{"id":727,"date":"2013-08-07T14:20:26","date_gmt":"2013-08-07T14:20:26","guid":{"rendered":"http:\/\/blog.chrislusto.com\/?p=727"},"modified":"2013-08-07T14:20:26","modified_gmt":"2013-08-07T14:20:26","slug":"whats-in-a-circle","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=727","title":{"rendered":"What's in a Circle?"},"content":{"rendered":"<p style=\"text-align: justify;\">Recently I had the good fortune to both attend <a title=\"Twitter Math Camp\" href=\"http:\/\/www.twittermathcamp.com\/tmc13\/\" target=\"_blank\">#TMC13<\/a> and share my very own personal favorite lesson during, appropriately, one of the \"My Favorite\" sessions. \u00a0And now I'll share it with you.<\/p>\n<p style=\"text-align: justify;\">I was getting ready to start a unit on conic sections with my Advanced Algebra kids. \u00a0I was planning to start, as I assume many of you do, with circles. \u00a0I imagined it going something like this, Lusto's 10 Steps to Circle Mastery:<\/p>\n<ol style=\"text-align: justify;\">\n<li>Get somebody, anybody in the room, to spout the definition of a circle , hopefully including a phrase such as, \"the set of all points a fixed distance from a given point...\"<\/li>\n<li>Talk about this distance business. \u00a0Do we know something about distance? \u00a0How do we measure it?<\/li>\n<li>The distance formula.<\/li>\n<li>Why the distance formula is fugly. \u00a0Remember the Pythagorean Theorem? \u00a0Of course you do. \u00a0It's awesome.<\/li>\n<li>In the plane: <span class='MathJax_Preview'><img src='http:\/\/blog.chrislusto.com\/wp-content\/plugins\/latex\/cache\/tex_9b73dd7b1d6feb00b9138143faf3cc26.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/li>\n<li>In the Cartesian plane, centered at the origin: <span class='MathJax_Preview'><img src='http:\/\/blog.chrislusto.com\/wp-content\/plugins\/latex\/cache\/tex_c2d59d2def6dd2ac1b32615c8a9805dc.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/li>\n<li>This distance,\u00a0<em>d<\/em>, has a special name in circles, right? \u00a0Right: <span class='MathJax_Preview'><img src='http:\/\/blog.chrislusto.com\/wp-content\/plugins\/latex\/cache\/tex_f516dc67efdfd0ffdabb0f63cda6db2d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/li>\n<li>Appeal to function families and translations by &lt;<em>h<\/em>,\u00a0<em>k&gt;<\/em>.<\/li>\n<li><span class='MathJax_Preview'><img src='http:\/\/blog.chrislusto.com\/wp-content\/plugins\/latex\/cache\/tex_8701b752fee8264bf07fe8ab9fba2f16.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/li>\n<li>Boom.<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">Based on my hopes for Step 1, and based on my need for like five uninterrupted minutes to take attendance, find my coffee mug, &amp;c., I hastily scribbled an extremely lazy and unimaginative warm-up discussion question on the whiteboard. \u00a0Four simple words that \u00a0led to some surprising and amazing mathematical conversation.<\/p>\n<blockquote>\n<p style=\"text-align: center;\"><strong>What is a circle?<\/strong><\/p>\n<\/blockquote>\n<p style=\"text-align: justify;\">That's it. \u00a0My favorite lesson. \u00a0The whole thing. \u00a0And here's how it went.<\/p>\n<p style=\"text-align: justify;\">I was walking around looking at\/listening to all the different definitions the groups had come up with. \u00a0And they were nuts. \u00a0There were dubious\u00a0claims about unquantifiable symmetry, sketchy sketches with\u00a0line segments of indeterminate provenance, rampant appeals to a mysterious property known as\u00a0<em>roundness<\/em>. \u00a0Most of the arguments were logically circular but, alas, mathematically not. \u00a0The word <i>curvy<\/i> appeared more than once. \u00a0It was a glorious disaster of handwaving and frustration. \u00a0I knew, deep in my reptilian brain, that this is what's known in the business as a \"teachable moment.\"<\/p>\n<div id=\"attachment_737\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/blog.chrislusto.com\/wp-content\/uploads\/2013\/08\/acircleis.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-737\" class=\"size-medium wp-image-737\" alt=\"What is not a circle.\" src=\"http:\/\/blog.chrislusto.com\/wp-content\/uploads\/2013\/08\/acircleis-300x278.jpg\" width=\"300\" height=\"278\" srcset=\"http:\/\/blog.chrislusto.com\/wp-content\/uploads\/2013\/08\/acircleis-300x278.jpg 300w, http:\/\/blog.chrislusto.com\/wp-content\/uploads\/2013\/08\/acircleis-500x464.jpg 500w, http:\/\/blog.chrislusto.com\/wp-content\/uploads\/2013\/08\/acircleis.jpg 555w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-737\" class=\"wp-caption-text\">Nailed it.<\/p><\/div>\n<p>At this point I was basically just walking around being a jerk. \u00a0I was drawing all kinds of crazy figures that minimally conformed to what they were telling me a circle was, and getting lots of laughs in the process. \u00a0And then I had the thought, even deeper in my reptilian brain, that transformed the whole experience from an interesting activity into a bonafide\u00a0<strong>lesson<\/strong>:\u00a0<em>Why the hell am <\/em>I<em> the one doing this?<\/em><\/p>\n<p>So here's what the lesson eventually became, Lusto's 6 Instructions to Humans on the Brink of Amazing Mathematical Discussion:<\/p>\n<ol>\n<li>In your groups, answer the question, \"What is a circle?\"<\/li>\n<li>Absolutely no book-looking or Googling. \u00a0If all goes well, you will be frustrated. \u00a0Your peers will frustrate you. \u00a0I will frustrate you. \u00a0Don't rob anybody else of this beautiful struggle. \u00a0If your definition includes the word\u00a0<em>locus<\/em>, you are automatically disqualified from further participation.