Is it meaningful to add a number to itself a fractional number of times?

Question

(Migrated from the math stack exchange, where I received an apt-seeming suggestion to pose the question here, at the math-educators stack exchange)

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I introduce my young kids to basic math concepts in the hopes that my armchair enthusiasm for math will rub off on them.

To this end, I've described multiplying as "a special kind of adding where you add a number to itself (however many times)."
In this regard, I've also described that $(n \times m)$ is always the same as $(m \times n)$, and I've explained why using items arranged in rows and columns, an egg carton being an easy illustrative example.
Putting this all together, I've been able to explain, for example, that $3 \times 4$ would be $(3 + 3 + 3 + 3)$ or $(4 + 4 + 4)$, i.e. "3 added to itself 4 times" or "4 added to itself 3 times."

I'm soon about to introduce fractions, and in my mind this is where my explanation, above, seems to falter.

Though my kids haven't formally learned fractions, they understand the concept of certain special, named fractions, such as a "half", and I can tie that to the "$\frac{1}{2}$"-style notation.
Therefore, I can reasonably explain that $(\frac{1}{2} \times 4)$ is "$(\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2})$"
I.e. this ties in nicely to my description of multiplying as "adding a number to itself (however many times)" because we're "adding $\frac{1}{2}$ to itself $4$ times."

But something falls apart when I try to circle back to the notion that $(n \times m)$ is always the same as $(m \times n)$.
I.e. "adding $\frac{1}{2}$ to itself $4$ times" is comprehensible...
but "adding $4$ to itself $\frac{1}{2}$ a time" doesn't quite make sense...at least grammatically.

Is my reluctance over the phrase "adding a number to itself a fractional number of times" merely a hangup over English grammar? Mathematically, is there any legitimacy to the notion of adding a number (to itself) $N$ times, where $N$ is not a whole number?

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Update: Since my question generated so many "off-by-one" responses, I thought deeper about what exactly I'd said to my kids, and I think key is my phrase "however many times". In my original post, I kind of hand-waved away that phrase, but during discussion with my kids I would have explained in a manner that conveyed that "however many times" is $(n-1)$ or $(m-1)$ "instances" of $m$ or $n$ in the "addition chain". Still clunky, no doubt, but I wanted to clarify that I didn't mislead my kids to a state of "off-by-one" confusion. (I think both the clunky phrasing and the grammatically "off" addition by a fractional amount are nicely addressed by the accepted answer)

Answer 1

For the product $a\times b$, I intentionally don't use the phrase "add $a$ to itself $b$ times", but rather I prefer something like "start with zero and add $b$ (copies) of the number $a$". This makes sense for $\frac{1}{2}\times 4$, since we start with zero and write four copies of the number $\frac{1}{2}$, giving us $0+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2$. Similarly, it makes sense for $4\times\frac{1}{2}$, since we start with zero and add one half (copy) of the number $4$, giving us $0+2=2$.

So, my "solution" for your issue is to just phrase what is happening a bit differently. Instead of saying we are adding something to itself a fractional number of times, indicating that the second factor counts the number of times the addition happens, I use that second factor to count the quantity of the first factor we are adding. It's subtle, but I've found it helps arithmetic students.

[Note: I started doing this when teaching positive integer exponents, when it seemed that some students were misapplying the interpretation of $9^1$ as "multiply 9 by itself 1 time", thinking this could mean $9\times 9$ (since we have, indeed, multiplied 9 by itself just once). I began explaining $9^n$ along the lines of "start with 1, and multiply $n$ factors of $9$. This has the added benefit of providing an explanation of how $9^0$ equals $1$. Of course, we later just use exponent properties.

On the note of fractional exponents (which could come next in the conversation), while I've never actually said this, I suppose "starting with 1 and multiplying a half of a factor of $9$ could be justified for $9^{\frac{1}{2}}$, though I don't think I'd want to have to also invent a convenient way to explain "half of a factor" as "the square root".]

Answer 2

Frame challenge: I think your verbiage "adding (whole number) to itself (whole number) times" is misleading and incorrect and exhibits an off-by-one error. Think about the example $(4×1)$. Would you call that "adding 4 to itself 1 time"? No, because you're not doing any addition at all. The result of "adding 4 to itself 1 time" should be the result of one addition, i.e. $(4+4)$, equivalent to $(4×2)$.

Answer 3

The Common Core State Standards "definition" of multiplication is that $A \times B$ represents the number of units when you have $A$ groups, each $1$ group containing $B$ units.

