tag:blogger.com,1999:blog-4227811469912372962Fri, 23 Mar 2018 17:46:05 +0000geometrydigressionbrainstormingmath club ideasgeneral philosophyPythagorean Theoremmanagementpuzzles ciphersalgebrabook reviewgamesnumber theorypi dayvirtual math clubworksheet homeworkCombinatoricscontestknights of pitriangle numbersdistributive propertymoemspurple comet15-75-90amcartdecodingdivisibilitydreamboxfractionsgraph theorymath club math circlemath countsolympiadpythagorean tripleKaprekar's operationamc 8arthur benjamincarnival of mathchesscontinuing fractionsegyptian fractionseuler characcteristiceuler characteristicexponentsfactorizationfibonaccifive trianglesfold and cutherons formulainversionsjulia robinsonlecturemagic squaremath nightmiddle schoolmomathnotice wonderpair and sharepascal's triangleplatonic solidsproblemquarticrecruitingresourcessangakuseriessierpinskisoftware reviewstatisticstilingtopologytwitterulam's spiralvarignonvnpsvoronoiMath off the gridRunning a middle school math club. This is part my planning process, part documentation and part a how to guide.http://mymathclub.blogspot.com/noreply@blogger.com (Benjamin Leis)Blogger205125tag:blogger.com,1999:blog-4227811469912372962.post-6374909480011441858Wed, 21 Mar 2018 20:56:00 +00002018-03-23T08:16:27.672-07:00artgeometry3/20 Visible MathThis week started with a walk through of the MathCounts problem that I gave out last week to do at home.<br /><br /><blockquote class="tr_bq"><span id="docs-internal-guid-d269c292-4a37-2390-e855-4a3042d79c12"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">Six standard six-sided dice are rolled, and the sum S is calculated. What is the probability that S × (42 – S ) < 297? Express your answer as a common fraction.</span></span></blockquote><br />This was the last question in the sprint round at Chapters. As I remember from the stats almost no one at the entire contest finished it correctly making it the hardest of the set. I decided this would make for a good communal walk through because so many of the kids had seen it once and it hits a couple of different themes. However, that's also the weakness of this problem. Conceptually its a bizarre hybrid of a counting problem and a quadratic inequality neither of which naturally goes with each other. I actually mentioned this to the kids. The phrase "franken-problem" might have been used.<br /><br />At any rate, I started with the basics and asked some background questions:<br /><br /><br /><ul><li>What is the range of values for the sum of the dice throws?</li><li>How many total combinations are there for 6 dice throws in a row? Why?</li><li>What is the most common sum / what would a probability graph look like?</li></ul><div>This part was very approachable and the kids easily supplied various answers. So it was time for the quadratic inequality. First I asked how many kids knew how to solve this algebraically? (Some of the room have not covered this at all) It turns out even those kids with Algebra actually used guess and check anyway. There are only 31 values after all and its not too hard to just plug them in and see what happens. The risk here is missing there is a range at both ends of the curve which I mentioned.</div><div><br /></div><div>I had one volunteer who brought the equation into almost standard form but no volunteers to finish the process. So I demoed the formal method myself.</div><div><ul><li>Factor to: (S-33)(S-9) > 0</li><li>Do a parity check: both factors are positive in which case S > 33 or both factors are negative in which case S < 9.</li><li>Notice the symmetry.</li></ul><div><i>This felt new to the room and the work with signs of the inequality also exposed some conceptual weakness. So something to look for more problems to do in another context.</i></div><div><br /></div><div>From here the problem becomes more standard and I had the kids do the case work on numbers of combinations for the 2 ranges. We've been doing small amounts but could also use more combinatorics exposure.</div></div><div><br /></div><div><br /></div><div>That covered, I was ready for the fun part of today. I've been looking at George Hart's <a href="http://makingmathvisible.com/">makingmathvisible.com</a> site and was fascinated by some of the constructions. So I chose the sample one: <a href="http://makingmathvisible.com/PaperTriangleBall/PaperTriangleBall.html">http://makingmathvisible.com/PaperTriangleBall/PaperTriangleBall.html</a> to try out. </div><div>Over the weekend I tested the templates and built my own ball:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-EIKFTzmtUGc/WrLCJZHZZII/AAAAAAAAM2c/5LqH6m615EMZ3Y7Cob_6lIME2jW2XfUogCKgBGAs/s1600/20180318_152654.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://1.bp.blogspot.com/-EIKFTzmtUGc/WrLCJZHZZII/AAAAAAAAM2c/5LqH6m615EMZ3Y7Cob_6lIME2jW2XfUogCKgBGAs/s320/20180318_152654.jpg" width="320" /></a></div><div><br /></div><div>It was a bit tricky, my ball almost fell apart at one time and I misplaced a few triangles leading to a dead end all of which gave me some ideas for how to guide when the kids tried it out. <b>Its really important to stress being precise when cutting the slots and also to work together when building the ball out to hold it together.</b></div><div><br /></div><div>Beforehand I pre-printed the templates at a copy shop on 110 lb card stock paper. I also bought some thicker colored card stock which couldn't go through a copy machine and required tracing. I then mostly followed the lesson suggested on George Hart's site. We worked through discovering combinations of 3, 4 and five triangles first before really working as group. It took the kids the entire rest of the hour to build the balls once in white and then again in a multicolored version. </div><div></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-7dEnEydv4AE/WrLDSQo0tMI/AAAAAAAAM2o/MHuOswF0BaQlaVKX83AbHoC1d9ltNNmiQCKgBGAs/s1600/20180320_163243.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://1.bp.blogspot.com/-7dEnEydv4AE/WrLDSQo0tMI/AAAAAAAAM2o/MHuOswF0BaQlaVKX83AbHoC1d9ltNNmiQCKgBGAs/s320/20180320_163243.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-UP_oGQgcbsw/WrLDSSpVUpI/AAAAAAAAM2o/2ui3S18he6M3naEcNP82A2bNqZEP4whbwCKgBGAs/s1600/20180320_163239.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://2.bp.blogspot.com/-UP_oGQgcbsw/WrLDSSpVUpI/AAAAAAAAM2o/2ui3S18he6M3naEcNP82A2bNqZEP4whbwCKgBGAs/s320/20180320_163239.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-nWHerflH-0g/WrLDSZ-J1GI/AAAAAAAAM2o/wdXP8oraUs4s2TIN4ID5AweAxFOW7L3xgCKgBGAs/s1600/20180320_163303.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://2.bp.blogspot.com/-nWHerflH-0g/WrLDSZ-J1GI/AAAAAAAAM2o/wdXP8oraUs4s2TIN4ID5AweAxFOW7L3xgCKgBGAs/s320/20180320_163303.jpg" width="320" /></a></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-c06layZX2kg/WrLDSd9E5kI/AAAAAAAAM2o/dSD3fS6OeDUDN0FfJIimHldYbXoyIm5ywCKgBGAs/s1600/20180320_165700.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-c06layZX2kg/WrLDSd9E5kI/AAAAAAAAM2o/dSD3fS6OeDUDN0FfJIimHldYbXoyIm5ywCKgBGAs/s320/20180320_165700.jpg" width="320" /></a></div><div><br /></div><div>This last one above was the most hard fought version. This group was the least focused and sloppiest cutters. So there were a few weakened triangles in their set. I kept coming over for a bit and helping them move forward with advice for kids to help hold the structure in place etc. But then in between when I went to work with others it tended to collapse. Finally, I decided I really wanted everyone to achieve success and I should stay in place until they finished. I had them substitute in some borrowed extra triangles from the other groups and basically guided them through the tricky middle stage when the ball is most unstable. They finished right at the end and there was a literal cheer from the group. (I was extremely relieved)</div><div><br /></div><div><br /></div><div>The other groups actually made it through the multi-colored version where I had them try to create a symmetry in their use of color:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-WSkIu2cD0Cg/WrLEnSEguCI/AAAAAAAAM20/V4yPG2m9-AwnKSPUQDkW5X_wvD0MbliNACKgBGAs/s1600/20180320_170146.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://3.bp.blogspot.com/-WSkIu2cD0Cg/WrLEnSEguCI/AAAAAAAAM20/V4yPG2m9-AwnKSPUQDkW5X_wvD0MbliNACKgBGAs/s320/20180320_170146.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-dOCJf_I5uK4/WrLEnV29VqI/AAAAAAAAM20/ani1JVD4qj8ewlYIsifzB6qXKmZtcBYbQCKgBGAs/s1600/20180320_165925.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://1.bp.blogspot.com/-dOCJf_I5uK4/WrLEnV29VqI/AAAAAAAAM20/ani1JVD4qj8ewlYIsifzB6qXKmZtcBYbQCKgBGAs/s320/20180320_165925.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-UZ9_n7bgO4w/WrLEnfYwMKI/AAAAAAAAM20/SgI7E8K_fbYPbPN2R3T8jr3X4WmW7xE5gCKgBGAs/s1600/20180320_165935.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://3.bp.blogspot.com/-UZ9_n7bgO4w/WrLEnfYwMKI/AAAAAAAAM20/SgI7E8K_fbYPbPN2R3T8jr3X4WmW7xE5gCKgBGAs/s320/20180320_165935.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-hIZec7N7j3s/WrLEnYy9d9I/AAAAAAAAM20/SgznOXO4oJQT-8opRF-D4aZ_B2QgPNO1wCKgBGAs/s1600/20180320_164758.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-hIZec7N7j3s/WrLEnYy9d9I/AAAAAAAAM20/SgznOXO4oJQT-8opRF-D4aZ_B2QgPNO1wCKgBGAs/s320/20180320_164758.jpg" width="320" /></a></div><div><br /></div><div>I was hoping to have enough time to discuss the extension questions about the combinatoric aspects of the colored balls but we ran the clock down. As usual for me, I worried about the exact opposite case and had printed out the next template for early finishers which no one needed to use. <a href="http://makingmathvisible.com/PaperSquareBall/PaperSquareBall.html">http://makingmathvisible.com/PaperSquareBall/PaperSquareBall.html</a> I'm currently testing this at home. (Someone has to use the card stock.) Based on that experience the second ball is quite a bit more difficult to assemble and I'd budget much more time for it / prepare for some dexterity challenges. That said, overall, I highly recommend this project. It was definitely a crowd pleaser!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-MPkLLLUS2ZA/WrMEmtskA0I/AAAAAAAAM3M/NA6hoqrvpcUIbjXXnkc_rS4rYfUTxhj9QCKgBGAs/s1600/20180321_181733.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-MPkLLLUS2ZA/WrMEmtskA0I/AAAAAAAAM3M/NA6hoqrvpcUIbjXXnkc_rS4rYfUTxhj9QCKgBGAs/s320/20180321_181733.jpg" width="320" /></a></div><br /><div style="text-align: center;">(Its a bit like the 2nd death star right now)</div><br /><br /></div><div><br /></div><div>P.O.T.W:</div><div>This one comes from Matt Enlow and is an interesting number theory experiment.</div><div><br /></div><div><a href="https://drive.google.com/open?id=1qIXjtkK-fpe21pFDtr5f0cLa10YABsy5e6n6iIZRLl0">https://drive.google.com/open?id=1qIXjtkK-fpe21pFDtr5f0cLa10YABsy5e6n6iIZRLl0</a></div>http://mymathclub.blogspot.com/2018/03/320-visible-math.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-3897318940052940575Wed, 14 Mar 2018 21:05:00 +00002018-03-14T21:16:52.602-07:00pi day3/13 Pi Day - 1This is my fourth experience with Pi Day or "Pi Day - 1" as I called it since we meet on Tuesdays.<br /><br />See:<br /><br /><ol><li><a href="http://mymathclub.blogspot.com/2017/03/314-pi-day.html">http://mymathclub.blogspot.com/2017/03/314-pi-day.html</a></li><li><a href="http://mymathclub.blogspot.com/2016/03/315-pi-day-2016-more-or-less.html">http://mymathclub.blogspot.com/2016/03/315-pi-day-2016-more-or-less.html</a></li><li><a href="http://mymathclub.blogspot.com/2015/03/310-pi-day-approximately.html">http://mymathclub.blogspot.com/2015/03/310-pi-day-approximately.html</a></li></ol><div><br /></div><div>In a nutshell, because there's pie to eat, the kids always have fun. But I was reminded of another perspective today from @evelyn_lamb</div><div><br /></div><div><a href="https://t.co/dJkmy20vaw">https://t.co/dJkmy20vaw</a></div><blockquote class="tr_bq"><br />"Pi Day bothers me not just because it celebrates the the ratio of a circle’s circumference to its diameter, or the number 3.14159 … It’s also about the misplaced focus. What do we see on Pi Day? Circles, the Greek letter π, and digits. Oh, the digits! Scads of them! The digits of π are endemic in math gear in general, but of course they make a special showing on Pi Day. You can buy everything from T-shirts and dresses to laptop cases and watches emblazoned with the digits of π."</blockquote><br /><img alt="Image result for larry shaw picture exploratorium" src="data:image/jpeg;base64,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" /><br /><br />I'm pretty much in total agreement with above. I've gently ranted in the past about pi digit memorization contests and other such trivialities. But as her article continues, there was a man behind the holiday, Larry Shaw the recently deceased director of the San Francisco Exploratorium. I think his vision was more than just eating pie but it was also an incredibly whimsical gesture which is why I believe its had as much cultural resonance.<br /><br />So I take the day partly in that spirit of whimsy and also with the mission to always ground it in circle geometry in some way and as said at the start, the kids always have fun celebrating. Mathematics doesn't have enough moments like this especially in school.<br /><br />This year I decided to go back to the basics. I had initially toyed with talking about the unit circle and the derivation of radians versus degrees but on reflection I found so much material that I couldn't fit that in. Instead, I started with a survey of student definitions of pi (while they were eating). This was surprisingly solid. The phrase "ratio of circumference to diameter" came up almost immediately. I then took a poll of how many kids had already done activities in class where they measured circular objects of some sort and divided them by their measured diameters to find pi approximations. Again, almost everyone had done so often several years ago in Elementary School.<br /><br />So with everyone convinced already pi existed and it had a value it was time for some deeper questions. The first one I posed was "Is measuring a single object a good way to prove pi's existence?" We chatted a bit about accuracy and sample sizes as well as whether from a mathematical perspective we can ever prove something from samples. My favorite version of this is<br />"What if only ordinary people sized circles have a ratio around pi and if we could measure microscopic or macroscopic versions we'd find something different?"<br /><br />One of the kids then suggested approximating the circumference of a circle with polygons so we then did that on the board for the hexagon version. I cold called in this case which I usually don't do to get a student to sum up the perimeter of the hexagon arriving at pi is approximately 3.<br /><br />From there we took a quick digression to also do the area of a circle visual proof where you cut the circle up and form a rough rectangle that is pi*r by r in size. Again I had the kids fill in and compute the area.