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When teaching addition and multiplication of fractions, I seem to recall some advice on this site that one should first treat the cases

$a \cdot \frac{c}{d}$ and $a + \frac{c}{d}$ before moving on to the more general $\frac{a}{b} \cdot \frac{c}{d}$ and $\frac{a}{b} + \frac{c}{d}$.

This made think about the case $a \cdot \frac{1}{b}$ first. I would like to demonstrate in some way, and preferably (but not necessarily) geometrically, that $\frac{a}{b} = a \cdot \frac{1}{b}$, but I am having trouble finding an actual difference between $a \cdot \frac{1}{b}$ and $\frac{a}{b}$.

Of course, the answer might just be that $\frac{a}{b}$ and $a \cdot \frac{1}{b}$ have the same representation, and therefore they are equal.

Question:

Is there an agreed upon difference between how we represent $\frac{a}{b}$ and $a \cdot \frac{1}{b}$?

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    $\begingroup$ Isn't $\frac ab$ just defined as $a\cdot\frac1b$? $\endgroup$ Nov 4 '21 at 21:27
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    $\begingroup$ I think I can understand what you're saying—consider a pre-defined length of $1$, dilate it to length $a$ then take $1/b$th of this on the one hand, and on the other take the length $1/b$ then sum $a$ copies together. Unfortunately I don't know how or whether to try and articulate such a distinction at an appropriate level. It might be subtle and hard for the already initiated to put a finger on, like with the quotative vs. partitive thing. $\endgroup$ Nov 5 '21 at 14:48
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    $\begingroup$ I'm not sure what should happen when this is first taught, but I have students in calculus who don't recognize that those are the same. $\endgroup$
    – Sue VanHattum
    Nov 5 '21 at 15:35
  • $\begingroup$ Related: matheducators.stackexchange.com/questions/14114/… $\endgroup$ Nov 6 '21 at 15:21
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    $\begingroup$ Similarly, one could ask for the difference between $a\;-\;b$ and $a\; +\; {}^-b$. In elementary-school math, $a-b$ is defined (when $a>b$) without mention of negative numbers. In more advanced math, $a\; +\; {}^-b$ is taken as the definition of $a-b$. $\endgroup$ Nov 18 '21 at 1:42
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The common core state standards definition of the fraction $\frac{N}{D}$ of a unit is to subdivide the unit into $D$ equal sized pieces. Each of these pieces is defined to be $\frac{1}{D}$ of the unit. Then $\frac{N}{D}$ is defined to be $N$ of these pieces.

Under these definitions, I think there is no difference between $N$ times $\frac{1}{D}$ and $\frac{N}{D}$.

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You should be aware that, in standard developments, the two expressions being considered here are true by definition. The quotient $\frac a b$ is just a shorthand ("syntactic sugar") way of writing $a \cdot \frac 1 b$.

For example, here it in Sullivan's Algebra and Trigonometry, Review Section R.1:

Definition of quotient, per Sullivan

You'll see that in any standard textbook on algebra or precalculus. Just looking at other books on my shelf: It's the same in Rosen, Discrete Mathematics, Appendix 1 (Axioms for the Real Numbers, in Exercises). Etc.

Of course, such a definition follows soon after the axiom for real numbers that observes any nonzero real number $b$ has a multiplicative inverse that we denote $\frac 1 b$. This is completely standard, and likewise follows the definitions in abstract algebra that a ring is an object with additive inverses, a division ring is an object with multiplicative inverses, etc. (Such objects including real numbers, obv.)

In short, in the standard development, this equality is not something you can prove, justify, or demonstrate; it's simply an invention of a new shorthand piece of writing. It's founded directly in the assumed multiplicative-inverse axiom for real numbers, and reflects the development at the level of abstract algebra.

However, the interesting thing (and possibly what the OP meant) is that there is a difference in the definition and meaning of $a \div b$ versus $\frac a b $, whose equivalence does indeed take a short proof. Steven Gubkin expanded more on this in his answer to my related question here.

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    $\begingroup$ Elementary classes will (in CCSS) define $a \div b = ?$ using $b \times ? = a$ or $? \times b = a$, but will define $\frac{a}{b}$ as $a \cdot \frac{1}{b}$, where $\frac{1}{b}$ is defined by the property that $b \times \frac{1}{b} = 1$. The challenge then is to explain why $a \div b = \frac{a}{b}$. This can be accomplished by seeing that it does solve the corresponding "missing factor" problem, or by appealing to visual intuition (one way to distribute $a$ cookies into $b$ groups is to break each cookie into $b$ parts, yielding $\frac{a}{b}$ cookies in each group). $\endgroup$ Nov 6 '21 at 17:09
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    $\begingroup$ Starting with an integral domain $R$, we define an equivalence relation on $R\times R^*$ and denote the equivalence class of $(a,b)$ as $\frac{a}{b}$. Next we define a multiplication on this set of equivalence classes. It is then a consequence that $a\cdot\frac1{b}=\frac{a}{b}$. So in this sense, the object $\frac{a}{b}$ logically precedes the characterization as multiplication of an inverse; it is not defined as $a\cdot\frac{1}{b}$. But yes--in standard developments not rooted in abstract algebra, the definition you cite as multiplication by an inverse is used. $\endgroup$
    – user52817
    Nov 7 '21 at 0:00
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    $\begingroup$ @StevenGubkin: Would you consider writing that comment up as an answer to my linked question? That's probably a better response than anything that's been posted there to date. (Noting my remaining issue that the terms "quotient" and "a divided by b" are used identically in both contexts.) $\endgroup$ Nov 7 '21 at 14:32
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Straightedge and compass construction is an interesting way to thing about this as an educator. This perspective would not be something you actually take into an elementary school classroom. Rather it is for the teacher to think about this from a higher viewpoint. Thinking about what we are teaching from higher viewpoints makes teaching more fun, and so it makes us better teachers.

