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Symmetry is an important concept in mathematics and it has a built-in aesthetic appeal. The following shows how different types of symmetry relate to one another. Start with the equation $f(x) = -x^2 + bx + c$. Without having to complete the square, we can find the axis of symmetry. Rewrite the equation as $f(x) = x(b - x) + c$. Now find $f(b - x) = (b-x)x + c = f(x)$.

Since $f(b-x) = f(x)$, we know that for every point $(x,y)$ on the curve, there is a point $(b - x,y)$ on the curve. The midpoint of these two points on the curve is $(b/2, y)$. The line $x=b/2$ is an axis of symmetry. We can generalize this by noticing that what made this work is that the equation for $f(x) = x(b-x) + c$ is symmetric in $x$ and $b -x$. We can get similar results for $f(x) = x^2 + bx + c$. $f(x) = -x(-b - x) + c$. Reasoning as before, it follows that $x = -b/2$ is an axis of symmetry.

Consider the function $f(x) = \arctan(x) + \arctan(6-x)$, which is symmetric in $x$ and $(6 - x)$. We can say immediately that the line $x=3$ is an axis of symmetry. One of the reasons that this works is that the function $g(x) = b - x$ is self-inverse, $g(g(x))=x$. If we take any function $f(x)$ that is symmetric in $x$ and a self-inverse function $g(x)$, we can say that $f(x) = f(g(x))$, although this will not usually result in an axis of symmetry.

Finally, there is a way of talking about self-inverse functions in terms of symmetry. The equation $y = b - x$ does not at first appear to have any symmetry, but if we write it as $x + y = b$, it is clear that it is symmetric in $x$ and $y$. The function for getting $y$ in terms of $x$ is the same as the function for getting $x$ in terms of $y$, causing $g(g(x)) = x$. For every point $(x, g(x))$ on a self-inverse curve there is a point $(g(x), x)$ on the curve. The midpoint is $((x+g(x))/2, (x+g(x))/2)$, which is on the line $y=x$. The line $y=x$ is an axis of symmetry for all self-inverse curves.

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    $\begingroup$ What's the question here? I suppose you have it in the title of the post, but it is worth reiterating what exactly you're asking, along with some important information: Who are the students you're teaching to? What is the context? $\endgroup$ – Brendan W. Sullivan Oct 11 '20 at 18:10
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    $\begingroup$ It sounds like you are enjoying thinking about symmetry this way. That's cool. I don't know if this would help someone else understand it or not. $\endgroup$ – Sue VanHattum Oct 11 '20 at 18:55
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    $\begingroup$ To me this seems more of a way of summarizing symmetry notions after a student has worked with them a lot in specific cases, spread out over at least several days (and don't toss every symmetry notion at them on the same day), than how you would introduce the topic. However, for that particular purpose this might work well, but you'll probably want to make it more into a guided instruction summary or a student-guided-discovery summary worksheet, than simply throwing all these assertions and connections at them like this. $\endgroup$ – Dave L Renfro Oct 11 '20 at 18:55
  • $\begingroup$ This was not intended as a single lesson. I was thinking that the material would be suitable for a ninth grade math class. I was hoping that looking at the equation for a parabola in a fresh way would catch the students' attention and act as a motivator. Surely at least some of the students would be impressed with how the plot of y=arctan(x)+arctan(6-x) is so clearly split in half by the line x=3, $\endgroup$ – user1153980 Oct 11 '20 at 22:08
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    $\begingroup$ arctan in 9th grade? $\endgroup$ – JTP - Apologise to Monica Oct 12 '20 at 12:22
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When you're designing instruction, the lesson is the last thing you create. You start with finding the standard that students need to meet. How will you assess whether students have met that standard? What do students need to know in order to succeed at that assessment? What do your students know coming in to this lesson? Only when you have answered all of those questions can you design a lesson that is custom-made for your students and their need to meet the specific standard.

I think that most teachers have discovered first-hand that there are consequences for skipping steps along that path. Probably fewer teachers have learned from those mistakes, because it's easy to blame students for not comprehending a lesson that was hard to understand and had no purpose. For myself, I remember teaching a similar lesson the first month I taught Calculus when I thought I had prepared my students to understand difference quotients. Lo and behold, I asked them what $f(x+h)$ was and they pretty much replied $fx+fh$ because they had never seen an expression inside function notation but they did know the distributive property.

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    $\begingroup$ I agree with you that the lesson should be the end point, but I do not see anything wrong with working backwards, by asking what the students need to know prior to the lesson. In the case of the set of lessons I described, the main requirement would be familiarity with the concept of what a function is and how to do simple manipulations with them. $\endgroup$ – user1153980 Oct 12 '20 at 5:46
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    $\begingroup$ Interesting, this is absolutely not how I teach. To my mind, you create the big story of the course, find how many lectures are needed, how to group concepts which fit the timeline of the term, and then, at the end, I hang assignments over that with the general aim of prompting the students to understand as much of the story which is told as possible. To my mind, Assignments are last, lessons are first. But, to each his own. $\endgroup$ – James S. Cook Oct 16 '20 at 0:28
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To your actual question (would this be a good way to teach about...) my serious but sounding mean answer is...no.

This is because when I read your explication, my eyes glassed over with all the detail, with the arguments inside sentences, etc. Furthermore, I also think you need to start at the simplest level first. y =x and y= -x maybe. Not with a quadratic. The second para is rather complex also.

In summary, I think you are making the classic mistake of thinking something that appeals to you (a sophisticate) is the right way to instruct a novice. Bad pedagogy.

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