius drawn from the origin and the point on the circle ]. In this case, cos(theta) is defined as the horizontal displacement of our point on the circle - it's definitely not a ratio.

We later learn that cos(x), sin(x), etc may be defined as infinite series of powers of x. In this case also, the trig values are definitely not ratios.

@Jen1812

You mention being confused by sin, cos, tan : as them being a "thing" by themselves.

I see this misconception regularly displayed by many students, including very able A-Level students (Y12 & Y13) who have mastered most (if not all) maths concerned with ratio & proportion.

Such students try to "divide by cos" , or "times by sin" as if cos, sin are a "thing", and they can verbally explain what they intend by this. It takes very little to convince these students that they're mistaken, and we spend only a moment to sit the trig functions in their place alongside other functions such as log(x), exp(x), sqrt(x), reciprocal(x), for example.
05 March 2019 00:29
@EmilySo

You say that we use proportion to solve direct variation problems, but we can do the same by using similar ratios.

In order to represent eggs : teaspoons of sugar : flour, you say that we should not use ratio, since we are not comparing like units. Later, you use a representation of ratio in order to show a solution to a given problem.

Going back to the original article, I was not (and am still not) happy with the statement that 1:2 is equivalent to 1/2. It seems that other contributors agree with this original claim. My issue here is the use of the word 'equivalent'. That would suggest that 1/2 is then equivalent to 1:2, since that's how the word 'equivalent' is intended to be used.

However, students presented with the following question:

Find P(rolling an even number using a regular, unbiased die)

would achieve full marks for 1/2, or 50%, or 0.5 , but zero marks for 1:2 (or for 1:1).

As for trig values being referred to as ratios, Euclid had a real issue with various types of arithmetic being carried out on ratios. Ratios compare relative sizes, so yes, I can see the reasoning behind labelling them as trig ratios. But in doing so, I think we are inviting students to be rather confused by this. Especially as we never, ever see written, for example, sin(30) = 1:2

In order to create the graphs of the trig functions, we use a unit circle, where the (x,y) coordinate of a point on the circle can be replaced by (cos(theta), sin(theta)) ... [ where theta is, of course, the angle made between 'East' , or positive x-axis, and the radius drawn from the origin and the point on the circle ]. In this case, cos(theta) is defined as the horizontal displacement of our point on the circle - it's definitely not a ratio.

We later learn that cos(x), sin(x), etc may be defined as infinite series of powers of x. In this case also, the trig values are definitely not ratios.

@Jen1812

You mention being confused by sin, cos, tan : as them being a "thing" by themselves.

I see this misconception regularly displayed by many students, including very able A-Level students (Y12 & Y13) who have mastered most (if not all) maths concerned with ratio & proportion.

Such students try to "divide by cos" , or "times by sin" as if cos, sin are a "thing", and they can verbally explain what they intend by this. It takes very little to convince these students that they're mistaken, and we spend only a moment to sit the trig functions in their place alongside other functions such as log(x), exp(x), sqrt(x), reciprocal(x), for example.
04 March 2019 22:13
Note that a ratio is used to compare two quantities of the same kind. Therefore, it bears no unit. If a recipe requires 3 eggs, 2 teaspoons of sugar and 170 grams flour, we should not write the ratio of eggs to sugar to flour = 3 : 2 : 170. It is not a ratio problem because the quantities are in different units. We can compare and work out any two of these quantities using rate or proportion mentioned below.

In connection with ratio, the term proportion has a specified meaning. If four quantities a, b, c and d have the relationship a : b = c : d, the quantities are said to be in proportion. We use proportion to solve direct variation problems.

e.g. Based on the above recipe, if there are 12 eggs, how many teaspoons of sugar are required?

Let the number teaspoons of sugar be n.

Then 12 : 3 = n : 2

n = 8.

Regarding to the difference and relationship between ratio and proportion, please read the article at
https://nrich.maths.org/4825
04 March 2019 22:10
My original comment was typed in Word. I found that this comment box does not support its Formula function. Therefore, I retyped some parts and made a mistake in the ratio of the price of the calculator to the price of the book. Please accept the following amendment.

