tag:blogger.com,1999:blog-53033074821589225652020-07-12T10:59:07.671-07:00Math Mama Writes...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger576125tag:blogger.com,1999:blog-5303307482158922565.post-22631731360228947042020-06-25T23:10:00.000-07:002020-06-25T23:29:48.242-07:00Playful Math Education Carnival #139 (formerly known as Math Teachers at Play or MT@P)<div style="text-align: center;"><i>"It’s like a free online monthly magazine of mathematical adventures." (Denise Gaskins)</i></div><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-Md2kd7DWlfM/XvWN_XKA4QI/AAAAAAAAF4s/hbVZ2-1xB28o06eKKZHh-hKeMck8x_pqACLcBGAsYHQ/s1600/juneteenth%2Bnumbers.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="960" data-original-width="671" height="320" src="https://1.bp.blogspot.com/-Md2kd7DWlfM/XvWN_XKA4QI/AAAAAAAAF4s/hbVZ2-1xB28o06eKKZHh-hKeMck8x_pqACLcBGAsYHQ/s320/juneteenth%2Bnumbers.jpg" width="223" /></a><b> </b><br /><br /><br /><br /><b>Black Lives Matter.</b> How does that idea and movement intersect with math and play? It's hard to imagine play intersecting with the painful history of racism in the U.S. We can <a href="https://apps.urban.org/features/school-funding-do-poor-kids-get-fair-share/" target="_blank">collect data</a> to show how pervasive anti-Blackness has been and is. We can discuss how <a href="https://www.nsf.gov/news/news_summ.jsp?cntn_id=101776" target="_blank">math courses have been used</a> to filter out students from desirable professions (doctors, engineers, lawyers). We can discuss how Black people are more involved in the history of math than you'd guess from the Eurocentric naming. (<a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle#History" target="_blank">Check out who knew Pascal's triangle before Pascal!</a>) None of that is playful. But celebration can be playful. Let's celebrate Juneteenth!<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><h2><b>139</b></h2>Every number is cool.* Here are some ways 139 is cool:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-eCnjz09_BvE/XvWMvB5EuxI/AAAAAAAAF4Y/HcGp4KBGtYgRzQwteH9jhHSFGN8-P2_6ACLcBGAsYHQ/s1600/139.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="200" src="https://1.bp.blogspot.com/-eCnjz09_BvE/XvWMvB5EuxI/AAAAAAAAF4Y/HcGp4KBGtYgRzQwteH9jhHSFGN8-P2_6ACLcBGAsYHQ/s200/139.jpg" width="200" /></a></div><ul><li>139 is the sum of 5 consecutive prime numbers (19 + 23 + 29 + 31 + 37). </li><li>139 is the smallest prime before a prime gap of length 10. </li><li>137 and 139 form the 11th pair of twin primes. </li><li>139 is the 34th prime number. </li></ul><br /><b>Puzzle</b>: The digit sum is the result after adding the digits repeatedly until you get down to one digit. 139’s digit sum is 4. If you write 139 in base two, you get 100 1011, which still has a digit sum of 4. Does this always happen? If not, does it happen in any other bases?<br /><br /><br /><br /><br /><b>New Homeschoolers </b><br />I have a hunch the quarantine has moved lots of families from school to homeschooling. If you’re new to homeschooling, get ready to have fun playing with math. Most mathematicians are in it at least partially for the fun of it. We like to play with numbers, shapes, and logic. The more you play with math with your kids, the more likely they are to enjoy it.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><a href="https://1.bp.blogspot.com/-2nOHeH_Mlpk/XvWOdRfHh3I/AAAAAAAAF40/LDGWjBhZo7A7C8vHWxXow_qQfrmMmRlOQCLcBGAsYHQ/s1600/beast.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="523" data-original-width="540" height="193" src="https://1.bp.blogspot.com/-2nOHeH_Mlpk/XvWOdRfHh3I/AAAAAAAAF40/LDGWjBhZo7A7C8vHWxXow_qQfrmMmRlOQCLcBGAsYHQ/s200/beast.png" width="200" /></a>There are vast resources online to help you. Until 3rd grade, just play games, cook, measure, read mathy stories, and have fun with it all. If your kid wants a curriculum before that because they love math, then check out <a href="https://beastacademy.com/" target="_blank">Beast Academy</a>. It has levels 2 to 5 (topics correspond to grades 2 to 5, <a href="https://artofproblemsolving.com/" target="_blank">Art of Problem Solving</a>. There are lots of other great resources; these are just my personal favorites.<br />difficulty levels are a grade or two higher). Some families never use a curriculum; if you’re interested, you may want to explore unschooling. Math lovers eventually want to take classes, which you can do either through your local community college (I’ll be teaching trigonometry, pre-calculus, and calculus I online this fall) or <br /><br />You might find ideas that work for you in my book, <a href="https://naturalmath.com/playingwithmath/" target="_blank"><i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i></a>. Or from other books from my publisher, <a href="https://naturalmath.com/goods/" target="_blank">Natural Math</a>. I also highly recommend Denise Gaskins’ <a href="https://denisegaskins.com/blog/" target="_blank">blog</a> (<a href="https://denisegaskins.com/2020/04/09/how-to-homeschool-math/" target="_blank">especially this post on homeschooling math</a>), <a href="https://denisegaskins.com/" target="_blank">website</a>, and <a href="https://tabletopacademy.net/playful-math-books/" target="_blank">books</a>. <a href="https://mathforlove.com/2016/04/how-to-help-your-kids-fall-in-love-with-math-a-guide-for-grown-ups/" target="_blank">Dan</a> and <a href="https://talkingmathwithkids.com/2015/08/31/let-the-children-play/" target="_blank">Christopher</a> have some good ideas about playing mathematically with kids too.<br /><br /><br /><br /><br /><b>Talking Math With Your Kids (#TMWYK)</b><br /><div class="separator" style="clear: both; text-align: center;"><b><a href="https://1.bp.blogspot.com/-QqQ8jlupCPg/XvWOoeQMLbI/AAAAAAAAF44/wRnlYp9TPFw9no1dIZ5BRIJSEStiYR5uwCLcBGAsYHQ/s1600/unschool.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="528" data-original-width="792" height="133" src="https://1.bp.blogspot.com/-QqQ8jlupCPg/XvWOoeQMLbI/AAAAAAAAF44/wRnlYp9TPFw9no1dIZ5BRIJSEStiYR5uwCLcBGAsYHQ/s200/unschool.jpeg" width="200" /></a></b></div><br /><br /><ul><li><a href="http://tjzager.com/2016/04/19/talking-math-in-ghirardelli-square/" target="_blank">Tracy Zager's daughter was thinking hard while mom sat there, exhausted.</a></li><li><a href="https://betweenthenumbers.wordpress.com/2019/03/08/tmwyk-overheard-on-muni-edition/" target="_blank">On the Muni, overheard by Breedeen Pickford-Murray.</a> </li></ul><br /><br /><br /><br /><br /><br /><b>Math & Language Play </b><br />One of my favorite math educators, <a href="http://www.marilynburnsmathblog.com/" target="_blank">Marilyn Burns</a>, invented a game where students look for $1 words. A=1¢, B=2¢, etc. You could combine math and any other subject by making $1 phrases. Sometimes kids like the simplest games. This might be a craze at your house. (My son used to love Shut the Box, a simple dice game that did nothing for me. It sure was good number practice for him.)<br /><br /><a href="https://aperiodical.com/2020/06/poetry-competition-%cf%80-ku/" target="_blank">π-ku, a competition</a>, in which all their favorites will be posted at the Aperiodical blog. I'll try:<br /><div style="text-align: center;">Three One Four.</div><div style="text-align: center;">Hmm.</div><div style="text-align: center;">Not very hard.</div><br /><br /><b>Games</b><br />So much of math is based on logic, any logic games you play will deepen your students' affinity for math. Here are a few:<br /><ul><li><a href="https://blog.tanyakhovanova.com/2020/06/set-tic-tac-toe/" target="_blank">Set Tic Tac Toe</a>, described by Tanya Khovanova, invented by her students. You may want to play the basic game of Set for a few months before attempting this. But if I could figure out a way to do this at a distance, I'd love to try this out. </li><li><a href="http://planarity.net/" target="_blank">Planarity game</a>. (This is connected to a field of math called graph theory.) </li><li><a href="http://math.hws.edu/eck/js/symmetry/wallpaper.html" target="_blank">Play with wallpaper symmetries.</a></li></ul><br /><br /><br /><b>Math History</b><br />Podcasts aren't my thing. Yet. But if this series is as good as it sounds, I'll just have to figure this newfangled genre out. <a href="http://intellectualmathematics.com/opinionated-history-of-mathematics/" target="_blank">Opinionated History of Mathematics</a>. With an <a href="https://aperiodical.com/2020/06/podcasting-about-opinionated-history-of-mathematics-podcast/" target="_blank">interview and glowing review at Aperiodical</a>. <br /><br /><br /><br /><br /><b>Online Events</b><br />This summer <a href="https://www.artofinquiry.net/" target="_blank">Art of Inquiry</a> is hosting free science webinars on space, astrobiology, and AI for school children and their families. The webinars are led by university professors and industry experts. You can register for the events on <a href="https://www.eventbrite.com/o/art-of-inquiry-5603297429" target="_blank">Eventbrite</a>. Here is their June-July 2020 schedule:<br /><ul><li>Living Through a Revolution: Multi-messenger Astrophysics - Dr. Roopesh Ojha, GSFC NASA, June 26th </li><li>Figuring out the Earth from inside out - Dr. Kanani Lee, Lawrence Livermore National Laboratory, June 30th </li><li>Mars Rovers - Dr. Allan Treiman, Lunar and Planetary Institute, July 3rd </li><li>The search for life on Mars in XXI century - Dr. Alex Pavlov, GSFC NASA, July 10th </li><li>Where in the Universe did we come from? - Dr. Ethan Siegel, science author, "Starts with a Bang" Forbes contributor, July 23rd </li><li>Why we should build a Moon base - Dr. Ian Crawford, University of London, July 31st </li></ul> If you know of other math-related online events, please mention them in the comments. <br /><br /><br /><br /><br />This series of blog carnivals was founded and is kept going by the fabulous Denise Gaskins. <a href="https://denisegaskins.com/mtap/" target="_blank">You can find out more at her blog.</a> <a href="https://mathhombre.blogspot.com/2020/05/playful-mathematics-carnival-138.html" target="_blank">Last month's carnival was hosted by John Golden, the Math Hombre. Check it out!</a> <br /><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-o2T1asrfc4w/XvWMEA0cXmI/AAAAAAAAF4Q/6pEQduaCf4Mn_19xw1mCYs-7C8kNDb7TQCLcBGAsYHQ/s1600/bad%2Bnumbers.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="952" data-original-width="1080" height="176" src="https://1.bp.blogspot.com/-o2T1asrfc4w/XvWMEA0cXmI/AAAAAAAAF4Q/6pEQduaCf4Mn_19xw1mCYs-7C8kNDb7TQCLcBGAsYHQ/s200/bad%2Bnumbers.jpg" width="200" /></a> <br />-----<br /><span style="font-size: x-small;">*Well, sometimes their coolness is in their bad reputation (Sounds like a few people I knew in high school) ... </span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com1tag:blogger.com,1999:blog-5303307482158922565.post-77760522413141998652020-06-18T19:44:00.001-07:002020-06-18T19:44:05.912-07:00<div class="_5pbx userContent _3576" data-ft="{"tn":"K"}" data-testid="post_message" id="js_aq4">The Math Teachers at Play Blog Carnival (aka Playful Math Education Carnival) will be a bit late this month.<br /><br /> I am looking for good posts now. If you can send me any links by Saturday, that would be great. I am hoping to put it together on Sunday.<br /><br /> Send your links to mathanthologyeditor@gmail.com, or post them here.<br /><br /> Want to know what a blog carnival is? Check out <a href="https://mathhombre.blogspot.com/2020/05/playful-mathematics-carnival-138.html" target="_blank">last month's, by my pal John Golden</a>.<br /><br /><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-55152477259805827432020-04-19T00:06:00.000-07:002020-04-19T00:06:37.770-07:00Corona Post #2: Teaching Online[#2 because my previous post in March on <a href="http://mathmamawrites.blogspot.com/2020/03/online-math-circle-pythagorean-triples.html" target="_blank">my online math circle</a> was due to people needing to take their math circles online when the shelter-in-place orders were just starting.]<br /><br /><a href="https://1.bp.blogspot.com/-tuXmtCoRr3c/XpvvoUB9jWI/AAAAAAAAF1E/p4Jb5KPO9QIjEwp1bb3tiaGIED4Ul81LQCLcBGAsYHQ/s1600/me%2Bwriting.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1600" data-original-width="1113" height="320" src="https://1.bp.blogspot.com/-tuXmtCoRr3c/XpvvoUB9jWI/AAAAAAAAF1E/p4Jb5KPO9QIjEwp1bb3tiaGIED4Ul81LQCLcBGAsYHQ/s320/me%2Bwriting.jpg" width="221" /></a>I've been teaching online for 4 weeks now, two before our spring break and two after. At first I was just trying to learn how to manage teaching on zoom. I bought a whiteboard that's still sitting on two chairs in my living room, and I sit in a tiny chair while I write on it. Not ideal, but I get to see my students, and I feel like I'm still working with them where they are, not just lecturing.<br /><br />(Some day I'll finally install it on my living room wall. I procrastinate with tasks like that. I'm not sure why it feels intimidating...)<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-Xt5CvgFsJ4I/XpvtyZWGSCI/AAAAAAAAF00/iK-bWlPA4pMjfsaFP4VMoipPNpJRF3d0ACLcBGAsYHQ/s1600/stars%2B%2526%2Bbars.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="448" data-original-width="868" height="165" src="https://1.bp.blogspot.com/-Xt5CvgFsJ4I/XpvtyZWGSCI/AAAAAAAAF00/iK-bWlPA4pMjfsaFP4VMoipPNpJRF3d0ACLcBGAsYHQ/s320/stars%2B%2526%2Bbars.png" width="320" /></a><br /><br /><br />A few weeks ago I made a google slides presentation for my Discrete Math course to explain a way of counting possibilities called <a href="https://docs.google.com/presentation/d/1a7uWiQrjy49chq1RxkvyHy6EB4ujwB4wR27g9rQyUGI/edit" target="_blank">Stars & Bars</a>. I had fun doing it. You're welcome to modify it and use it in your teaching.<br /><br /><br /><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-zE7Uwgh6zxo/XpvvfniJybI/AAAAAAAAF1A/hpxvm8yId_cmUCFkcaU3Mpw9qclJHpQAgCLcBGAsYHQ/s1600/maclaurin.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="370" data-original-width="664" height="178" src="https://1.bp.blogspot.com/-zE7Uwgh6zxo/XpvvfniJybI/AAAAAAAAF1A/hpxvm8yId_cmUCFkcaU3Mpw9qclJHpQAgCLcBGAsYHQ/s320/maclaurin.png" width="320" /></a><br /><br />Just now I made another. This one is for Calculus II, on <a href="https://docs.google.com/presentation/d/1nCUcTNIbgXQ5c5SJtH352WbYMLEMWQ1F1kRJMClrwOQ/edit#slide=id.g83b4e189f7_0_58" target="_blank">Taylor & Maclaurin Series</a> (really just a Maclaurin series). I was motivated by knowing that there would be too much writing for my little whiteboard. This presentation has a <a href="https://drive.google.com/file/d/1JtrcbZ7hLlxHQj3QIZOwUFiofgbVky6d/view?usp=sharing" target="_blank">handout</a> to go with it.<br /><br /><br /><br /><br /><br /><br /><br /><br />I'm also teaching Calculus I. I haven't made any cool new materials for that course yet. But I will...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-89413990039697147382020-03-23T15:00:00.000-07:002020-03-29T15:05:16.464-07:00Online Math Circle: Pythagorean Triples<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-G4UUOAGkiYw/XnkviZR20gI/AAAAAAAAFzw/68ZRFSJkKPQt3q5QxSZw8cV2KXNgMGMMACEwYBhgL/s1600/3-4-5-triangles.