{"id":253,"date":"2012-02-10T15:16:35","date_gmt":"2012-02-10T21:16:35","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=253"},"modified":"2012-02-10T15:16:35","modified_gmt":"2012-02-10T21:16:35","slug":"smashmouth-mathematics","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=253","title":{"rendered":"Smashmouth Mathematics"},"content":{"rendered":"

If it were physically possible to fold a piece of paper in half 50 times (it's not), how thick would the resulting origami sculpture be?\u00a0 Quick!\u00a0 No fair calculating.\u00a0 What does your gut say?<\/p>\n

If you have absolutely no idea, I'll tell you that a standard piece of printer paper, folded six times by high a school student with very little concern for symmetry or crease definition, has an average thickness somewhere between six and eight millimeters.\u00a0 How much will that increase over the next 44 folds?\u00a0 Any ideas?<\/p>\n

Out of roughly 100 students, all guesses but one were under 10 feet.\u00a0 One girl said 100 feet, and she just about got laughed out of the classroom.\u00a0 The actual answer: a little over 100 billion meters.\u00a0 No, that's not a typo.\u00a0 In fact, if you could manage to fold the paper in half one more time, for 51 total, you'd have a stack that's taller than the distance from the Earth to the sun.<\/p>\n

The math is pretty straightforward.\u00a0 Every time we fold our idealized sheet of paper, the thickness doubles.\u00a0 After 50 folds, that's 250<\/sup> times the original thickness, which, for the printer paper stashed in my room, is about 0.1 mm, or 0.0001 m.\u00a0 Multiply those two values together and be amazed.<\/p>\n

I chose this exercise because it's a nice way to introduce recursive geometric sequences, and because it's a nice change of pace from the standard penny-doubling problem<\/a>.\u00a0 But I chose it mostly because it's so incredibly counterintuitive.\u00a0 And that<\/em>, I submit, makes a powerful argument for mathematics.<\/p>\n

Every time we smack our students' intuition in the mouth, we implicitly answer the nagging question, What is any of this good for?<\/em>\u00a0 Well for one thing, Mr. Thinks-the-answer-is-two-feet, It's good for critically examining our preconceptions, which are often comically off base.\u00a0 When we can spectacularly highlight the fallibility of the human gut, we create a tiny void that has to be filled with something <\/em>useful.\u00a0 And the best candidate: mathematical inquiry.\u00a0 A surprising result forces students to shove a problem across the corpus callosum and give the analytical self an at-bat.\u00a0 Do that enough times in the classroom and we start to build mathematical habits of mind.\u00a0 Math starts to edge out intuition as the default setting.<\/p>\n

So next time you're looking for an interesting problem to build a lesson around, give some extra weight to the ones with the most startling solutions.\u00a0 I conjecture that the dividends grow exponentially.<\/p>\n","protected":false},"excerpt":{"rendered":"

If it were physically possible to fold a piece of paper in half 50 times (it's not), how thick would the resulting origami sculpture be?\u00a0 Quick!\u00a0 No fair calculating.\u00a0 What does your gut say? If you have absolutely no idea, I'll tell you that a standard piece of printer paper, folded six times by high […]<\/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":[16,22,32,55],"class_list":["post-253","post","type-post","status-publish","format-standard","hentry","category-math-teaching","tag-estimation","tag-intuition","tag-math","tag-teaching"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/253","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=253"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/253\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=253"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=253"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}