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.
Answer
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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