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="332" data-original-width="691" height="191" src="https://1.bp.blogspot.com/-G4UUOAGkiYw/XnkviZR20gI/AAAAAAAAFzw/68ZRFSJkKPQt3q5QxSZw8cV2KXNgMGMMACEwYBhgL/s400/3-4-5-triangles.jpg" width="400" /></a></div><br /><br />The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a<sup>2</sup>+b<sup>2</sup> = c<sup>2</sup>. Usually that makes at least one side something ugly like square root of 2. But a few combinations make all three sides whole numbers. Those are called Pythagorean triples. Here are a few of them: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 20-12-29.<br /><br /><br />Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)<br /><br /><br /><br />Edited to add:<br /><br />This <a href="https://4cd.zoom.us/rec/play/vsUvJeD5_Do3GtDAtwSDB6AqW9XrKq6sgCAar6JbzUe2BnMKMFfyb7YWMbOY7Ulru-Cg7C8-GK2QCS85?continueMode=true&_x_zm_rtaid=Ttrr4k29TpW-Z7JJMVK5Ug.1585518711595.2dc48300251a4ba1f7e3cdf8ef175510&_x_zm_rhtaid=898" target="_blank">online math circle happened on <b>Friday, March 27, at 10am PDT</b> (1pm EDT)</a>. [This link is to the zoom recording, along with its automatically produced (therefore hilariously bad) audio transcript.]<br /><br />I promised to write up some of it here.<br /><br />Way back in 2007, I read Bob and Ellen Kaplan's book, <a href="https://www.betterworldbooks.com/product/detail/Out-of-the-Labyrinth--Setting-Mathematics-Free-9780195147445" target="_blank"><i><b>Out of the Labyrinth: Setting Mathematics Free</b></i></a>, about the math circles they lead. It was such a discovery for me! I went to their first Summer Math Circle Teacher Training Institute, held at Notre Dame, and fell in love with this community. I kept going back for years, craving a discussion of math among equals, figuring out new ways of seeing. One summer we discussed Pythagorean triples, and that December I tried to rebuild what I had learned. I am blessed with a very bad memory, so what I did in December looked very different from what we had done in the summer.<br /><br />I was also exploring online, and ended up putting together a book that collected some of the best resources I had found: <a href="https://naturalmath.com/playingwithmath/" target="_blank"><i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i></a>.<br /><br />Our circle was prompted by Rodi Steinig's request for help learning how to use zoom for online math circles. I offered one of my favorite topics, and off we went. Participants came from as far away as Colombia (and farther?).<br /><br />We proved a few things, and explored a bunch more. I hope some participants went home eager to prove more on their own.<br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-87101425623023152232020-01-04T19:31:00.001-08:002020-01-04T19:31:20.882-08:00Multiplication Chart with PicturesIn the story I'm writing, Althea remembers a multiplication chart that was posted in their bathroom. It had cool pictures around the edges for many of the facts.<br /><br /><ul><li>2x3 was a 6-pack of soda.</li><li>2x6 was a carton of eggs.</li><li>8x8 was a chessboard.</li><li>The fives were sometimes collections of nickels, but 5x12 was the 60 minutes on a clock, and 5x6 was time too.</li></ul><br />I thought I knew of more iconic sets like these, but I can't think of any more as good as these. I'm hoping for help. Do you have images in your head for any of the multiplication facts?<br /><br />Maybe threes will be 3-leaf clovers. 6 of them have 18 petals. That doesn't seem nearly as iconic as the ones above, though.<br /><br />Fours could be legs on dogs. 6 dogs have 24 legs. Twos could be eyes on friends...<br /><br /><br /><b>What are your favorite images for multiplication facts?</b>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-46379242717000628332019-09-21T20:10:00.000-07:002019-09-21T22:15:35.717-07:00The History of Imaginary Numbers is a Soap OperaI first read about this history in <a href="https://www.abebooks.com/servlet/SearchResults?isbn=0471500305&cm_sp=mbc-_-ISBN-_-all" target="_blank"><i><b>Journey Through Genius</b></i></a>, by William Dunham, in the chapter titled Cardano and the solution of the cubic. It reads like a soap opera in which the math is done for glory, not for any possible connection to the real world. The people who came up with imaginary numbers thought they were fictitious elements in the process of solving a cubic, and never expected them to have any real meaning. Turns out they do. Imaginary numbers help scientists describe electrical current and probability distributions, among other things.<br /><br />Years ago, on Living Math Forum, a mom wrote in to ask for help. Her son had asked: “The square root of 1 is 1, so what's the square root of -1 ?” That inspired me to write a math poem, <a href="https://mathmamawrites.blogspot.com/2009/03/math-poems.html" target="_blank">Imaginary Numbers Do the Trick</a>.<br /><br />Recently I was having a lovely discussion with my editor, <a href="https://naturalmath.com/goods/" target="_blank">Maria Droujkova</a>, and another author, about math storytelling. I realized this topic might possibly make for a good children's story. I'm working on it now. As I think about it, I'm not sure how to find the right age range. The math seems like it requires high school, but the story could interest younger kids, I think.<br /><br /><br /><br /><b>The History</b><br />Here's a short version:<br /><ul><li><b>Scipione del Ferro</b> solves equations of the form <i>x<sup>3</sup> + mx = n</i> (called depressed cubics). On his deathbed, in 1506, he passes his method on to his student, <b>Antonio Fior</b>.</li><li><b>Niccolo Tartaglia</b> boasts that he can solve cubics of the form <i>x<sup>3</sup> + mx<sup>2</sup> = n</i>, so in 1535, Fior challenges him with 30 depressed cubics. (These challenges were a common feature of life as a mathematician in 1500's Italy, and provided a way for mathematicians to get more recognition and paying students.) Tartaglia's return problem list to Fior has a variety of problems. Tartaglia does not yet have a solution for the depressed cubic, and sweats it, working feverishly to try to figure it out. At the last moment, he succeeds, and solves all 30 problems. Fior does not do so well, and is humiliated. </li><li><b>Gerolamo Cardano</b> comes to Tartaglia, asking him to disclose his method. He begs repeatedly, and Tartaglia, now Cardano's guest in Milan, finally concedes. Cardano takes an oath of secrecy. Tartaglia writes his solution in cipher, <a href="https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-poem" target="_blank">as a poem</a> (!). </li><li>Cardano takes on a brilliant student, <b>Ludovico Ferrari</b>, with whom he shares the secret. Together, they solve the general cubic, and then Ferrari goes on to solve the quartic. But all their work depends on reducing these to the depressed cubic, which Cardano has sworn not to tell about.</li><li>Cardano and Ferrari travel to Bologna, and are able to inspect the papers of ... Scipione del Ferro, where they find the solution. Cardano figures that relieves him of his oath and publishes, in his 1545 book, <a href="https://www.maa.org/press/periodicals/convergence/mathematical-treasure-cardanos-ars-magna" target="_blank"><i>Ars Magna</i></a>. He gives both del Ferro and Tartaglia credit, but Tartaglia is furious.</li><li>In the book, Cardano lays out the steps for solving the general cubic. But in doing so, he introduces a mystery. The depressed cubic <i>x<sup>3</sup> - 15x = 4</i> clearly has solutions x = 4 and x = -2+-√3. And yet the formula found by Ferro, Tartaglia, Cardano, and Ferrari includes a √-121 for this equation. Cardano threw up his hands at the mystery. It was explored but not truly understood 30 years later by Rafael Bombelli. It took another almost two centuries for Euler to finally solve the mysteries of complex numbers.</li></ul><br /><ul></ul><a href="https://www.quora.com/What-were-the-renaissance-mathematics-competitions-in-Italy" target="_blank">Here's a nice write-up</a> I found online, but it suggests different facts than the version in Dunham's book. I will keep reading while I write, so I can hopefully get my facts right.<br /><br /><br /><br /><b>My Request</b><br />I'm looking for kids who would like to read my draft versions and tell me what parts they like. If you have kids who understand (at all) the notion of a square root and the idea of what a cubic equation is, would you ask them if they'd like to read my story? (I would, of course, mention them in my book if it gets published.) You, or your kids, can email me at mathanthologyeditor on gmail.<br /><br /><br /><br /><b>Just a Beginning</b><br /><div style="text-align: center;"><b>The Saga of the Imaginary Numbers</b></div><div style="text-align: center;"></div>“Mom, I’ve been thinking… If the square root of 1 is 1, what is the square root of -1?”<br /><br />“What a fun thing to think about, Althea! What have you figured out so far?”<br /><br />“I know that when I square 1 I get 1, and that’s why the square root of 1 is 1. But when I square negative 1, I get 1 too, so shouldn’t the square root of 1 be negative 1 too? But how can it be two things?”<br /><br />“Hmm, that’s a strange one, isn’t it? I think there are too many ones in this for me to keep track of things. Let’s switch to 3.<br /><br />"I’m going to try to say what you said, but with 3 and 9. 3 squared is 9, so the square root of 9 is 3. But negative 3 squared is still 9, So why isn’t the square root of 9 equal to negative 3 also? Is that basically the same question you asked?”<br /><br />“Yes. Except the square root of 9 can’t have two answers. Can it?”<br /><br />“Well, somebody a long time ago decided that there should be just one answer for the square root of a number. And so we say that there is the square root of 9 and also the negative square root of 9.” Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com10tag:blogger.com,1999:blog-5303307482158922565.post-35465726222910762332019-08-25T21:46:00.004-07:002019-08-25T21:48:16.916-07:00Another Semester, Starting with Good IntentionsMy handouts are copied, the piles are organized on my desk. My rosters are printed. (I entered student names into excel, so I can organize things my way.) I've looked over my computer folders and found a few more things to share tomorrow. And I'm getting better at using Canvas' features - I plan to have students evaluate the new activities online, to help me decide whether each activity stays, goes, or gets improved.<br /><br />Will I manage to set up a new student survey in Canvas for each new activity?<br />Will I blog about my classes, like I'd like to?<br />Will I do more activities and less lecture in each class?<br /><br />Once the semester gets rolling, it's hard for me to change things up. It's so much easier to do what I have done before. May my passion keep me improving, all through the term. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-41195648085784163842019-08-17T11:28:00.004-07:002019-08-17T20:01:21.697-07:00First Day, Once AgainI've seen some great advice for the first day of class. (<a href="https://www.chronicle.com/interactives/advice-firstday" target="_blank">Here's the one I've read more than once.</a> I've seen other great ideas, but I don't see them now.) I'd summarize my take on this article as:<br /><ul><li>(<i>Community</i>) Start learning names, and get them learning each other's names. </li><li>(<i>Expectations</i>) Don't spend much time on the syllabus; there are more important things to do. (Have them read it, and then you can quiz them on it the next day, or just ask for their questions.)</li><li>(<i>Learning, Expectations</i>) If you use group activities (which are a very effective way to help groups of students learn), then you want to introduce students to this on day one.</li><li>(<i>Curiosity</i>) An activity that helps them see what's coming in the course would be especially nice.</li></ul><br />Our classes start in <strike>ten</strike> nine days. I am getting ready...<br /><br /><br /><b>First day activity for Calculus I </b><br />I have them graph a parabola (y=x<sup>2</sup>), then draw a tangent to it at x=3. (Some don't know what that means, so I walk around checking.) And then <i><b>estimate</b></i> its slope. After they're done, I get to talk about what makes actually finding the slope hard - such a good intro to the course. And they've had time to review graphing a parabola.<br /><br /><br /><b>First day activity for Linear Algebra </b><br />I have them solve a system of 3 equations in 3 variables. I ask them to:<br /><ul><li>Write down a description of the process, </li><li>Solve the system, </li><li>Now figure how to check whether your answer is correct. (Naysayers, has the group done enough to be sure that the answer is correct?) </li><li>Extra: What does each equation represent geometrically? What does the solution represent geometrically? </li></ul>Once again, a great introduction to the themes of the course.<br /><br /><br /><b>First day activity for Geometry</b><br />I taught geometry earlier this summer for the first time. I had them draw a triangle (and make sure it was different than their neighbor's). Find the midpoints of each side (they could measure or fold). Connect each midpoint to the opposite vertex. I hoped most would do it well enough that the 3 connecting lines would intersect at one point. My goals were to highlight: vocabulary, shapes, construction (which we were not doing with straightedge and compass - yet), conjecture, and the possibility of proof.<br /><br /><br /><br /><b>First day activity for Precalculus</b> <br />I've been thinking for the last few weeks about what I'd like to do for Precalculus. I have found exciting activities in the past that turned out to be way too hard, and intimidated the students. I have a lovely fractions activity, but that doesn't represent what we'll do going forward.<br /><br />I am working hard to create an activity that looks at functions (and circles too) from 4 perspectives: equations, graphs, tables of values, and stories. I have 7 types of relationships (linear, quadratic, polynomial, rational, exponential, periodic, and circles). I don't have stories yet for the polynomial and rational. (My eternal gratitude to anyone who can give me a story I like for either of these.) And I won't show an equation for the periodic. (A trig function wouldn't make sense yet. But we'll get to discuss that.) So that makes 25 "clue sheets".