<br /><br />Finally I noted that we don't actually compute pi to a billion digits using geometry and asked if anyone knew of other ways to get it. This was a new idea for the room and a good setup for the 2 videos I chose for the day.<br /><br />The first was this amusing (there were a lot of genuine laughs while watching) video of Matt Parker computing pi by hand using the alternating series 1 - 1/3 + 1/5 - 1/7 ....<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/HrRMnzANHHs/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/HrRMnzANHHs?feature=player_embedded" width="320"></iframe></div><br />But of course this doesn't really explain why this works only that it appears to do so. So I also picked the very ambitious following one by 3blue1brown:<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/d-o3eB9sfls/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/d-o3eB9sfls?feature=player_embedded" width="320"></iframe></div><br />Its about as approachable as its going to get with this amount of background knowledge but still a stretch. I stopped several times to ask questions about some of the background concepts. There are several potential stumbling blocks here:<br /><br /><br /><ol><li>law of inverse squares</li><li>Inverse pythagorean theorem</li><li>The general abstraction model used</li><li>The number line can be thought of as a curve.</li></ol><div><br /></div><div>The last one was the one I chose to focus on the most and I framed it as a thought experiment "What if the number line isn't really a line at all but a curve, we're just at a small portion of it and just like with a curve if you magnify enough it appears to be straight." My hope is that if nothing else stuck that idea was interesting and thought provoking (hello Calculus in the future) My informal survey is that most kids found it interesting but I may have had one where this pushed too far. So I am planning to do a little preamble next week "Its ok to give me feedback if you found anything too confusing and I also sometimes want you to focus on the big ideas in moments like this even if the details aren't accessible yet"</div><div><br /></div><div>P.O.T.W:</div><div>I gave out the last problem from MathCounts this year now that it was released: <a href="https://drive.google.com/open?id=1mvYa9rWcU04MMcykZixWhdg8ehHJIPHoniVSEBdQrPA">https://drive.google.com/open?id=1mvYa9rWcU04MMcykZixWhdg8ehHJIPHoniVSEBdQrPA</a> </div><div><br /></div><div>Its actually a fairly awkward merge of quadratic inequalities and dice counting problem but I wanted to provide a capstone to the kids experience there and dig into how to solve it.</div><div><br /></div>http://mymathclub.blogspot.com/2018/03/313-pi-day-1.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-8671383036316226650Thu, 08 Mar 2018 21:03:00 +00002018-03-08T13:03:14.938-08:00artdigressionCoCa Photo Diary - Art Math IntersectionI had a chance during lunch to look at Dan Finkel's brainchild at the Center on Contemporary Art.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-ubbYw5955Po/WqGkWEjGgKI/AAAAAAAAMsY/HTWMteOWwBwHhAEMGj0etDNJeDbPx0pOACKgBGAs/s1600/20180308_122655.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="640" src="https://3.bp.blogspot.com/-ubbYw5955Po/WqGkWEjGgKI/AAAAAAAAMsY/HTWMteOWwBwHhAEMGj0etDNJeDbPx0pOACKgBGAs/s640/20180308_122655.jpg" width="360" /></a></div><br /><br />Its a small space but they filled it with a lot of math related art. Bonus, I recognized several of the mathematicians who participated.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-SnI7xr33_Eo/WqGkpY2xFyI/AAAAAAAAMsc/fFU0p7F4Y8wH09hUbSvTBnyr3UOnamIXQCKgBGAs/s1600/20180308_122419.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://4.bp.blogspot.com/-SnI7xr33_Eo/WqGkpY2xFyI/AAAAAAAAMsc/fFU0p7F4Y8wH09hUbSvTBnyr3UOnamIXQCKgBGAs/s400/20180308_122419.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-4gNNsWAN9Hw/WqGkpZU3pKI/AAAAAAAAMsc/IiNaPhxqkbUvmzzrzfKxiWNZj5OZQQmPQCKgBGAs/s1600/20180308_122432.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://1.bp.blogspot.com/-4gNNsWAN9Hw/WqGkpZU3pKI/AAAAAAAAMsc/IiNaPhxqkbUvmzzrzfKxiWNZj5OZQQmPQCKgBGAs/s400/20180308_122432.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-N6K2sExco7w/WqGkpc-UQEI/AAAAAAAAMsc/bb389N3hghkuSTbnPN6xA2O-I8TUIepSgCKgBGAs/s1600/20180308_122617.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://1.bp.blogspot.com/-N6K2sExco7w/WqGkpc-UQEI/AAAAAAAAMsc/bb389N3hghkuSTbnPN6xA2O-I8TUIepSgCKgBGAs/s320/20180308_122617.jpg" width="180" /></a><a href="https://2.bp.blogspot.com/-F-Quy3gs4wg/WqGkpVhn9wI/AAAAAAAAMsc/-ST-U2JjJCIIqftYWOI1lG0FoAUoXuv4ACKgBGAs/s1600/20180308_122439.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://2.bp.blogspot.com/-F-Quy3gs4wg/WqGkpVhn9wI/AAAAAAAAMsc/-ST-U2JjJCIIqftYWOI1lG0FoAUoXuv4ACKgBGAs/s320/20180308_122439.jpg" width="180" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-F2bO3NOkn7k/WqGkpTG8yOI/AAAAAAAAMsc/efjAxwDZhxsB3eT-OEyUfAT4YbvJue6-gCKgBGAs/s1600/20180308_122448.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://1.bp.blogspot.com/-F2bO3NOkn7k/WqGkpTG8yOI/AAAAAAAAMsc/efjAxwDZhxsB3eT-OEyUfAT4YbvJue6-gCKgBGAs/s320/20180308_122448.jpg" width="180" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-5oEUTt6mBLI/WqGkpSU0LDI/AAAAAAAAMsc/MVm93tb9aFwfJu5Rc4fT4Ee-tYyd3X4xgCKgBGAs/s1600/20180308_122457.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="640" src="https://4.bp.blogspot.com/-5oEUTt6mBLI/WqGkpSU0LDI/AAAAAAAAMsc/MVm93tb9aFwfJu5Rc4fT4Ee-tYyd3X4xgCKgBGAs/s640/20180308_122457.jpg" width="360" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-xf4JrioHMkg/WqGkpbaZoEI/AAAAAAAAMsc/0WPRHVtPHEcCtsEK4qt8ixBot1a5X3mlgCKgBGAs/s1600/20180308_122511.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://2.bp.blogspot.com/-xf4JrioHMkg/WqGkpbaZoEI/AAAAAAAAMsc/0WPRHVtPHEcCtsEK4qt8ixBot1a5X3mlgCKgBGAs/s320/20180308_122511.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-sPcQSE8Lkl4/WqGkpcHs8HI/AAAAAAAAMsc/dMgOB16jQgwBVo238JCqS94TouqcIYPKwCKgBGAs/s1600/20180308_122518.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-sPcQSE8Lkl4/WqGkpcHs8HI/AAAAAAAAMsc/dMgOB16jQgwBVo238JCqS94TouqcIYPKwCKgBGAs/s320/20180308_122518.jpg" width="320" /></a><a href="https://1.bp.blogspot.com/-Uz83T0GgTVI/WqGkpcWckBI/AAAAAAAAMsc/5iB9kF2PoOU8vnSvGcC40WDYZgPl9jnLwCKgBGAs/s1600/20180308_122609.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://1.bp.blogspot.com/-Uz83T0GgTVI/WqGkpcWckBI/AAAAAAAAMsc/5iB9kF2PoOU8vnSvGcC40WDYZgPl9jnLwCKgBGAs/s320/20180308_122609.jpg" width="180" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-vF-j-slCw5Y/WqGkpRtUVlI/AAAAAAAAMsc/D7cFzk5cfSA-XzKD8wdBfnPKPoVPIU3vQCKgBGAs/s1600/20180308_122526.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://4.bp.blogspot.com/-vF-j-slCw5Y/WqGkpRtUVlI/AAAAAAAAMsc/D7cFzk5cfSA-XzKD8wdBfnPKPoVPIU3vQCKgBGAs/s400/20180308_122526.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-IkjbbEdsJ3g/WqGkpdw3miI/AAAAAAAAMsc/wSTxUW6Zf5wT_8lQ5aKJDil3__vnOKq2ACKgBGAs/s1600/20180308_122535.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://3.bp.blogspot.com/-IkjbbEdsJ3g/WqGkpdw3miI/AAAAAAAAMsc/wSTxUW6Zf5wT_8lQ5aKJDil3__vnOKq2ACKgBGAs/s400/20180308_122535.jpg" width="400" /></a></div><br /><br />http://mymathclub.blogspot.com/2018/03/coca-photo-diary-art-math-intersection.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-5252022970334923410Thu, 08 Mar 2018 05:39:00 +00002018-03-07T21:41:15.568-08:00fractions3/6 Infinite Countable Sets or more fractionsIt was a big weekend for the Math Club or should I say team. We finally participated in the rescheduled MathCounts chapter contest. I was very lucky the new date worked for me personally since we were out of town the prior two weekends and all of the students who had already signed up amazingly also still came out. Overall, I had a great time and from what I can tell debriefing the kids they did as well. The format was fairly intimate. There were 10 schools participating with around 80 sixth to eight graders. During most of the rounds I hung out in the coaches room and chatted. This was a lot of fun. I met teachers from St. Ann's, Lakeside, Kellog M.S. and Hamilton. I actually told the kids later when we were talking about the day that this was my favorite part. As the kids finished and burst into the hall, I checked how everything was going and how the difficulty level went. Finally after a nerve wracking countdown round we had one overall 7th place winner and a 4th place team. That was good enough to let us go to the state competition this weekend! What's gratifying is most of the team members were 6th graders and the teams that placed above them were all eight graders so I think there is headway to grow over the next few years (another message I gave everyone)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-b9kphZS5-CI/WqDGA_eIJWI/AAAAAAAAMrE/hyOj-PGIvEcxUnEmRLWv34EI13gR9DHlwCKgBGAs/s1600/20180303_144507.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://1.bp.blogspot.com/-b9kphZS5-CI/WqDGA_eIJWI/AAAAAAAAMrE/hyOj-PGIvEcxUnEmRLWv34EI13gR9DHlwCKgBGAs/s320/20180303_144507.jpg" width="180" /></a></div><br /><br /><br />During the actual club meeting today I had all the kids talk about their experiences at MathCounts to encourage each other. Again everyone even those who hadn't won anything seemed upbeat. I also gave my long but true speech about focusing on the fun parts of the competition and not the absolute outcomes and finding the joy in the math. Its true but perhaps hard to see in Middle School that the kids who keep going will ultimately benefit regardless of trophies. So as usual, I still worry about the discouraging aspects of these meets but it seems to have gone well.<br /><br /><br />After this talk, we briefly went over the old problem of the week. I only had one student really work on it so I'm thinking about what to do to refresh. This weeks problem is quite a bit easier and approachable <a href="https://mathforlove.com/2018/02/a-mathematician-at-play-puzzle-9/">https://mathforlove.com/2018/02/a-mathematician-at-play-puzzle-9/</a> which may help. I'm also thinking about different types of problems and to remember to talk about participation at the beginning and end of each session for the immediate future, I'm hoping to get back to near half of the kids working on this.<br /><br />For the main task of the day I chose a topic from the recent Math Teacher's Circle magazine<br /><a href="https://www.mathteacherscircle.org/news/mtc-magazine/s2018/touching-infinity/">https://www.mathteacherscircle.org/news/mtc-magazine/s2018/touching-infinity/</a>, counting the set of rationals. In addition to looking interesting, this tied in well with 2 weeks ago on Farey Sequences: <a href="http://mymathclub.blogspot.com/2018/02/213-farey-sequences.html">http://mymathclub.blogspot.com/2018/02/213-farey-sequences.html</a><br /><br />Before starting though I wanted to warm up a bit with a small problem I'd seen on twitter. So I had all the kids work on the whiteboard with a number line to find where to place 1/4 between 1/3 and 1/5. I didn't supply any hash marks or much more than a simple explanation of the problem. I was gratified this time that almost everyone came up with an accurate answer. Universally kids chose to convert the fractions into the GCD denominator of 60 and place 1/4 = 15/60 between 20/60 and 12/60. On review as a group, I also asked since 1/4 is not there what is the number in the exact middle which was a good followup question. Note: for some reason when I did it myself beforehand I chose to calculate the difference between each endpoint and 1/4 and then find the ratio of the two distances which no one else did.<br /><br />With that covered we dove in and I described the hyperbinary system. It uses the binary place values but in addition to 0 and 1 you may also use 2 in every digit. i.e. 27 = 0x11011 AND 0x2211 This took a few examples to make clear. From there I had everyone start to make a chart of the first 15 numbers in hyperbinary and how many representations each had.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-yCw7Ukl8_MY/WqDI64rf32I/AAAAAAAAMrQ/G8mZY6Gqtm4__Gu5BaRHBkBnAxOtGB8GQCKgBGAs/s1600/20180306_161559.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://3.bp.blogspot.com/-yCw7Ukl8_MY/WqDI64rf32I/AAAAAAAAMrQ/G8mZY6Gqtm4__Gu5BaRHBkBnAxOtGB8GQCKgBGAs/s320/20180306_161559.jpg" width="320" /></a></div><br /><br />The next step was to get everyone to find a pattern in the chart. This was only partly successful. The kids eventually identified the left hand rule where <em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">b</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">(</span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">n</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">) = </span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">b</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">(2</span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">n</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">+1) </span> but finding <em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">b</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">(</span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">n</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">) + </span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">b</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">(2</span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">n</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">+1) = </span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">b</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">(2</span><em style="background-color: white; color: #333333; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">n</em><span style="background-color: white; color: #333333; font-family: "helvetica neue" , "helvetica" , "arial" , sans-serif; font-size: 14px;">+2)</span> proved more difficult. So in the interest of time I showed them this one. I'm still brainstorming how to do this a bit better in the future (and I really want to repeat this again since the whole activity is fascinating). One note here: its super important to keep these 2 rules in mind yourself and practicing write beforehand is very helpful.<br /><br />We then moved onto the big discovery the Calkin-Wolf Tree<br /><br /><img src="http://www.mathteacherscircle.org/wp-content/themes/mtc/assets/CalkinWilfTreeOfFractions.png" height="137" width="320" /><br /><br />Again I described the rules for the tree and had everyone generate them on the whiteboard. I asked kids who finished to see if they could find a relationship between the tree and the previous hyperbinary numbers chart. We had just enough time for one student to discover they were identical and wrap up a bit as a group. So I pointed out a few more interesting facts about the tree we didn't have time to work on like the presence of every rational reduced fraction.