8(c2-4d2) / 12(c2+4d2-4cd) = ...Did I get it right? ;-) Now, if he's another Lagrange, it's not quite obvious because his talent must be profoundly hidden, so he would have "F" all the time. This grade has been recently brought to the "C/D" range but that's probably it. ;-)
... = 2(c+2d) / 3(c-2d)
You know, such lessons are not about the simplification of rational expressions only. We had to deal with many gaps that are more elementary - the difference between addition and multiplication (especially when numbers are substituted for variables), priorities of mathematical operations and the question which parentheses may be forgotten, the difference between 2(c) and 2(c+1) when "+1" is left after pulling a factor out, the difference between "u" and "1/u" when "nothing" is left in the numerator, the difference between "mn^2" and "(mn)^2", the reason why "1/5" is equal to "0.2", and so on, and so on.
Off-topic: Sonic boom visualized. See also genus one bubbles and other slow motion videos including popcorn pop. Hat tip: Rae Ann
I am confident that many of these idiosyncrasies are rather generic among children and teenagers and they should be taken into account when these things are being taught. More generally, I am convinced that the existing process of teaching doesn't sufficiently reflect the "natural" expectations of children (and adults).
What do I mean? Most people build and increase their knowledge using some kind of generalized wavelets. You begin with a rather simple rule that may be adopted in a wide range of contexts. For example, when you begin to learn about the origin of species, you may start with the assumption that "all animals were created by God". Or if you are equally "sensible" as the creationists and learn something about physics, you may begin with "all of modern theoretical physics is not even wrong".
These are not the best examples. I should have started with some insights that are at least remotely correct, like "weather and life are changing all the time". Fine. Such insights resemble a big wavelet that covers a big portion of the JPEG picture. Once you know these things, you know more than the people who don't know "anything at all" - for example the people who believe that the climate shouldn't be naturally changing or that the animal species were created to be constant forever.
However, this knowledge is clearly not terribly refined and detailed. So we keep on adding smaller wavelets. We are learning which things are changing and which things are not changing, how quickly they're changing, what they're actually changing into, and so on. In each case, a new "blob" or "wavelet" is added into the JPEG picture (or MPEG movie) of our knowledge. It's important to notice that we rarely remember things in the same way as the BMP bitmap image - even though, in some cases, it should be acknowledged that pure memorization is the best approach.
On one hand, when a pupil constructs a "blob" or "wavelet" in her head that allows her to solve certain problems correctly without extra learning, she shouldn't spend too much time with learning the "new" stuff that is not quite "new". On the other hand, when previous, nearby "blobs" and "wavelets" are likely to influence her thinking and lead her to erroneous conclusions, some lessons should focus on these particular "frequent mistakes". It must be explained why the previous "blob" no longer works in the special context and why (and how) the new rules replace it.
The teaching should be organized in such a way that the (generalized) students are able to reproduce as much correct information as possible after a given amount of time and effort that has to be invested to the teaching process. In this sense, the architects of textbooks face a similar challenge as JPEG compressing algorithms.
Also, various "big" or "ideological" disputes are actually about the size of the wavelets - i.e. about the area covered by one "wavelet" or another one: such ideological "wavelets" describe the reality in a simplified (or oversimplified) way. Now, these "ideological" questions are partially about social conventions rather than objective reality: the exact image is more than the oversimplified "wavelet". Nevertheless, when the real picture includes a big and dark spot, the simplified picture with a "big wavelet" could be better than others: so these social conventions about the way how to present simplified (and oversimplified) rules are at least partially rooted in reality.
It brings me to another point. The facts presented in the classroom are almost always phrased in a "positive" way: children always hear what is true but they never hear what is not true. This fact makes most classes in the world both boring as well as inefficient as methods to transfer the information. Moreover, children often do not realize that two things contradict each other when they do.
For example, when it comes to creationism, children hear almost nothing about it at schools. Now, I am surely not going to defend a "positive" teaching of creationism because it's all crap. However, I do agree with the statements that the kids should be "taught the controversy". What do I mean?
While there is no controversy among sensible evolutionary biologists, there is surely a controversy in the society. It is unrealistic to think that every student will end up "believing" the evolutionary story about the origin of species. And in fact, there is even no good reason to dream about this uniformity. What would it be good for?
So children should naturally hear about these thought-provoking topics. Those who are getting it are likely to understand the structure of the ideas and arguments more properly and they will be more able to explain it to others, too. The other children who are not getting it will realize that there actually exist some arguments that the Darwinists honestly believe. And the "evolutionary" children will hear some "better" arguments from the creationists rather than the hateful and dishonest caricatures.
But the current situation in which the kids are taught complicated trees of hundreds of kingdoms, superclasses, classes, and species of animals and plants - and many (or most) of them end up believing that all this hierarchy is unphysical bogus unrelated to the origin of the life forms - shows that the current focus is unbalanced. Does it make sense to teach stunning details about evolutionary biology if you can't guarantee that most children will believe the very basic pillars of this very field, evolutionary biology?
Similar comments apply to other disciplines, too. For example, students learn how to do a lot of things correctly in quantum mechanics or quantum field theory. But they are almost never led to understand why wrong methods and opinions are wrong (not even the highly popular wrong methods and opinions). In principle, learning the correct answer is equivalent to learning all the wrong answers. However, in reality, it doesn't work in this way.
For example, if a graduate student is unable to see that and why virtually every paper by Lee Smolin about quantum mechanics, quantum field theory, or quantum gravity is wrong and scientifically ludicrous - i.e. why physicists refer to Lee Smolin as a crackpot - he or she has not learned quantum field theory and related subdisciplines well. If someone really knows the correct answer and why it is correct, he should also know why all the incompatible answers are incorrect. Much like in the case of biology discussed above, students are being taught much more detailed - and generally less important - things than the insights they need to have in order to see why a wrong paper about some "big questions" is wrong. In other words, students are often taught things that are less important than those that they should learn.
Although I have mentioned biology and particle physics only, the same comment applies to all of human knowledge.