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I am not sure whether to give this as a comment or an answer, but sometimes symmetry can give us clever simple solutions to things. Understanding symmetry together with angles, for example, gives this nifty technique for building a really cool-looking cabinet. (Video: a method using complementary/supplementary angles to get a cabinet that fits together ...

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I recently used symmetry to work out the coordinates of the vertices of an archimedian-ish solid. This significantly reduced the number of variables I needed when setting up and solving the equations. A simpler example might be using symmetry to work out the coordinates of vertices of a hexagon or octogon. Once you've determined the x and y coordinates of a ...

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Some very simple examples: The commutative property of multiplication is a symmetry. When you learn your times table in grade school, this symmetry cuts the amount of memorization roughly in half. In STEM, students learn lots of equations that have squares in them. Almost always, there is a nice concept that can be harvested by observing that the square gets ...

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It's a key aspect of chemical structure determination. I once saw someone prove that a posited isomer was the same molecule (there was a subtle C2 axis), to the flummoxing of a seminar speaker. For similar reasons it is a critical aspect of NMR and IR spectrum analysis (affects peak numbers). Cotton's Group Theory Chemical Applications is the classic ...

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Answer 1: Ease of use. Some things are easier to use if they are symmetrical. Eg a box lid, bricks, shape sorter toy, device connections such as HDMI. (Think of the frustration using a USB device because of the standards limited symmetry). Answer 2: Safety of use. Some things are safer to use if they have limited or no symmetrical. Eg 3 pin plug, SIM cards, ...

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I say that the story of quadratics -- their algebra and their graphing -- is a beautiful story of the of power of symmetry. If I have a rectangle of area 36, you know little about that rectangle: it was area 36. But if I add I have a (maximally) symmetrical rectangle of area 36, then you know everything about it: it must be a 6-by-6 square. The mention of ...

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Note: as in the originally referenced article, I'll take $a=1$ to keep it less complicated. I think it's important to first establish two key issues about the graphical setting: What does it mean to "be a square" in a graph. What information are we assuming we can "read" from the graph. For (1), in the geometric setting this notion is ...

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I really wish people would stop teaching the Pythagorean Theorem as $a^2 + b^2 = c^2$, for the following reason: Give your students the diagram below, and ask them to solve for $c$. At least 1/3 of a typical high school class will write $a^2 + b^2 = c^2$ and report back to you that $c = 5$. The problem is that the equation $a^2 + b^2 = c^2$ is so ...

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Whatever you choose, make sure to follow some basic rules: Clearly state the preconditions and make sure they are understood (right triangle in your case). Clearly state the meaning of the symbols (e.g. which symbols stand for the sides adjacent to the right angle, and which for the third one). Use the symbols consistently (don't make e.g. the same symbol &...

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Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a picture instead For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...

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