For instance $4 \times \frac{1}{2}$ could represent the number of pounds of flour you get from $4$ bags when each $1$ bag contains $\frac{1}{2}$ of a pound. Note that this is "backwards" from the way you are interpreting multiplication. Now I am aware that multiplication is commutative, but this is not obvious from either your definition or the CCSS definition so it needs to be argued.

Similarly $\frac{1}{2} \times 4$ could represent the number of pounds of flour you get from $\frac{1}{2}$ of a bag when each $1$ bag contains $4$ pounds.

The CCSS adopted this definition (rather than a "repeated addition" perspective) exactly to solve this issue: it provides a unified mental model which applies equally well across all number system representations. We can equally well make sense of $\frac{5}{4} \times 1.7$: If each bag of flour has 1.7 pounds of flour, how many pounds of flour would be in $\frac{5}{4}$ of a bag?

I will make two more observations:

(1) The "repeated addition" perspective is really limited to integers. The "proof" would be something like $A \times B = A \times (1 + 1 + 1 + ... + 1) = A +A+A + ... + A$. In other words, multiplication by an integer $B$ is related to repeated addition because of the distributive property.

(2) The CCSS definition is very much related to unit conversion and measurement. One could say that $A\times B$ represents the following situation: you measure an object using $\textrm{unit}_1$ find that the object is $A \textrm{ unit}_1$ . When you measure $\textrm{unit}_1$ using $\textrm{unit}_2$ you find that each $1$ $\textrm{unit}_1$ is $B \textrm{ unit}_2$. Then $A \times B$ is defined to be the number of $\textrm{unit}_2$ you would obtain when measuring the same object.

Answer 4

Putting this all together, I've been able to explain, for example, that 3×4 would be (3+3+3+3) or (4+4+4), i.e. "3 added to itself 4 times" or "4 added to itself 3 times."

No, 3 added to itself 4 times is something like: 3+3, 3+3, 3+3, 3+3. To describe 4 times 3, you should say: four threes all added up. Then there is no problem explaining half times 4, since it is just half a four (all added up).

Answer 5

I think it is important to consider your audience. Are these kids going to immediately jump to asking about "adding 4 to itself $\frac{1}{2}$ times?" I'd guess not. I think this is a perfectly fine definition of multiplication for positive integers. Let them discover extensions to larger sets of numbers once they have mastered this.

Answer 6

You can try going back to the way you may have first taught them the concept of addition: combining collections of things -- "if you have 3 apples, and I give you 4 apples, how many apples do you have?". So 4 means a collection like

X X X X

When you multiply, you duplicate it that many times. So 4x3 looks like:

X X X X
X X X X
X X X X

Fractions are similar, but you instead of duplicating the whole collection, you stop short. 1/2 means you stop half-way through the collection.

X X

This allows them to understand mixed fractions like 3 1/2. This has 3 complete repetitions, followed by a partial repetition.

X X X X
X X X X
X X X X
X X

Answer 7

Go back to your egg-carton analogy. Then, move it to the area of a rectangle as number of unit squares Cutting the rectangle in half, either horizontally or vertically, produces the same answer.

Answer 8

I think you have to teach progressively. Even with bright kids, advanced kids. Even when not really establishing mastery, but giving exposure. So, let the kids have more exposure to fractions themselves, writing, rationalizing, adding subtracting, etc. before worrying about this conundrum.

So instead of concerning this conundrum and how to contort yourself to push it with a kid that is very young, I would concern yourself more with getting other parts of their number sense mastered. And mastered does NOT mean "heard the rule", it means going for walks and asking them little (easy) questions like 5+8, 13+ 7. (Or whatever level they are on...I ask an 11 year old relative of mine, first degree in x questions as we walk/scooter. Kids enjoy hanging with you and showing off a little with stuff of a little difficulty, but still within their grasp.)

Also, it is very natural to think of half times four as 1/2 * 4. Once they are more familiar with the "easy way" (fraction first), you can introduce the reverse. Hopefully after they've also had experience with 3 * 2 = 2 * 3 from extensive single digit quizzes, learning the times table, and even hearing the rule, etc.

I would also do some writing of the numbers. Both verbal drill and written drill are helpful for learning things. Multiple modes...

Answer 9

Half times something is the same as half of it. When students are used to that, and also used to the distributive law, then you can put them together like $2\frac{1}{2}\times a$ is $a + a + \frac{1}{2} \times a$.