(c) two quantities of the same kind

e.g. If the price of a calculator is £10 and the price of a book is £6,

the ratio of the price of the calculator to the price of the book

= £10 : £6

= 10/6 = 5/3

= 5 : 3.
03 March 2019 23:46
In the 1940s and 1970s UK textbooks The Essentials of School Algebra (P. 385) and The Tutorial Algebra Volume 1 (P. 41), a ratio of two quantities containing respectively x units and y units of the same kind, is represented by x : y and its value is equal to x/y. In the current publication Oxford Concise Dictionary of Mathematics, the term ratio is defined as the quotient of two numbers or quantities giving their relative size. In the good web site Wolfram Mathworld, the ratio is defined in the same way.

http://mathworld.wolfram.com/Ratio.html

Therefore Shanghai Mathematics team is not the only source that defines the ratio x : y = x/y . In fact, it is a universal definition. This is also the reason why simplification of ratio is the same as simplification of fractions.

By this definition, a ratio can be used to compare:

(a) a part to a part

e.g. If a class has 9 boys and 11 girls,

the ratio of boys to girls = 9 : 11

= 9/11 .

(b) a part to a whole

e.g. In the above case,

the ratio of boys to the total number of students = 9 : 20

= 9/20.

We can also say that the fraction of boys in the class = 9/20 .

(c) two quantities of the same kind

e.g. If the price of a calculator is £10 and the price of a book is £6,

the ratio of the price of the calculator to the price of the book = £10 : £6

= 6/10

= 3/5

= 3 : 5.

e.g. In a right-angled triangle, if the opposite side of an acute angle A is 24 cm and the adjacent side is 18 cm,

the ratio of the opposite side to the adjacent side = 24 : 18

= 24/18

= 4/3 .

This is the trigonometric ratio tan A.

Other examples are aspect ratio, gear ratio, student-teacher ratio, map scale, etc.

Besides, Essentials Algebra mentions that in the ratio a : b, a is called the antecedent and b the consequent. However, these two technical terms have “now” (in 1940s) become almost obsolete. It is not necessary to teach these two terms nowadays.

References:

Mayne, A.P. (1938) The Essentials of School Algebra. London: Macmillan Education Limited.

Walker G., Briggs and Byran (1940) The Tutorial Algebra Volume I. Great Britain: University Tutorial Press Ltd.

Clapham, C. and Nicholson, J (2014) Oxford Concise Dictionary of Mathematics 5th Edition. United Kingdom: Oxford University Press.
02 March 2019 14:52
(Continuing from previous comment - sorry it's so long):

Or perhaps the issue is that we tend to think of ratios (usually part:part) as being written with a colon, rather than in fraction form. Maybe we need to just think of ratios (and fractions, decimals and percentages too?) as being different and interchangeable ways of expressing the relative size of two or more things, as long as we clearly define which things we are comparing? So we can consider the trig ratio of sin(30) as the comparison of the length of the opposite side to the length of the hypotenuse, which is ½ as a fraction and 0.5 as a decimal. This could be regarded as a part to part ratio (the opposite length, which is part of the total perimeter, compared to the hypotenuse, which is also part of the perimeter), but written as a fraction or decimal.
02 March 2019 14:49
Some really interesting points raised here... as a student in secondary school, I remember getting really confused about the concept of trig ratios - I could use sin, cos and tan, but could not comprehend how they could be regarded as ratios. As far as I was concerned, from what I had been taught previously, a ratio was, by definition, 2 (or 3 or 4 etc.) numbers with two dots (i.e. a colon) in between them. Telling me that sin is the opp/hyp, cos is the adj/hyp and tan the opp/adj did not help, as as far as I was concerned, these were fractions, not ratios! (To be fair, I think I was also confused about the concept that sin, cos, tan etc. are functions of the angle, rather than an actual objective 'thing' themselves - sorry, not sure how best to describe that.)

Although I have been willingly using trig ever since, it wasn't until a couple or so years ago, when (now as a maths teacher) I saw the topic of part: part and part: whole ratios on the year 7 scheme of work we were using and decided to investigate what this was all about, that this seemed to make a lot more sense to me conceptually.

