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Question: how to explain the undefinitions $0^0$ and $\frac{0}{0}$ for Middle school students??

I am a math teacher and I don't know how to answer properly when studens ask me why some operations give undefined/indetermination (the most frequent are $0^0$ and $\frac{0}{0}$) or why division by zero result infinity. So most of time a avoid to answer such because I am affraid to confuse them more with too complicated explanations. Some student understand most explanations, but others have more difficulties.

To explain division by zero I try to use their intuition, making divisions by factor every time smaller, so I get a kind of limit without mentionig it (I say: "dividing by a number every time smaller, what you get is always a bigger one, tendind to a huge number, the $\pm\infty$"). But still, some continue to asking me: "I understand that dividing nothing by any number, the result is nothing for each" ($\frac{0}{n} = 0$, division of finite by zero). "Why is that, if I divide any number by no one, I should get infinity?" (these students continue to think that we should expect no change after such a division).

I tried once to explain $\frac{0}{0}$ with an simple equation like $\frac{a}{0}=b$. In this case we can use algebrism to write $a = 0\cdot b$, which means $a$ was already known ($~=0$), and we can say nothing about $b$, that is, it is undefined (in fact, I am not quite sure this is an satisfactory answer).

And what about the other indeterminations if some clever student asks me? Can someone help me out with this doubt? I hope I made myself clear.

I was searching for other similar questions but didn't find what I was looking for. Some interesting posts related are: Ways to solve indeterninations; Solving indetermination in limit; Two square roots in an indeterminate. See also Mathematics in Wikipedia.

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You should also consider participating in the public beta for Mathematics Educators for questions like these. They would appreciate your support. – J. W. Perry May 25 '14 at 4:13
$0^0$ is just $1$. The thing that's indeterminate is not $0^0$, it's $\lim_{x \to a} f(x)^{g(x)}$ where $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$. – Qiaochu Yuan May 25 '14 at 4:23
@J.W.Perry: oh nice, actually I did not know about this plataform.. i Will search also enter there and interact. I thought "Area 51" was a Q&A for enthusiasts of science fiction, and maybe "X-files" :b, but certainly I was wrong. – Claudia May 25 '14 at 4:26
It is new, and it is in public beta, so we have no direct path to migrate the question there. You may get responses here, but if you post on Mathematics Educators, we will want to delete it here assuming no one provides answers here (cross posting is not desirable). Welcome to StackExchange. If I were you I would wait for responses here before cross posting :) – J. W. Perry May 25 '14 at 4:28
@J.W.Perry Yes, so far I won't duplicate my post. I have one answer already. Ty – Claudia May 25 '14 at 4:46
up vote 1 down vote accepted

Here is what I would suggest as an informal explanation for some kinds of indeterminate forms, though it may be less helpful for others.

If you try to evaluate $0^0$ by concentrating on the exponent, you would probably say, "anything to the power $0$ is $1$, therefore the answer is $1$". On the other hand, if you concentrated on the base, you would probably say "$0$ to any power is $0$, therefore the answer is $0$". The fact that you can get contradictory answers in this way is what makes it an indeterminate form.

Similarly, for "$\frac00$", concentrating on the numerator suggests an answer of $0$ while concentrating on the denominator suggests an answer of $\infty$. In this case however, I would be very careful not to let the students believe that $\infty$ is ever a sensible answer to an arithmetic question.

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thaks for the answer... you gave me good points to think of. – Claudia May 25 '14 at 4:54

Refer to the following link from "Dr. Math" for some reasonable and pretty simple explanations that refrain from mentioning limits: answer I like is this: Consider a physical description of division in the positive integers. For instance, we can consider $10 \div 2 = 5$ as a statement that if you divide, for instance, $10$ blocks into groups of $2$ blocks, you end up with $5$ groups. Consider doing this with $10 \div 0 = ?$ many groups of zero blocks can we divide $10$ blocks into?

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yes, in fact it is a good way to see it. Thanks for the answer! – Claudia May 25 '14 at 6:28

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