<br /><br />I have 40 students, who I'll put in 10 groups of 4. Each group will start with two clue sheets. [So 5 of the sheets will not be handed out at first. I can label those as graphs on the back, keep them at the front, and let student turn them over once they're pretty sure they didn't find a graph match to their set.] Each will describe a different type of function/relation from one perspective and ask them to do a few things. Then they pair up with a clue sheet for each pair, and go looking for the matching clue sheets (same function/relation, different perspective). They go back to their group and explain to each other what they found. (I'll have extras up front, so anyone done early can work on a 3<sup>rd</sup> function/relation.) When we're done, we'll have a summary of the function types we'll be studying all semester.<br /><br />I dreamed some of this up late last night. When I started working today, I worried that it would be too hard. (I make up some crazy stuff sometimes when I'm falling asleep.) So my goal as I put this together has been to scaffold it enough. I am assuming some comfort with linear functions, and some familiarity (but not comfort) with quadratics and exponentials. They may not have encountered the others. (And most will not know any trig.)<br /><br />I put my <strike>first</strike> second draft into <a href="https://drive.google.com/file/d/1u93faKFuoc7UbTJCSK7QETdUBBX-H844/view?usp=sharing" target="_blank">a google doc here</a>. Your suggestions may help me improve it. (I decided to leave out the rational function. 6 functions with 4 clues each would be 24. One story and one equation are left out. That's 22. The last two clues will sit up front.)<br /><br />Edited (8/17): This is a great activity, but too complex for day one. I will do it on day two. On day one, we will review linear functions in a similar, but much simpler way. <a href="https://drive.google.com/file/d/119M2fMb-7qfptog4d2ldfguGHbeqswoa/view?usp=sharing" target="_blank">Here is my handout</a>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-VFwpR4i0q0w/XVi8jvTc8_I/AAAAAAAAFbc/x0-yMuiaE0k_1zMmFd_zjfmHyEdZJCn2gCLcBGAs/s1600/day%2Bone%2Bhandout.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="647" data-original-width="499" height="640" src="https://1.bp.blogspot.com/-VFwpR4i0q0w/XVi8jvTc8_I/AAAAAAAAFbc/x0-yMuiaE0k_1zMmFd_zjfmHyEdZJCn2gCLcBGAs/s640/day%2Bone%2Bhandout.png" width="492" /></a></div><br /><br /><br /><b>My Goals</b>:<br /><ul><li>Review plugging values in for x to find y. (Do they remember that b<sup>0</sup>=1? What else might trip them up?)</li><li>Review graphing.</li><li>See functions/relations in the context of modeling a situation.</li><li>Identify functions/relations by type.</li><li>See precalculus as a place to strengthen their understanding of all of this. </li></ul><br />If you use this activity, please let me know what changes you decide to make and how it goes.<br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-34334853681258422332019-06-17T20:02:00.001-07:002019-06-17T20:02:29.449-07:00Teaching Geometry - Journal Post #1I usually try to write blog posts with a point. I don't often have the energy for that lately. But I do want to keep track of how this class goes... <br /><br />Today was the first day of my geometry class. (It is the first time ever for me to teach geometry. I have done lots and lots of prep.) It goes for 6 weeks, and meets 3 1/2 hours a day. (Yikes!) My students are mostly high school students, though this is a college class. The college screwed up registration. I had 28 students on my list, but a bunch thought they were on a waitlist (only happens when a class is full with 40 students), and came in not yet registered. I now have 35 or 36 students. Of course I ran out of handouts.<br /><br />I gave what I thought were good directions for our first activity, and students managed to do strange other things.<br /><br /><br />My directions:<br /> <br /><div align="center" class="MsoNormal" style="text-align: center;"><b style="mso-bidi-font-weight: normal;">Welcome and Triangle Activity</b></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This activity is an introduction to the themes of the course. The more thought and care you put into it (the activity and the course), the more learning and joy you will get out of it.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">With a straightedge, draw a big triangle on this page. Make it as weird as you want as long as it is a triangle. Make sure it looks different than the triangle your neighbor draws.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Now you are going to draw three lines inside it. Each line will go from a corner (aka <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">vertex</i></b>) to the middle of the opposite side (aka <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">midpoint</i></b>). These lines are called the <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">medians</i></b> of the triangle. How will you find the midpoints? Do this as accurately as you can. Now look at what others have drawn. What do you notice?</div><div class="MsoNormal"><br /></div><div class="MsoNormal">[Can you guess how this activity connects with the themes of the course?]</div><div class="MsoNormal"><br /></div><div class="MsoNormal">***</div><div class="MsoNormal"><br /></div><div class="MsoNormal">I should have had them find the midpoints first. And <b><i>then</i></b> talk about connecting those to opposite vertices. (Fixing the handout now.) Some people connected the midpoints to each other. Some drew a new triangle inside the first triangle. (Pretty but nothing like what I asked for.)<br /><br />So the activity took lots longer than I expected. Sadly, I didn't note the time. (I have a tutor working with me in class. I will ask her to keep track of how long things take.)<br /><br />I thought I could manage to lecture on 4 sections of our textbook in about an hour and a half. No way. I enjoyed what I did with the students, but we only got through 2 sections.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Some of what I did was impromptu. I asked them to each write down their own definition of angle. Then I asked for volunteers to read their definitions. I talked about courage. I waited. Finally I got a volunteer. And then another. I did not criticize their definitions, even though I didn't like them much. The third one was closer. He was struggling for a word for how far the one side is from the other. I repeated his definition, with a big pause where he had been stuck. And then I said rotation. We talked about things that rotate.<br /><br />One of the book problems we did just asked them to measure a line segment against a ruler printed in the book, and angles on a protractor. It was hard for them. And that was good for me to see. It tells me where we are.</div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"><br /></div><div class="MsoNormal">I hated our textbook when I first started reading it, but eventually it seemed fine to me. I think the first chapter has more problems than the rest of the book.<br /><br />1.1 Reasoning. They talked about types of reasoning and included 'intuition' (along with inductive and deductive reasoning). No. Intuition helps us guess what we might want to try, and helps us see things that might be true. But I don't like them calling it one of 3 types of reasoning.</div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"><br /></div><div class="MsoNormal">1.2 on measurement turns out to be more useful than I expected. Looking it over now, there's a lot more there that I haven't gotten to. Good thing I've asked them to turn in their notes on their reading. I hope that helps. (They are supposed ot read the first 4 sections.)</div><div class="MsoNormal"></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The last hour and a half is devoted to labtime. We will be using lots of labs from Henri Picciotto's <a href="https://www.mathed.page/geometry-labs/index.html" target="_blank">Geometry Labs</a> (free). I had hoped to do 3 of them today. We didn't even finish one. (Lab 1.1 involves putting different shapes of blocks around a point.) I had also hoped to introduce them to <a href="http://www.euclidthegame.com/" target="_blank">euclidthegame</a>. We'll do that tomorrow.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Quiz first thing tomorrow. I am hoping to keep them on top of things by quizzing daily.<br /><br />We are part of a program called High School STEM Connection. 20 of my 35 students are part of that program, and are required to come to tutoring after class. Most of them did. They got most of their homework done. Yay them. I love spending time with them where I'm not in charge. I'm just there to help out when they need me.</div><div class="MsoNormal"></div><div class="MsoNormal"><br />I am hoping to complete 5 sections of the book tomorrow. I'm tempted to throw out the cool group activities, but I know I shouldn't. I think I'll just see how far we get, like today. And then tomorrow evening I'll have to think again about how to adjust.<br /></div><style><!-- /* Font Definitions */ @font-face {font-family:"ＭＳ 明朝"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:1 134676480 16 0 131072 0;} @font-face {font-family:"ＭＳ 明朝"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:1 134676480 16 0 131072 0;} @font-face {font-family:Cambria; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-1610611985 1073741899 0 0 159 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --></style>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-85576729872421086122019-05-19T20:49:00.000-07:002019-05-19T20:49:39.165-07:00Teaching Basic Geometry<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-yfiK_AiPLwE/XOIiYs3u_CI/AAAAAAAAFOM/T8MRUuXu7tQcTYheZ0_3LKWAsibGK93ZQCLcBGAs/s1600/cat%2Bgeometry.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="514" data-original-width="600" height="274" src="https://4.bp.blogspot.com/-yfiK_AiPLwE/XOIiYs3u_CI/AAAAAAAAFOM/T8MRUuXu7tQcTYheZ0_3LKWAsibGK93ZQCLcBGAs/s320/cat%2Bgeometry.jpg" width="320" /></a></div><br /><br /><br />Our college offers a course that is pretty much the same as high school geometry. I have never before taught it because we only offer it in the summer, and summer semester is squished into 6 weeks. I haven't wanted to teach math on that sort of schedule because it seems like it would be hard for students to learn well that quickly.<br /><br />But I'm under load and I love geometry, so I will be teaching this course in a month. I've been preparing ever since January - reading the text, making plans and handouts, doing constructions. Geometry doesn't have the same flow for me that algebra does. I thought that might be because I've been teaching algebra for 30 years. But other math educators have also noticed how much harder geometry can be. Brain teasers everywhere. I like that, but I also need to figure out how to make this doable for the average student. I have gone through the book twice now, thinking about what they'll be learning, and hoping to notice the places they'll get stuck.<br /><br />We will be doing lots of compass and straightedge constructions. Even though I loved geometry in high school, I hated that compass. I'd press too hard and open it further in the middle of an attempted circle. Or not hard enough, and then the pencil wouldn't draw. Arghh. Computer tools make that a problem of the past. I've mentioned two sites in previous posts, <a href="http://sciencevsmagic.net/">sciencevsmagic.net</a> and <a href="http://euclidea.xyz/">euclidea.xyz</a>. But euclidea won't let you unlock a higher level until you succeed at the lower level, and I found that frustrating. <a href="https://www.mathed.page/" target="_blank">Henri Picciotto</a> pointed out that <a href="http://euclidthegame.com/">euclidthegame.com</a> is much like euclidea, but doesn't require you to solve one level before doing the next. After checking it out, I'm not sure I have any reason to go back to euclidea; euclidthegame is much more satisfying.<br /><br /><br /><b>Translating a Line Segment </b><br />So one of the tools I'll be using with my class is euclidthegame. I had a blast working through the levels back in January, and wanted to redo them now to see which sorts of things might be hard for the students. I got stuck on level 7, translating a line segment. I replicated two solutions given in the comments, but I didn't understand why the steps worked.<br /><br />When I looked up 'translate a line segment', all of the sites I found said to hold your compass at the right size and move it over. That doesn't work on euclidthegame, and wasn't part of the original Greek compass and straightedge protocol. I almost gave up on understanding this construction, and thought about telling students they could skip this one, since our book does it the easy way too.<br /><br />People say believing you can do something helps you do it, but I often succeed right after giving up. Ornery, I guess. I doodled a bit, and realized that thinking about a parallelogram was the key. After that, I was able to solve it quickly. I want to share my solution here. (Turns out, this idea was in the comments at euclidthegame, I just hadn't seen it.)<br /><br />The idea is to make a line segment congruent and parallel to AB, with C as one endpoint of the new segment. If you imagine making a parallelogram with the old segment and the new one, and if you know that diagonals of a parallelogram intersect at their midpoints, it becomes relatively straightforward.<br /><ol><li>Sketching parallelogram ABDC, </li><li>We know that we can construct the diagonal BC, </li><li>And find its midpoint (E). </li><li>Then make a ray from A through E, </li><li>And a circle centered at E through A. </li><li>Where that circle and ray meet will be D. (Because E is now the midpoint of AD.)</li><li>CD is the translated line segment we wanted. </li></ol> Lovely.<br /><br />The students won't have enough information when they get to this construction in the book (chapter 1) to do it the right way. We'll have to wait until chapter 4, where they will find out why those diagonals bisect each other. I sure am glad to know that ahead of time.<br /><br /><b><br /></b><b>Planning the Course</b><br />We will be meeting from 8 to 11:35 am four days a week. Who is going to get there on time at 8am if they don't have to? So I will start with a quiz every day. If they don't do well, they can retake it (new questions, same topics) outside of class time. [So a real sleepyhead could still ace the quizzes, even if they came late. But that will take some serious dedication. That's cool.]<br /><br />Then I'll do my usual combination of lecture, pairwork, groupwork, and guided discussion from 8:15 to 9:45. After they take a break they will have lab time. Partly because I don't believe anyone can concentrate for 3 1/2 hours in a regular classroom, and partly because I think they really need to get their hands on some of these tools. I'll have geoboards and dot paper to record their geoboard results. I'll be using most of Henri Picciotto's lovely <a href="https://www.mathed.page/geometry-labs/index.html" target="_blank"><i>Geometry Labs</i></a> book. (Free. Thank you, Henri!) And they'll be using both euclidthegame and <a href="https://www.geogebra.org/" target="_blank">geogebra</a> to do their constructions. Lab projects are 20% of their grade. They'll have to do constructions, geoboard activities, proofs, and one activity of their choice. <br /><br />Anyone teaching geometry who'd like more details, please ask. And anyone who would like to share tips, I'm all ears.