Timeline for Is This Trick Helpful?
Current License: CC BY-SA 4.0
28 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2023 at 7:49 | comment | added | Brahadeesh | For some reason, the image in Bill Dubuque's comment above does not display for me on my mobile device. Just in case there are others with this problem, here is the link to the image on Imgur: i.sstatic.net/uJphi.png | |
| Sep 10, 2018 at 23:20 | comment | added | Steven Gubkin | @Number To answer your question, I still do not understand what the phrase "looking from the left" is trying to get at. Your observation about the symmetries of the square is a nice one for sure, and could certainly be turned into a memorable and enlightening experience for students at this level. | |
| Sep 10, 2018 at 21:14 | comment | added | Mr Pie | @Number Regarding your first comment, the explanation was unnecessary; I understood what the image described, but that's okay. Regarding your second comment, thank you for telling me of the transformation names (I just called 'em "fractions", but not anymore)! And finally, regarding your third comment, thank you for the link, I will check it out! Edit: Wow! That is very in-depth, particularly this other link! Thank you!! :D | |
| Sep 10, 2018 at 14:30 | comment | added | Bill Dubuque | There are many (web) expositions on these Symmetries of the Square | |
| Sep 10, 2018 at 14:30 | comment | added | Bill Dubuque | It is clear that the rigid rotation/reflection transfomations on the wheel don't alter the diagonal connections (spokes), so they preserve the two cross-products (up to order) so the new fractions are also equal. The method in this answer is equivalent, but it instead fixes the square wheel and rotates/reflects the glass pane (= fraction viewpoint, or frame of reference). They are inverse transformations since their product (composition) is the identity (e.g. fix the pane to the wheel and rotate the entire contraption). | |
| Sep 10, 2018 at 14:30 | comment | added | Bill Dubuque | @user477343 A simple real world (manipulative) model is a (Flintstone!) square wheel, with 4 spokes representing the diagonally-related numbers in the cross products). The numbers hang on the rim corners at the ends of the spokes. Placed in front of the wheel is a glass pane which displays the fraction bars (vinculums) and the equal sign (in front of the wheel hub), which tells us how to read the numbers as fractions. | |
| Sep 10, 2018 at 2:33 | comment | added | Mr Pie | @Number Well, I know both those puzzles you mentioned as examples. Thank you very much!!!!! You have been of great help, and is very noble of you to not use any of your comments as an official answer. Looking at your profile, how you got your Math.SE account suspended is beyond me. Tell 'em mods to put it back :D | |
| Sep 10, 2018 at 2:32 | comment | added | Bill Dubuque | @user477343 It's just an image. Alas, I don't have enough spare time at the moment to compose an answer that does the matter justice. Hopefully the hints in the comments help lead you along the correct path. Maybe you can employ these and other beautiful simple connections between arithmetic and geometry to motivate your brother, e.g. since you enjoy puzzles you could show him some very simple applications of group theory to puzzles, e.g. the fifteen puzzle or Rubik's cube (google them with "group theory"). | |
| Sep 10, 2018 at 2:08 | comment | added | Mr Pie | @Number what the — that is all from $\LaTeX$ ??? Where's the answer?! That is a BEAUTIFUL explanation! $(+++++++\,1)$ $\ast$takes screenshot$\ast$ | |
| Sep 10, 2018 at 1:56 | comment | added | Bill Dubuque | @user477343 The basic idea is quite simple. If we view the equal fractions as 4 numbers in a square, then fraction equality is equivalent to equality of the diagonal (cross) products. But the diagonals don't change under symmetries of the square (rotations & reflections - see below). They may swap the order of terms in products, or swap sides on the equality, but this doesn't alter the truth of the equality of diagonal products. $$\style{ display: inline-block; background: url(//i.sstatic.net/uJphi.png?s=515&g=1) no-repeat center;}{\phantom{\Rule{515px}{30px}{327px}}} $$ | |
| Sep 9, 2018 at 23:22 | comment | added | Mr Pie | @Number yes, it is clearer to me, thanks to your help, especially. The maths that I know thus far has mostly been taught and discovered by myself (e.g. in Grade 6, I taught myself how to find $x$ in "typical" algebraic equations such as $2x+1=5$), but I have had some friends who helped me in my high-school years, so I agree with you on how they might be more discovered than taught. Once again, though, thank you for your help. You can combine your comments in one answer, if you like :D | |
| Sep 9, 2018 at 22:59 | comment | added | Bill Dubuque | @StevenGubkin Do you really not understand how the new equality is derived by "looking from the left", or do you mean to imply that it might not be clear to some readers so could use further elaboration? Also, the author never claimed that the proofs of these transformations should be omitted. It is truly a shame that this answer have been so heavily downvoted (now -4+3) when in fact it is the only answer that sheds any light on the essence of the matter. | |
