Ok, so I got an answer wrong on my exam because my teacher says that the function $f(x)=\frac{(x+2)x}{x+2}=x$ but I insist that it isn't defined for x=-2. If it was then $\frac{x}{x}=1$ for all reals and so $\frac{0}{0}=1$. However this doesn't seem to do the trick with my teacher. How can I use the latter fact to prove something outrageous and convince her?
[Math] How to convince the math teacher
algebra-precalculuseducation
Related Solutions
You can prove that all the numbers are equal ;-)
Let's assume that for all $a,b,c,d \in \mathbb{R}$, $b \neq 0$, $d \neq 0$ we have
$$ \frac{a}{b} = \frac{c}{d}\quad \text{ implies }\quad a = c\ \text{ and }\ b = d. \tag{$\spadesuit$}$$
Now take any two numbers, say $p$ and $q$, and write
$$\frac{p}{p} = \frac{q}{q}.$$
Using claim $(\spadesuit)$ we have $p = q$. For the special case, where one of them equals zero (e.g. $q$), use $$\frac{2p}{2p} = \frac{p+q}{p+q}.$$
I hope this helps ;-)
My question is: Is there any disadvantages to thinking about Algebra like this? Is there anything later in my math education that will require me to know that I am subtracting or adding 2x to get rid of it on this side?
As a college algebra instructor, I'm involved with remediation efforts for hundreds of students each year who have graduated high school but can't get started with college math, mostly due to incorrect concepts picked up in their prior schooling. So I would say "yes". There are some shortcuts that teachers can take to get students to pass some specific tests or programs that they are involved in; but the incorrect concepts definitely make things more difficult for students, sometimes overwhelmingly so, later on. (A majority of students that land in college remediation programs never get college degrees.)
The first thing that I would point out is that the "apply inverse operations to both sides" idea is generalizable to any mathematical operation; this allows you to cancel additions, subtractions, multiplications, divisions, exponents, radicals... even exponential, logarithmic, and trigonometric functions. (With appropriate fine print: no division by zero, square roots to both sides creates two plus-or-minus solutions, trigonometric inverses creates infinite cyclic solutions, etc.)
In contrast, the "move over and change the sign" method is not generalizable, as it only works for addend terms. This sets students on a course that requires memorizing many apparently different rules, one for each operation, which is much harder. When solving $2x = 10$, how is the multiplier of 2 canceled out? Must we remember to move it and turn it into the reciprocal 1/2? Will the students mistakenly change the sign and multiply by -1/2? Or add or multiply by -2 (I see this a lot)? How do we remove the division in $\frac{x}{2} = 5$ (probably some other rule)? How will we remember the seemingly totally different rule to solve $x^2 = 25$?
By way of analogy, I have college students who never memorized the times tables; they did manage to get through high school by repeatedly adding on their fingers, and can get through perhaps the first part of an algebra course that way. But then we start factoring and reducing radicals: "What times what gives you 54?" I might ask; "I have no idea!" will be the answer (this happened this past week; and here's a student who has effectively no chance of passing the rest of the course).
In summary: There are shortcuts or "tricks" that can get a student through a particular exam or test, which prove to be detrimental later on, as the "trick" fails in a broader context (like in this case, with any operations other than addition or subtraction). This then sets a student on a road to memorizing hundreds of little abstract rules, instead of a few simple big ideas, and at some point that complicated ad-hoc structure comes crashing down. Be polite and don't fight with your teacher to change things; but make sure to pick up a broader perspective for yourself, and share it with other students if they're willing, because you will need it later on. Take the opportunity to think about how you could improve on teaching the material, and then you may be on the path to being a master teacher yourself someday, and helping lots of people who need it.
Best Answer
$$\begin{align}\frac00&=\frac{2\times0}{1\times0}=\frac21=2\\\\\\=\frac00&=\frac{3\times0}{1\times0}=\frac31=3\\\\\\\implies2&=3\end{align}$$ NOPE.