<br /><br />Overall, this went well but I could definitely improve the experience. I think I had between 50-75% engagement over the whole session which was a bit too low to my taste. The sustained effort by the time we were working on the tree was definitely at the limit for some students. On the bright side those who stayed engaged were very excited. I think what might work better would be to do both parts simultaneously and let the kids move between white board stations. At the end we could then look for the patterns between the two parts as a group.http://mymathclub.blogspot.com/2018/03/36-infinite-countable-sets-or-more.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-6276084193603241943Thu, 01 Mar 2018 17:43:00 +00002018-03-03T21:25:05.804-08:00carnival of mathCarnival of Mathematics 155<div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">Welcome to the 155th Carnival of Mathematics which collects a sampling of interesting math(s) related posts from around the web. This is my first time hosting and as my passion is topics for middle school math clubs you'll see a few of my personal choices. For all those interested in Carnival of Mathematics future and past, visi<span style="background-color: transparent; color: #222222; font-family: sans-serif; font-size: 14px;">t</span><span style="background-color: transparent; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="color: blue;"><a href="http://aperiodical.com/carnival-of-mathematics/" style="background-color: transparent; color: blue; font-family: sans-serif; font-size: 14px;">The Aperiodical</a><span style="font-family: sans-serif;"><span style="font-size: 14px;"><span style="color: blue;"> </span></span></span></span>where you can also submit future posts.</div><blockquote class="twitter-video" data-lang="en"><div dir="ltr" lang="en">Chains of circles<a href="https://t.co/RPfPhnLP9R">https://t.co/RPfPhnLP9R</a><a href="https://twitter.com/hashtag/math?src=hash&ref_src=twsrc%5Etfw">#math</a> <a href="https://twitter.com/hashtag/maths?src=hash&ref_src=twsrc%5Etfw">#maths</a> <a href="https://twitter.com/geogebra?ref_src=twsrc%5Etfw">@geogebra</a> <a href="https://twitter.com/hashtag/mtbos?src=hash&ref_src=twsrc%5Etfw">#mtbos</a> <a href="https://twitter.com/hashtag/iteachmath?src=hash&ref_src=twsrc%5Etfw">#iteachmath</a> <a href="https://twitter.com/hashtag/mathart?src=hash&ref_src=twsrc%5Etfw">#mathart</a> <a href="https://twitter.com/hashtag/mathchat?src=hash&ref_src=twsrc%5Etfw">#mathchat</a> <a href="https://t.co/5NvXW9BwrA">pic.twitter.com/5NvXW9BwrA</a></div>— Daniel Mentrard (@dment37) <a href="https://twitter.com/dment37/status/969250003561713664?ref_src=twsrc%5Etfw">March 1, 2018</a></blockquote><script async="" charset="utf-8" src="https://platform.twitter.com/widgets.js"></script> <br /><div style="line-height: inherit; margin-bottom: 0.5em;"><div style="background-color: white; color: #222222; font-family: sans-serif;"><span style="font-size: xx-small;"><b><i>"Chain of Circles - Daniel Metrard @dment27"</i></b></span></div><br />To start off here's a few facts about the number 155 I found on the wikipedia:<br /><br /><div><b>155</b> is:</div></div><ul><li>a <a href="https://en.wikipedia.org/wiki/Composite_number">composite number</a></li><li>a <a href="https://en.wikipedia.org/wiki/Semiprime">semiprime</a>. </li><li>a <a href="https://en.wikipedia.org/wiki/Deficient_number">deficient number</a>, since 1+ 5 + 31 = 36 < 135 </li><li><a href="https://en.wikipedia.org/wiki/Odious_number">odious</a>, since its <a href="https://en.wikipedia.org/wiki/Binary_expansion">binary expansion</a> 10011011 has a total of 5 ones in it.</li></ul>There are 155 primitive <a href="https://en.wikipedia.org/wiki/Permutation_group">permutation groups</a> of degree 81. <a href="https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/11px-OEISicon_light.svg.png" /></a> <a href="https://oeis.org/A000019">A000019</a><br /><br />If one adds up all the primes from the least through the greatest prime factors of 155, that is, 5 and 31, the result is 155. (sequence <a href="https://oeis.org/A055233">A055233</a> in the <a href="https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences">OEIS</a>) Only three other "small" semiprimes (10, 39, and 371) share this attribute.<br /><br /><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><h2>Posts</h2><div><br /></div><div>Patrick Honner's Favorite Theorem</div></div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">Evelyn Lamb<br /><a href="https://blogs.scientificamerican.com/roots-of-unity/patrick-honners-favorite-theorem/">https://blogs.scientificamerican.com/roots-of-unity/patrick-honners-favorite-theorem/</a><br /><div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"><span style="font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span></div></div><blockquote class="tr_bq">"In this episode of My Favorite Theorem, Kevin Knudson and I were happy have Patrick Honner, a math teacher at Brooklyn Technical High School, as our guest. You can listen to the episode here or at <a href="https://kpknudson.com/my-favorite-theorem/2018/2/17/episode-13-patrick-honner">kpknudson.com</a>. I rarely have cause to include a spoiler warning on this podcast, but this theorem is so fun, you might want to stop the episode around the 4:18 mark and play with the ideas a little bit before finishing the episode. Parents and teachers may want to listen to it alone before sharing the ideas with their kids or students."</blockquote><div>This entire series at Scientific American has been really fun to read/listen to. This month's exploration of Varignon's theorem may be the best one yet.<br /><div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><br /><div style="color: black; font-family: "Times New Roman"; font-size: medium;"><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="text-align: center;" width="200" /></div><div><br /></div></div>Pythagorean Proof </div><div>Loop Space</div><div><a href="http://loopspace.mathforge.org/CountingOnMyFingers/FavouriteProof/">http://loopspace.mathforge.org/CountingOnMyFingers/FavouriteProof/</a><br /><blockquote class="twitter-tweet" data-lang="en"><div dir="ltr" lang="en">I came up with this question today for Yr8 who haven't done pythagoras yet, inspired by <a href="https://twitter.com/UKMathsTrust?ref_src=twsrc%5Etfw">@UKMathsTrust</a> 2017 junior paper (qu8) <a href="https://twitter.com/hashtag/mathschat?src=hash&ref_src=twsrc%5Etfw">#mathschat</a> <a href="https://twitter.com/hashtag/mathscpdchat?src=hash&ref_src=twsrc%5Etfw">#mathscpdchat</a> <a href="https://twitter.com/hashtag/mtbos?src=hash&ref_src=twsrc%5Etfw">#mtbos</a> <a href="https://t.co/tKlJrz4Bso">pic.twitter.com/tKlJrz4Bso</a></div>— Mark Horley Maths (@mhorley) <a href="https://twitter.com/mhorley/status/961950184560263168?ref_src=twsrc%5Etfw">February 9, 2018</a></blockquote><br />An interesting twitter thread from above led me to this post. I've experimented with how to teach the Pythagorean therorem in the <a href="http://past/">past</a> several times and like how this approach based on similarity differs from some of the more commonly used algebraic techniques.<br /><br /><div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><img height="320" src="https://pbs.twimg.com/media/DV6gQQaX4AAcmXW.jpg" width="250" /><br /><br /><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="color: black; font-family: "Times New Roman"; font-size: medium; text-align: center;" width="200" /><br /><br /></div>Fun with Fractions—from elementary arithmetic to the Putnam Competition the first 1/2<br />Dan McQuillan<br /><a href="http://voices.norwich.edu/daniel-mcquillan/2018/02/25/fun-with-fractions-from-elementary-arithmetic-to-the-putnam-competition-the-first-1-2/">http://voices.norwich.edu/daniel-mcquillan/2018/02/25/fun-with-fractions-from-elementary-arithmetic-to-the-putnam-competition-the-first-1-2/</a><br /><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><blockquote class="tr_bq">"Elementary discussions and good questions in grade school can prepare students for far more difficult challenges later. This post provides an example, by starting with simple fraction questions and ending with a Putnam Mathematical Competition Question (intended for stellar undergraduates). It also features atypical ways of comparing fractions. A much shorter discussion of these problems is possible; this discussion reflects an attitude of starting from little and gaining quickly."</blockquote>We had recently been working with Farey Sequences: <a href="http://mymathclub.blogspot.com/2018/02/213-farey-sequences.html">http://mymathclub.blogspot.com/2018/02/213-farey-sequences.html</a> so this article had special resonance for me. The extension at the end is particularly good.</div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><br /></div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="background-color: white; text-align: center;" width="200" /><br /><br />Triangulations and face morphing<br />David Orden<br /><a href="https://mappingignorance.org/2018/02/21/triangulations-face-morphing/">https://mappingignorance.org/2018/02/21/triangulations-face-morphing/</a></div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><blockquote class="tr_bq">"This post talks about one of the easiest mathematical tools for morphing, using triangulations, and explains recently published results about morphing planar graph drawings."</blockquote><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">A very nice overview and perhaps a starting point for further reading.<br /><br class="Apple-interchange-newline" /><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="text-align: center;" width="200" /><br /><br /><div style="text-align: center;"><br /></div>Fun Not Competition the story of My Math Club<br />Dr. Jo Hardin<br /><a href="http://scholarship.claremont.edu/jhm/vol8/iss1/17/">http://scholarship.claremont.edu/jhm/vol8/iss1/17/</a></div><div class="separator" style="clear: both; text-align: center;"></div><div style="margin-left: 1em; margin-right: 1em;"></div><br /><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><blockquote class="tr_bq">"For almost three years, I have spent most of my Sunday afternoons doing math with my daughters and a group of their school friends. Below I detail why and how the math club is run. Unlike my day job, which is full of (statistical) learning objectives for my college students, my math club has only the objective that the kids I work with learn to associate mathematics with having fun. My math club has its challenges, but the motivation comes from love of mathematics, which makes it fun, and worth every minute."</blockquote>This is a lovely personal account of Dr Hardin's experiences working with young children. I'm a very strong believer in the power of Math Circle's to impact students so hopefully this will motivate someone else.<br /><br /></div><h2><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="background-color: white; text-align: center;" width="200" /></h2></div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">The many faces of the Petersen graph<br />Mark Dominus<br /><a href="https://blog.plover.com/math/petersen-graph.html">https://blog.plover.com/math/petersen-graph.html</a></div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><br /><img alt="The Petersen graph has two sets of five vertices each. Each set is connected into a pentagonal ring. There are five more edges between vertices in opposite rings, but instead of being connected 0–0 1–1 2–2 3–3 4–4, they are connected 0–0 1–2 2–4 3–1 4–3." src="https://pic.blog.plover.com/math/petersen-graph/Petersen-fivecycle-jolly.svg" /><br /><br /><blockquote class="tr_bq">"The Petersen graph is a small graph that is an important counterexample to all sorts of things. It obviously has a fivefold symmetry. Much less obviously, it _also_ has threefold, fourfold, and sixfold symmetries! You can draw it in many ways and it can be really hard to tell that they are all drawing of the same thing!"</blockquote><br /><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="background-color: white; text-align: center;" width="200" /><br /><br /></div><div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">Parameterizing the Space of 3D Rotations<br />Arvind Rao<br /><a href="http://www.rao.im/mathematics/2017/12/30/parametrizing-the-space-of-3d-rotations/">http://www.rao.im/mathematics/2017/12/30/parametrizing-the-space-of-3d-rotations/</a></div><br /><blockquote class="tr_bq">In game development and 3D image processing it is common to represent 3D rotations not as 3 x 3 matrices but as quaternions. I wrote a somewhat long read at the end of last year describing the relationship between SO(3), the space of 3D rigid rotations, and the unit quaternions. I think readers will enjoy the use of heuristic visualizations to uncover the true 'shape' of SO(3). Also, with SymPy, a wonderful symbolic computation library, I compute representations that give coordinates on SO(3). The calculations are really involved, so SymPy is super helpful; all code is linked within the post.</blockquote><img alt="Image result for line separator" src="http://www.authorbrianjackson.com/gimpAndHTML/images/filigree.png" height="25" style="background-color: white; text-align: center;" width="200" /><br /><div><br /></div>DIY Pattern Maker<br /><a href="http://linescurvesspirals.blogspot.co.uk/2018/02/diy-pattern-maker.html">http://linescurvesspirals.blogspot.co.uk/2018/02/diy-pattern-maker.html</a></div><div><br /></div><div><img src="https://1.bp.blogspot.com/--dYDb7JQGQM/WnxRGBtGXpI/AAAAAAAACz4/Vl9_jb8XeJQ3lNSLfQGtFm7Vor3wgNHOwCEwYBhgL/s320/IMG_1484.JPG" /></div><div><br /></div><div><br /></div><div>This is a visual exploration of patterns as well as inventive recycling that looks fun to use in a classroom. </div><div><br /><h2>Request</h2>If you've made it this far and are involved in Mathematics Research I would love it if you would consider contributing some answers to <a href="http://mymathclub.blogspot.com/2017/05/questions-for-mathematicians.html">Questions for Mathematicians</a> that I've been compiling for my kids and thanks for reading this post. Either just add a comment on the page or email me.</div>http://mymathclub.blogspot.com/2018/03/carnival-of-mathematics-155.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-9118612297266805938Mon, 26 Feb 2018 19:43:00 +00002018-03-01T15:01:47.412-08:00algebradigressionIn praise of the Rational Roots Theorem<script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); </script><script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> First some personal historical background. In my school district, you could do Algebra in middle school but unlike a standard class it only covered linear equations. When I was in High School after doing Geometry in 9th grade, I entered a 3 year accelerated math sequence that terminated with AP Calculus BC. For the first year we did a semester going over quadratics leading up to the derivation of the quadratic formula and a semester of trigonometry. So for all intents and purposes, I didn't learn anything from the standard Algebra II curriculum. Interestingly, this didn't have any particular consequences and as time went by I learned some of the topics when necessary and required for something else. I remember thinking in College, "I wish I had covered more Linear Algebra/Matrices" but never really "What's Descartes' rule of signs?"