Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$? I was on the tube and overheard a dad questioning his kids about maths.  The children were probably about 11 or 12 years old. 
After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually $∞$.
I thought the Dad's statement was a little irresponsible.  Does that seem like reasonable attitude? I suppose this question is partly about morality.
 A: The usual meaning of $a/b=c$ is that $a=b\cdot c$. Since for $b=0$ we have $0\cdot x=0$ for any $x$, there simply isn't any $c$ such that $1=0\cdot c$, unless we throw the properties of arithmetic to the garbage (i.e. adding new elements which do not respect laws like $a(x+y)=ax+ay$).
So "undefined" or "not a number" is the most correct answer possible.
However:
It is sometimes useful to break the laws of arithmetic by adding new elements such as "$\infty$" and even defining $1/0=\infty$. It is very context-dependent and assumes everyone understands what's going on. This is certainly not something to be stated to kids as some general law of Mathematics.
Also:
I believe that the common misconception of "$1/0=\infty$" comes from elementary Calculus, where the following equality holds: $\lim_{x\to 0^+}\frac{1}{x} = \infty$. This cannot be simplified to a statement like $\frac{1}{0}=\infty$ because of two problems:


*

*$\lim_{x\to 0^-}\frac{1}{x} = -\infty$, so the "direction" of the limit matters; moreover, because of this, $\lim_{x\to 0}\frac{1}{x}$ is undefined.

*By writing $\lim f(x)=\infty$ we don't really mean that something gets the value "$\infty$" - in Calculus $\infty$ is what we call "potential infinity" - it describes a property of a function (namely, that for every $N>0$ we can find $x_N$ such that $f(x_N)>N$ and $x_N$ is in some specific neighborhood).

A: This has been treated extensively both in this question and in others. But I think the clearest formulation possible is this : for me the most correct answer is "NaN", Not a Number.
It is not wrong to say that $1/0 = \infty$ from a topological point of view (be it an Alexandrov compactification, or a projective space), and it's actually very useful and natural in some contexts.
However, as pointed out by everyone, it cannot be made coherent with arithmetic laws, addition and multiplication. So it's "not a number".
That said, for pedagogical purposes I think it's best to not go into these things and just illustrate the arithmetic problems arising when defining division by 0 (and maybe say something like "one might define 1/0 in some contexts, but we will not because of these problems"). Indeed arithmetic notions are much more intuitive than abstract, topological constructions at this stage. 
A: You ask:

Is it wrong to tell children that 1/0 = NaN is incorrect, and should be $\infty$.