<br /><br /><br /><br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-48025788058788050992019-04-01T14:36:00.000-07:002019-04-01T14:36:30.225-07:00Playful Math Ed Blog Carnival (aka Math Teachers at Play) #126<a href="https://4.bp.blogspot.com/-wUcPqYc5Jso/XKJDh1DwWgI/AAAAAAAAFGw/NRyX9aBF_Vg4RxSRpR1P0pmErDEbOwxXgCEwYBhgL/s1600/baby-chicks-3-300x198.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="171" data-original-width="187" src="https://4.bp.blogspot.com/-wUcPqYc5Jso/XKJDh1DwWgI/AAAAAAAAFGw/NRyX9aBF_Vg4RxSRpR1P0pmErDEbOwxXgCEwYBhgL/s1600/baby-chicks-3-300x198.jpg" /></a>126 = 6*21 = <span style="color: red;">2</span>*<span style="color: blue;">3</span>*<span style="color: blue;">3</span>*<span style="color: lime;">7</span><br /><br /><br />If you want to choose 4 chicks randomly from 9 total chicks, there are 126 ways to do it.<br /><br /><br />Students learn more if they make up the stories for story problems themselves. Can your students make up stories for these ways of making 126?<br /><br />126 = 2<sup>7</sup> - 2<sup>1</sup> (difference of powers of 2)<br />126 = 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> + 7<sup>2</sup> (sum of consecutive squares) <br />126 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers) <br /><div style="text-align: right;"> (from <a href="http://www.archimedes-lab.org/numbers/Num70_200.html">http://www.archimedes-lab.org/numbers/Num70_200.html</a>)</div><br /><div style="text-align: right;"><br /></div><div style="text-align: left;">This blog carnival has evolved from being mainly contributions to being mainly items the blog host has discovered. Since my passion lately is geometry, this issue is dedicated to <span style="color: purple;"><b><span style="font-size: large;">geometry</span></b></span>. (Which of the 3 ways of making 126 above has a geometric interpretation? Hint: There's a picture of it here... somewhere...)<br /><br /><br /><br /><b>Constructions</b><br />I have been intrigued for the past few years with Archimedes' method of finding pi. He didn't have the square root symbol, so he approximated using fractions, getting pi between 3 10/71 and 3 1/7. But if we follow his steps, and keep the square roots, we get a lovely pattern for our answer.You can try it. Construct a hexagon in a circle. If the radius of the circle is 1, then the hexagon's perimeter is 6. Perimeter over diameter = 6/2 = 3. Now create a dodecagon (12-sided polygon) from the hexagon. You can find the side lengths from repeated use of the Pythagorean theorem, and then find perimeter over diameter. Your result will be closer than for the hexagon. You can repeat this process until a pattern emerges. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-3jDmRswYY4g/XKJQ6o6nOMI/AAAAAAAAFHE/uxQd1CiSEuAO3OqGwiUCCWktUuDQ8XEkwCLcBGAs/s1600/construction2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="368" data-original-width="516" height="228" src="https://4.bp.blogspot.com/-3jDmRswYY4g/XKJQ6o6nOMI/AAAAAAAAFHE/uxQd1CiSEuAO3OqGwiUCCWktUuDQ8XEkwCLcBGAs/s320/construction2.png" width="320" /></a></div><br /><br />If you want to get better at geometric construction (straightedge and compass style), play with it at <a href="https://sciencevsmagic.net/geo/" target="_blank">sciencevsmagic</a> or <a href="http://www.euclidthegame.com/" target="_blank">euclidthegame.</a><br /><br />You can improve your geometric reasoning skills with the puzzles in <a href="https://solvemymaths.com/2017/12/03/geometry-snacks/" target="_blank"><i><b>Geometry Snacks</b></i></a> (and <i><b>More Geometry Snacks</b></i>), by Ed Southall and Vincent Pantaloni. There are more puzzles at his blog. If you like them, the book is a treasure trove.<br /><br />Because I've fallen in love with geometry, I decided to teach it this summer, for the first time ever. So I'm doing a lot to prepare. Henri Picciotto is an expert geometry teacher who graciously offered me his time over breakfast. He advised me to download his <a href="https://www.mathed.page/geometry-labs/index.html" target="_blank">Geometry Labs</a> book (free) from his Math Ed Page site. There is so much more there than this. But this alone was a huge gift. I think it may transform my course.<br /><br /><br /><a href="https://4.bp.blogspot.com/-nA-3Mqx03V0/XKJ9O4CY8II/AAAAAAAAFHQ/3IiOaHeetE0D4rHw_ic0JBS9_jDJCis2gCLcBGAs/s1600/pentagram.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="559" data-original-width="591" height="188" src="https://4.bp.blogspot.com/-nA-3Mqx03V0/XKJ9O4CY8II/AAAAAAAAFHQ/3IiOaHeetE0D4rHw_ic0JBS9_jDJCis2gCLcBGAs/s200/pentagram.png" width="200" /></a>I've been collecting geometry mysteries. Medians are the lines from midpoints of the sides of a triangle to the opposite vertices. The 3 medians seem to always cross at one point. Why is that? I tried for weeks to prove it, and just couldn't. I finally gave up and looked at the proof. (And told my students how much fun I had failing!) I then found another proof that followed a very different path. Can you prove it?<br /><br />Here's a simpler mystery: If you make a 5-pointed star (perfectly even, I can't do that without digital help...), what is the angle at each point?<br /><br /><br />One of my favorites for seeing the geometry in math topics you didn't know were geometric is <a href="https://www.facebook.com/magicpi2/videos/1286450834841623/?__xts__[0]=68.ARDmt_2oyE4H0IuYYZQFo4t4sQmCtQBFtLesqoGzsheGcF7sbtzMLAUlzbF1S_E1CedPAr5tiu6pTAprqj4U3AssRvyHfneOkXFr-2ETSBv1GAjUX0NCOQaCwyT4oIbtP2-T2wbzSPCosCtuxXAI5uzk55POVqqvGdEbjZEv0c4TFRXMbQ496JKm6h0eIqii-10c-s5FLNm_9FnMkuS9xyuGwFgYgNnp7827AmcIvw_jxH1T08_fz7xVnZ7zPG2XkyErx9dRXcLyZ-Hc85_Pm0wdN07l5VlbTni2XPVIu_OtLmG0zdbIXIjnf96lmQNQbsrQOTM3ND2iekLJZdi5H7ctbcMTpW24A-rHwA&__tn__=-R" target="_blank">Magic Pi - math animations</a>. I hate that they're only on Facebook because I am not comfortable linking to facebook in class. But they are amazing. (I linked to one that's pure geometry. So cool.) They apparently do most of their animations in <a href="http://geogebra.org/" target="_blank">geogebra</a>. I am a complete novice next to them. Here's a geogebra sketch I made today. It might be my first in their 3D mode.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MF9H7HV0css/XKJOHU-CUKI/AAAAAAAAFG4/-fvEBbMQj_wttCk2fINZYO9hAtCxZ5xuACLcBGAs/s1600/stack%2Bof%2Bsquares.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="479" data-original-width="709" height="216" src="https://1.bp.blogspot.com/-MF9H7HV0css/XKJOHU-CUKI/AAAAAAAAFG4/-fvEBbMQj_wttCk2fINZYO9hAtCxZ5xuACLcBGAs/s320/stack%2Bof%2Bsquares.png" width="320" /></a></div><br /><br /><br /><br /><b>Making Your Own Math</b><br />At the beginning, I mentioned having students make up their own story problems. <a href="https://www.arithmophobianomore.com/starting-points-building-story-problems/" target="_blank">Here's a lovely post from <i>Arithmophobia No More</i></a> about just that. <a href="http://mathstateofmind.blogspot.com/2019/03/numberless-word-problems-and-bar-models.html" target="_blank">Here's another angle on teaching story problems</a>, from Jen at Math State of mind. Leaving out the numbers helps students to slow down. <br /><br /><a href="https://whenlifegaveuslemons.wordpress.com/2019/02/21/create-your-own-math-game/" target="_blank">This blog post, by Amy at <i>When Life Gave Us Lemons</i></a>, is about her son making up his own math games. And <a href="http://mathhombre.blogspot.com/2012/04/multiplying-game-possbilities.html" target="_blank">John Golden has a whole class make up variations</a> on a game he shared with them.<br /><br /><br />Denise Gaskins, founder of this carnival, pulls together so many books and ideas I love in <a href="https://denisegaskins.com/2014/08/13/fractions-15-110-180-1/" target="_blank">this post</a>. I don't know how she does it! The (surface) topic is fractions, but more than that, it made me think about how we can help students learn by saying less. The video she includes, with a teacher asking the two boys questions, and never telling them they're wrong, is fabulous. One of the commenters at Denise's post linked to <a href="https://established1962.wordpress.com/2014/08/14/a-productive-conversation-with-susie/" target="_blank">a discussion of his own with a student</a>. And that made me think about <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxaTdjODd6cFkxMjQ/view" target="_blank">Bob Kaplan's guide to 'becoming invisible'</a> (or not giving away the math). (What math delights have you found lately by following your nose? Bunny hops rock!)</div><div style="text-align: left;"><br /><br /><br />You can check out the <a href="https://aperiodical.com/2019/03/carnival-of-mathematics-167/" target="_blank">Carnival of Mathematics here</a>. And if you'd like to host <i>this</i> carnival (we need help next month!), you can <a href="https://denisegaskins.com/mtap/" target="_blank">learn more and sign up here</a>.<br /><br /><br /><br /> </div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-13491867549079422772019-03-22T11:43:00.000-07:002019-03-22T11:43:05.595-07:00Coming soon: Math Teachers at Play (aka Playful Math Ed) blog carnivalI'll be posting the blog carnival here sometime late next week. Right now I'm beginning to gather links to lovely, playful math ed posts (and sites and videos and ...). If you know of something I should include, please email me at mathanthologyeditor on gmail.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-dkKfh1Uj1M8/XJUsgIJwf5I/AAAAAAAAFFI/bNsUIA1Y4pkmNZK4QzY8yuYzEZh4YWrrACLcBGAs/s1600/fireworks.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="183" data-original-width="276" height="424" src="https://4.bp.blogspot.com/-dkKfh1Uj1M8/XJUsgIJwf5I/AAAAAAAAFFI/bNsUIA1Y4pkmNZK4QzY8yuYzEZh4YWrrACLcBGAs/s640/fireworks.jpg" width="640" /></a></div><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-88159732612540827282019-02-12T12:07:00.000-08:002019-02-12T12:10:05.160-08:00When Math Tells a Story<div class="separator" style="clear: both; text-align: center;"><a href="https://media1.tenor.com/images/3a620033bd9d5725d53b542b00accea4/tenor.gif?itemid=4852871" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="498" data-original-width="498" height="320" src="https://media1.tenor.com/images/3a620033bd9d5725d53b542b00accea4/tenor.gif?itemid=4852871" width="320" /></a></div><br /><br /><br />On the <a href="http://groups.yahoo.com/group/LivingMathForum/" target="_blank">Living Math Forum</a> group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):<br /><br /><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">The equation for a circle is (x-h)<sup>2</sup> + (y-k)<sup>2</sup> = r<sup>2</sup> . Many students see each equation like this as separate from any other equations/formulas they know. I try to get them to look at this deeply. From the structure of it (square plus square equals square), I see that it's really the <a href="https://mathmamawrites.blogspot.com/2012/10/proving-pythagorean-theorem.html" target="_blank">Pythagorean Theorem</a>. Why would something for right triangles show up in the equation of a circle?! (That blew me away a few years back. I've been teaching math for 30 years, but that question seemed deep.) </div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">It's because our coordinate system has the two axes perpendicular to each other. So the distance from the x-coordinate of a point on the circle to the x-coordinate of the center is measured horizontally and the similar y distance is measure vertically. You can build a right triangle from the center to (almost) any point on the circle. The constant radius is the hypotenuse of that right triangle. <br /><br />So this equation tells a little story.</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">How does a whole course tell a story?</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">Algebra is about solving equations and about graphing. We want to see how real life situations (anything with data that has two components, like time and height) can be represented with equations and with graphs. In Algebra 1 students learn to solve simple equations and to graph. And hopefully they learn how the two skills are connected. There's a bit more. Systems of equations allow us to model slightly more complex situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after factoring) is graphing and recognizing equations of parabolas. <br /><br />Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">The course can tell the tree's story, or any story that can be told through data, graphs, and/or equations. It tells the story of using math to help us think quantitatively about problems we care about. (And of course there are plenty of things we care about that cannot be quantified.)</div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-57200343759117636702019-02-10T21:58:00.001-08:002019-02-10T21:58:13.032-08:00Links to Good Math PostsI am closing tabs, so that <i>maybe</i> my browser will move faster. Here are all the links I couldn't stand losing:<br /><ul><li><a href="https://www.youtube.com/user/misterwootube" target="_blank">Mr. Woo's Math Channel on youtube, aka WooTube</a>. This came recommended by the amazing Julie Brennan (of <a href="https://www.livingmath.net/" target="_blank">Living Math</a>).</li><li>From Math with Bad Drawings, <a href="https://mathwithbaddrawings.com/2018/08/01/the-teacher-who-only-says-gimme/" target="_blank">The Teacher Who Only Says Gimme</a> (part of the <a href="https://samjshah.com/mathematical-flavors-convention-center/" target="_blank"><i>Convention of Mathematical Flavors</i></a> blogfest), on getting students to think creatively in math.</li><li><a href="https://www.mathteacherscircle.org/news/mtc-magazine/f2018/we-the-people/" target="_blank">We the People</a>, the math behind various preferential voting schemes. (And why there is not a best way.) From the Math Teacher's Math Circle Network.</li><li><a href="https://www.nytimes.com/interactive/2018/06/13/upshot/boys-girls-math-reading-tests.html" target="_blank">Sometimes boys and girls do equally well, other times not. What is involved? New York Times article.</a></li><li><a href="http://busynessgirl.com/favorites/contemporary-algebra-collection/" target="_blank">Busyness Girl's Contemporary Algebra Collection</a></li></ul>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-18783376296012761832019-01-10T19:38:00.000-08:002019-01-10T19:40:36.019-08:00Geometry and Visual Thinking Puzzle Collection - Help Needed<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-7ra_rFJvtJg/XDgN28BCMkI/AAAAAAAAE9U/hTP68t-gLdIzrDccvuS9EBa0BI2u9ZYxgCLcBGAs/s1600/4%2Brectangles.