| Sep 9, 2018 at 22:52 | comment | added | Bill Dubuque | @user477343 I asked because you wrote in your first comment above that you didn't understand the proposition briefly sketched in this answer. Is it clearer to you now, including the symmetry-based approach that I alluded too? These are not topics typically taught in grade 9 or earlier (though they certainly are capable of comprehension at that level if presented by a talented teacher). Indeed, they are probably independently discovered much more frequently than they are taught. | |
| Sep 9, 2018 at 21:57 | comment | added | Mr Pie | @Number Yes, I am sure. I am doing Maths Methods currently in my Year 9 class, so I think my knowledge suffices. And by the way, that is some good experience you got there! Well done :P | |
| Sep 9, 2018 at 13:55 | comment | added | Bill Dubuque | @user477343 To teach something well you must know it well. Are you sure that you do? I've taught algebra for almost 4 decades, to students at all levels, from elementary to research level (e.g. assisting Artin, Gosper, Knuth, Wolfram with Macsyma). I know from this extensive experience that anyone proficient at fraction arithmetic knows by heart this basic transformation (not "trick") - just as they know well other basic algebra (e.g. Binomial Theorem, Difference of Squares factorization, quadratic formula). We learn them early then commit them to memory for rapid subconscious application. | |
| Sep 8, 2018 at 23:55 | comment | added | Mr Pie | @inéquation I wrote $a=b\div c$ instead of $a=\dfrac bc$ to make things look neater, which might have served confusion, but either way, you must be right then; though if your trick was easier, my brother would never have come up to me and asked for a trick. He is learning about what you wrote and finding it difficult; he might just need more time to comprehend it. | |
| Sep 8, 2018 at 16:55 | comment | added | user5402 |
@user477343 I'm sure my trick to read an equation from right or from left is easier than the sundown swaps around and "sunrise" since I, as an teacher since 13 years, had to read it more than once to understand what you're saying.
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| Sep 8, 2018 at 16:44 | comment | added | Mr Pie | @Number if you read my comment above, I do not think my brother will notice this symmetry you speak of in inéquation's answer. I have posted this for the sake of my brother, not myself. | |
| Sep 8, 2018 at 16:33 | comment | added | Mr Pie | @inéquation well, to be honest, this answer deemed rather partial. I asked for feedback, and you provided a different equation to consider. I know how it works, it is just that my brother does not and cannot stand maths. He hates it, and in his own words, "it is for nerds like me." All he wants to do is pass with something above a $C$ to impress my mum. This is definitely not the right attitude he should have toward his learning, but I do not want him to give up and fail, either. StevenGubkin's answer might just convince my brother to think otherwise, but I am unsure about yours :\ | |
| Sep 8, 2018 at 16:18 | comment | added | user5402 | @StevenGubkin You're missing the point. The OP ask about a trick, I answered with another (hopefully simpler) trick that I use. Of course one should teach the fundamentals first but that's not what the question is about nor it is how students usually solve equations. | |
| Sep 8, 2018 at 16:12 | comment | added | Bill Dubuque | @StevenGubkin The innate symmetry that governs the transformations (which at least is hinted above - but completely obfuscated in your answer) | |
| Sep 8, 2018 at 16:11 | comment | added | Steven Gubkin | @Number I would love if you would answer this question as well, so I can see what intuition I am lacking. | |
| Sep 8, 2018 at 16:08 | comment | added | Bill Dubuque | @StevenGubkin Your answer is just as lacking - more seriously so in my opinion since it provides no intuition whatsoever. | |
| Sep 8, 2018 at 16:07 | comment | added | Steven Gubkin | @Number I also have no idea what inequation means by "from the left/right/bottom". This answer really says little more than just "do it the right way". | |
| Sep 8, 2018 at 16:04 | comment | added | Bill Dubuque | @StevenGubkin One needs both, i.e. the combination of your answer and this answer, along with some motivation (e.g. by symmetry). Your (unexplained) downvote could have left the OP with the wrong impression, e.g. wrongly inferring that the above algebra is incorrect, or it is not how experts perform such transformations, etc. It is usually better to have constructive dialog before downvoting something which is correct but needs elaboration. | |
| Sep 8, 2018 at 15:52 | comment | added | Steven Gubkin | I downvoted because I believe that students should be thinking through reasons why things are true until they reach this "mechanical reasoning" through enough practice. Training them to perform mechanically first, without the thought, is damaging. | |
| Sep 8, 2018 at 13:48 | comment | added | Mr Pie | I do not understand what "proposition" you are referring to. The only equation I am considering is $a=b\div c$ and not any other equation. I can let $a=x\div y$ and have the equation $x\div y = b\div c$ and now I can literally do the same trick from all angles, for there are now two such "horizons" and another pair of "sun positions". I believe you are saying that there are many ways to manipulate the equation, but as far as I am aware, my brother was only concerned with the equation strictly involving $a$, $b$ and $c$. If I show him your answer with $d$, I am sure it will complicate matters. | |
| Sep 8, 2018 at 13:41 | history | answered | user5402 | CC BY-SA 4.0 |
