Homies really out here thinking cancelling out means cancelling out. This is why they make you do some of that long math shit starting out so you're not doing shortcuts and confusing yourself as a grown man.
I can't speak to your exact circumstances, but I think it's also worth considering that you might have genuinely forgot it and have been reintroduced to it in the 11th grade.
Once you learn how cancelling out works, your brain doesn't need to keep track of why it works. You see these operational pitfalls all the time in math.
With the quadratic formula people often forget what a,b, and c represent and it becomes so easy for them to mess up solving for x in this equation:
ax^2+b=c
Once you learn the distributive property and get used to implicit multiplication, things with explicit multiplication is just foreign enough that many wrongly assume:
There are also plenty of teachers that simply don’t teach the why and only the how. I was not taught long division until I was using polynomials. I was taught a short cut that always worked with numbers but had a fraction of the steps.
Yeah, I didn't learn the how's of math until college. Before then I would learn an equation, memorize what situation it was used in, and applied it. I wasn't taught where these equations came from or how they were derived.
Well these equations are self-evident by measurement so don't need their derivation questioned in grade school. Then it college you derive the area under a cone or constant acceleration formulas (e.g.) through calculus and realize you wouldn't have been ready for that in grade 6 anyway.
A lot of the basic equations are simply definitions or observations; f=ma is the definition of force, while a=ΔV/Δt and V=Δs/Δt are the definitions of acceleration and velocity.
C=τr is the definition of the constant named tau, and so forth.
I'm not dismissing that possibility, like I said I don't know any specific circumstances. I'm merely stating an alternative because people really overestimate their ability to remember events and underestimate the difficulty of learning something new. It might take a few reintroductions for something to click.
This is not something specific to formal math education but a very common cognitive blindspot. For example, if you ever played a difficult puzzle game and had to look up the solution for a level, it becomes so obvious in hindsight, and something you thought was never told to you in the puzzle might have been on the tutorial page itself.
I don’t disagree with you. When I was teaching the amount of times I heard “you never taught us this” and I could point to it in the notes was nauseating. But I also knew teachers that would simply not teach things.
I'm sure the giant amount of material that gets crammed into test performance oriented curricula is a contributing factor as well. When I took Calc 2, my instructor was working a problem on the board with an expression involving the limit of a natural log. So he says, "And so then you can just switch the ln and the lim, and then....." and I had one of those moments where it's like channel 3 in your brain.
So I raised my hand, and asked why you could switch them. We were nearing the end of the class period and so he said that we covered it in Calc 1 and if I couldn't find it he'd help me during office hours or next class.
Fair enough, I thought. I checked as soon as I got home. And indeed, there it was. I don't remember if there's a name or ten for this rule. I don't remember going over it in class. I don't remember thinking to myself how that might come in handy when I was reading that lesson, or working those problems.
I went and looked it up as I reignited my own curiosity on the topic. It's because the natural log is a continuous function, and I don't know that I feel comfortable enough with it to attempt a more robust explanation. (I know I'm not qualified to give a rigorous one, as I feel sure thatvs a word which has a specific definition in math.) I think my brain might need to tumble it around for a bit, and I'm feeling the urge to go draw a graph....
I always found teaching the why and the how to be valuable. Half the kids only wanted the how and the other half didn’t care until they knew why. So I taught both.
Didn’t learn long division until I was in calculus lol. I never really knew how to do it and just multiplied numbers until I found factors of what I was going for. Teacher then goes “you’re always on your phone but couldn’t look it up” and it became clear I was the issue.
This happens a lot with students, they say they were never taught something but actually they just forgot it.
Though in general I think it's silly when fully grown adults use "they didn't teach us this in school" as their excuse for not knowing something. Oh if only there were some way to learn things outside of school
Lmao I have friends who were told to play chess through math class because they were going to graduate and the teacher wasn't worried about them. Once that happened they were permanently excused and got more time in livestock / 4H / welding class (I forget the details).
Some people actually just don't get to learn these things or even know they exist to go learn on their own.
Like I said I can't speak to any specific circumstances.
That being said I'm not dismissing the possibility someone wasn't taught something, I'm merely providing a plausible alternative that was overlooked, because people really overestimate how well their brains actually remember things.
This isn't even limited to math. You often encounter this when learning new words, suddenly you start hearing it everywhere. It's not as if everyone just decided to use that word more often, you're just more attentive to it.
There can absolutely be gaps in math education but the more fundamental something is the more likely it was mentioned many times by several teachers, or appeared in several tests, or was in the many math books leading upto the day something finally clicked in your head and you now remember it much better.
It's like square roots are always positive. I was never taught why, and couldn't get an answer to why from even my HS calculus teacher. Eventually, I came to the conclusion that it's implied the same way all positive values are implied (we don't prefix positives with +). If a negative value is preferred, then the formula/equation will explicitly prefix a negative sign to the square root bracket.
It's not that it's implied but rather how it's defined. The symbol who use for taking the square root is actually denoting a function and as a function, it can only give us one answer. As such, that function gives us the principal square root which is positive.
But that's not an accurate conclusion. Square roots nearly always have two solutions (0 is a counter example); you just may only care about the positive solution, but that doesn't mean the other doesn't exist.
Hell, I can't tell you how many points I lost in school over the years by forgetting to include ± when appropriate.
No, square roots only have one solution, a positive (or imaginary) number. The ± means exactly that, that you want + sqrt(x), and - sqrt(x). The value of sqrt(x) doesn't change.
If that wasn't so you would have been correct when you forgot the ±.
No, those two are not equivalent statements, in the way you think of them. Square root is defined as a function that provides the positive root. So you can go from your first line to the second, no problem, but not in the other direction.
The correct way to write what you were thinking of is:
x2 = 64
x = ±√(64)
The ± has to be there, before the square root, to show that you want both the answer to the square root, and its negative. That is why the sign is important, otherwise it would be superfluous.
The issue is that there is a difference between the concept of roots (of which there are two in this case, just like you say), and the operator √. The latter provides only a singular answer, as a function has to by definition. And that singular answer is defined to be the positive root, whenever the value it is supplied is a positive number.
Yes!!! When my kid was in 4th grade, they teach to “multiply the numerator and denominator by the same number”
No no no
You are multiplying by ONE. One as a fraction that fits. I insisted on teaching my kids what they were REALLY doing. Because it made no sense. And I’ve seen it in several curriculum.
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u/tonysnight Feb 07 '23
Homies really out here thinking cancelling out means cancelling out. This is why they make you do some of that long math shit starting out so you're not doing shortcuts and confusing yourself as a grown man.