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="290" data-original-width="311" src="https://4.bp.blogspot.com/-7ra_rFJvtJg/XDgN28BCMkI/AAAAAAAAE9U/hTP68t-gLdIzrDccvuS9EBa0BI2u9ZYxgCLcBGAs/s1600/4%2Brectangles.png" /></a></div><br /><br /><br /><br />Math lovers, I need your help. (Math-likers welcome too. Math-haters who are into torturing themselves are also quite welcome, if you know any.) If you love puzzles, that might be enough to suck you in. I'm hoping...<br /><br /><br />I am pulling together a collection of problems meant to entice students who aren't all that into math. (So, relatively easy puzzles.) I think we focus too much on algebra and not enough on the visual skills that come along with geometry. So each problem in this collection include<span class="text_exposed_show">s visualizing. Most of them involve geometry.</span><br /><br /><div class="text_exposed_show">I'm calling them PPODs - Puzzle Problem Of the Day. I plan to post them in my classrooms and in the math department of the community college I work at.<br /><br />I have found quite a few good problems in <i>Geometry Snacks</i>, by Ed Southall and Vincent Pantaloni. (I bought the e-version of the book yesterday.) I just now discovered Ed's blog (<a data-ft="{"tn":"-U"}" data-lynx-mode="asynclazy" href="https://l.facebook.com/l.php?u=http%3A%2F%2Fsolvemymaths.com%2F%3Ffbclid%3DIwAR1vkFgv7KbIBWKvUCmNmx9feUiBTQwd8sTFLA5c46CSJtcex5q2gEgFpKs&h=AT2L_1Xq97XC_3LyTdBFc4hqB0BVSjKgm5Xw9rxjungeeAaK6m8Xu-gs300OxOtULWWm8L6Y2kkhQp1oIAFfPBXkLzXNxXYtsB80Y479iSgFCOiFo6VQYCFzXeF6D80C6b-q9gHcRu9waeDRTyGpOOSuoVKnPAaH0RyF" rel="noopener nofollow" target="_blank">solvemymaths.com</a>), and found more goodies. My other favorite source is the beautiful by-hand drawings done by Catriona Shearer, @cshearer41. I couldn't find who to credit for about a quarter of the problems I've collected.<br /><br />I'm aiming for 60 problems, one for (almost) each day (M-Th) of the 16-week semester. I have about 50 already. What I need is help determining the difficulty level of each one. I need volunteers to help me figure that out. I'm using a scale of 1 to 5 myself, and anything I'd call a 4 or 5, I'm throwing out. (I've gotten stuck on a few problems. Those are not in the collection, of course.)<br /><br />If you're interested, I'll send you my collection. And then you post your ratings <a href="https://docs.google.com/spreadsheets/d/1MttNarw13gRA9UMkBVRGbpVKyOXJ3kK5yyiZ1vuQZV4/edit#gid=0" target="_blank">here</a>.<br /> Just let me know in the comments if you'd like to check these out. (Or email me at mathanthologyeditor@gmail.com.) Also, I'm interested in your ratings and thoughts even if you just end up doing a few of the puzzles.</div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-39529956116951815772018-11-28T16:32:00.001-08:002018-11-28T16:32:24.174-08:00Logarithms: How Math Helps Science<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-RV7f8jCIUIQ/W_8vtfc__HI/AAAAAAAAE0k/2Hefz9_QXigjSn5pWHtIWPzW8yJE2aHcACLcBGAs/s1600/kepler2law.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="249" data-original-width="398" height="199" src="https://2.bp.blogspot.com/-RV7f8jCIUIQ/W_8vtfc__HI/AAAAAAAAE0k/2Hefz9_QXigjSn5pWHtIWPzW8yJE2aHcACLcBGAs/s320/kepler2law.JPG" width="320" /></a></div>I'm teaching my algebra students about logarithms today. It is likely the hardest algebra topic there is. When I started at Contra Costa College, after having taught full-time at Muskegon Community College for 6 years, I think I still was not 100% solid on logs. (I could do the problems, but proving those properties was a tangled mess.)<br /><br /> I had developed <a href="https://mathmamawrites.blogspot.com/2010/04/murder-mystery-project-for-logarithms.html" target="_blank">my murder mystery project</a> because my officemate at MCC was a chemistry teacher, and she had complained that teachers weren't teaching logs well enough in Intermediate A<span class="text_exposed_show">lgebra.I hoped that project would help. I think it does, but I don't have anything solid to base that on.</span><br /><span class="text_exposed_show"><br /></span><div class="text_exposed_show"> For many years now, I've told students a bit of history in my introduction to logarithms. John Napier invented them in 1614 as a way of making hard calculations easier. (Can you imagine finding the square root of 192.7 without a calculator? You'd have to guess, check, and modify repeatedly...)<br /><br />Around that time Kepler came up with his three laws of planetary motion. Over the years, I have repeatedly wondered out loud whether Kepler was using logarithms while he figured these things out. <br /> Today when this came up, I was able to turn to our tutor and ask him to look it up. He found a <a href="https://www.mathpages.com/rr/s8-01/8-01.htm?fbclid=IwAR3bUzHl523g6gm-U97PVLLZfXuE5Un2Z2OcxerfjjfU6NsOOJ-Hxf6rwdE" target="_blank">fascinating article</a> on it.<br /><br /> Turns out Kepler came up with the first two "laws" before playing with Napier's new idea. But his third came after.<br /><br /> I. Planets move in ellipses with the Sun at one focus.<br /> II. The radius vector (line from sun to planet) describes equal areas in equal times.<br /> III. "The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances."<br /><br /> This is an improper use of the word proportion. He was making a proportion from the logs of the ratios, and saw that log(T1/T2) = 1.5*log(r1/r2). In modern terms, we say something like: The square of the ratio of the times (that it takes two planets to go around the sun) equals the cube of the ratio of their mean distances from the sun. The fact that he was thinking in terms of logs shows how helpful the new (in 1614) concept was for helping scientists see patterns. (<a href="https://www.mathpages.com/rr/s8-01/8-01.htm?fbclid=IwAR3bUzHl523g6gm-U97PVLLZfXuE5Un2Z2OcxerfjjfU6NsOOJ-Hxf6rwdE" target="_blank">Read that article</a> for more on Kepler and Napier.)<br /><br />I love that math can help us see new things. </div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com1tag:blogger.com,1999:blog-5303307482158922565.post-31119513984073600172018-09-08T16:17:00.002-07:002018-10-04T19:46:13.164-07:00Geometric Construction of the Regular Pentagon<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-oYXNhVqh9us/W4ywV5RDpEI/AAAAAAAAEnc/FMxwu0EqB-k8HFhzUjIQkZE0Fy27evQXgCLcBGAs/s1600/1st%2Bpentagon.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="721" data-original-width="985" height="292" src="https://2.bp.blogspot.com/-oYXNhVqh9us/W4ywV5RDpEI/AAAAAAAAEnc/FMxwu0EqB-k8HFhzUjIQkZE0Fy27evQXgCLcBGAs/s400/1st%2Bpentagon.png" width="400" /></a></div><br />I never particularly enjoyed geometric constructions when I was younger. It may be because I had a tendency to press down too hard, and mess up the width my compass was set to. Like so many things, technology has a fix for that.<br /><br />You can do geometric constructions on a number of sites, and in <a href="https://www.geogebra.org/" target="_blank">geogebra</a>. My favorite site, which <a href="https://mathmamawrites.blogspot.com/2013/07/euclidean-construction-science-vs-magic.html" target="_blank">I discovered about 5 years ago</a>, is <a href="http://sciencevsmagic.net/">sciencevsmagic.net</a>. I having been telling students about it for years, and decided this summer to play with it again, to see how much I still like it. (A lot, it turns out.)<br /><br />There are 40 challenges, in sets of 4. This time around I got every shape but the pentagon. I asked for hints (from math friends on facebook) which I then avoided reading because I wanted to do it on my own. I got nowhere that way, and finally followed instructions for how to construct it. Every other shape I have constructed makes sense on its own, but the construction of a regular pentagon must be proved with algebra. Ok, I don't feel so bad about 'cheating'.<br /><br />I kept constructing it over and over, as I tried to understand it better. The first time (which you see above) I used <a href="http://www.opentutorial.com/How-to-Construct-a-Pentagon/construction" target="_blank">these instructions</a> (which I found by following links from wikipedia) and constructed it at sciencevsmagic, achieving my final shape challenge. Then I did it again using geogebra. I needed to figure out how it worked, and was at a loss. <a href="http://www.cut-the-knot.org/pythagoras/pentagon.shtml" target="_blank">Cut-the-knot</a>, a site I've gone to many times with my math questions, had a different construction - which I followed on both sciencevsmagic and geogebra - and an explanation (by Scott Brodie) that I kept working through as I wrote this.<br /><br />[Sadly, Alexander Bogomolny, the creator of the cut-the-knot site, has died. I hope the math blogging community can find a way to maintain his amazing site. We miss you, Alexander.]<br /><br />I will explain this as I understand it. My explanation will mostly just restate what I learned from these sites. But it may help some people, since some of the reasoning steps in Scott's explanation were hard for me to follow.<br /><br />The constructions are worth doing first, so you can get a feel for what's happening as you do it. But when you're done, the question is still open - is this <i>really</i> a regular pentagon? The following explanation proceeds in two major steps. First it looks at relationships in the regular pentagon and the pentagram (which is a regular pentagon with a five-sided star inside it). Then it looks at the relationships built by the construction method, and we finally see that the side length given in the two ways is the same. Bingo!<br /><br /><br /><br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-leVPM5Oo39Y/W49YYua6XFI/AAAAAAAAEoM/iAakg5reoMYIvOFTrUdzP7a_KvtsFc5kACLcBGAs/s1600/pentagon%2Bv2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="696" data-original-width="709" height="314" src="https://3.bp.blogspot.com/-leVPM5Oo39Y/W49YYua6XFI/AAAAAAAAEoM/iAakg5reoMYIvOFTrUdzP7a_KvtsFc5kACLcBGAs/s320/pentagon%2Bv2.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><b>The Construction</b><br />If you're ready to follow this, I trust that you already know how to construct the perpendicular bisector of a line segment. I will leave out the construction details for each of those.<br /><ul><li>Starting with a circle with center O, construct a horizontal diameter, AB.</li><li>Now construct its perpendicular bisector, CD.</li><li>Construct the perpendicular bisector of AO, intersecting it at E. </li><li>Construct the circle with center at E, through C, labeling its intersection with AB as F.</li><li>Construct the circle with center at C, through F, labeling its intersection with the original circle (above B) as G.</li><li>Go around the original circle, constructing this same size circle from each new point. (Center at G, through C, gives H at the other intersection with the original circle. Etc.)</li></ul>The question remains: Does this really construct a regular pentagon, or is it maybe just pretty close?<br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Blx1DUz_W-Q/W49ZLf3UsPI/AAAAAAAAEoU/wSKrp1CNzqocUpUTfxEqgN2ZX1oXaeMHwCLcBGAs/s1600/pentagram%2Bv2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="700" data-original-width="692" height="400" src="https://4.bp.blogspot.com/-Blx1DUz_W-Q/W49ZLf3UsPI/AAAAAAAAEoU/wSKrp1CNzqocUpUTfxEqgN2ZX1oXaeMHwCLcBGAs/s400/pentagram%2Bv2.png" width="395" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br /><b>Relationships in a regular pentagram</b><br /><ul><li>The angles in a regular pentagon are each 108<span class="ILfuVd yZ8quc">°, so each external angle, like </span>∠HMN, is 72<span class="ILfuVd yZ8quc">°. That makes each angle at a point, like </span><span class="ILfuVd yZ8quc">∠NHM, 36</span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°. Since </span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠HMG is 108</span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°, </span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠MHG and </span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠MGH must each be 36</span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°. </span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"> Since angles </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠GLH and </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠LGH are each 72</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°</span></span>, </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Δ</span></span></span></span></span></span></span></span></span></span></span></span></span>HGL has two equal sides, HL and HG (which also equal CM, HI, etc).</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">ΔCHI is similar to </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Δ</span></span></span></span></span></span></span></span></span></span></span></span></span>IHM. So corresponding sides are in proportion. We get CH/HI=HI/HM, giving us HI<sup>2</sup> = CH*HM = CH(CH-CM) = CH(CH-HI).</span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Let CH = x*HI. Then the equation above gives us </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">HI<sup>2</sup> =x*HI(x*HI-HI), which gives us a (perhaps) familiar equation: 1 = x</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>-x, whose solution (in this case, where x is clearly greater than 1) is x = (1+</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2 aka φ. (At this point, I get excited. φ, aka the golden ratio, shows up in such different contexts!)</li><li>So CH = φ*HI, and (similar triangles) HI = φ*HM and HM = φ*MN.</li><li>We now have relationships between all the sides of the pentagon, star, and line segments on the star, but these are not yet connected to the radius of the circle.</li><li>Now consider ΔODH. Two sides are radii, and the angle at O is 36°. Another triangle similar to all the others we've found. So OD = φ*DH.</li><li>Also note that ∠DHC (inscribed angle on the diameter) is a right angle.</li><li>Let's switch to simple variables names now. Let r=OD, s=CG, t=DH, and d=CI=CH.</li><li>Then d= φ*s, r= φ*t, and t<sup>2</sup>+d<sup>2</sup> = (2r)<sup>2</sup>, giving (r/φ)<sup>2</sup> +(φ*s)<sup>2</sup> = 4r<sup>2</sup>.</li><li>This gives us s<sup>2</sup> = 1/φ<sup>2</sup>*(4r<sup>2</sup> - (r/φ)<sup>2</sup>) = (4/φ<sup>2</sup> - 1/φ<sup>4</sup>)*r<sup>2</sup>. </li></ul><br /><br /><br /><b>Analyzing our construction</b><br /><ul><li>We will find the length of s = CG = CF.</li><li>OE=1/2*r. And EC = <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√(OE</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>+OC</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>) = </code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√(5/4)*r = </code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2*r = EF. OF = (<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2- 1/2)*r.</li><li>So <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">s<sup>2</sup> = </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>CG<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = CF<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = OC<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> + OF<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = (1+(<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2- 1/2)<code><sup>2</sup>)*r<sup>2</sup>.