<\/li>\n<li>Each group will have one representative present your definition to the class. \u00a0No clarification. \u00a0No on-the-fly editing. \u00a0No examples. \u00a0No pantomime. \u00a0Your definition will include, and be limited to, English words in some kind of semantically meaningful order. \u00a0Introduce variables at your own risk.<\/li>\n<li>If you're going to refer to some other mathematical object (and I suspect you will), make sure it's <strong>not<\/strong> an object whose definition requires the concept of\u00a0<em>circle<\/em> in the first place. \u00a0(Ancillary benefit: you will be one of the approximately .01% of the population who learns what \"begging the question\" actually means.)<\/li>\n<li>Once a group presents a definition, here is your new job: construct a figure that meets the given definition precisely,\u00a0<strong>but is not a circle<\/strong>. \u00a0Pick nits. \u00a0You are a counterexample machine. \u00a0A bonus of my undying respect for the most ridiculous non-circle of the day.<\/li>\n<li>When you find a counterexample, make a note of the loophole you exploited. \u00a0What is non-circley about your figure?<\/li>\n<\/ol>\n<p>After giving the instructions, I could pretty much just sit back for a while and watch things get awesome. \u00a0If there's one thing that's easy to do, it's get teenagers to argue with each other. \u00a0Granted, it's a little harder to get them to argue about <em>math<\/em>, but not much. \u00a0(They're basically ready to fight at all times; the <a title=\"MacGuffin\" href=\"http:\/\/en.wikipedia.org\/wiki\/MacGuffin\" target=\"_blank\">MacGuffin<\/a> is largely unimportant.) \u00a0So that's one thing that makes this lesson my favorite. \u00a0Another thing is that we ended up with a pretty bullet-proof circle definition by the end of the exercise. \u00a0When you spend a whole lot of time crawling through loopholes in the hopes of beating up on your peers, you find an awful lot of loopholes to close. \u00a0Talk about a fantastic mathematical habit of mind. \u00a0Yet another thing, and maybe the coolest, is that it led to some of the best questions\/observations my kids ever came up with. \u00a0Here is a brief, paraphrased sampling, with my annotations as to why they're so great:<\/p>\n<ul>\n<li><strong>Wait, if the circle is just the points on the edge, then how can a circle have area? \u00a0<\/strong>There's nothing that gets kids thinking precisely about mathematical language faster than the realization that every teacher in the world is using it incorrectly. \u00a0We should really be saying,\u00a0<em>the area of the region bounded by a circle<\/em>, or\u00a0<em>the area of the circle's interior<\/em>... \u00a0I had never thought twice about that, but now I sure as hell do.<\/li>\n<li><b>What does it mean to be \"inside\" a circle<\/b><strong>?<\/strong>\u00a0 Similar to the above, but even more amazing. \u00a0The fact that a circle divides the plane into two disjoint regions is a completely nontrivial result. \u00a0It's basically a statement of the <a title=\"Jordan Curve Theorem\" href=\"http:\/\/en.wikipedia.org\/wiki\/Jordan_curve_theorem\" target=\"_blank\">Jordan Curve Theorem<\/a>, which was proved pretty recently in the history of mathematics.<\/li>\n<li><strong>If a radius is a distance, then a \"circle\" depends on how you measure distance. \u00a0<\/strong>This one is on my Holy Shit List. \u00a0We had spent like a half hour one day talking about the <a title=\"Taxicab Geometry\" href=\"http:\/\/en.wikipedia.org\/wiki\/Taxicab_geometry\" target=\"_blank\">Taxicab Plane<\/a>, just because I thought it was cool and made the distance formula seem mildly less boring. \u00a0But somebody pointed out that circles would look totally different if we measured distances that way. \u00a0And yes! \u00a0They would! \u00a0At that point, I felt like I should probably just retire.<\/li>\n<\/ul>\n<p>The final reason this lesson is my favorite is probably also the reason that you should care. \u00a0There's nothing particularly special about the word\u00a0<em>circle<\/em>. \u00a0You can take the sentence,\u00a0<em>What is a [widget]?\u00a0<\/em>and pick just about any mathematical widget that kids have some nascent intuition about. \u00a0Give it a shot. \u00a0Maybe it won't be your favorite, but it'll be pretty great.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Recently I had the good fortune to both attend #TMC13 and share my very own personal favorite lesson during, appropriately, one of the \"My Favorite\" sessions. \u00a0And now I'll share it with you. I was getting ready to start a unit on conic sections with my Advanced Algebra kids. \u00a0I was planning to start, as [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[62,60,63,61],"class_list":["post-727","post","type-post","status-publish","format-standard","hentry","category-math-teaching","tag-circles","tag-definitions","tag-favorite-lesson","tag-teaching-math"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/727","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=727"}],"version-history":[{"count":10,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/727\/revisions"}],"predecessor-version":[{"id":750,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/727\/revisions\/750"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=727"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=727"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=727"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}