To take an example, we can consider the case of having a set of 10 counters, of which 4 are red, 3 are blue, 2 are green and 1 is yellow. We can then write the ratio to compare any two of these colours in any order, e.g. the ratio of red to green is 4:2 (or 2:1) and the ratio of blue to red is 3:4. We can call these part: part ratios as the red, blue and green counters are each part of the whole set. Similarly, we can write the ratio of any three of these colours in any order, e.g. the ratio of blue to red to yellow is 3:4:1 (this is a part: part: part ratio), and we can write the ratio of all four colours in any order e.g. green: blue: red: yellow as 2: 3: 4: 1 (this is a part: part: part: part ratio).

However, in this scenario, we can also write the ratio of any colour (or combination of colours) as a ratio of the whole. For example, the ratio of red counters to the total number of counters is 4:10 (or 2:5). This is a part: whole ratio, as we are comparing the number of red counters with the total number of counters (the whole set). Of course, we would normally write this as a fraction i.e. that the number of red counters as a fraction of the total is 4/10 (or 2/5). However, we can also write fractions as part/part if we wish to e.g. the number of green counters as a fraction of the number of red counters is 2/4 (or ½). This can be harder to understand, but we can explain (similar to in the apple and orange example above) that the number of green counters is half of the number of red counters.

I think that part of the issue here may be that we almost always think of ratios as being part:part and fractions as part/whole. When we teach converting between fractions and ratios we also tend to think this way. However, I have found that, having introduced ratios and ratio to fraction conversion normally (i.e. converting from part:part ratios to part/whole fractions) my Yr 7 students have been quite receptive to the idea that ratios and fractions can be both be written as part to part and part to whole. (In my experience, this distinction between part to part and part to whole ratios and fractions does not tend to be made explicit in English classrooms, though I think that in the USA they do make this distinction. I have shown my students this video which seems to be quite helpful: https://www.youtube.com/watch?v=pTuVs-dfnoA)

Of course, we then have the idea that a fraction is actually a division, and we can also write this as a decimal (or even percentage?). I think introducing this concept of part/part and part:whole in Key Stage 3 could be really helpful in avoiding misconceptions with ratio, fractions and trigonometry later on.
26 February 2019 13:56
@ Gwen

Thanks for the response to the 3 previous comments.

Seeing what Caitlin has written about the semantics of this piece, I'd have to agree. This is why I had such a difficulty accepting the part in your article about 1:2 being 'equivalent' to 1/2.

When thinking about the words ratio and proportion, I link them to Scale and Scale Factor.

The scale of a map is 1:25000. Scale is the word we use to replace the word ratio.

If I want to use my map to estimate the real distance btwn two points, I'll measure the distance on the map, then apply the Scale Factor to work out the real distance.

Scale Factor is the term we use to replace proprtion (in your article example this is the '1/2')

When teaching trig, I never refer to trig ratios. I call them trig values.

I link speed = distance over time, Pressure = Force over Area, Current = Voltage over Resistance & Density = Mass over Volume when treating the manipulation of trigformulae. Many teachers use a 'triangle' to help students manipulate such formulae. I encourage effective algebraic rearranging, which always leads to better handling of algebra (which the triangle way does not).

I've never heard speed called a ratio, nor pressure, current nor density. So why would we refer to trig values as 'ratios'?
26 February 2019 13:55
@ Gwen

Thanks for the response to the 3 previous comments.

Seeing what Caitlin has written about the semantics of this piece, I'd have to agree. This is why I had such a difficulty accepting the part in your article about 1:2 being 'equivalent' to 1/2.

When thinking about the words ratio and proportion, I link them to Scale and Scale Factor.

The scale of a map is 1:25000. Scale is the word we use to replace the word ratio.

If I want to use my map to estimate the real distance btwn two points, I'll measure the distance on the map, then apply the Scale Factor to work out the real distance.

Scale Factor is the term we use to replace proprtion (in your article example this is the '1/2')

When teaching trig, I never refer to trig ratios. I call them trig values.