<br /><br />However, over the last few years my affection for two particular tools from there has grown quite a bit: The Rational Roots Theorem and Polynomial division. First, these are often under attack and dropped (just as in my own experience). Its not unusual to see people wonder online: what are the real world applications of these or will they ever be used again? In High School, I might have said you can always graph and use approximation techniques like Newton's Method when these come up. More significantly, the existence of Wolfram Alpha has made generalized solutions to cubic and quartic equations easily accessible (if not derivable) From my perspective, they are two basic polynomial analysis techniques that offer a gateway to understand higher degree polynomials. That understanding is valuable in itself but in addition they offer a fairly general technique for a lot of algebraic puzzles that I try out and I find it extremely satisfying to be able to analyze these with just pencil and paper.<br /><br /><h4><br />Example 1:</h4><br />$$x^2 - 13 =\sqrt{x + 13}$$<br /><br />This looks not to hard at first until you square both sides to get rid of the radical and realize its a quartic in disguise:<br /><br />\(x^4 - 26x^2 + 169 = x + 13\) => \(x^4 - 26x^2 -x + 156 = 0\)<br /><br />A common strategy at this point is to look for clever factorizations. But its often really hard to see where to start. In fact, I find these are often easier to derive backwards after you know the roots anyway.<br /><br />So let's start with a quick graph of the functions. This could be done by hand but I'll use geogebra here. The left hand side is a parabola with vertex at (0,-13) while the the right hand side is half of the rotated 90 degree version of the same parabola with a vertex at (-13,0). If you're looking for factorization this symmetry is something that provides an avenue of attack. But for our purposes it also shows us there are only 2 real roots in the quartic and approximately where they lie.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-2Wugb8mFEOs/WpRRctyqOLI/AAAAAAAAMlY/CLQxOmyvRvM0Aq4BOtbV6Im8w3Dx6Qs6gCLcBGAs/s1600/graph.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="793" data-original-width="817" height="619" src="https://1.bp.blogspot.com/-2Wugb8mFEOs/WpRRctyqOLI/AAAAAAAAMlY/CLQxOmyvRvM0Aq4BOtbV6Im8w3Dx6Qs6gCLcBGAs/s640/graph.png" width="640" /></a></div><br /><br /><br />Here's where the Rational Roots test comes in. Since 156 = \(2^2\cdot 3 \cdot 13\) It says that if there is rational root its going to be either: \( \pm 1, \pm 2, \pm 3, \pm 4\, \pm6, \pm 12, \pm 13, \pm26, \pm 39, \pm 52, \pm 78\) or \( \pm156 \) That's a bit daunting but looking at the graph or the behavior of the functions indicate we really only need to test smaller values and 4 is probably the most promising.<br /><br />I just plugged that back into the original problem rather than doing the quartic and indeed -4 works out. (This is a bit of a cheat since not all the quartic solutions are also solutions to the original problem due to sign issue with the radical but if it does work then you're golden.)<br /><br />At this point we know know x+4 is a factor of the original quartic and we can divide it out to get a simpler cubic equation.<br /><br />Apply polynomial division \(\frac{x^4 - 26x^2 - x + 156 }{x+4} = x^3 -4x^2 - 10x + 39\) to get the remaining cubic part of the equation.<br /><br />Now once again we can apply the rational roots test but on the much smaller set {1,3,13,39}. Its clear from the graph that none of these are going to be a solution to the original problem and that again the smaller ones are more likely. So starting at 1, I find that 3 works out (27 - 36 - 30 + 39 = 0). That mean x -3 is another factor. Interestingly you can see why it doesn't work: 9 - 13 = -4 while the square root of 13 + 3 = 4. So the inverse sign changes have interfered (but if the bottom of the sideways parabola were present that would be an intersection point). <br /><br />Once again apply polynomial division \( \frac{x^3 -4x^2 - 10x + 39}{x - 3} = x^2 - x +13 \) Having factored the quartic down to an approachable quadratic we can now apply the quadratic formula to find two more solutions: \( \frac{1 \pm \sqrt{53}}{2} \). Either by testing or looking at the graph we can see \( \frac{1 - \sqrt{53}}{2} \) is the second solution while its converse again lies on the intersection of missing bottom half of the sideways parabola.<br /><br />Extension for another time: We have 3 and 4 wouldn't it be nice if 5 also showed up (and this is tantalizingly close to the generator function for pythagorean triples in the complex plane)? Is there a general form to the intersections of this type i.e. a parabola and its rotated counterpart?<br /><br /><br /><h4>Example 2:</h4><br />Find the integer solutions to: \(x^3y^3 - 4xy^3 + y^2 + x^2 - 2y - 3 = 0\)<span style="background-color: white; color: #444444; font-family: sans-serif; font-size: 15px;"></span><br /><br />This again looks fairly complex and of degree 6 on first glance. But lets try experimenting with values of x and see what falls out: [I'm going to only consider the x >= 0 for simplicity here but somewhat similar logic applies for the negatives.]<br /><br />if x = 0 this simplifies to:<br /><br />$$y^2 - 2y - 3 = 0$$ which has 2 integer roots.<br /><br />if x = 1 this simplifies to:<br />$$-3y^3 + y^2 -2y - 2 = 0$$<br />We can applies the rational root test and check the possible integers \(\pm1, \pm2\) with no hits.<br /><br />if x = 2 this simplifies to:<br />$$y^2 -2y + 1 = 0$$ which has one integer solution.<br /><br /><b>Note the constant term flipped from negative to positive at this point and now something interesting happens</b><br /><br />if x = 3 this simplifies to<br />$$15y^3 + y^2 -2y + 6 = 0$$<br />The rational roots test now is only going to give positive candidates and the higher degree terms start to dominate making it impossible for this to reach 0 with an integer. I.e. \(15y^3 + y^2 > 2y\) for all integers > 1.<br /><br />Continuing on the same trend continues<br />if x = 4 this simplifies to<br />$$48y^3 + y^2 -2y + 13 = 0$$ For the same reason this is even worse \(48y^3 + y^2 > 2y\) for all integers > 1.<br /><br />So we can infer with this logic that no integer solutions exist above x = 2.http://mymathclub.blogspot.com/2018/02/in-praise-of-rational-roots-theorem.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-2260490419549384660Thu, 22 Feb 2018 04:09:00 +00002018-02-21T20:09:19.024-08:00Aloha<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-3mA6kD3f3Jc/Wo5Bu2j183I/AAAAAAAAMjM/d1fULzd8OEYOwFa-oH3MLcR_iIiR3ebsACKgBGAs/s1600/20180218_143508.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://4.bp.blogspot.com/-3mA6kD3f3Jc/Wo5Bu2j183I/AAAAAAAAMjM/d1fULzd8OEYOwFa-oH3MLcR_iIiR3ebsACKgBGAs/s320/20180218_143508.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>I'm on mid winter break with the kids for the week in Oahu.<br /><br />In the meantime check out my collected problems which are almost at 30 in total: <a href="http://mymathclub.blogspot.com/p/collected-problems-2.html">http://mymathclub.blogspot.com/p/collected-problems-2.html</a><br /><br /><br /><br />http://mymathclub.blogspot.com/2018/02/aloha.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-1162481924502703447Fri, 16 Feb 2018 20:30:00 +00002018-02-16T12:30:12.639-08:002/13 Farey Sequences<div class="separator" style="clear: both; text-align: center;"><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Farey_diagram_square_9.svg/220px-Farey_diagram_square_9.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="233" data-original-width="220" height="400" src="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Farey_diagram_square_9.svg/220px-Farey_diagram_square_9.svg.png" width="377" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Today was a special occasion for Math Club. Instead of just me or a vicarious video, we once <a href="http://mymathclub.blogspot.com/2017/05/59-dating-for-elementary-students.html">again</a> had a guest lecture from the UW Applied Math department. </div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><a href="https://2.bp.blogspot.com/-qM6So_zvgeE/Wocu15H9rOI/AAAAAAAAMeE/Q1sg7sWntQwXH4tSS9zrYcF9mPcvOrlAQCKgBGAs/s1600/20180213_163416.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://2.bp.blogspot.com/-qM6So_zvgeE/Wocu15H9rOI/AAAAAAAAMeE/Q1sg7sWntQwXH4tSS9zrYcF9mPcvOrlAQCKgBGAs/s400/20180213_163416.jpg" width="400" /></a><br /><div class="" style="clear: both; text-align: left;"><br />This time, <a href="https://faculty.washington.edu/jathreya/">Professor Jayadev Athreya</a> came out to the middle school to give a talk on Farey Sequences. That was fairly propitious, since I had meant to get to this subject during this session: <a href="http://mymathclub.blogspot.com/2017/11/1128-egyptian-fractions.html">http://mymathclub.blogspot.com/2017/11/1128-egyptian-fractions.html</a> but quickly realized I didn't have enough time to cover even Egyptian Fractions. So there was a good thematic fit with some of the other things we've done.<br /><br />My favorite moment of the day came early on when Jayadev had each of the kids talk about why they came to math club. (I usually do this on the first session too) There were a smattering of "I like competitive math" responses but then we reached a girl who roughly said "I don't know why I came originally but I like it so I keep coming." That's victory in my book!<br /><br />What's also interesting here is a chance to more closely observe all the kids and another person's teaching style. Jayadev's basic structure was fairly similar to what I might have done.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><li>Start with having the kids map out all the reduced form fractions where the denominator was less than or equal to 10 and then arrange them by size from smallest to largest. He then graphed this on a number line as a group.</li><br /><ol></ol><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-wgWQG_82fIw/Wocu19bM59I/AAAAAAAAMeE/alEq5RO0POk7XqFUW-adNE4oyctAaCudgCKgBGAs/s1600/20180213_160104.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-wgWQG_82fIw/Wocu19bM59I/AAAAAAAAMeE/alEq5RO0POk7XqFUW-adNE4oyctAaCudgCKgBGAs/s320/20180213_160104.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>Closely investigate the numerators of the fractions (in suitable common denominator form) when comparing them to notice a trend: they always differed by one.</li></ul><br /><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span><a href="https://2.bp.blogspot.com/-pWwj4oYvzdM/Wocu18LGtyI/AAAAAAAAMeE/7AIECo0aLTIbfwXFOwZzO7GkiPjGPgCsgCKgBGAs/s1600/20180213_162847.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://2.bp.blogspot.com/-pWwj4oYvzdM/Wocu18LGtyI/AAAAAAAAMeE/7AIECo0aLTIbfwXFOwZzO7GkiPjGPgCsgCKgBGAs/s320/20180213_162847.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span><a href="https://2.bp.blogspot.com/-oG4Hnm-lS94/Wocu140ojiI/AAAAAAAAMeE/MrXoJVXDHHoYU79vids5XrOP_pwbRM37QCKgBGAs/s1600/20180213_162840.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://2.bp.blogspot.com/-oG4Hnm-lS94/Wocu140ojiI/AAAAAAAAMeE/MrXoJVXDHHoYU79vids5XrOP_pwbRM37QCKgBGAs/s400/20180213_162840.jpg" width="400" /></a></div><ul><li>Build up the definition of the mediant: <a href="https://en.wikipedia.org/wiki/Mediant_(mathematics)">https://en.wikipedia.org/wiki/Mediant_(mathematics)</a> and see how it relates to the fractions already found.</li></ul><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-gF45QLhauyw/Wocu14d1_7I/AAAAAAAAMeE/7c0E-Mv-7uciW1CG17eN_yjpY8kGlqXIQCKgBGAs/s1600/20180213_164443.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://1.bp.blogspot.com/-gF45QLhauyw/Wocu14d1_7I/AAAAAAAAMeE/7c0E-Mv-7uciW1CG17eN_yjpY8kGlqXIQCKgBGAs/s320/20180213_164443.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"></div><ul><li>Do a formula proof that the mediant is always 1 apart from its generators if they are 1 apart.</li></ul><br /><div><br /></div></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-0Tu14cfizgM/Wocu13doKLI/AAAAAAAAMeE/wCZKZVQKQJwnvGX5ktQZTcIP2saY0f2RACKgBGAs/s1600/20180213_161743.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-0Tu14cfizgM/Wocu13doKLI/AAAAAAAAMeE/wCZKZVQKQJwnvGX5ktQZTcIP2saY0f2RACKgBGAs/s320/20180213_161743.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-h5BbyDGrPQc/Wocu1xRaPlI/AAAAAAAAMeE/Z3RDa_WXx3gvRBR6x3pohmKmb3jSj-Z3QCKgBGAs/s1600/20180213_160449.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://2.bp.blogspot.com/-h5BbyDGrPQc/Wocu1xRaPlI/AAAAAAAAMeE/Z3RDa_WXx3gvRBR6x3pohmKmb3jSj-Z3QCKgBGAs/s320/20180213_160449.jpg" width="320" /></a></div><br />A lot of this was structured as group work at the tables with discussions after a few minutes where things were consolidated. I always find these transitions a bit tricky to time so it was useful watching someone else. I would also probably have done a few of these parts as group work on the whiteboard itself and then gallery walked for the discussions but there was good work done at everyone's seats.<br /><br />Like last time, I also noticed some unexpected hesitancy with operating on fractions. It took a bit more time to draw out the kids and have them explain how to compare fractions with different denominators. Although create common denominators was mentioned as well as variety of numeracy instincts ("for unit fractions, the fraction become smaller as the denominator increases" or "you can also create common numerators to do informal comparisons"). Again, I felt like this was a useful practice/review for some. Alternate hypothesis: the kids were more reluctant to volunteer at points which was more social rather than indicating any gaps. If this is the case, I'd like to work on activities to bring out more questions. One idea I have toyed with in the past is, is selecting one student to be "the skeptic" during any demo and come up with at least one question about the logic. If I do go this way, I'll probably start with having them do this with something I discuss and depending on how it goes try it also during all whiteboard discussions.<br /><br />Overall I was really pleased. We now have an invitation to visit the Applied Math Center on the UW campus. I have to investigate whether the logistics are workable.<br /><br />http://mymathclub.blogspot.com/2018/02/213-farey-sequences.htmlnoreply@blogger.com (Benjamin Leis)1tag:blogger.com,1999:blog-4227811469912372962.post-7723245742904162003Thu, 08 Feb 2018 21:07:00 +00002018-02-09T09:35:34.842-08:002/6 Olympiad #3 and AMC 10<div>I almost cancelled this week's math club due to feeling ill the night before. But in the end I was well enough and the activities were straightforward so I went ahead with the meeting. We started with the candy I had forgotten to bring last week. My wife picked up some red vines for me, the reception of which I was curious to see. They were all eaten by the end so there may be more licorice in the future.