</code><code></code></li><li>We have two very messy expressions for s<sup>2</sup>, one from the regular pentagon/pentagram and one from our construction. Are they equal? It may help to write out positive and negative powers of <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">φ </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>to help simplify the first expression. Yes! They are both equal to (5-<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2*r<sup>2</sup>, making the side length of the regular pentagon inscribed in a circle of radius 1 equal the square root of (5-<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2.</li><li>And that proves that our construction created a regular pentagon. (Whew!) </li></ul><br /><br />I've completed 32 of the 40 challenges at <a href="https://sciencevsmagic.net/geo/" target="_blank">sciencevsmagic</a>. I still haven't figured out how to make some of the shapes within the original circle, and still haven't found the least moves for some of the shapes. I'm so glad there's still something to work toward the next time I revisit this.<br /><br /><br /><br />[Blogger doesn't deal well with superscripts and square roots. Once again, I am noticing that I ought to learn some LaTex. Sigh.] <br /><ul></ul><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com5tag:blogger.com,1999:blog-5303307482158922565.post-29660198845747253032018-07-01T18:20:00.000-07:002018-07-01T18:20:23.674-07:00Math Teachers at Play #118<br /><a href="https://2.bp.blogspot.com/-teS_QhiW1AI/Wzl15QPAiEI/AAAAAAAAEWw/8Iq6MsiunOQyBUY6MtTgB1orslYj3CtegCLcBGAs/s1600/118%2Bv4.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="225" data-original-width="225" height="320" src="https://2.bp.blogspot.com/-teS_QhiW1AI/Wzl15QPAiEI/AAAAAAAAEWw/8Iq6MsiunOQyBUY6MtTgB1orslYj3CtegCLcBGAs/s320/118%2Bv4.jpg" width="320" /></a>In two more months, we'll hit MTAP #120, which <i>should</i> be the ten-year anniversary mark. Is it?<br /><br /><br /><br />The number 118 ...<br />... factors to 59*2.<br />... is 1110110 in base two, 11101 in base three, 1312 in base four, 433 in base five, and 226 in base 7.<br />Not particularly exciting...<br /><br /><br /><br />Aha! Can you make 118 using four 4's? (I did.) I wonder if you can make it using five 5s, or ...? <br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /><b>Summertime Learning</b><br /><a href="https://www.mathmammoth.com/lessons/summer_math.php" target="_blank">Summer Math Resources</a> from Math Mammoth's Maria Miller. <br /><br />Denise Gaskins offers us one of her FAQs, on <a href="https://denisegaskins.com/2018/06/15/faq-forgetting-what-they-learned/" target="_blank">forgetting what they learned</a>. My favorite of all her suggestions? Play games! <br /><br />On <a href="https://emergentmath.com/2018/06/25/routines-the-driving-beat-of-your-class/" target="_blank">Routines</a> and <a href="https://emergentmath.com/2018/06/28/lessons-the-stuff-were-used-to-but-better/" target="_blank">Lessons</a> <br />I like Geoff's perspective: "What’s the shortest amount of time you could possibly do the talking? Go with that. And maybe subtract a few more minutes." After 30 years of teaching, I am still learning, and I still get excited when I see something that might make a difference for my students. This looks helpful, and perfect for my summer meditations on teaching.<br /><br /><br /><br /><b>Young Math </b><br />Can you imagine doing a college math lesson with 2nd graders, and having it work out well? Manan Shah did it! <a href="http://mathmisery.com/wp/2018/06/14/set-theory-for-second-grade/" target="_blank">His lesson was on set theory.</a><br /><a href="https://www.mathmammoth.com/lessons/summer_math.php" target="_blank"><br /></a>Ahh... A whole blog on magical math books. And she wrote <a href="https://mathbookmagic.com/2018/06/18/uncovering-the-complexities-of-counting-with-a-magical-counting-book/" target="_blank">about Christopher Danielson's new book, How Many?</a> If you like this, she also has <a href="https://mathbookmagic.com/blog-book-list/" target="_blank">a list of every book she has posted about</a>. Thank you, Kelly Darke!<br /><br />Remember playing the card game War? If you're a parent, those memories may not be so fond. Kids love it and parents can get sooo bored. Kent wrote <a href="https://www.gamesforyoungminds.com/blog/war-card-game" target="_blank">a great post about his interactions with his son around Addition, Subtraction, and Multiplication War</a>. (If you want a good summary of even more mathy variations on war, <a href="https://denisegaskins.com/2006/12/29/the-game-that-is-worth-1000-worksheets/" target="_blank">check out Denise Gaskins' classic post.</a>)<br /><br /><a href="https://medium.com/@jamestanton/the-global-math-project-uplifting-mathematics-for-all-9253d383e5e6" target="_blank">The first ever Global Math Week</a> went well, with its exploding dots exploding around the world. Thank you, James Tanton, for getting people all over the globe into math.<br /><br />How do you talk about numbers with young children? There are so many ways! Here's another:<a href="http://realteachingmeansreallearning.blogspot.com/2018/06/dice-chats.html" target="_blank"> Counting with Dice</a>, from Dave Martin. <br /><br /><br /><br /><br /><a href="https://3.bp.blogspot.com/-cZfAlzi1CUs/Wzl15qHyafI/AAAAAAAAEXI/lObvsKEgyEUFA1zUBDilQt9NOkawImIeQCEwYBhgL/s1600/118%2Bv3.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="768" height="180" src="https://3.bp.blogspot.com/-cZfAlzi1CUs/Wzl15qHyafI/AAAAAAAAEXI/lObvsKEgyEUFA1zUBDilQt9NOkawImIeQCEwYBhgL/s320/118%2Bv3.jpg" width="320" /></a><b>Calculus</b><br />Which is bigger, e<sup><span class="st">π</span></sup> or <span class="st">π<sup>e</sup>?</span> Sure, you can check it on a calculator. Or <a href="http://blog.mrwaddell.net/archives/1835" target="_blank">you can use areas to see why it's true</a>. Lovely! (from Glenn Waddell)<br /><br />The Fundamental Theorem of calculus says that you can figure out areas by using anti-derivatives. I do a project to help students understand it. Sam Shah does even more. <a href="https://samjshah.com/2018/06/21/part-iii-the-final-turn/" target="_blank">This is Part III of his lessons for this topic.</a> I recommend clicking on the links to Parts I and II first.<br /><span style="background-color: #ead1dc;"><br /></span><br /><br /><b><br /></b><b><br /></b><b>A Few More Goodies...</b><br />Dan McKinnon shares his notes from his <a href="http://www.mathrecreation.com/2018/06/origami-workshop-again.html" target="_blank">Origami Workshop</a>. For those who enjoy doing origami, can you find something new here? <br /><br />Logic and Math go together so well. Check out this <a href="https://mathsvenns.com/nearest-integer-one-decimal-place-two-decimal-places/" target="_blank">blog full of Venn Diagrams to fill in</a>.<br /><br />In his post <a href="https://pballew.blogspot.com/2018/06/from-surds-to-ab-surds.html" target="_blank">From Surds to Ab-surds</a>, Pat Bellew looks at an interesting relationship, which looks like bad simplifying but is still correct. He suggests it as a challenge in an algebra course to produce more correct 'bad simplifying' equations. Hmm, I want to think about how to do that.<br /><br /><br /><br /><br /><b>Until Next Month...</b><br />Our sister blog carnival, <a href="http://aperiodical.com/2018/06/carnival-of-mathematics-158/" target="_blank">The Carnival of Mathematics</a>, often includes posts that are above my head. This month, our big sister is quite approachable. Enjoy!<br /><br />If you have suggestions for next month's MTAP, share via the form at the <a href="https://denisegaskins.com/mtap/" target="_blank">Carnival home page</a>. Sharing in the carnival, or hosting, is a great way to increase connections in the #mtbos/#iteachmath community.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-31276521474713485332018-01-07T22:00:00.001-08:002018-01-07T22:00:40.043-08:00Logic Puzzle - What Does Your Friend See?<div><div><div><div><div>I love logic puzzles, and was drawn by <a href="http://nautil.us//blog/the-logic-puzzle-you-can-only-solve-with-your-brightest-friend" target="_blank">the title saying this was a hard one</a>. Usually Nautilus is well-written, but their version of this puzzle isn't as good (in my opinion) as <a href="https://mindyourdecisions.com/blog/2016/03/20/the-seemingly-impossible-escape-sunday-puzzle/" target="_blank">the original, blogged about by Presh Talwalkar</a>.</div><div> </div><div>The Nautilus version of the puzzle says to imagine your brightest friend. I imagined a friend I know likes logic puzzles (Sharon), but since I wanted a different initial than mine for notation purposes, I imagined another smart friend who <i>might</i> like logic puzzles (Linda). And I began scribbling away with S's (for Sue) and L's.<br /></div>The answer I got is different than the answer the author got because we made different assumptions. Mine were based on his wording, his were based on Presh's wording.</div><div><br /></div><b>Puzzle #1 </b>(with Sue's interpretation):</div>You’ve been caught snooping around a spooky graveyard with your best friend. The caretaker, a bored old man fond of riddles (and not so fond of trespassers), imprisons each of you in a different room inside the storage shed, and, taking your phones, says, “Only your mind can set you free.” <br /><br />To you, he gestures toward a barred window. Through it, you can see 12 statues. Out of your friend’s window, which overlooks the opposite side of the graveyard, she can see eight. Neither of you know the other’s count.The caretaker tells you each, individually, that together you can see either 18 or 20 statues. Unfortunately, there’s no way to tell your friend how many you can spot. <br /><br />The only way for you both to escape is for one of you to give the total number of visible statues. Get it wrong, and neither of you ever leave. The caretaker asks you each once a day [Sue assumed neither person knew who was asked first], and you can choose to answer or to pass. If you both pass on a given day, the question—are there 18 or 20?—is posed to each of you again the next day, and the next, and so on, until you get it right or wrong.<br /></div><b>Puzzle #2</b> <a href="https://mindyourdecisions.com/blog/2016/03/20/the-seemingly-impossible-escape-sunday-puzzle/" target="_blank">is at Presh's blog</a>. (No way to copy-paste that one.) It's now Alice and Bob, and they know that Alice gets asked first, so they'd both be released before Bob is asked if she has it right.<br /></div>I think I have found a solution to (my version of) Puzzle #1. I would love to hear other folks reasoning before posting anything. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-57196971611274875702017-12-31T21:33:00.003-08:002017-12-31T22:27:21.455-08:00Arithmetic, by Paul Lockhart<div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;"><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-hXXBQJafQOw/WknIG70g7lI/AAAAAAAAD78/Rta2KpJ6PWAB3zn2mUdmjKeV2DyerhPcwCLcBGAs/s1600/arithmetic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="192" data-original-width="128" src="https://4.bp.blogspot.com/-hXXBQJafQOw/WknIG70g7lI/AAAAAAAAD78/Rta2KpJ6PWAB3zn2mUdmjKeV2DyerhPcwCLcBGAs/s1600/arithmetic.jpg" /></a></div><br /></div><div style="margin-bottom: 0; margin-top: 0;"><br /><br />This book is part history and part philosophy of arithmetic. He also includes a few exercises in italics (one every few pages). </div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">I love his easygoing tone (arithmetic is not mathematics, it's an art or craft, something like knitting, I enjoy it, and I hope you will), his low-key sense of humor, and his perspective.</div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">I loved his book <i>Measurement</i>, and look forward to someday working through all the challenging problems in it. This one is much easier, and yet it's not boring for me.</div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">It would be a lovely book to read to kids and think through together.</div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">(It came out this year in hardcover. $22.95. I love the cover, though I know I shouldn't judge a book by that...)</div><div style="margin-bottom: 0; margin-top: 0;"><br /><br /><br /><br />[Edited to add: <br /><div style="margin-bottom: 0; margin-top: 0;">I wrote this review before finishing the book, because I was so excited. I've read a few more pages now. Sadly, Lockhart is sometimes sexist. On page 45 he mentions the (Japanese) emperor's concubines, and says "Now, <i>this</i> is why people do arithmetic!" (To please the concubines.) No. It's not. And I thought better of you, Paul Lockhart. </div><div style="margin-bottom: 0; margin-top: 0;"><br /></div>I still think I'm going to love 99% of this book...]</div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-89776316923837675272017-11-04T12:27:00.000-07:002017-11-04T12:28:14.288-07:00Re-reading Archimedes CodexI am re-reading <a href="https://www.bookfinder.com/search/?ac=sl&st=sl&ref=bf_s2_a1_t1_1&qi=29KqANHAecX,oVkeOt4wCqLVadg_1497963026_1:6:5&bq=author%3Dreviel%2520netz%26title%3Darchimedes%2520codex%2520how%2520a%2520medieval%2520prayer%2520book%2520is%2520revealing%2520the%2520true%2520genius%2520of%2520antiquity%2527s%2520greatest%2520scientist" target="_blank"><i>The Archimedes Codex.</i></a> It is fabulous. A detective story of history, science, and math. (I wrote <a href="http://mathmamawrites.blogspot.com/2015/06/book-review-archmedes-codex.html" target="_blank">my first review of it two years ago</a>.)<br /><br />Both authors have a sweet nerdy guy sense of humor, gently self-deprecating, piercing when it should be.<br /><blockquote class="tr_bq">"What are readers today [of science] afraid of? They are afraid of equations. With good reason: they were force-fed such equations for several, terrible years of their childhood and adolescence." </blockquote><br /><div class="text_exposed_show">I actually think he (Netz) is wrong here. I love math, and I love the beauty of some equations, but equations are still intimidating when you don't yet see what they're saying.<br /><br />Here's another math-related quote, pointing in a different direction: <br /><blockquote class="tr_bq">"Archimedes wrote out this problem in verse! A poet-mathematician! - the thought seems to us absurd, but it was natural for Archimedes, whose entire science was built on a sense of play and beauty, on hidden meanings."</blockquote>Not surprisingly, I disagree with him here too. <a href="http://mathmamawrites.blogspot.com/2010/02/math-poetry-wiki.html" target="_blank">Some of us</a> like <a href="http://mathmamawrites.blogspot.com/2009/03/math-poems.html" target="_blank">mixing poetry and math</a>. And the problem written up this way was a silly one. Archimedes was kind of like Martin Gardner with a few of his problems, and it makes sense he'd be playful in his presentation of it. Kind of like <a href="http://mathmamawrites.blogspot.com/2009/12/holiday-logic-puzzles.html" target="_blank">this one by Mike Shenk</a> (sorry for the xmas reference this early).