# How is $\frac{a}{b}$ interpreted?

I was having a discussion with a colleague who is in the process of writing some curriculum, and we ended up having a discussion about what $$\frac{a}{b}$$ (with all the standard restrictions) meant. We couldn't quite agree. Here's where each of us landed.

1. One of us thought that $$\frac{a}{b}$$ was strictly a fraction. That this is, strictly speaking, one number. If needed, and depending on the context, you could use the binary operation of division by using the numerator and denominator as inputs to map $$\frac{a}{b}$$ to its decimal representation.
2. One of us thought that $$\frac{a}{b}$$ could be interpreted as either a fraction or as division, it depends on context.

I haven't been able to find any clear distinction between the nuance of these two. I'm curious to know if:

1. Anyone knows of a spot where a distinction has been made in the literature

and/or

1. Which of the two definitions (or third definition not listed) of $$\frac{a}{b}$$ above do you adhere to?
• You can also think of it as a proportion. I basically think of it as a fraction first, but it's context dependent and I can move my frame pretty readily. – guest Jan 16 at 23:05
• @guest Proportion is an equality of two ratios: $\frac{a}{b} = \frac{c}{d}$. What we have above is one ratio: $\frac{a}{b}$. – Rusty Core Jan 16 at 23:11
• I guess I just mean a ratio then. A comparison of two items. – guest Jan 17 at 17:50

A US specific answer: The Common Core State Standards define $$\frac{1}{b}$$ by saying it is one of $$b$$ equal parts making up a whole $$1$$. $$\frac{a}{b}$$ is then defined as $$a$$ of these. Connecting $$\frac{a}{b}$$ to $$a \div b$$ requires some reasoning.

For instance $$\frac{5}{3}$$ of a candy bar means you take your one candy bar, divide it into 3 equal sized pieces. 5 pieces of that size (which requires more than one candy bar to form!) is $$\frac{5}{3}$$ of a candy bar.

$$5 \div 3$$ might come up in this context: I have 5 candy bars, and three children. I want each child to receive an equal share. One way to think about it is to break each candy bar into thirds, so I have $$15$$ thirds. Then each child can receive $$\frac{5}{3}$$ of the candy bar.

While we take a lot of these basic equivalences for granted, I do think that having a single consist meaning, and arguing the equivalences (at first) is likely to lead to deeper understanding than just declaring that the same set of symbols can have a whole host of (apparently) distinct meanings.

• one thing that made it click for a lot of my [highschool] students was pointing out that the division symbol is just the abstraction of a fraction: one number, a line, over another number (each number, in the case of the symbol, represented as dots). – AmagicalFishy Jan 16 at 21:12
• Sure, it's an easy fix. But your initial phrasing is pretty much the literal meaning of "five thirds of a candy bar," taking "a" strictly to mean "one." I wonder if this is a real or a trivial cognitive difficulty for first learners. The proper interpretation has been second nature to me for so long I can't remember. And I don't know of any research. – user1027 Jan 17 at 12:15
• @user1527 It is definitely a real problem, and one which continues to plague students even through college. I teach future elementary school teachers, and assigning meaning to "improper fractions" is very hard for them for exactly this reason. Also many of them harbor the false belief that all fractions are less than 1, and so will plot $\frac{5}{3}$ between $0$ and $1$ on a number line. – Steven Gubkin Jan 17 at 18:29
• @StevenGubkin "Many of them harbor the false belief that all fractions are less than 1, and so will plot $\frac{5}{3}$ between 0 and 1 on a number line" - this explains the opinion "if you cannot get an engineering job, become an elementary school teacher". Also explains chronic shortage of high school math, chemistry and physics teachers, because anyone worth their salt can get a better position at an engineering company. – Rusty Core Jan 17 at 19:14
• @RustyCore You get what you pay for. I wish we, as a society, valued education enough to properly compensate those who pursue this career. – Steven Gubkin Jan 17 at 19:17

A key idea in maths education, that at least becomes more visible at university level, is that people tend to start thinking of new ideas as processes, but to do more advanced maths they need to move on to thinking of the same things as objects.

That is, we are taught to think of $$2+3$$ as an instruction to take $$2$$ and add $$3$$ to it. Most people will write $$2+3=5$$ to mean 'when I follow the instruction given the thing I end up with is $$5$$'.

But mathematically it is more correct to say that $$2+3$$ is a number, and $$5$$ is a number, and $$2+3=5$$ is the statement that these two numbers are in fact the same number. This interpretation is rarely taught at school, meaning most people never get past thinking of maths as 'following some instructions to arrive at a number'.

When you are thinking that way, it is very hard to ask questions about the properties of $$+$$. Instructions aren't things we tend to think about the properties of. But if $$+$$ changes to being an object (a binary operation), it makes more sense to ask about it being associative, commutative etc.

Going back to the original question, $$\frac{a}{b}$$ is both a fraction and a division. More precisely, fractions and division are two different ways of thinking about the same mathematical thing. The key word you use is interpreted. There is one mathematical object, but more than one way of thinking about it, and it is good to be prepared to think about it in different ways in general (although in a specific case there may be a good reason to pick one interpretation over the other).

As an aside, you might like to consider $$\frac{2x+3}{4-x}$$. This is a number if $$x$$ is a number (ignoring details). But if $$x$$ is a formal variable then this is not a number. It's still a fraction, just in a different ring of fractions. It is still the result of division, or an instruction to divide. For practical purposes though, what will mostly help is to be prepared to think of it as one thing, so you don't have to use up too much brain space thinking about it (see cognitive load theory).

• I'm familiar with APOS theory – Andrew Sanfratello Jan 21 at 13:09

By definition, a rational number is a number that can be expressed as the quotient of two integers. This quotient is called fraction and is written as $$\frac{a}{b}$$. Hence, division and fraction are the same, at least in the context of dividing integers and turning them into rationals. This concept is expanded in middle school for dividing irrationals and polynomials.

Everyone whom I know uses division and fractions interchangeably depending on whichever is simpler in a particular case, but fractional form is the preferred one. I do the same, in fact, this is how I was taught in school.

Say, $$5 \div 3 \times 6 \div 4 \div 10 + 3 \div 8 \times 2$$ turns into $$\frac{5 \times 6}{3 \times 4 \times 10} + \frac{3 \times 2}{8} = \frac{1}{4} + \frac{3}{4} = 1$$

• This answer is very rational. – James S. Cook Jan 20 at 15:46

Adding another view to the point, one could view a fraction $$\dfrac{a}{b}$$, as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. $$7$$ mean? It is $$7$$ of some unit/unitary quantity. For instance, $$7$$ cats, or $$7$$ pies etc. This can lead us to introduce the notion of $$\dfrac{7}{1}$$, as the symbol that denotes exactly the above; 7 instances of some (whole) unit (1). Then, we extend this notation to $$\dfrac{7}{2}$$, meaning that we talk about 7 objects of a unit seperated in two parts. Similarly, we can write $$\dfrac{7}{3},\dfrac{7}{4}$$ etc.

In this context, a fraction is not seen as the result of division but more as a shorthand for a process - changing measurement units. Under this view, division is "hidden" as a process within the fraction symbol.

One advantage of viewing fractions as shorthands for the "measurement unit shift" process is that it this definition stands in the middle of a purely procedural and a purely declarative perception of the fraction. It is neither entirely a process nor entirely a number - so, it remains easy to lean towards one of the two views of the fraction (number or division operator) whenever needed.