</div><div><br /></div><div>Participation in the problem of the week was lighter that I would like but I had enough kids to still demo solutions. In particular with this problem: <a href="http://www.cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWD-17-NA-16-S.pdf">http://www.cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWD-17-NA-16-S.pdf</a> the key is to count the total number of pips in the set<br />of dominos.<br /><br /><img alt="Image result for domino image" height="200" src="data:image/png;base64,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" width="100" /><br /><br />The students demonstrated two different methods which was good. Most approaches end up with a triangular table since when you calculate the combinations you often end up with n pips | m pips and m pips | n pips which are the same domino. I'm trying to elicit more questions from the other kids. This time it worked well when I asked "Does anyone have any questions about X's diagram and how X did ...." I also spent some time modelling asking questions about their strategies and why they had created the triangles to draw this out.<br /><br />Once done we participated in the third MOEMS olympiad. My feeling is this was the most unbalanced of the set so far. The starter questions were all fairly easy and they gave a hint that unwound most of the complexity of one of them and then it ended with a really interesting<br />Diophantine fraction equation that was quite a bit more difficult to do. One followup question I have for myself: is even in cases that simplify is it enough to consider just the factors of the denominator of a sum i.e. if K1 / A + K2 / B = K3/K4 where all the cases are constant.<br /><br />Finally I chose another AMC based question for the Problem of the Week:<br /><br /><a href="https://drive.google.com/open?id=1N4HaNYRjwhhSSYq9UqMZNyBvqaL2w65UFIjV83z3G6Y">https://drive.google.com/open?id=1N4HaNYRjwhhSSYq9UqMZNyBvqaL2w65UFIjV83z3G6Y</a><br /><br />Overall everything ran well but it was not my most inventive day which was probably just as well since I felt very low energy at points.<br /><br />The next day, one of the teachers at Lakeside graciously let me send a few students over to take AMC10. I couldn't justify the cost to do this on site for so few students. In the future, I'm hoping with more eight graders this might change. At any rate, this was fun for me. I had the three kids take a practice test first to make sure this was a reasonable move. My goal was for everyone to get at least 6-10 answers correct. What I don't want to happen is for kids to go and get so few questions correct that the entire experience is discouraging. I've also been feeding more sample questions from AMC10 as problems of the week. Generally, given enough time they often make really good exercises. The kids reported this year's test was a bit harder than the practice versions so I'm cautiously awaiting the official results.<br /><br />Looking forward: next week is going to be real fun. One of the professors from UW, Jayadev Athreya is coming to give a guest talk to the kids on Farey sequences (which by coincidence we didn't quite get to on: <a href="http://mymathclub.blogspot.com/2017/11/1128-egyptian-fractions.html">http://mymathclub.blogspot.com/2017/11/1128-egyptian-fractions.html</a>. <br /><br /><h4>Resource Investigations</h4>I just learned about the MoMath Rosenthal prize winners <a href="https://momath.org/rosenthal-prize/">https://momath.org/rosenthal-prize/</a> I'm going to look through the sample lessons to see if there is anything that is usable in our context. By that I mean far enough off the beaten curriculum track.<br /><br />I also really like this investigation from the Math Teacher's Circle Network: <span id="goog_1236653583"></span><a href="https://www.blogger.com/"></a><span id="goog_1236653584"></span> <a href="https://www.mathteacherscircle.org/news/mtc-magazine/s2018/touching-infinity/">https://www.mathteacherscircle.org/news/mtc-magazine/s2018/touching-infinity/</a> on countable infinite sets. I have to spend some time thinking about it but it looks quite promising.</div><div><div><br /></div><div><br /><div><br /></div><div></div></div></div>http://mymathclub.blogspot.com/2018/02/26-olympiad-3-and-amc-10.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-4380262148718457787Wed, 31 Jan 2018 20:32:00 +00002018-03-01T13:35:09.841-08:001/30 Math Counts Prep DayWe're only two weeks out from MathCounts and I've been so busy with various topics and activities that I haven't really specifically focused on it. For the most part we're doing interesting problems that will overlap anyway and it will all work out but I wanted to spend one day going over the format before the kids go so they know what to expect.<br /><br />So I went to the MathCounts site and printed out last year's contest questions:<br /><br /><a href="https://www.mathcounts.org/programs/competition-series/past-competitions">https://www.mathcounts.org/programs/competition-series/past-competitions</a><br /><br />I knew I would go over the basic format and rules i..e how many questions, can you use a calculator what do you do as a team? I also wanted to try out a little bit of everything. Immediately, I decided that I couldn't really do the countdown rounds. Those are run like a quiz bowl and I have neither the equipment nor desire to to replicate that. For one, I have a few kids who I think would find it too high pressure and secondly it only allows a few kids to participate at a time which I dislike for class management reasons as well as on general principle that I want every kid doing math for as much of the scant hour that we have. So hopefully that won't have any impact on the performance at the contest.<br /><br />Instead I decided to focus on the individual and team sections. (I printed the target round but knew even going in we wouldn't have time to try those out.)<br /><br />Thinking about this ahead of time, I decided to try out a new strategy with the individual round: speed dating.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-vP15GaLARY4/WnIji3oX_gI/AAAAAAAAMY4/5I1tL1ScousUHDOThP9J_JtWaLGnWE3zACKgBGAs/s1600/trollface.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="821" data-original-width="1460" height="223" src="https://3.bp.blogspot.com/-vP15GaLARY4/WnIji3oX_gI/AAAAAAAAMY4/5I1tL1ScousUHDOThP9J_JtWaLGnWE3zACKgBGAs/s400/trollface.jpg" width="400" /></a></div><br /><br />Basically I had the kids setup a large row of tables in the center of the room and had everyone face someone else. To start I gave out one of the even numbered problems to each kid. My instructions were: this is your problem, you will solve it and then for everyone else you will be the expert and double check their answer as well as help with any problems. We then rotated every few minutes. Every rotation the kids told each other their respective problems and then worked on them.<br /><br />I was worried going in that the rotation timing would be tricky especially since the problems varied in difficulty. That turned out to not be an issue because they were generally "simple enough" that everyone could finish within a few minutes and I just had to survey where everyone was. It also let me point out that the difficulty varied and that different people would take different amounts of time depending on which problem they were on. That had a useful effect on expectations.<br /><br />Overall, I would use this format again for easier problems/review. It seemed to keep kids working over a larger set of problems and I liked how it farmed out answer checking. There are 4 issues to keep in mind<br /><br /><ul><li>In a complete rotation everyone will only see half of the problems. So you need to swap the problems at that point if you want to have everyone to do everything.</li><li>Timing can be still be tricky. The problems should be varied in difficult but not by "too much".</li><li>I didn't stress the ownership as much as I need to initially. If I reuse I will emphasize that role and go around and check for any questions at that point about the problems.</li><li>I suspect this falls apart the more complex the questions are.</li></ul><div><br /></div><div>Coincidentally, one of the teacher's running the yearbook wandered in to take photos in the middle of all this. So we'll definitely be in the yearbook looking studious. As my son remarked afterwards, the club hasn't gotten any school paper mentions and I should work on this in the future. For one, I'll take a team photo at MathCounts and submit it.</div><div><br /></div><div>For the second half, I handed out the team tests and just group everyone based on where they had landed at the end of all the seat rotations. (coincidental Visible Random Grouping) During this section I floated a lot, asked hopefully helpful questions, answered any of theirs, and pointed out problems that were not correctly done yet. I was actually pleased that this went very smoothly. I didn't really need to do any prompting to keep everyone engaged.</div><div><br /></div><div>Finally, because in my excitement I had jumped in I had to reserve 5 minutes at the end to go over the problem of the week. Interestingly there were two programmatic solutions submitted this time. If this trend continues I'm going to start handing out explicit problems aimed all the kids who want to program.</div><div><br /></div><div>New P.O.T.W:</div><div><br /></div><div><a href="http://www.cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWD-17-NA-16-P.pdf">http://www.cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWD-17-NA-16-P.pdf</a></div><div><br /></div><div>A domino pip problem from UWaterloo. I've liked these type problems in the past.</div><br />http://mymathclub.blogspot.com/2018/01/130-math-counts-prep-day.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-3347979812581077852Wed, 24 Jan 2018 18:36:00 +00002018-01-25T09:23:11.867-08:00fold and cut1/20 Fold and Cut IIToday started with an interesting whiteboard demo for the <a href="http://mymathclub.blogspot.com/2018/01/117-graphs-and-paths.html#potw">Problem of the Week</a>. This is a fairly straight forward combinatorics problem on a small 2^9 total set of possibilities. One of my students just went ahead and wrote a python program to brute force check for the answer. While this won't work in a contest setting, I really like the use of computational math. If I had access to a computer lab and I knew everyone could program I'd love to do a whole session around the <a href="https://projecteuler.net/">Project Euler</a>. It would also make a really cool class structure to learn programming over a period of time.<br /><br />But the other thought experiment this generated was what is the purpose of some of these problems in the age of cheap computing? This is well trod territory. Open Middle problems as they are commonly formulated often make me think this is better done as a brute force search.<br /><br /><blockquote class="twitter-tweet" data-lang="en"><div dir="ltr" lang="en">Checking an open middle type problem for my son <a href="https://t.co/vN202f1Kek">pic.twitter.com/vN202f1Kek</a></div>— Benjamin Leis (@benjamin_leis) <a href="https://twitter.com/benjamin_leis/status/915060957998493696?ref_src=twsrc%5Etfw">October 3, 2017</a></blockquote><script async="" charset="utf-8" src="https://platform.twitter.com/widgets.js"></script> <br />My current thinking is that computational math is more interesting if its quicker to write a program than a formal method or if essentially you need to search a wide domain for the answers and there isn't much structure to help out. Also problems can be modified to make the computational requirements more interesting. But this is obviously a fuzzy standard and I'm not sure how to align this with my general ambivalence about calculators.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-MRIDSkTSXhU/WmjB2c7yrtI/AAAAAAAAMVQ/UHpFHOramLofaBlcK5q_6FY6BMNpB8VVgCKgBGAs/s1600/20180124_092446.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="360" src="https://3.bp.blogspot.com/-MRIDSkTSXhU/WmjB2c7yrtI/AAAAAAAAMVQ/UHpFHOramLofaBlcK5q_6FY6BMNpB8VVgCKgBGAs/s640/20180124_092446.jpg" width="640" /></a></div><br />The problem was also an opportunity to hand out some geeky stickers I bought on a lark from <a href="https://mathsgear.co.uk/">https://mathsgear.co.uk/</a>. As an aside I went back and forth if the black sticker at the bottom should be read "No change in learning (bad) or peak learning (good)"<br /><br />For the main activity, I've been meaning to do another day focusing on the <a href="http://fold%20and%20cut%20theorem/">Fold and Cut Theorem</a> since it went so well <a href="http://two%20years%20ago./">two years ago.</a> At this point I only have 3 or 4 kids left from that time and I thought I could provide enough different tasks and/or they had not reached the end the first time that it wouldn't be boring for them.<br /><br />This time around I went with a part of Erik Demaine's lecture @ MIT.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/K0GuKDSX1FA/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/K0GuKDSX1FA?feature=player_embedded" width="320"></iframe></div><br />The choice was motivated by the fact Demaine developed a lot of the algorithms and includes some historical notes on the first examples in Japan. But also I'm terrible at folding and there are a bunch of great demos in the first 10 minutes which the kids really liked. That saved me from a lot of practice at home.<br /><br />I paused at around the 9 minute mark and handed out worksheets I've used before from Joel Hamkins:<br /><a href="http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cut/">http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cut/</a><br /><br />These work great even for older kids. While circulating I just made sure to periodically have everyone throw out their scrap paper and to emphasize the role of symmetry in any of the solutions.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-iIz7ZiPm0uQ/WmjRRonP81I/AAAAAAAAMWE/zJpl-NdzkdguW03HyjoH4NM-gt8ewPW9wCKgBGAs/s1600/20180123_162515.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://2.bp.blogspot.com/-iIz7ZiPm0uQ/WmjRRonP81I/AAAAAAAAMWE/zJpl-NdzkdguW03HyjoH4NM-gt8ewPW9wCKgBGAs/s320/20180123_162515.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-fKY-HmCNhC8/WmjRRoW2uhI/AAAAAAAAMWE/zsxPspl2kwIzOe6bS55ubDGaMLfQRKH3wCKgBGAs/s1600/20180123_162520.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-fKY-HmCNhC8/WmjRRoW2uhI/AAAAAAAAMWE/zsxPspl2kwIzOe6bS55ubDGaMLfQRKH3wCKgBGAs/s320/20180123_162520.jpg" width="320" /></a></div><br /><br />(Some handiwork)<br /><br />Finally I reserved 10 minutes at the end to go further in the video and watch the explanation of the straight-skeleton method.<br /><br /><br />P.O.T.W:<br /><span id="docs-internal-guid-409590d9-2974-2c34-ac30-d90c24d5370a"><br /></span>Another slightly modified AMC problem.