<br /><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-60303062256520126332017-08-14T21:38:00.000-07:002017-08-14T21:47:18.356-07:00First Day, Again<br />First day of classes. I was not as excited as usual. But I had my prep done, and once I got in the classroom, I loved it like always.<br /><br />25 people in beginning algebra. I tell them how math is not about memorizing but about making sense. I get them talking in groups about the first time math didn’t make sense and got past them.<br /><br />I ask them to estimate the percentage of the population that’s uncomfortable with math. First one says 100%. I almost laugh, but manage not to. Percentages all over, 32% up to another at 95%. I tell them I don’t know either but I guess 70 to 80%. So that means most elementary teachers are nervous when they teach fractions, and then they pass it on. I think I see a few nods.<br /><br />I ask how to make meaning of 3 -5. Someone says you go past 0. I say “You’re talking about a number line, right?” And I draw it. I say I like that, but how can we give some real world meaning to this problem. Someone else says “Debt”, but I hear “Death” at first. I shake my head at my bad hearing. (I hope it doesn’t interfere with my teaching.) And I flesh it out. “Yeah, you’ve got $3 in your pocket. And you want to buy a $5 something. What does the 3 -5 tell you?” A student says, “How much you need to borrow.” I add in the temperature model, which I tell them might have worked better for my students back in Michigan who had experienced 3 degrees, and going down 5 degrees from there.<br /><br />Then I give them 31 – 52. They discuss in their groups. It’s easier than I meant it to be.<br /><br />We discuss the syllabus in between other things. I give them a sheet full of magic squares that use negative numbers. Some use fractions. I’ll need to check in tomorrow to see how they did with those.<br /><br />I think the class went well. If they really feel good about it, they’ll end up thinking I’m their best teacher ever. That only happens about a third of the time with this course. I can hope…<br /><br />Then I had to run to my next class. Statistics. I had them average the ages of 5 other students, and type their average onto my computer. I averaged their averages. Which is not the same as just averaging everyone. But it often comes close. Then I put a number line up, and we each put a dot at our age. That’s a dot plot. I showed how it’s skewed right. And talked about how median is a better way to show the center than mean (which is the average they had done).<br /><br />I had an hour to eat lunch and chill.<br /><br />Then I taught calculus. I love that class! Every time. Draw y = x squared. Draw a tangent line at x=3. Estimate the slope of the tangent line. What makes this different from algebra is that we need the idea of infinity here. Ahh… Happy me. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-60599483224795629622017-06-30T22:12:00.000-07:002017-07-01T15:43:31.988-07:00Math Teachers at Play #109 (a blog carnival)<span style="font-size: small;"><b>Who is 109?</b> 109 is a twin prime, twinned with 107. <span style="font-size: x-small;">(from numbergossip.com)</span></span><br /><br /><br /><span style="font-size: small;"><br /></span><br /><a href="https://2.bp.blogspot.com/-dP-LUEneJY4/WVKXWU4NMWI/AAAAAAAADck/7eib_zjkWHUSgC5jJz1_bA-xeJVmxS4IACLcBGAs/s1600/pipe%2Borgan.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1234" data-original-width="1600" height="246" src="https://2.bp.blogspot.com/-dP-LUEneJY4/WVKXWU4NMWI/AAAAAAAADck/7eib_zjkWHUSgC5jJz1_bA-xeJVmxS4IACLcBGAs/s320/pipe%2Borgan.jpg" width="320" /></a><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/949.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="949" /></a> If 109 is written in Roman notation (CIX), then it becomes reflectable along the line it is written on. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/949.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="949" /></a> The pipe organ at the Cathedral of Notre Dame in Paris has 109 stops. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/950.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="950" /></a> When chilled below minus 109°F, CO<sub>2</sub> becomes a solid, called dry ice. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/951.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="951" /></a> 109 equals the square root of 11881 or 118 - 8 - 1. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/15936.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="15936" /></a> The only three-digit prime formed by concatenation of consecutive numbers. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Silva">Silva</a>] </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/18156.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="18156" /></a> 109 = 1*2+3*4+5*6+7*8+9. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Silva">Silva</a>] </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/12541.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="12541" /></a> The Sun is just over 109 times the diameter of the Earth. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Friedman">Friedman</a>] </span><br /><span style="font-size: x-small;"> (from https://primes.utm.edu/curios/page.php/109.html)</span><br /><span style="font-size: small;"><br /></span><span style="font-size: small;"> </span><br /><br /><span style="font-size: small;"><b>A Puzzle</b>: Can you make 109 from four 4's? (I don't promise that it's possible...)</span><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-I91iL-YpHOM/WVKYIv7WLlI/AAAAAAAADcs/09NTs37-iHk1eHOlnkyE9oozhllTLBaCgCLcBGAs/s1600/gggg%2Bgranny.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="624" data-original-width="1024" height="195" src="https://4.bp.blogspot.com/-I91iL-YpHOM/WVKYIv7WLlI/AAAAAAAADcs/09NTs37-iHk1eHOlnkyE9oozhllTLBaCgCLcBGAs/s320/gggg%2Bgranny.jpg" width="320" /></a></div><br /><br /><br /><br /><span style="font-size: small;"><a href="https://twitter.com/tweetsauce/status/661169211310518272?lang=en" target="_blank">At age 109, Augusta Bunge became</a> the youngest living great great great great grandmother. Is that mom to the fifth power? </span><br /><br /><br /><br /><br /><br /><br /><br /><i><b>Math Teachers at Play</b></i> is a monthly blog carnival, hosted at a different blog each month. <span style="font-size: small;">I was hoping to give you 109 math links this month, but life intervened (parenting...) long before I got through my storehouse of cool stuff. There are plenty of goodies here, but not as many as I'd hoped.</span><br /><br /><span style="font-size: small;"> </span><br /><div class="MsoListParagraph" style="margin-left: 0.25in; text-indent: -0.25in;"><span style="font-size: small;"><span style="font-family: "times";"></span></span></div><span style="font-size: small;"> </span><br /><div class="MsoListParagraphCxSpFirst" style="margin-left: 0.25in; text-indent: -0.25in;"><br /></div><span style="font-size: small;"> </span><br /><br /><b> Books</b><br />There has been an explosion of super cool mathy books since I last hosted MTAP. Here are some I know about. I am embarrassed to admit that I haven't read most of them, and so I can't guarantee how cool they are. Let me know in the comments.<br /><ul><li><a href="http://learnersinbloom.blogspot.com/2016/07/upcoming-book-club-and-math-circle-math.html" target="_blank">Math and Magic in Wonderland</a>, by Lilac Mohr (This is volume 1, there is now a volume 2. I've been meaning to buy this for the past year, and finally did, just now.) </li><li><a href="http://naturalmath.com/avoid-hard-work/" target="_blank">Avoid Hard Work</a>, by Maria Droujkova, James Tanton, and Yelena McManaman</li><li><a href="https://beastacademy.com/" target="_blank">Beast Academy</a> (Fabulous curriculum, don't believe the grade levels (3 to 5), adults can find challenges in these)</li><li><a href="http://bentobooks.com/mathgirls/" target="_blank">Math Girls 3</a>, by Hiroshi Yuki (I love the first two. Turns out the 3rd one is out.)</li><li><a href="https://new.myubam.com/p/5003/this-is-not-a-math-book" target="_blank">This is Not a Math Book</a>, by Anna Weltman</li><li><a href="https://tabletopacademy.net/playful-math-books/" target="_blank">Any of Denise Gaskins books</a></li></ul><br /><br /><b>Animations</b><br /><ul><li><a href="http://www.businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5" target="_blank">GIFs for Trig concepts </a></li><li><a href="http://www.decisionsciencenews.com/2017/06/19/counterintuitive-problem-everyone-room-keeps-giving-dollars-random-others-youll-never-guess-happens-next/" target="_blank">A simulation of money moving randomly</a> </li><li><a href="https://undergroundmathematics.org/divisibility-and-induction/picture-this" target="_blank">Explore Euclid's Algorithm for finding GCF,</a> using a diagram instead of numbers. </li><li>I do not yet understand <a href="http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/" target="_blank">this animation of 'involutes'</a>, but I want to read it and think about it again. </li></ul><br /><b>Puzzles & Games </b><br /><ul><li><a href="https://solveme.edc.org/Mobiles.html" target="_blank">Solving equations as represented by mobiles (online)</a></li><li><a href="https://mathhombre.blogspot.com/2017/01/pentiremeter.html" target="_blank">Reduce the perimeter game</a> (John Golden) </li><li><a href="https://mathlesstraveled.com/2016/06/17/the-route-puzzle/" target="_blank">Find the number: Primes, Triangle Numbers, more</a></li><li><a href="http://www.numbrcise.com/numbrcise-puzzles/" target="_blank">Equation Crossnumber puzzles</a></li><li><a href="http://blog.tanyakhovanova.com/2016/05/a-travel-puzzle/" target="_blank">Rate, Time, and Distance Puzzle</a> </li><li><a href="http://musingmathematically.blogspot.com/2016/05/min-max-imize-classroom-game-for-basic.html" target="_blank">Highest / Lowest Result Game</a></li><li><a href="https://sonatamathematique.wordpress.com/2016/04/22/favorite-problems-1-pirates/" target="_blank">Pirate Puzzle</a> </li></ul><br /><br /><b>Early Math</b><br /><ul><li>Skip counting and lots more: If you like <a href="http://gregtangmath.com/index" target="_blank">Greg Tang</a>'s books, he's provided a page full of lovely resources for you to play with.</li><li><a href="http://tjzager.com/2017/06/17/representations-and-manipulatives-and-tools-and-things/" target="_blank">A good problem to help resistant adults take a fresh look at multi-digit multiplication</a></li><li><a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">Why Number Sense is the most important math skill</a></li><li><a href="http://blog.mathedpage.org/2016/05/fractions.html" target="_blank">Fractions</a> (Henri Picciotto) </li><li><a href="https://kristinlabeau.com/number-sense-what-is-it-and-why-is-it-so-important/" target="_blank">Number Sense</a> (Kristin LaBeau) </li><li><a href="https://denisegaskins.com/2017/06/20/10-ways-to-play-math-with-play-doh/" target="_blank">Play Math with Play Doh</a> (Lucy Ravitch)</li></ul><br /><br /><b>Geometry</b><br /><ul><li><a href="http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html" target="_blank">Triangles and Pythagorean Triples in Math Club</a> </li><li><a href="https://sonatamathematique.wordpress.com/2016/05/01/favorite-problems-2-triangles/" target="_blank">Solution process for a hard problem involving three triangles</a></li><li><a href="http://blogs.ams.org/blogonmathblogs/2016/04/20/fold-your-way-to-glory/" target="_blank">What shapes can you get with one cut on a folded paper? </a></li><li><a href="https://samjshah.com/2016/04/20/a-new-insight-on-the-famous-painted-block-problem/" target="_blank">Insight: How deep can you go? Painted Block Problem</a> (Sam Shah)</li><li><a href="http://matematicas11235813.luismiglesias.es/2017/06/18/18617-teselado-mosaicos-matematicas-arte-geometria-worldtessellationday/" target="_blank">Some lovely tesselations</a> (Luis Iglesias) </li></ul><br /><br /><b>Probability & Statistics</b><br /><ul><li><a href="https://matheducators.stackexchange.com/questions/1325/impressive-common-misleading-interpretations-in-statistics-to-make-students-awar?newsletter=1&nlcode=478309%7c101e" target="_blank">Misleading Interpretations (Stack Exchange) </a></li><li><a href="https://matharguments180.blogspot.com/2016/06/514-probability-game.html" target="_blank">Fun Probability Question</a></li><li><a href="https://onegoodthingteach.wordpress.com/2016/05/06/an-interesting-math-club-problem/" target="_blank">Math Club Probability Problem</a> </li><li><a href="http://blogs.cofc.edu/owensks/2016/04/22/modeling-fun-with-paper-fish/" target="_blank">Estimating Population Size with Capture / Recapture</a> </li></ul><br /><br /><b>Writing in Math Class </b><br /><ul><li><a href="http://busynessgirl.com/elaborations-for-creative-thinking-in-stem/" target="_blank">Problem Elaborations: One way to get students writing in math class (Maria Andersen)</a></li><li><a href="http://mrhonner.com/archives/16315" target="_blank">Math Haiku</a></li><li><a href="https://denisegaskins.com/2017/06/12/math-humor-and-copywork/" target="_blank">Math Copywork and Humor </a>(Denise Gaskins) </li></ul><br /><br /><b>Math and ...</b><br /><ul><li><a href="https://www.nytimes.com/2017/06/10/opinion/sunday/cuny-ending-the-curse-of-remedial-math.html" target="_blank">Ending the Curse of Remedial Math (NYT)</a> </li><li><a href="https://blog.supplysideliberal.com/post/31224211784/the-logarithmic-harmony-of-percent-changes-and" target="_blank">How logarithms explain economics</a> </li><li><a href="https://rootsoftheequation.wordpress.com/2016/12/21/vietnamese-age-and-the-sat" target="_blank">Is it hard for you to understand how math tests can be culturally biased? Here's an example.</a> (James Cleveland) </li><li><a href="http://www.johndcook.com/blog/2016/05/03/family-tree-numbering/" target="_blank">Family Tree Numbering</a> (John D. Cook) </li></ul><br /><b>On Teaching </b><br /><ul><li><a href="http://hyperbolicguitars.blogspot.com/2012/07/zen-and-fermats-last-theorem.html" target="_blank">'Showing' doesn't work. (Mike Thayer)</a></li><li><a href="https://www.nature.com/articles/nchem.2788.epdf?author_access_token=GkqcEB_h_vNGj_nd4juBdNRgN0jAjWel9jnR3ZoTv0PKX0DnafBppEkrPagykiwxk3FuydjDvDwk4TDN-Jwo1WHUwM_avG3XaTIIJ8o3x7OahEvhhwtSSKs30N4vYdOU" target="_blank">Playing with slide rules improves estimating skills.