I link speed = distance over time, Pressure = Force over Area, Current = Voltage over Resistance & Density = Mass over Volume when treating the manipulation of trigformulae. Many teachers use a 'triangle' to help students manipulate such formulae. I encourage effective algebraic rearranging, which always leads to better handling of algebra (which the triangle way does not).

I've never heard speed called a ratio, nor pressure, current nor density. So why would we refer to trig values as 'ratios'?
25 February 2019 11:14
Thank you all for taking the time to respond to my piece – your comments have given me much food for further development in my thoughts.

@BW_2012: Yes, good point about the limitations of the terminology used. Still, I found it useful to have terminology to hang the ideas on, so maybe you are right about it needing a rethink!

I’m not sure that I understand the difference you draw between a ‘single-reference whole’ and the ‘yet-to-be-built whole’ – can you explain further? Which of these do trig ratios fit into?

I like your idea of ratios taught as lengths on similar polygons, with a shared centre of enlargement – sounds like it might make a good follow up article?

@stefn: Understanding that 1:2 is equivalent to ½, when the whole being considered is 2 (as in my apples and oranges example – there are half as many oranges as apples), is exactly what is needed for trig. When we say that sin 30 = opp/hyp, we are saying that sin 30 = 1/2, that is, the ratio of opp:hyp = 1:2

And this is the understanding that I had previously glossed over because of not understanding it explictly myself. Your claim that 1:2 is equivalent to ½ :1 is also true, but when the whole is the sum of the two parts. I don’t think this is a straightforward difference, but a really complex one – but I suspect we make it more complex by not expressing it explicitly to students.

There is no attempt by myself, or the NCETM, to hold Shanghai teachers up as God-like creatures. Simply there is a recognition of just how good and confident Chinese children are at maths (across the ‘ability’ range), and a question of what we can learn from this. We could put it all down to cultural difference, more time spent learning maths, teachers with more time to plan and develop their lessons, etc. and then go home. Or we could ask if there’s anything we can learn by watching a Shanghai lesson (the same question we should ask when watching any colleague teach, I believe). My observations are simply what I learned in the space of an hour – the discussion generated has taken this further.

@caitlinob: If I understand you correctly then, trig ‘numbers’ are ‘proportions’ not ‘ratios’? Can this really be correct? In which case, the language we all use of ‘trig ratios’ is really unhelpful – I wonder how you deal with this when you teach trig? And how do you deal with tangent ratios greater than 1? The distinction you draw between a ratio and a proportion is not a distinction I have seen in maths before, though the dictionary does seem to agree with you!
25 February 2019 08:22
Apologies: there is a typo in my post. The ratio I quote should be 6 : 12 : 15 : 18 rather than 6 : 12 : 15 : 30.
23 February 2019 23:00

To be explicit a "ratio" express a comparison of the parts (to each other) of a whole. The parts need not necessarily have the same dimensions and there may be more than two parts. So a recipe might have 1 egg to 300 g of flour to 250 ml of milk, which would be expressed 1 : 300 : 250, with the dimensions implicit. In some ratios the dimensions might be the same but this is not strictly necessary.

However, when a part is compared with a whole then we dealing with a "proportion". In this case the dimensions of the part and the whole must necessarily be the same. So a metal alloy might be made up of aluminium, copper and zinc in the ratio 1 : 2 : 7 - where all the dimensions are the same (and it is actually irrelevant what the dimensions are - grammes, kg, ounces). Since a proportion is a part compared with a whole then it is in fact dimensionless and lies between 0 and 1. Proportions can be expressed as fractions, decimal numbers or percentages. So in the above metal alloy example the alloy is made of 10% aluminium.

In the article the **proportion** of oranges in the collection of fruit is 1/3.

Yet the **ratio** of oranges to apples is 1 : 2.

There is a slight problem in recipe type problems you might be given a "whole" which has a different dimension to the parts. So three eggs might be needed to make eight scones. This should be treated as a ratio since the dimensions are different.

I think it is a grave mistake to start writing ratios as fractions - at any point in the proceedings. The way I teach ratios is to point out that ratios can be simplified to a basic form by dividing by the highest common factor of ALL the numbers in the ratio. So the numbers in the ratio 6 : 12 : 15: 30 have an HCF of 3 hence the most basic form is 2 : 4 : 5 : 6.