<br /><br /><br /><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; white-space: pre;">In 1998 the population of a town was a perfect square. Ten years later, after an increase of 150 people, </span><br /><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; white-space: pre;">the population was 9 more than a perfect square. Now in 2018, with an increase of another 150 people </span><br /><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; white-space: pre;">the population is once again a perfect square. What was the population in all three years?</span></div><span id="goog_1487229636"></span><a href="https://www.blogger.com/"></a><span id="goog_1487229637"></span><br />Planning:<br /><br />MathCounts Prep 1/30<br />Olympiad #3 2/6<br />UW Lecture 2/13<br /><br />http://mymathclub.blogspot.com/2018/01/120-fold-and-cut-ii.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-7765220305404588820Fri, 19 Jan 2018 19:05:00 +00002018-01-24T09:10:32.340-08:00graph theory1/17 Graphs and PathsThis week I saw a numberphile video with a fairly charming problem that inspired me:<br />Can you find a way to arrange the numbers 1 through 15 in sequence such that every pair sums to a perfect square?<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/G1m7goLCJDY/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/G1m7goLCJDY?feature=player_embedded" width="320"></iframe></div><br /><br />I decided I wanted to do a graph theory day around this. This goes well on a whiteboard so I had all the kids work on it for about 10-12 minutes. Most found a solution faster than I expected. In retrospect this seems more difficult than it really is since there are only four square sums to consider 4,9,16 and 25 and its clear there are lots of pairs that sum to 25 1 + 24, 2 + 23 etc and very few that sum to 9: 1 + 8. To keep pacing on target I had groups that finished early try adding numbers on. I also asked the kids to consider why was this happening at all.<br /><br />[<b>If I repeated I definitely would stress this question: Is it expected that this is possible and why or why not? What patterns related to the square sums affect the likelyhood?]</b><br /><b><br /></b>After stretching to allow most kids to find the solution we had a group discussion. No one had considered this in terms of graphs so after all the kids were done explaining I showed Matt Parker's solution. This was a good bridge to do a quick discussion about what is a graph, what is an edge, node and degree.<br /><br />Next I introduced the classic Bridges of Konigsberg <a href="https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg">problem</a>.<br /><br /><img src="https://upload.wikimedia.org/wikipedia/commons/5/5d/Konigsberg_bridges.png" /><br /><br />This was a risk because I assumed some kids had seen it before. So I just outright asked who already had worked on it at the beginning. Interestingly most hadn't so I had everyone satisfy themselves like the townspeople that they couldn't find a path for a few minutes. Then we had a discussion about whether there is a way to prove its impossible. No one made the jump to counting the degree of the nodes. So I talked through Euler's logic. I think this could be broken apart more formally by asking the kids to create the graph equivalent themselves and then creating other graphs, classifying them and looking for patterns.<br /><br /><br />From there I had less luck creating the problem sequence. So I went with a few problem sets from the chapter on graphs in Jacobs "Mathematics a Human Endeavor". I liked the problems in the sets but I knew from experience the format was less than ideal. So I gave a packet to each group and had them focus on finding their favorite problem to show to the group at the next break. By circulating among groups I was mostly able to keep forward progress going through questions but its hard work. I'm continually tempted to do a deep dive on a topic but I'm usually still better off creating a coherent problem set stream that come in a few chunks on the whiteboard with discussion interleaved.<br /><br /><br />Of the set, there was a Classic Hamiltonian Path problem (find the loop that visits each node below):<br /><br /><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/60/Hamiltonian_path.svg/220px-Hamiltonian_path.svg.png" /><br /><br /><br />That I think works well and another maze problem that I would probably break out.<br /><br />So overall I think this day was decent but with one or two more Euler/Hamiltonian problems added on (and I'll keep my eyes out for them) I think this could be really tightened.<br /><br /><div id="potw">Problem of the Week:<br />I'm feeling the AMC10 problems more recently so I went with this probability one:<br /><br /><a href="https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10A_Problems/Problem_20#Problem">https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10A_Problems/Problem_20#Problem</a><br /><br /></div><br />For the future:<br /><a href="https://plus.maths.org/content/graphs-and-networks">https://plus.maths.org/content/graphs-and-networks</a><br /><br /><br />http://mymathclub.blogspot.com/2018/01/117-graphs-and-paths.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-922998066187329184Sun, 14 Jan 2018 04:23:00 +00002018-01-14T14:52:18.754-08:00digressiongeometryFun with Pentagons<script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); </script><script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> <br /><div class="separator" style="clear: both; text-align: left;">I'm in the mood for a geometry walk-through. I'll start out by saying this one has tons of solutions. I've thought of 3 or 4 and seen several additional ones (one of my favorite parts of geometry.) I tend in this case to prefer the synthetic to trigonometric solutions but if you add that \( cos(36) = \frac{\phi}{2} \) or any variant rather than blindly calculating a decimal I'm good.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-_yByiPjozbw/WlrQ5IlFoII/AAAAAAAAMSU/LuK3ypiSrvgpWiZOtclmHF1DnVlCDZcmACLcBGAs/s1600/problem.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1101" data-original-width="1080" height="640" src="https://2.bp.blogspot.com/-_yByiPjozbw/WlrQ5IlFoII/AAAAAAAAMSU/LuK3ypiSrvgpWiZOtclmHF1DnVlCDZcmACLcBGAs/s640/problem.jpg" width="625" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;">[@<span class="username u-dir" dir="ltr" style="background: rgb(230 , 236 , 240); color: #657786; direction: ltr; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; font-weight: 700; outline: 0px; unicode-bidi: embed;"><a class="ProfileHeaderCard-screennameLink u-linkComplex js-nav" href="https://twitter.com/jldavilaa01" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #657786; outline: 0px; text-decoration-line: none !important;">@<span class="u-linkComplex-target" style="font-weight: normal; text-decoration-line: underline !important;">jldavilaa01</span></a>]</span></div><br />This is the second interesting pentagon problem I've seen in a week or so. With this one, I immediately thought I'll be disappointed if the golden ratio is not embedded somewhere in the answer. When playing around I spent some time angle chasing and looking for similar triangles. This led to several different ways to find the ratio. I've included the simplest one below.<br /><br />First I assume a regular pentagon of side length 1 for the rest of this discussion. Secondly, I'm going to briefly discuss how the golden ratio is found within the figure.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-DlhlommNjmA/WlrR4XNjvYI/AAAAAAAAMSg/BWGiXXeHN5QvNgVQdI-GQb3K1hX_Mk2fwCLcBGAs/s1600/lemma.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="655" data-original-width="717" height="365" src="https://2.bp.blogspot.com/-DlhlommNjmA/WlrR4XNjvYI/AAAAAAAAMSg/BWGiXXeHN5QvNgVQdI-GQb3K1hX_Mk2fwCLcBGAs/s400/lemma.PNG" width="400" /></a></div><br /><br />If you look at 4 points on the pentagon (A, C, D and E) its clear they form a cyclic quadrilateral with three sides of length 1. Further all the other sides and diagonals have the same length since they are all in congruent triangles.<br /><br />Let \( d = \overline{CE} = \overline{AD} = \overline{AC} \)<br />Using Ptolemy's theorem: \(1^2 + 1\cdot d = d^2\) Solving you get \(d = \frac{1+\sqrt{5}}{2} = \phi\) also know as the golden ratio.<br /><br />With that result in hand I now did some angle chasing:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-hj4gA7osQnk/WlrT02PQFRI/AAAAAAAAMSs/FjmH_tqVLGoTO7nAPghOyH77WWazEYyxgCLcBGAs/s1600/main.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="684" data-original-width="779" height="560" src="https://2.bp.blogspot.com/-hj4gA7osQnk/WlrT02PQFRI/AAAAAAAAMSs/FjmH_tqVLGoTO7nAPghOyH77WWazEYyxgCLcBGAs/s640/main.PNG" width="640" /></a></div><br />I found three 36-54-90 triangles: DHK, EDI and ACG (which are outlined in red above). In addition we already know that:<br /><br /><br /><ul><li>\(\overline{EI} = \frac{\phi}{2}\)</li><li>\(\overline{AC} = \phi \)</li><li>\(\overline{DE} = 1 \)</li><li>\(\overline{HI} = \overline{HK} = b\)</li><li>\(\overline{AG} = 2a \)</li></ul><div>So now we can apply the similar triangles:</div><div><br /></div><div>From DHK and EDI:</div><div>$$\frac{\overline{DH}}{\overline{HK}} = \frac{\overline{DE}}{\overline{EI}} $$</div><div>$$\frac{\overline{DH}}{b} = \frac{1}{\frac{\phi}{2}} \text{ or } \overline{DH} = \frac{2b}{\phi}$$</div><div><br /></div><div>Then \(\overline{DI} = \overline{DH} + \overline{HI} = \frac{2b}{\phi} + b = b\cdot(\frac{2}{\phi} + 1) \)</div><div><br /></div><div>Now look at EDI and ACG:</div><div>$$\frac{\overline{DI}}{\overline{DE}} = \frac{\overline{AG}}{\overline{AC}} $$</div><div>$$\frac{b\cdot(\frac{2}{\phi} + 1)}{1} = \frac{2a}{\overline{\phi}} $$</div><div><br /></div><div>Rearranging:</div><div><br /></div><div>$$\frac{a}{b} = \frac{\phi}{2} \cdot (\frac{2}{\phi} + 1) = \frac{2 + \phi}{2}$$</div><div><br /></div><div>Note: there was a fun alternative presented online by @asitnof using areas rather than similar triangles:</div><div><br /></div><div><a href="https://pbs.twimg.com/media/DTbyl81W0AI8CP0.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="570" data-original-width="800" height="454" src="https://pbs.twimg.com/media/DTbyl81W0AI8CP0.jpg" width="640" /></a></div><div><br /></div><div>Again we start with the cross diagonals being phi in length but instead find 2 different expressions for the length of the triangles. One based on the incircle and the second on the base and height.</div><div><br /></div><div><br /></div><div><br /></div>http://mymathclub.blogspot.com/2018/01/fun-with-pentagons.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-3595951086764477438Wed, 10 Jan 2018 22:35:00 +00002018-01-10T14:44:55.137-08:001/9 Olympiad #2Recruitment<br /><br />By today, I was up to 11 boys and 6 girls. So I'm beyond my target size of 15. I also had a bit of a challenge in that 2 kids hadn't shown up the previous week when I focused on introductions and I was primed to do an Olympiad today. My main strategy here was to be honest with the newcomers via email and send them a practice Olympiad ahead of time as well as stressing that we'd be more "math circle" oriented in future weeks.<br /><br />New Largest Prime<br /><br />To start up, I decided to start this session with a quick mention of the recent discovery of a new largest prime: <span style="background-color: white; color: #222222; font-family: "roboto" , "arial" , sans-serif; font-size: 16px;">2</span><b style="background-color: white; color: #222222; font-family: Roboto, arial, sans-serif; font-size: 16px;"><sup>77,232,917</sup></b><span style="background-color: white; color: #222222; font-family: "roboto" , "arial" , sans-serif; font-size: 16px;">-1 which has </span>23,249,425 digits. My main point was to reinforce that new mathematical discoveries are occurring all the time and the field is evolving. But in the ensuing discussion one student brought up the factoring in public / private key <a href="https://en.wikipedia.org/wiki/Public-key_cryptography">encryption.</a> (As an aside, someday I'd love to do a numerical computing activity like implement some of the RSA algorithm.) This was a great coincidence since I had been planning to talk about that anyway.<br /><br />Stealthy Skills Practice<br /><br />Thinking about the new prime and the connection between factoring and encryption beforehand I came up with the following quick activity.<br /><br />1. Breakout into pairs. I had everyone choose someone they didn't know well as a partner.<br />2. One person is the encoder and picks two numbers less than 200 and multiplies them together.<br />3. He or she then gives the result to their partner.<br />3. The second person then is "the hacker" and has 5 minutes to see if they could find a way to non-trivially factor this product.<br /><br />This was meant to serve several purposes. One I wanted the kids to build relationships especially with the two new students. Secondly, it was a great quick demo of the difficulty of factoring and why its so useful for encryption (I had I think only 3 pairs crack the code out of the group). This led to a few interesting followup conversations. But also equally important just like last week with some of the 2018 problems this was a chance to practice factoring/multiplication/division in disguise. Watching kids work through basic computations, I'm always looking for more chances to practice skills which in theory they know but in practice could use a little reinforcement. If I were running a real class I might buckle down and use a review worksheet like those on <a href="https://www.kutasoftware.com/">https://www.kutasoftware.com</a>. But in this context I worry about keeping the kids engaged and maintaining the separation between recreation and school.<br /><br />The main activity for today was the 2nd round of the MOEMS Olympiads. If you've been following along, you'll remember I've been moving these around quite a bit to fit our schedule and am about one test behind the official schedule. Overall on first glance, I believe the kids did a bit better than the first time even though most of them took longer to complete. My only disappointment was that after going through a speech about reading the directions and making sure to answer the question that was asked I still had a few kids still answer a question asking for a whole number less than 1000 with values that were (much) larger than it. On the bright side when I had everyone demo answers on the board, I had tons of volunteers and was able to have almost everyone of the new students come up to the whiteboard. So we're already on a great start.<br /><br /><br />As usual, I'm not allowed to directly discuss the problems but I by coincidence saw a very similar problem in AMC10 to my favorite one from the set today that I'm going to discuss instead.<br /><br />2016 AMC10B problem 18:<br /><b><br /></b><b>"In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?"</b><br /><br />What I find interesting in these problems is the different behaviors for odd and even numbers.<br /><br />First for odd series with 2n + 1 members, if you write the sum as:<br /><br />(x - n) + (x - (n + 1)) + ... + (x - 1) + x + (x + 1) .... (x + n) its easy to see the sum is just (2n+1)x<br /><br />That implies for all the odds (2n+1) saying that such a sum exists is equivalent to saying that number is a multiple of 2n+1.