</a></li><li><a href="http://blog.mathedpage.org/2016/06/forward-design.html" target="_blank">Forward</a> and <a href="http://blog.mathedpage.org/2016/06/the-perils-of-backward-design.html" target="_blank">Backward Design</a> (Henri Picciotto) </li><li><a href="https://blog.peerinstruction.net/2016/06/07/increasing-student-engagement-by-design/" target="_blank">Designing a Course with Peer Instruction and Project-Based Learning </a></li><li><a href="https://samjshah.com/2016/05/29/pitching-college-math-courses/" target="_blank">Getting students excited about higher math</a> (Sam Shah)</li><li><a href="https://rootsoftheequation.wordpress.com/2016/05/04/shear-madness/" target="_blank">Improvisational Theater and Improvisational Teaching</a></li><li>One of the principles of improv is "Yes, and...". <a href="https://deltascape.blogspot.com/2016/05/how-does-mathematician-see-world.html" target="_blank">Here's a way to use that</a> to get students playing with math. </li><li><a href="http://fawnnguyen.com/always-sometimes-never/" target="_blank">Encouraging discussion with Always, Sometimes, Never</a></li></ul><br /><b><br /></b><b>Random Stuff</b><br /><ul><li><a href="https://opencalculus.wordpress.com/2016/04/29/last-day-at-aim-meet-some-of-my-new-friends" target="_blank">Links to people doing good work in Open Resources</a></li><li><a href="http://spot.pcc.edu/math/clm/section-velocity.html" target="_blank">Calculus Lab Manual </a></li><li><a href="http://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/" target="_blank">Using Newton's Method for a game</a></li><li>Fermat's Last Theorem was proved by Andrew Wiles in 1994. Though Fermat thought he had proved it, we are pretty sure he didn't. James Propp writes <a href="https://mathenchant.wordpress.com/2016/05/16/fermats-last-theorem-the-curious-incident-of-the-boasting-frenchman" target="_blank">a lovely summary of what Fermat probably knew</a>. </li></ul><br />I have more but it's bedtime and June is ending. Would you like to see your favorite blog post in next month’s playful math blog carnival? <a href="https://denisegaskins.com/mtap/mtap-submission-form/" target="_blank">Submissions are always open!<br /></a><br /><br /><br /><br /><span style="font-size: x-small;">(Note: Edited on 7/1 to add a few forgotten links, and fix a few broken links.)</span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com7tag:blogger.com,1999:blog-5303307482158922565.post-52881048698576622012017-06-06T09:37:00.000-07:002017-06-06T09:37:05.175-07:00What I learned at CAP's Community of PracticeCAP is <a href="http://accelerationproject.org/" target="_blank">California Acceleration Project</a>. Check out their <a href="http://accelerationproject.org/Publications" target="_blank">publications</a>. The first time I attended one of their conferences, I struggled with the word acceleration. It does not mean getting through the material faster. It means getting to the good stuff faster - shortening the pathway of required prerequisite courses students must take before taking a college level course. Their work is mainly with math and English, the two subjects that generally hold students back.<br /><br />In math, the college level course for someone not interested in STEM is statistics. Students take a placement test, and the large majority (86% at my college) are placed in remedial courses, anywhere from 1 to 4 levels below the statistics course. Imagine a student starting 3 levels below, at pre-algebra, which is where over half of our students are put by the placement test. If we had phenomenal success rates, with 90% passing each course, and phenomenal persistence rates, with 90% going on to the next course, we'd still only get 43% of these students finishing statistics (.9^8 = .43). What happens to the other 57%? Usually they give up on college, for at least a while.<br /><br />Because housing is pretty segregated in the U.S., and that makes k12 education pretty segregated, with people of color getting less resources dedicated to their schools, this becomes a civil rights issue. CAP is dedicated to: changing the way we place students (many who do badly on the placement test can still pass a college statistics course), developing models for co-requisite courses that students can take with statistics to improve their success rates, and developing radically shortened and improved remedial pathways (creating a pre-statistics course that prepares students with just enough algebra and lots of data analysis).<br /><br />I have been attending their workshops whenever I can for the past few years. This past weekend I went with two other math faculty and 5 English faculty. Even though I've seen much of the information before, I still got a lot out of it. (Maybe I'm a slow learner!)<br /><br />Here's something I put together yesterday at the request of our dean for equity, which summarizes some of the important points I learned...<br /><br /><br /><div dir="ltr" id="docs-internal-guid-c9ed0335-7e43-0c7c-03e5-e9d94866dc11" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Planning a High-Impact Course</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">More important than any one course are these 3 principles:</span><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Create separate pathways for STEM and non-STEM.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Place students as high in the sequence as possible.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Shorten the sequence as much as possible.</span></div></li></ul><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">CAP’s 5 design principles</span></div><ol style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Backward design</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Low-stakes, collaborative practice</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Relevant, thinking-oriented curriculum</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Just-in-time remediation</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Intentional support for affective needs</span></div></li></ol><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">[For more detail, see: </span><a href="http://accelerationproject.org/Publications/ctl/ArticleView/mid/654/articleId/12/Toward-a-Vision-of-Accelerated-Curriculum-and-Pedagogy-High-Challenge-High-Support-Classrooms-for-Underprepared-Students" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: underline; vertical-align: baseline;">http://accelerationproject.org/Publications/ctl/ArticleView/mid/654/articleId/12/Toward-a-Vision-of-Accelerated-Curriculum-and-Pedagogy-High-Challenge-High-Support-Classrooms-for-Underprepared-Students</span></a><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> ]</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">High Performing Math Classrooms (Internationally)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">James Stigler on high performing math countries. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">All have these things in common:</span></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Productive struggle</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Explicit connections</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Deliberate practice, increasing variation and complexity over time</span></div></li></ul><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">[For an NPR piece on Stigler’s work, see: </span><a href="http://www.npr.org/sections/health-shots/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: underline; vertical-align: baseline;">http://www.npr.org/sections/health-shots/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning</span></a><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> ]</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Lesson Planning (CAP)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Given a topic you want students to learn through groupwork,</span></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Identify the prerequisite skills needed,</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Decide whether these will be addressed through productive struggle (ie not addressed overtly), targeted group activity, or just-in-time mini-lecture, and how you’ll do that,</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Plan main activity,</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Plan closure (vital for making explicit connections)</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Note: Over-scaffolding brings down the thinking level required.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">(Sue has a form from CAP. If this link works, it’s to all the CAP materials: </span><a href="https://app.box.com/s/965xg12luwsgjgmeq86px8oonsr9yolm" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: underline; vertical-align: baseline;">https://app.box.com/s/965xg12luwsgjgmeq86px8oonsr9yolm</span></a><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> )</span></div></li></ul><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Thinking Levels</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">The thinking levels mentioned above come from a study by Quasar. Here’s the relevant info:</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 12pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">“This research yielded two major findings: (1) mathematical tasks with high-level cognitive demands were the most difficult to implement well, frequently being transformed into less-demanding tasks during instruction; and (2) student learning gains were greatest in classrooms in which instructional tasks consistently encouraged high-level student thinking and reasoning and least in classrooms in which instructional tasks were consistently procedural in nature.” (Stein p. 4)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">QUASAR Task Analysis Guide (adjusted slightly to address statistical thinking as well as mathematical thinking)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Lower-Level Cognitive Demand</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Memorization Tasks </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Involve either reproducing previously learned fact, rules, formula, or definitions; </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure; </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Not ambiguous; clear and direct instructions to reproduce previous material; </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">No connection to the concepts or meaning that underlie the fact, rules, formula, or definitions. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Procedures Without Connections Tasks </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Algorithmic; direct instructions to use a procedure or the use of the procedure is evident based on prior instruction, experience, or placement of the task. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require limited cognitive demand for successful completion. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">There is little ambiguity about what needs to be done and how to do it. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">No connections to concepts or meaning that underlie the procedure being used. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Focused on correct answers rather than developing mathematical or statistical understanding. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require no explanations, but may require students to “show work”.</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Higher-Level Cognitive Demand</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Procedures With Connections Tasks</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Focus students’ attention on the use of procedures or concepts for the purpose of developing deeper levels of understanding of mathematical or statistical concepts and ideas. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Usually are represented in multiple ways (e.g. graphs, tables, numerical summaries, verbal descriptions). </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Making connections among multiple representations helps to develop meaning. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require some degree of cognitive effort. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Although general procedures may be followed, they cannot be followed mindlessly. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Doing–Mathematics or Doing–Statistics Tasks</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require complex and non-algorithmic thinking (i.e. there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example). </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require students to explore and understand the nature of mathematical or statistical concepts, processes, or relationships. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Demand self-monitoring or self-regulation of one’s own cognitive processes. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require students to access and make appropriate use of relevant knowledge and experiences </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-33812288731554717822017-03-04T15:20:00.001-08:002017-03-04T15:20:31.818-08:00At the Julia Robinson Math Festival Today<a href="http://jrmf.org/index.php" target="_blank">Julia Robinson Math Festival</a>s invite kids to play with math puzzles that start easy and offer harder questions as you go along. Today's festival was at Bentley School in Lafayette. (Some festivals are open to the public, and are much bigger.)<br /><br />I was working the Pilgrim's Puzzle table. We had this puzzle to work on.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-SvcnOr6eJw4/WLtJaOJoBzI/AAAAAAAAC60/g0CceMG04kwKajMbE3oHmG7CODnP-Ze-wCLcB/s1600/pilgrim%2527s%2Bpuzzle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-SvcnOr6eJw4/WLtJaOJoBzI/AAAAAAAAC60/g0CceMG04kwKajMbE3oHmG7CODnP-Ze-wCLcB/s1600/pilgrim%2527s%2Bpuzzle.png" /></a></div>It was really fun watching kids and parents get engaged with it. Some paths give you fractions, and then taking away 2 can give you something like 1/8 - 2, which can be pretty confusing for a 3rd grader.<br /><br />The first time I tried to help a kid with a problem like that I was not able to find an image that made this sensible. When B was stuck with a problem like this, I came up with anti-matter apples. It worked! We imagined 1/8th of an apple, and imagined two anti-matter apples. We cut the 2nd one into 8 pieces, took one of those pieces and exploded it with the regular 1/8th slice to make a poof and then nothing. So we had one anti-matter apple and ... 7/8ths of another, which we wrote as -1 7/8. Done.<br /><br />I will be teaching beginning algebra in the fall. I don't think I've ever found an image for negative fractions that worked as well as I think this one will. I'm excited.<br /><br />Here's B and me.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-C1bzhBTsp7I/WLtLja-XC7I/AAAAAAAAC7A/tLxU3DehExI3uzlhWLHsqIQ6fLKa8PyPgCLcB/s1600/me%2Bat%2Bjrmf.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://3.bp.blogspot.com/-C1bzhBTsp7I/WLtLja-XC7I/AAAAAAAAC7A/tLxU3DehExI3uzlhWLHsqIQ6fLKa8PyPgCLcB/s320/me%2Bat%2Bjrmf.jpg" width="320" /></a></div><br /><br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3