J.R.
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 Feb22 comment Visually stunning math concepts which are easy to explain @Peter - I'll admit that I'm using the word derive a bit loosely for a mathematics forum (it's a soft answer to a soft question), but it's too bad you see this as "dumbing down" and "a waste of time." Most 7th-grade teachers charged with teaching volume would simply write the formula on the board, and it would be well-forgotten by the end of summer. Her technique might have been weak insofar as mathematical rigor goes, but the pedagogy was very strong. I assure you, this woman was not one to "dumb down" anything; I remember re-learning concepts in 11th-grade that she taught us in Jr High. Feb15 awarded Great Answer Dec19 awarded Constituent Dec15 awarded Caucus Oct19 comment Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? RE: "I think there is only one infinity, we can think it as the biggest number..." You might want to get a better grasp on what infinity means; there is no "biggest number." Incidentally, a better way to phrase your title would be to refer to the string of 0's as infinitely many zeroes, not an infinite number of zeroes. Sep24 awarded Autobiographer Aug28 comment Visually stunning math concepts which are easy to explain @DanielM - You're right, the experiment isn't a proof. Then again, neither is the formula. I just wanted to draw attention to how my math teacher's technique left a lasting impression that the formula by itself could never manage to do. Apr22 revised If there are obvious things, why should we prove them? deleted 1 character in body Apr20 answered If there are obvious things, why should we prove them? Apr20 comment If there are obvious things, why should we prove them? Not my downvote, either, but: Immoral? Sacred? For someone professing such strict adherence to rigor, those words are a bit over-the-top. Apr20 comment Are there 3 trig functions or are there 6 trig functions? The bottom line answer always seems to be "it depends." Great link, btw. Apr13 comment Visually stunning math concepts which are easy to explain @Travis - That's why I gave up and constructed my own image. (Hopefully one that illustrates my point a little better.) I left the other image in my answer so as not to render all these comments obsolete. Apr13 awarded Editor Apr13 revised Visually stunning math concepts which are easy to explain added 411 characters in body Apr13 comment Visually stunning math concepts which are easy to explain @Travis - Yes, a few folks have made that observation. Perhaps I didn't choose the best examples. I'll stand by my point, though: diagrams showing little more than a polygon, some labels, and an equation often lead a student toward a plug-and-chug mindset that isn't as instructive as it could be. I still think first diagram has plenty of room for improvement; it could be drawn in a way that would do a better job of nudging a student toward your line of reasoning. Apr7 awarded Good Answer Apr7 comment Visually stunning math concepts which are easy to explain @LaC - I'd think the first picture would be fine, too, if it had a rectangle drawn around it, so that it's plain to see that the four triangles that form the rhombus cover half the rectangle. My beef with the first drawing is that most middle schoolers would not think to do that; I'm afraid they'd simply follow the formula without any thought about why it works. Apr7 awarded Nice Answer Apr7 comment Visually stunning math concepts which are easy to explain @WillOrrick - You make a good point. Yet I'm still disappointed in the first image because it doesn't really nudge the learner toward thinking about the area of a rhombus as the area of two adjacent triangles with related dimensions. It just shows the formula; a two-color rhombus might have worked better. (As for the second image, I agree, it doesn't look as square as it should.) Apr7 comment Can a coin with an unknown bias be treated as fair? Even if the SD card comes up as "heads" 2/3rds of the time, you only have a 50-50 chance of picking the favorable side, assuming you have no data to nudge you toward the correct choice. I think this answer helps answer the O.P.'s question: Does not knowing contribute to "fairness"?