<br /><br />What's also fun is that looking at the series another way you get:<br /><br />x + (x + 1) + (x + 2) .... (x + n - 1) which is equivalent to nx + T(n -1) where T is the triangle number function. So putting that together you have an informal proof that all the odd triangle numbers are also divisible by their index.<br /><br />Then looking at the evens (2n) which are bit more tricky:<br /><br />(x - (n - 1)) + ... + (x - 1) + x + (x + 1) .... + (x + (n - 1)) + (x + n) you get 2nx + n or n(2x +1)<br />In other words the even series (2n) are always a multiple of n and some odd number.<br /><br />Returning to the original question, this all means its really at heart a question of factoring!<br /><h4>P.O.T.W</h4><div>I saw a similar problem online somewhere in the last few weeks and although I couldn't find the original, I decided to construct my own version. This is actually a fairly straightforward linear system once you deal with the fact the lines continue on beyond the page so I'm hoping for a lot of participation.</div><div><br /></div><br /><br /><a href="http://4.bp.blogspot.com/-T2GsTY-6Ry0/WlU0mumiOfI/AAAAAAAAMPw/_TszO8rrxtcdHiEoWrxfut0sds0XuyrrACK4BGAYYCw/s1600/Selection_050.png" imageanchor="1"><img border="0" height="640" src="https://4.bp.blogspot.com/-T2GsTY-6Ry0/WlU0mumiOfI/AAAAAAAAMPw/_TszO8rrxtcdHiEoWrxfut0sds0XuyrrACK4BGAYYCw/s640/Selection_050.png" width="526" /></a>http://mymathclub.blogspot.com/2018/01/19-olympiad-2.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-5528462137132062034Wed, 03 Jan 2018 07:30:00 +00002018-01-08T11:35:59.833-08:001/2 Math for a New Year (2018)It was a good winter break. I took off some time from work to spend with my family, my parents flew in, we hosted a New Year's Party for a large group of friends and I recruited new students and problems for the start of this term.<br /><br />First, recruitment went really well. Beyond a student I lost after the first few weeks, I only had one not return and I picked up five new students including two girls. I'm now at 15 [<b>Update: 17 because its hard to say no] </b>which was my target level for this year. As of today, I finally have two seventh graders, my really great core group of eighth graders and a significant chunk of the sixth grade Algebra I class. I think this is a great springboard for next year. Based on my previous experience when a club starts up you usually have spotty upper grades initially but the younger kids stick with you and and a few years later the kids keep bubbling up to form a solid program.<br /><br />To start things off, I went through my normal first day routines. I had every kid introduce themselves and talk about either why they had joined or what their favorite activity was from last quarter. Interestingly this time, the most mentioned item was the MOEMS contest followed by several of the puzzles we did (which is a big change from previous years.) I'm happy with that but my goal is the next time to have one of more math circle topic days be picked. We also went through my abbreviated discussion of the club charter and procedures.<br /><br />Over the last few weeks I've been collecting 2018 themed problems and planned to mostly work on that vein. (This is the second time I've done this: see <a href="http://mymathclub.blogspot.com/2017/01/110-new-years-celebration.html">http://mymathclub.blogspot.com/2017/01/110-new-years-celebration.html</a>) But to start things off I wanted to do a game. Since I didn't find anything new that struck my fancy I went back to Buzz, a choral counting game. <a href="http://rules/">Rules</a> As usual we did some demos as a large group to learn the rules and then I split the group in half and let the kids run two circles at once. Everyone seemed to be having fun so once this was going I turned my back to write some problems on the whiteboard to prep for the next section. To my chagrin, a few minutes later I sensed something was wrong and turned around to see one of the students was worked up and unhappy. The stress of getting the numbers right even in completely no-stakes group situation was just too much. So I took her aside and after trying to console her decided I'd bring the whole thing to a close after after the current round ended. I'm still processing what happened since I've done this before and everyone usually really enjoys the game. At this point, based on my experience with her over the fall, I decided to breakout the art/calendar project I had brought along and had the student work on getting a demo one setup for everyone else. Fortunately, that worked as well as I had hoped and by the end of the day there were some smiles again.<br /><br />The calendar is actually a pretty cool dodecohedron. Source: <a href="https://t.co/VYU7IsPzPL">https://t.co/VYU7IsPzPL</a><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://folk.uib.no/nmioa/kalender/deskcal.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://folk.uib.no/nmioa/kalender/deskcal.jpg" data-original-height="256" data-original-width="256" /></a></div><br />I didn't have enough scissors to have everyone do it at once so I always had planned to have a craft station going during the next section of time with kids dropping in when there were tools available to use.<br /><br /><h4>New Year Math</h4><div>For the main activity we started by factoring 2018. Before the kids got going I had everyone volunteer how to go through the process to make sure the kids had a solid idea of what to do. Once I let them start, they broke out into several self organized groups mostly working on the whiteboard. I had one new student who wanted to work alone which while totally fine I'm also a little nervous about when I don't know them well. So I tried to consciously drop in a lot and check on what they were doing to make sure everything was going okay. </div><div><br /></div><div>Here are the problems I chose:</div><div><br /></div><div><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://pbs.twimg.com/media/DSewtxdX0AAgjoH.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://pbs.twimg.com/media/DSewtxdX0AAgjoH.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;">[@matematick_man]</div><div class="separator" style="clear: both; text-align: center;"><img src="https://pbs.twimg.com/media/DSa1K5TXkAAav4v.jpg" style="text-align: start;" /></div>The Red Square has an area of 2018, the Blue has an area of 900 What is dimension of the green rectangle?<br /><br />[@five_triangles]<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><img src="https://letsplaymath.files.wordpress.com/2017/12/365fraction.png?w=648&h=109" style="text-align: start;" /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;">This one is done as a mental math challenge only.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;">[@denise gaskins]</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-W_vJAekrQwA/WkyES-Ah9WI/AAAAAAAAMNw/xRnmHgIjDWAH4XRKHhYs3A_CJbb2EJ3wgCKgBGAs/s1600/20180102_165802.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="360" src="https://4.bp.blogspot.com/-W_vJAekrQwA/WkyES-Ah9WI/AAAAAAAAMNw/xRnmHgIjDWAH4XRKHhYs3A_CJbb2EJ3wgCKgBGAs/s640/20180102_165802.jpg" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;">(one of the whiteboards with several kids work on it)</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">I also wrote out a few of the following fun facts from the set below and stopped everyone in the middle to show some of the interesting 2018 trivia:</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><img src="https://pbs.twimg.com/media/DSZ6j3pXUAApVvE.jpg" /><br /><br /><br /><span style="background-color: #f5f8fa; color: #14171a; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; white-space: pre-wrap;">((2+0+1)×8×(2⁰+1+8)+2⁰-1+8+2+0+1)×8+2⁰+1+8 = 2018 10×9×8×7÷6÷5×4×3+2×1 = 2018 1×2×34+5×6×(7×8+9) = 2018* 98+7×6+5⁴×3+2+1 = 2018* (*via </span><a class="twitter-atreply pretty-link js-nav" data-mentioned-user-id="163700738" dir="ltr" href="https://twitter.com/IJTANEJA" style="background: rgb(245, 248, 250); color: #1da1f2; font-family: "Segoe UI", Arial, sans-serif; font-size: 14px; text-decoration-line: none; white-space: pre-wrap;">@IJTANEJA</a><span style="background-color: #f5f8fa; color: #14171a; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; white-space: pre-wrap;"> </span><a class="twitter-timeline-link" data-expanded-url="https://arxiv.org/pdf/1302.1479.pdf" dir="ltr" href="https://t.co/m19yvhmCvw" rel="nofollow noopener" style="background: rgb(245, 248, 250); color: #1da1f2; font-family: "Segoe UI", Arial, sans-serif; font-size: 14px; text-decoration-line: none; white-space: pre-wrap;" target="_blank" title="https://arxiv.org/pdf/1302.1479.pdf"><span class="tco-ellipsis"></span><span class="invisible" style="font-size: 0px; line-height: 0;">https://</span><span class="js-display-url">arxiv.org/pdf/1302.1479.</span><span class="invisible" style="font-size: 0px; line-height: 0;">pdf</span><span class="tco-ellipsis"><span class="invisible" style="font-size: 0px; line-height: 0;"> </span>…</span></a><span style="background-color: #f5f8fa; color: #14171a; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; white-space: pre-wrap;">) • 2018 = 2×1009 • 2+1009 = 1011 • 1011₂ = 11₁₀ • 11 = 2+0+1+8</span><br /><span style="background-color: #f5f8fa; color: #14171a; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; white-space: pre-wrap;"><br /></span><span style="background-color: #f5f8fa; color: #14171a; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; white-space: pre-wrap;"><br /></span><span style="background-color: #f5f8fa; color: #14171a; font-family: "segoe ui" , "arial" , sans-serif; font-size: 14px; white-space: pre-wrap;"><img src="https://pbs.twimg.com/media/DSaaib7WAAAmIjY.jpg" /></span> <br /><br /><br /><br />Overall this portion flowed really well. The new group dynamics seem to be gelling. Finally, I gave out a sample MOEMS test for the problem of the week since I need to do the next round next week.<br /><br />http://mymathclub.blogspot.com/2018/01/12-math-for-new-year-2018.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-6599737438499224266Wed, 20 Dec 2017 06:23:00 +00002017-12-20T07:57:12.119-08:00amc 82017 AMC 8 QuestionsWe finally received our scores. Overall I always have to remind myself that "comparison is the death of joy". But really I think the kids did very well and when the final stats come out most will be at or above average. I also hope that I get a chance to see some of the same students take it next year and that I'll have evidence how much everyone has grown.<br /><br />Art of Problem Solving has posted the 2017 AMC 8 problems and solutions at: <a href="https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems">https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems</a><br /><br />To get a feel for this year's set, I tried them out and timed myself. It took me about 50 minutes to complete them carefully. Since I don't find it as interesting, I didn't use guess and check to speed things up or look at the multiple choice answers unless the question required it. The kids however only had 40 minutes which made it fairly difficult in my mind. You had to work really quickly and do some intelligent strategic guess work from time to time to finish everything. I definitely will mention my own timing when we talk about it. From what I can gather the test was a bit harder this year than 2016 with overall cut scores about 2 points lower for the top 1 and 5 percent overall.<br /><h4></h4><h4>Favorite Parts:</h4><br /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">22) In the right triangle </span><img alt="$ABC$" class="latex" height="13" src="https://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png" style="background-color: #f9f9f9; border: 0px; box-sizing: border-box; font-family: sans-serif; font-size: 15px; vertical-align: inherit;" width="42" /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">, </span><img alt="$AC=12$" class="latex" height="13" src="https://latex.artofproblemsolving.com/b/b/8/bb890e788c451fed032c8b221192c5ac43790206.png" style="background-color: #f9f9f9; border: 0px; box-sizing: border-box; font-family: sans-serif; font-size: 15px; vertical-align: 0px;" width="69" /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">, </span><img alt="$BC=5$" class="latex" height="12" src="https://latex.artofproblemsolving.com/6/e/1/6e156331c13aae7e86644da6fa391b4389b826d9.png" style="background-color: #f9f9f9; border: 0px; box-sizing: border-box; font-family: sans-serif; font-size: 15px; vertical-align: inherit;" width="61" /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">, and angle </span><img alt="$C$" class="latex" height="12" src="https://latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png" style="background-color: #f9f9f9; border: 0px; box-sizing: border-box; font-family: sans-serif; font-size: 15px; vertical-align: inherit;" width="14" /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"> is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?</span><br /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"><br /></span><img alt="[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy]" src="https://latex.artofproblemsolving.com/1/7/f/17f66d4cef8b689eb626ec8c226380e1e72d8719.png" /><br />[My personal preference is always for the geometry ones plus this had a 5-12-13 in it.] The 3 different ways to solve listed on the site are part of why I like this. Note: its always simpler to look for similar triangles rather than jumping straight to the Pythagorean theorem.<br /><br /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">24) Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?</span><br /><br />A stealthy number theory problem. Its actually fun to make out a chart for the first 60 days after which it repeats anyways. And as often is the case its easier to find the inverse than the positive case i.e. 2/3 * 3/4 * 4/5 = 1 - (1/3 + 1/4 + 1/5 - (1/12 + 1/15 + 1/20) + 1/60)<br /><br /><br />11) <span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;">A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?</span><br /><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"><br /></span><br />I just enjoyed visualizing this one.<br /><br /><h4></h4><div><br /></div><br /><h4></h4><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />http://mymathclub.blogspot.com/2017/12/2017-amc-8-questions.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-204137120984417025Wed, 13 Dec 2017 22:30:00 +00002017-12-20T07:58:44.814-08:0012/12 End of QuarterThe end of this quarter really snuck up on me. In my planning process I decided to finally use the video below on Schumann's enumerative Geometry. What I particularly liked was the triangle puzzles discussed in the video and the fact they linked to a modern discovery. But on looking through the video for a last time I became a little worried. The end part gets quite complex and I wasn't sure if the kids would be able to follow it. So I started to look back at old Julia Robinson Festival questions and assemble a min-festival we could do in Math Club. Then I remembered that it was the last session and I wanted to do game day to celebrate. So in the end I decided to go with video, take the problems with me in case it looked like a dud as well as the games.