There are standard constructions for $a\cdot b$ and $\frac{a}{b}$.

multiply divide

Now let's use notation in the answer posted by Steven Gubkin and compare $\frac{N}{D}$ and $N\cdot\frac{1}{D}$, which strongly suggests an assumption that $D$ and $N$ are positive integers.

  1. For $\frac{N}{D}$, we start with a unit length $1$ and construct a segment of length $N$. Then we use the division construction to construct $\frac{N}{D}$.
  2. For $N\cdot\frac{1}{D}$ we use the division construction to construct $\frac{1}{D}$ and then use the multiplication construction to construct $N\cdot\frac{1}{D}$. Since $N$ is assumed to be an integer, we can do the multiplication by marking off $N$ copies of $\frac{1}{D}$.

It is a proverbial "exercise for the reader" to reconcile that (1) and (2) result in the same length.

More generally, we can start with segments of length $a$ and $b$ and a unit, where we do not assume $a$ and $b$ are integers. We use the division and multiplication constructions to construct segments of length $\frac{a}{b}$ and $a\cdot\frac{1}{b}$. Then we step back and try to see that the two segments are equal in length.

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As an abstract concept, I don't think they have different meanings. As with many situations, though, different notations can represent different situations in stories. As an example:

$N \cdot \frac{1}{D}$ could represent having $\frac{1}{D}$ of $N$ different things, while $\frac{N}{D}$ could represent having $\frac{N}{D}$ of the same thing. We're discussing the same quantity, but the organization is different.

For example, let our story problem be about pizzas with $N=3$ and $D=8$. The amount $N \cdot \frac{1}{D}$ suggests to me that I might have had one slice ($\frac{1}{8}$ of the pizza) of 3 different pizzas, while $\frac{N}{D}$ might suggest to me that I had 3 slices of the same pizza.

This is certainly not a requirement to think in these terms, and some people may view them reversed -- but if you are trying to communicate to students who can't conceptualize them as being the same, this sort of example might bridge the gap for them. In both cases, regardless of what else happened, I had three slices of pizza, whether $3\cdot\frac{1}{8}$ or $\frac{3}{8}$.

(For a more abstract geometric representation, you could use the standard rectangles and circles common for discussing fractions; the comparison would still be one fraction from many shapes vs multiple fractions from the same shape.)

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For most everybody here, the expressions $\frac{a}{b}$ and $a \div b$ look identical, but to elementary students they are not. I mention this because I think the more salient distinction here is not between $a \cdot \frac 1b$ and $\frac ab$, but rather between $a \div b$ and $a \cdot \frac 1b$.

$a \div b$ is usually understood as the answer to one of the two following questions:

A segment (or other quantity) of length (or other magnitude) $a$ is divided into $b$ pieces of equal size. What is the size of each piece?

or:

A segment (or other quantity) of length (or other magnitude) $a$ is divided into pieces, each of size $b$. How many pieces are there?

The first question is called a partitive model for division, and the second question is called a quotative model for division.

In contrast, $a \cdot \frac 1b$ is usually represented as the answer to the question:

A segment (or other quantity) of length (or other magnitude) $1$ is divided into $b$ equal pieces. If we take $a$ of those pieces together, what is their combined size?

Understanding why $a \div b$ and $a \cdot \frac 1b$ are equal seems to me a non-trivial question. It amounts to an understanding that it does not matter whether you first multiply and then divide, or whether you first divide and then multiply.

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I think students need to become used to thinking, reading and computing with various representations of fractions: a/b, a 1/b's, a:b, etc.

Mathematically, your two examples are the same and even not that hard to resolve, conceptually. But, sort of as you discuss, a/b versus a*(1/b) is a simpler example of what could be more complex representations and calculations, like those involved in multiplication, division (the whole cross multiplication thingie), addition (the whole common terms thingie), etc.

However, pedagogically, I think you are better off just teaching and building familiarity, conventionally. Rather than trying to anticipate and preclude every edge case and conceptual hurdle before it occurs. For instance, I think (as with most research mathematicians), you show a false belief that some sort of proof will build comprehension for very young learners. It's not like learning to count is enhanced by learning the Peano axioms.

Also, I think the effort to brainstorm possible hurdles and then analyze them and come up with correctives is a much weaker approach to improving pedagogy than working with students, trying different things, and observing them. Educational methodology is a natural science, like birdwatching or chemistry or the like. It's not a Euclidean derivation.

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    $\begingroup$ As an educator of 17 years, it takes exploration in both directions. While yes, a fair amount of decision about curriculum comes from seeing students interact with the material and forming conclusions about how to improve that interaction, there is also an attempt to be proactive by teachers and have answers prepared for potential problem areas. While I might not say "Let's add this new lesson" based on nothing but brainstorming, I might have any examples discussed here ready "in my back pocket" in case students express not understanding how they are the same. $\endgroup$
    – Vaekor
    Nov 5 '21 at 15:51

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