(My) answer: 
Absolutely. When we talk about multiplication and division with numbers, I think that it important to get things right. If people would only grow up learning that $\infty$ is not a (real) number, then I think that it would relieve a lot of confusion about how math treats infinity. And it is definitely not right to say that the statement "$1 / 0$ is not defined" is incorrect. That is what is taught in all basic calculus classes. Why would this be incorrect?
Now, if the question is whether one can ever come in a situation where it is appropriate to write $\frac{1}{0} = \infty$, then the answer is definitely yes. This is illustrated in some of the other answers to your questions. But to say that $1$ divided by $0$ is infinity and then just stop, then I would agree that this is wrong (I might even go so far as to call it irresponsible). 
When people ask me why we can't divide by $0$ and if they don't want to "leave it", then I would simply start talking about groups and rings and fields as sets with the operations. When you do that carefully you clearly see why it doesn't make sense. 
The wonderful thing about math is that we never have to answer why something isn't defined, the burden of proof is in the hands of the person who claims that something is true.
A: The teacher is right: $1{\color{red}/}0$ is undefined. (if he said 'doesn't have an answer', then he is being somewhat sloppy)
However, the father is right: $1{\color{green}/}0 = \infty$.
But also, the title is right: $1{\color{blue}/}0 = \mathrm{NaN}$.
The problem is that the topic fits into what I believe to be a significant gap in mathematics education: people aren't taught syntax and mathematical grammar at all, so they don't have the ability to make precise statements about what they mean -- or even to know that it's an issue!
(I've added color to emphasize that I mean three different things in those three statements!)
The first version of division is what is taught in elementary school; the teacher is right on that point. $1{\color{red}/}0$ is a syntax error: $(1,0)$ isn't in the domain of ${\color{red}/}$, and so it is illegal to write the expression evaluating ${\color{red}/}$ at $(1,0)$.
The second version, however, is the division of projective numbers. The projective numbers are very useful for algebraic purposes, and even for some analytic purposes: e.g. it can be convenient to have $\tan$ be projective-valued, so that $\tan(\pi/2) = \infty$. I was being a little forgiving when I said the father was right, though -- I find it more likely he was thinking about the extended real numbers (but not knowing that by name!), and simply made a common mistake.
The third version is back to ordinary division, but in a syntax/semantics based on something like partial functions or composition of relations. A rough description is that in so far as functions $\{ * \} \to \mathbb{R}$ correspond to elements of $\mathbb{R}$, the partial function $\{ * \} \to \mathbb{R}$ with empty domain corresponds to $\mathrm{NaN}$.
On this last point, note that to some extent we force students to actually think in terms of this family of concepts with notation like $1 \pm \sqrt{2}$ and $x^3/3 + C$, and questions like "What is the domain of $1/(1-x)$?". But IMO, these notions are somewhat incongruous with what students are actually taught about functions.
A: The real numbers line can be extended in multiple ways. One way is to add one element, unsigned infinity $\infty$ (this is called projective extended real numbers), another way is to add two elements, negative and positive infinities $-\infty$ and $+\infty$ (this is called affinely extended real numbers). When moving to complex numbers, the things become even more complicated, one can add complex infinity and infinities corresponding to any direction on the complex plane.
Thus there is no universally-accepted way to extend the real line and complex plane with infinities. For example if one only adds positive and negative infinities $-\infty$ and $+\infty$, the expression 1/0 still has no answer.
Claiming that something is equal to $\infty$ may be ambiguous depending on whether the real line is extended with signed or unsigned infinity, so when using $\infty$ without a sign one should specify whether unsigned infinity or positive infinity is meant.
It should be noted that in calculus "$\infty$" is usually used to designate positive infinity, so teaching the kids that this means unsigned infinity may complicate their future experience with calculus.
It should be noted that finding value for 1/0 can be viewed as solving the equation $\frac 1x=0$. In that case both positive and negative infinities fit. So we face a similar problem as with expression $\sqrt{2}$. While there are two real numbers which squared give 2, by convention only the positive one is considered the value of the expression $\sqrt{2}$. But for dividing there is no such convention.
A: Well I think people are taking your quest as a math rather than the demand you have made. I see you actually have asked two questions together:
Q#1) What is the actual answer for 1/0 to tell a 11/12 year oldie?
Q#2) Was father's attitude towards answering his daughter wrong?
So I would rather choose to answer something like this:
A#1:
Since the question is very simple so is the answer (Gadi's way???) . But the main thing involved is that how to tell such answers to your little ones who are trying to understand things which become important in their upcoming lives. I dont know what was the actual answer from the "teacher" but whatever teacher answered I consider it correct because s/he answered the kid since s/he knew that it was better to explain them that it has nothing. The girl must have skipped all explanation the the teacher would have made at the time when the question was answered but she told her dad what she understood from that.
I guess, teachers and parents should answer their students and kids by keeping the age in mind.
A#2:
I consider the "teacher taught it wrong" attitude from father was NOT good. Instead he could have explained it politely with a better way so that the daughter would not get a feeling that the teacher dont misguide but there are better ways to explains such things. And parents are the next best choice to get best answers.
Being parent, I am going through this phase. Its tough, its difficult how to teach your kids everything when there are many things having different answers on different ages.
We know Newton's laws are not 100% perfect, but they are taught. Why dont we explain theory of relativity from the beginning instead? Because they are more complex than understanding Newton's laws.
Anyway, I think instead of answering (Gadi's) mathematically (which a 11/12 age group would not understand) we should explain it in words with as many examples as one could reach a better understanding level depending on their age group so that in coming years they understand it more as they advance.
A: I'm assuming "NaN" means "not a number".  Certainly $\infty$ is not a number.  There are indeed various different sorts of infinite numbers (cardinalities, non-standard reals, surreals, ordinals, the "infinities" of generalized functions like Dirac's delta and its derivatives, . . .), but $\infty$ itself is not a number.
In some contexts $1/0=\infty$ makes sense.  In thinking about projective geometry, trigonometric functions, or rational functions, it makes more sense to have just one $\infty$ at both ends of the real line that $+\infty$ and $-\infty$ as separate entities.  But in other contexts, one should distinguish between the latter two things.
A: I'm sorry but I think there is a little misconception here:
1/0 is not infinity, never was, never will be. This would imply that $$0\cdot\infty = 1$$ which is absolutely wrong.
The correct mathematical formula is $$\lim_{x \to 0+}(1/x) = +\infty$$ 
The teacher is therefor right, and it's actually the parent that was wrong. 1/0 is undefined, and so is $$0*\infty$$ even in term of limits. 
A: First, some assumptions
Children were 11-12 years old. In the U.S. public school system, that is equivalent to last year of elementary school (6th grade) to perhaps 7th or even 8th grade, which are middle school/ junior high. Most 6th, 7th and 8th grader's are not learning first year algebra yet. Some basic number theory (primes, integers), as well as fractions, exponents, roots, probably reinforcing how to do long division. Also, very basic geometry, like polygon names and area of quadrilaterals and circles. And units of measurement, Celsius to English conversion, scientific notation.
Given that context, with no knowledge of algebra or calculus, only simple number theory, it seems reasonable that the teacher described 1/0 as "undefined". "Not a number" is equally acceptable.
Big picture
School teachers have degrees in education, with specialty training by subject matter. Instruction is not done in isolation, but as part of a coherent curriculum. If the teacher defined it in this particular way, it is likely part of a longer term mathematics instruction plan which will redefine the meaning of 1/0 in more appropriate ways when the time is right, i.e. when the accompanying content is at a sufficiently advanced level so that it will make intuitive sense.
Unintended consequences
The father needs to be very careful here, because he might undermine the teacher's credibility to his children. Worse yet, his children will potentially give incorrect answers on tests, based on the father's comments. This puts the children in an awkward situation which they should not be subject to at this point in their lives. If the father has concerns about the quality of instruction, he should take it up with the teacher directly, not in the way he did. At least, not with children who are only 11 or 12 years of age.
A: Claiming $\frac 10=\infty$ is wrong (if $\infty$ means just positive infinity, as usual). Look yourself, $\frac10$ should be equal to $-\frac10$ because $0=-0$, but $\infty\ne-\infty$.
A more correct answer to your question would be that $\frac10=\tilde{\infty}$, where $\tilde{\infty}$ is complex infinity. It can be expressed in terms of positive infinity though:
$$\frac10=\tilde{\infty}=\infty(-1)^\infty=\infty e^{i\infty}$$
This answer will be the correct one given we have extended complex numbers with $\infty$.
A: As @hardmath indicated there is indeed a floating point representation for infinity.  
added:

To address the non-math side of this question. IMHO, it was bad form for the father to contradict authority to his child and sets a bad example without setting a better attitude such as "there may be another answer". I think... it's infinity, what do you think? Here he shows his child he can be vulnerable have a different opinion, even if it technically wrong, and that can be OK if you consider the possibility that someone could be wrong and find a way to verify supposition of facts and confirm or deny them in a rational non-confrontational way. That also demonstrates tolerance and rational objectivity to determine truth by asking if she knows the reason in a non-confrontational way and learn to discuss differences of opinions without fear of criticism as overheard in this example.

" If 'a' is zero and the payload is zero, then it represents infinity."  Is the technical reason why the teacher was correct.
However for teaching purposes such as the concept of the distance of the universe, the Big Bang and the weight of an atom, it may be useful to consider teaching in "metaphors" so that approaching a zero numerator or denominator, which may become a Singularity when you approach infinity with a real number.
It is also useful to define the order of rules when the paradox of;


*

*"anything multiplied by zero is zero" which contradicts...

*"anything divided by zero is infinity"


*

*...Actually (2) really bad boo boo ;) (2.) is not true, as it is only a conditionally true if the numerator is also zero, otherwise, it is NaN.



So explain to the child that 0/0 is a special case with religion. ;) or something of 
greater than human intelligence.
Fun Fact: Thanks to John Wallis who introduced the infinity symbol to mathematical literature with his works on the collisions of heavenly bodies at university in the women's dorm. Consider how fortunate we are that we dont have to call it a Wallis instead of ∞ infinity. However John eventually got his immortality with an asteroid in 2000 and his name assigned to it.
A: Computers are so ubiquitous and important, I'm not sure that one shouldn't use the IEEE floating point notation as the "default" to teach in middle school (where I assume you'd be talking about some representation of real numbers):
scala> val x = 1.0/0.0
x: Double = Infinity

The mathematician in me cringes, but if you're teaching things in order for them to be maximally useful, I think there's a pretty good argument for this.  However, saying that
scala> val y = 1/0
java.lang.ArithmeticException: / by zero

is the wrong answer is itself wronger than giving this as an answer; as this question and associated answers demonstrate, the actual situation is more involved than any simple declaration of fact.
