wish i actually knew any history of anything, this kind of maths stuff can be done by a program in seconds but history takes humanness…
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I suck at maths, so it’s okay. But history, well you can always read about that and learn something new! I can always recommend something in a topic of your interest… everything has a history, after all (though know more books in some fields better than others)! 
This is the right way to do it
I couldn’t find the post you’re quoting, so …
Trig in Grade 10 is a complete prerequisite for pre-calc in Grade 11. Miss that and you’re going to delay freshman calc (calc 1, calc 2) and those are in turn prerequisites for the entire physics curriculum and a good bit of electrical engineering, mechanical engineering, aeronautical engineering, and I forget what else. Skip trig at (latest) Grade 11 and you’ve just tacked on a year to the bachelors’ degree program for a lot of people. Which is, at best, expensive.
Should we be teaching stats in the basic HS math program? Totally. However, trig no later than Grade 11 is a gate that keeps far too many bright people out of the whole STEM option. I’m far from “STEM or GTFO” but there’s plenty of research showing that the rather unforgiving maths track in middle school on is a significant contributor to female attrition in STEM by those who would love to go that way (check Society of Women Engineers, a really great organization.)
I reiterate my observation above that the quality of instruction makes a huge difference.
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I’d add one more, because I wind up using it so often to solve trigonometric equations or integrate trigonometric expressions: If you make the substitution u = tan(x/2), then sin x = 2u/(1+u²), cos x = (1-u²)/(1+u²), dx = 2 du/(1 + u²). That reduces expressions with trig functions in x to rational functions of u.
You’ve nailed it, both in content and delivery.
Long ago I taught a bunch of remedial students math for a semester. These were kids who had been labelled as “bad at math”. After a couple of months, these students were thriving. The key was creating a class culture where math was seen as a game, a puzzle. We fixed the individual gaps in knowledge that prevented students from understanding the rules of the game. And we ensured that the students had ongoing successes that built confidence.
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They are lucky they ran into you! If they’re discouraged (or neglected) early and often enough with respect to math skills, it generally sticks. They attach the idea that “I’m not good at math”/“I don’t like math”/“math doesn’t like me”, and they carry it through with them their entire lives. Often, later in life, they’ll say it with an almost sincere pride. Somehow, this doesn’t carry the equivalent shame that “I can’t read” carries.
Other option for a mnemonic so you don’t have to re-derive, if you need more speed:
Write the integers 0 to 4. Take the square root of each. Divide each by 2. Read left to right, it is sin 0, 30, 45, 60, 90. For cosine, read right to left
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That works but (and please don’t take offence), I would never recommend a technique like this to a student. While the better students will understand that this is simply a memorization aid, the weaker students will latch onto it in lieu of actual understanding. This leads them down the route of “math as a recipe”. If I was teaching basic trig, and if I was forced to test student understanding, I’d instead get them to draw a 1-1-c right-angled triangle, and a 2-2-2 equilateral triangle with a vertical line down the middle. They’d complete the unknown lengths and internal angles, we’d confirm that everyone had it right, and then they’d proceed onto the second part of the test. Calculating sin, cos and tan of 0,30,45,60,90 is something anyone can do in their head. Even now, twenty years since I was last in school, I can conjure up those triangles in my head and fill them out. In real life – if the calculation actually matters – speed is never going to be an issue. If you have any doubt, you’ll fall back on a calculator … but that fundamental understanding will always be there.
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I disagree. Math is beautiful but to get to the point where you can properly appreciate this beauty you need to slog for a while. You need to learn to manipulate symbolic systems so well that manipulating symbolic systems is something you do without conscious thought just like, say, you write without much thought about the mechanics of pressing keys or moving the pencil. You think at a higher level and all the crude mechanical stuff just happens.
Well, that’s required for math too.
Is there room to enhance this heavy lifting with more insight? Absolutely. On one hand we can take the brilliant approach of channels like 3Blue1Brown to inspire, and on the other we can try honesty. I generally tell my students when something’s a slog. I tell them that if they are bored silly of it after five minutes, that merely shows they still have a pulse and it’s nothing to be ashamed of. Push through it and develop both dexterity of thought and callouses that let you tolerate slogs and the sunny uplands of truly getting whatever we are learning will be open to you.
It seems to work reasonably well.
(Oh, and the problem itself took maybe… a minute or so?)
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None taken, you’re absolutely right.
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One ready answer to the people who say, “math is hard for me, I must not be any good at it”: Of course it’s hard for you. It’s hard for everyone. The people who are good at it worked hard at getting good at it.
A lot of kids never even considered that! It encourages some of them to persevere.
With practice you get good at it and this stuff isn’t so hard. But by then you’ve moved on to harder problems, because in the end everything that’s worth doing is hard.
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This is an important point not to be missed. We often hear the work “knack” used to explain someones mathematical or mechanical knowledge, as if it is somehow a gift from the gods, and some people just don’t have that "knack’. This is quite insulting to those who put the effort in to gain that knowledge.
I have never never heard anyone say of a top notch cardiac surgeon, :“He or she has a knack for hearts”
When Arthur Fonzerelli hit the jukebox just right he knew there is a 4 pole latching relay mounted on a 45 degree angle just below the bezel and if he can just momentarily break the contact points the machine will register a free play.If he presses other buttons to light up the front display before doing this, that may drop the output voltage of the transformer thereby reducing the holding force of the relay somewhat.
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Hmm, I use the word, but I think with a slightly different connotation: “I can do MIG, but I’ve never done enough stick welding to get the knack of it.” It’s not that I’m not favoured of the gods, I just never found it worth my while to put in the work to acquire that particular skill.
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