From this point you can say that any example of the object made up of these parts is a multiple of the ratio numbers. i.e. 2x : 4x : 5x : 6x. (In effect if x=3 then you are reversing the simplification. This is a nice example of the opposit effects of factors and multiples - you can link all this into earlier work on factors and multiples.) Then ANY problem reduces down to finding x which takes part in an equation.

EXAMPLE: A metal alloy bar is made up of aluminium, copper and zinc in the ratio 3 : 4 : 8. If there is 35kg more zinc than aluminium in the bar calculate the weight of copper in the bar.

SOLUTION. The actual weights are 3x, 4x and 8x respectively.

Thus 8x - 3x=35

Hence x=7.

Thus the weight of copper in the bar is 4 x 7 = 28 kg.

I use (and have used) this with a bottom set Year 9 class and they are perfectly happy (including such things as currency exchange rates). They know the difference between a ratio and a proportion, they never mix them up and when handling ratios they never have to mess about with fractions.

EXAMPLE One GBP is worth 1.50 USD. If I change 20 GBP into USD how many dollars do I get?

SOLUTION: The basic ratio is 1 : 1.5. Now express the actual amounts as multiples of this basic ratio. So it is 1x : 1.5x.

We are told that 1x = 20 hence x = 20.

Put this into dollars and we get 1.5 x 20 = 30 USD.

The trick is to be CONSISTENT and PRECISE with your terminology. I ask my pupils: Are we looking at a proportion or a ratio? If it's a proportion then it's going to be a number between 0 and 1. If it's a ratio then start thinking multiples of the ratio numbers: use x as the common multiple. And to repeat: when dealing with ratios fractions NEVER appear because they are utterly misleading (their units are bewildering if nothing else: eggs/gram. And what is worse 1/300 means one three hundredth of an egg for each gram of flour.)
22 February 2019 14:43
I have an issue with the "equivalent ratio" shown. Namely : '1:2 can also be written as 1/2 '.

These are not equivalent.

But , "1:2 is equivalent to 1/2 : 1" is correct.

I think the fad of portraying teachers from Shanghai as some sort of God-like group of maths teaching experts needs to be questioned. BW_2012 highlights some inconsistencies with Vivian's approach, even suggesting a much more logical presentation (of similar polygons ... possibly triangles?) which many of us more seasoned teachers use when introducing more advanced ratio ideas.

And this is the real point. Year 8s are introduced to ratio. Equivalent ratio. Dividing an amount by a given ratio. Algebraic manipulations involving ratio. These are the BASICS of ratio.

Once these foundations of understanding ratio are solid, we can then ask students how ratios interconnect with fractions, building on good understanding of these ideas, rather than confuse students with both sets of ideas at the same time.

How can 1/2 be the same as 1/3 ??
22 February 2019 12:25
Technical names for parts of ratio are certainly worthy of thought by the highest minds. The history books may provide assistance.

However, I would urge caution. "Antecedent" and "Consequent" (or "preceding item" and "latter item") are fascinating; but may collapse entirely when facing a ratio such as 3:4:5 or 5:12:13. Which term(s) are "Consequent"? It all sniffs of the whole "!?vertically?! opposite" angles thing: contextualised terms sneaking out of their specific (limited) contexts...

Nonetheless, the terms (& indeed symbols!) of ratio and fractions (a little like the Julian calendar) would benefit from discussion and upgrade to enable greater (wider) understanding by the masses.

Is the NCETM the right place to start to look at the symbols for the different *kinds* of numerators? (i) the type that are part of a single reference "whole" [1/3 + 1/4 = 7/12]; (ii) the type that are, themselves, part of a yet-to-be-built whole [15/80 + 24/80 + 79/80 = 118/240?].

Control surfaces. Name it (properly) and one can control it.

P.S. I was expecting your article to feature ratios redrawn as lengths on similar polygons with a shared, central(?), centre of enlargement... #ScaleFactor is one technical term one can *definitely* use for all ratios... isn't it?