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Discussion: LeveragingExplorations
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%3 N_20200203143136a_CDW    By CDW 2020/02/03 @ 14:31:36a -------------------------------- Recently there was a tweet about a different and, to some, new method for bisecting an angle with straight-edge and compass.        (select only this node)     N_20200203143136b_CDW    By CDW 2020/02/03 @ 14:31:36b -------------------------------- It's a lovely method and deserves to be more widely know.        (select only this node)     N_20200203143136a_CDW->N_20200203143136b_CDW N_20200203143625a_CDW    By CDW 2020/02/03 @ 14:36:25a -------------------------------- Original discussion: https://www.solipsys.co.uk/Chitter/BisectingAnAngle.svg        (select only this node)     N_20200203143136a_CDW->N_20200203143625a_CDW N_20200203143136c_CDW    By CDW 2020/02/03 @ 14:31:36c -------------------------------- Some said that their students make fewer errors when using it, and that's always a welcome outcome.        (select only this node)     N_20200203143136b_CDW->N_20200203143136c_CDW N_20200203143136d_CDW    By CDW 2020/02/03 @ 14:31:36d -------------------------------- I mentioned that it doesn't work well *as given* when the angle is large, close to 180 degrees.        (select only this node)     N_20200203143136c_CDW->N_20200203143136d_CDW N_20200203143136e_CDW    By CDW 2020/02/03 @ 14:31:36e -------------------------------- Ed Southall gave a modified version that then *does* work.        (select only this node)     N_20200203143136d_CDW->N_20200203143136e_CDW N_20200203143136f_CDW    By CDW 2020/02/03 @ 14:31:36f -------------------------------- A potential problem is then that students will want the "One True Method", and might be confused by having multiple versions.        (select only this node)     N_20200203143136e_CDW->N_20200203143136f_CDW N_20200203143839a_CDW    By CDW 2020/02/03 @ 14:38:39a -------------------------------- Here: https://twitter.com/edsouthall/status/1224275584316067840        (select only this node)     N_20200203143136e_CDW->N_20200203143839a_CDW N_20200203143136g_CDW    By CDW 2020/02/03 @ 14:31:36g -------------------------------- Ed pointed out that explorations are a Good Thing(tm).        (select only this node)     N_20200203143136f_CDW->N_20200203143136g_CDW N_20200203143431a_CDW    By CDW 2020/02/03 @ 14:34:31a -------------------------------- I've lost count of the number of times I've seen brilliant things that draw gasps from teachers who then say - "That's amazing, I'll be using that from now on" only for it then to disappear for ever.        (select only this node)     N_20200203143136g_CDW->N_20200203143431a_CDW N_20200203150856a_CDW    By CDW 2020/02/03 @ 15:08:56a -------------------------------- It bothers me.        (select only this node)     N_20200203143431a_CDW->N_20200203150856a_CDW N_20200203143431b_CDW    By CDW 2020/02/03 @ 14:34:31b -------------------------------- I've lost track of the number of times I've seen myself, or heard of, some simple, elementary thing I know, and that I assume pretty much everyone would know, be heralded with: "Why have I never seen this ?!?"        (select only this node)     N_20200203143431a_CDW->N_20200203143431b_CDW N_20200203143431b_CDW->N_20200203150856a_CDW N_20200203143431c_CDW    By CDW 2020/02/03 @ 14:34:31c -------------------------------- There is some amazing stuff that I think can and should be integrated into scheme of works, not because they are directly examinable, but because they will reduce the subsequent work that has to be done to teach those things that will be examined, and which will be useful in the future lives of the students.        (select only this node)     N_20200203143431b_CDW->N_20200203143431c_CDW N_20200203145847a_CDW    By CDW 2020/02/03 @ 14:58:47a -------------------------------- One example is the observation that all odd primes greater than 3 are of the form 6k-1 or 6k+1.        (select only this node)     N_20200203143431b_CDW->N_20200203145847a_CDW N_20200203145649a_CDW    By CDW 2020/02/03 @ 14:56:49a -------------------------------- The exploration of methods to bisect the angle is exactly one example of this.        (select only this node)     N_20200203143431c_CDW->N_20200203145649a_CDW N_20200203143431d_CDW    By CDW 2020/02/03 @ 14:34:31d -------------------------------- And it surfaces, goes viral, then sinks without a trace.        (select only this node)     N_20200203143431c_CDW->N_20200203143431d_CDW N_20200203143431d_CDW->N_20200203150856a_CDW N_20200203143431e_CDW    By CDW 2020/02/03 @ 14:34:31e -------------------------------- Is all this true?        (select only this node)     N_20200203143431d_CDW->N_20200203143431e_CDW N_20200203143431f_CDW    By CDW 2020/02/03 @ 14:34:31f -------------------------------- Is there something here that we should fix?        (select only this node)     N_20200203143431e_CDW->N_20200203143431f_CDW N_20200203143431g_CDW    By CDW 2020/02/03 @ 14:34:31g -------------------------------- If so, where should we start? What should we do?        (select only this node)     N_20200203143431f_CDW->N_20200203143431g_CDW N_20200203145649b_CDW    By CDW 2020/02/03 @ 14:56:49b -------------------------------- For students to see one method, compare it with another, and then create and devise their own, has benefits (I'd claim) beyond this single task.        (select only this node)     N_20200203145649a_CDW->N_20200203145649b_CDW N_20200203150642a_CDW    By CDW 2020/02/03 @ 15:06:42a -------------------------------- A lovely extension problem ... Given two non-parallel lines, find the line that is the angle bisector when the point of intersection is not accessible.        (select only this node)     N_20200203145649b_CDW->N_20200203150642a_CDW N_20200203145847b_CDW    By CDW 2020/02/03 @ 14:58:47b -------------------------------- This is beautifully demonstrated by highlighting primes in a grid of numbers that's of width 6.        (select only this node)     N_20200203145847a_CDW->N_20200203145847b_CDW N_20200203145847c_CDW    By CDW 2020/02/03 @ 14:58:47c -------------------------------- Or 12.        (select only this node)     N_20200203145847b_CDW->N_20200203145847c_CDW