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/U8sq3BplCfI/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/U8sq3BplCfI?feature=player_embedded" width="320"></iframe></div><br /><br />What's especially nice is that there are several natural breaks in the video for pausing and trying the math out yourself. I took advantage of the ones in the beginning and had the kids try out assembling triangles and looking for patterns:<br /><br /><br />Base shapes:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-7spXSuLIx-U/WjGm2qGIQaI/AAAAAAAAMCQ/N-r9ik5Y4voZqQcP4Qw2bYNtlWrDCBbAACLcBGAs/s1600/base%2Bshapes.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="203" data-original-width="586" height="110" src="https://4.bp.blogspot.com/-7spXSuLIx-U/WjGm2qGIQaI/AAAAAAAAMCQ/N-r9ik5Y4voZqQcP4Qw2bYNtlWrDCBbAACLcBGAs/s320/base%2Bshapes.PNG" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">I brought lots of colored pencils and had the kids draw versions on their own paper. If I had more time I might have precut out base triangles out instead. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Examples:</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-6mELQn0MVPc/WjGm4LPAccI/AAAAAAAAMCU/G1ZMB5RG_tkjGTv9Jt3wd2zMqKNMBWGAACLcBGAs/s1600/tripuz.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="231" data-original-width="605" height="244" src="https://2.bp.blogspot.com/-6mELQn0MVPc/WjGm4LPAccI/AAAAAAAAMCU/G1ZMB5RG_tkjGTv9Jt3wd2zMqKNMBWGAACLcBGAs/s640/tripuz.PNG" width="640" /></a></div><br /><br />Overall this went pretty well after all. The kids wanted to see the end of the video when I offered them the chance to move on so we did watch the whole thing. The only other mistake I made was pausing a hair early the first time and having to explain the rules more than I expected.<br /><br />In the back half of the day I brought in my usual assortment of board and card games:<br /><br /><ul><li>pente</li><li>set</li><li>prime climb</li><li>terzetto</li><li>rush hour</li></ul><div>These are still popular with the middle schoolers although I really need to pick up a new one before next time. My favorite moment here was one student pulled out last week's skyscraper puzzle to finish working on it today. I really like this display of persistence.</div><div><br /></div><div>Finally, we also had a club discussion about recruiting. The kids decided to talk to their friends and in front of their math classes as well as one is going to make a PA announcement. We'll see how this effort works. I like that I'm offloading some of this to the kids and hopefully I'll find some 7th graders next quarter.</div><br /><br /><br />http://mymathclub.blogspot.com/2017/12/1212-end-of-quarter.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-5894325425883784981Thu, 07 Dec 2017 22:04:00 +00002018-01-10T08:35:49.197-08:00managementmath club ideasMotivating KidsI recently saw this tweet<br /><br /><br /><br /><blockquote class="twitter-tweet" data-lang="en"><div dir="ltr" lang="en">US Students Aren’t Bad at Math—They’re Just Not Motivated<a href="https://t.co/w5Af6kgsfs">https://t.co/w5Af6kgsfs</a><br />It will be very interesting to see where this goes.<a href="https://twitter.com/hashtag/math?src=hash&ref_src=twsrc%5Etfw">#math</a> <a href="https://twitter.com/hashtag/mathchat?src=hash&ref_src=twsrc%5Etfw">#mathchat</a> <a href="https://twitter.com/hashtag/edchat?src=hash&ref_src=twsrc%5Etfw">#edchat</a></div>— Patrick Honner (@MrHonner) <a href="https://twitter.com/MrHonner/status/938200439308288007?ref_src=twsrc%5Etfw">December 6, 2017</a></blockquote><script async="" charset="utf-8" src="https://platform.twitter.com/widgets.js"></script>The rather interesting gist of the research was how much better US students performed on PISA when given a monetary incentive. That made me immediately think of my recent success and failures in getting everyone in the Math Club engaged.<br /> <br />If you haven't run an after school club, you might be excused thinking its nothing like a Math class. Of course, everyone is there voluntarily and excited to work on Math problems. The truth is a bit more complicated. For one, kids show up for a variety reasons including the dreaded "my parents made me do this." Secondly, a student's temperament varies day to day. The club meets after six long hours of school has already taken place. Some days even the best of kids are already worn out. Moreover, a teacher in a classroom has a whole set of tools to leverage to make kids participate such as grades. The club is a purely voluntary affair, buy in on everything from talking in front of the rest of the kids, to doing a problem of the week at home is a hard fought battle. Each day I need to find a way to create flow and draw kids into the topic I want us to explore.<br /><br />There is no perfect answer to the problem and I continue to evolve in how I think about this issue. That by itself, is the first and foremost principle. After each session I try to be critically honest with myself about how well it went and what I could do to improve. In practice, I almost always find I do better presenting a topic the second time. Since I'm continually searching for new material this is something I have to keep in mind. For every really new activity, leverage whatever connection it has to previous ones to inform how it will be done and fall back to more tried and true formats/topics after experimenting. I don't want to always be on the bleeding edge.<br /><br />The culture of the club builds on itself. First that means I always try to emphasize and reinforce when I see notable participation. I'm also ambitious in the sense I want the kids to engage with complex Math that requires a lot of focus. In my ideal vision we would just do a challenging problem set that I'd print out each week. That would in reality be a recipe for disaster. Instead I'm very mindful of the need to thread in puzzles/games/activities that are particularly playful. This is especially true when starting up with new kids I haven't worked with.<br /><br /><div><br /></div><div>There are several general strategies I'm currently following that are working reasonably well</div><div><br /></div><br /><br /><ul><li>Games and Puzzles are always great as long as they are mathematically relevant. Often they can be repeated multiple times and kids will develop more insightful strategies.</li><li>Leverage media. I'm super careful not to show a video most days. But sometimes after working really hard one week, a numberphile video is just the right change in tempo to keep everyone going. </li><li>Have the kids use the whiteboard as much as possible. I've written about VNPS before: <a href="http://mymathclub.blogspot.com/2017/03/328-vnps.html">http://mymathclub.blogspot.com/2017/03/328-vnps.html</a> This remains an excellent strategy. </li><li>I utilize a very minimal common routine to get everyone into a Math frame of mind. Mostly this consists of an introduction and talk about what we're planning to do for the day and a group review of the problem of the week.</li><li>If things don't go as well as I want one week - move on and change things up next time. </li><li>Use competition from time to time. I'm also super cagey about this but official contests bring out a lot of energy and focus in most kids. </li><li>Shamelessly bribe them with treats. I'm still giving out candy for homework participation. I only give one problem a week and the goal is to have time to think about something interesting over more than a few minutes. When enough work is handed back as a group I bring in treats. The ends seem to justify the means.</li><li>Talk candidly about where I think things are with kids. If I see a problem or direction I want the kids to go, I'll usually mention it up front. For example, last week I knew we were going to walk through student solutions to the the MOEMS contest. So at the start I told everyone that was coming up and I wanted to focus on listening to each other.</li></ul><div>Overall lest this paint a picture of perfection, I still work on motivation from week to week. I'm always looking for other people's ideas on what works and what I might adapt. Engagement is very near the heart of mathematics teaching, its complex and its not easy.</div><div><br /></div><div>Looking forward:</div><div>Now that I've experienced 4th-8th graders I can definitely see the growth in maturity as kids get older. Right now I only have 3 eighth graders. If I can recruit more of them, I'm hoping to leverage their leadership potential more. </div><br /><br /><br />http://mymathclub.blogspot.com/2017/12/motivating-kids.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-6758250576475730150Wed, 06 Dec 2017 02:54:00 +00002017-12-08T10:32:13.517-08:0012/5 Olympiad #1Today was the very delayed first <a href="http://moems.org/">MOEMS</a> middle school contest day. As I mentioned before this contest was supposed to be on the same day as AMC 8 so we had to push it out and then I needed some buffer. Fortunately, MOEMS is a low key organization and as long as you get all the contests into the system at the very end in March you can move individual dates around. I was really curious going in how the kids would do and react to the contest. In looking at the questions before hand I thought this was slightly easier than any of the ones from last year. My great worry was actually that it would be too easy for everyone.<br /><br />That turned out to not be the case. When I polled at the end although the kids thought it was easier than AMC 8, they also generally all enjoyed it. That's great since I think its a good format: 5 questions over a half hour gives enough time for most kids to solve what they are capable of solving. And the split over 5 different weeks allows you to parcel the questions out and discuss them in manageable chunks as a group.<br /><br />Which brings me to the other win for the day. This was probably the best whiteboarding session I've done yet this year. Almost everyone volunteered and there were multiple solutions presented for each of the 5 problems. There was just a ton of enthusiasm. Sadly, I'm not allowed to discuss any of the details of the problems but the kids came up with a lot of good problem solving solutions and really listened to each other. I'm hoping to extend this streak to next week's whiteboarding and have some more interesting details to record here.<br /><br />As usual to occupy everyone who finished early I brought a low-key puzzle. In this case I went back to the skyscraper puzzles from <a href="https://www.brainbashers.com/skyscrapers.asp">https://www.brainbashers.com/skyscrapers.asp</a> and printed an easy and hard 6x6 one.<br /><br />P.O.T.W<br />This is my absolute favorite linear systems problem:<br /><br /><div id="potw" 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; white-space: pre;">Assume that </span><a href="https://www.codecogs.com/eqnedit.php?latex=x_1%20%2C%2C%2C%20x_7%0" style="text-decoration: none;"><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; white-space: pre;"><img height="9" src="https://lh4.googleusercontent.com/ifBVAwSO471M73DmHFp1VCEsrMID0eP7hesCORVV3zhu3Xnol7kzdDN6SHogufYkYfgA4A0HPjI-3sDbbeKT0gdlccOzr4i-9Qn8xVS8rwkASBoSNm9Btuhd5L6WRgtAeS9O9o3q" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="49" /></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; white-space: pre;"> are real numbers such that</span></div><b id="docs-internal-guid-ea59f7ca-29bd-7387-e045-20279bf4b00d" style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><a href="https://www.codecogs.com/eqnedit.php?latex=x_1%20%2B%204x_2%20%2B%209x_3%20%2B%2016x_4%20%2B%2025x_5%20%2B%2036x_6%20%20%2B%2049x_7%3D%201%0" style="text-decoration: none;"><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; white-space: pre;"><img height="13" src="https://lh3.googleusercontent.com/UzRAdvcHl8VqgXhcDGkenc0JJJbqKQR2JHGEws-b67CDKpR2QChWnPNq3k5Rz0vKKXolK2u-FT07yPZfvO-1kZu5TFc17lzRxZjSfNVRtZgyqGKyVIwBgOjeQsT2hAImxnmJ_8Us" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="345" /></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; white-space: pre;"><br class="kix-line-break" /></span></a><a href="https://www.codecogs.com/eqnedit.php?latex=4x_1%20%2B%209x_2%20%2B%2016x_3%20%2B%2025x_4%20%2B%2036x_5%20%2B%2049x_6%20%2B%2064x_7%20%3D%2012%0" style="text-decoration: none;"><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; white-space: pre;"><img height="13" src="https://lh3.googleusercontent.com/aVi-ORJ8pAHsuvIzcnIGBsDNdsypre1f5NwinySfLdBv058nVgQ_I-E5eVFiupDR9JtQzLzTi7QQBtpieHsFC7mG5AFH74sn3r41irnFbqhruocnr_t9chdhIgRrywcgD4f0f2yp" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="371" /></span></a></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><a href="https://www.codecogs.com/eqnedit.php?latex=9x_1%20%2B%2016x_2%20%2B%2025x_3%20%2B%2036x_4%20%2B%2049x_5%20%2B%2064x_6%20%2B%2081x_7%20%3D%20123%0" style="text-decoration: none;"><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; white-space: pre;"><img height="13" src="https://lh3.googleusercontent.com/keZ7R2gqfqLsOkS7qnnW62bCVtdDPifgukObrqv_gykIWkLQTLNNq0nicD_0M5wMZstBTNBGgr25BL9Op8dN58oFSJMrcBJ9vz3yOcSpV7DJIHi5dFhGfqy6hlcheHhOfHnRQ8Pc" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="387" /></span></a></div><b style="font-weight: normal;"><br /></b><br /><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; white-space: pre;">Find the value of </span><a href="https://www.codecogs.com/eqnedit.php?latex=16x_1%20%2B%2025x_2%20%2B%2036x_3%20%2B%2049x_4%20%2B%2064x_5%20%2B%2081x_6%20%20%2B%20100x_7%0" style="text-decoration: none;"><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; white-space: pre;"><img height="13" src="https://lh3.googleusercontent.com/SRfrzUAUNKAAbic-SwdJIJSzWW-Zb2Xp8cbEJJkZFFN-Y10Q7kDNixx1wXq0jV1D4g4_2Qj5irkcKKY60hmkHJZ9lN6ySIUIMy8hZacLplfgMjV1V-x-gFhcOVEV1wVV9MJWNBFV" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="356" /></span></a></div><br />http://mymathclub.blogspot.com/2017/12/12-5-moems-contest-1.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-3787201526396091018Wed, 29 Nov 2017 22:35:00 +00002017-12-05T09:46:52.270-08:00egyptian fractions11/28 Egyptian Fractions<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/images/ahmesPapyrus.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/images/ahmesPapyrus.gif" data-original-height="298" data-original-width="800" height="236" width="640" /></a></div><br /><br />Brainstorming this week I became interested in <a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=13&cad=rja&uact=8&ved=0ahUKEwiysse0x-XXAhVslVQKHY4DBdUQFghbMAw&url=http%3A%2F%2Fmathworld.wolfram.com%2FEgyptianFraction.html&usg=AOvVaw0fS8njLjWukrzdV8SwIBWO">Egyptian Fractions</a> because they dovetail nicely with the math history from <a href="http://mymathclub.blogspot.com/2017/11/the-intersection-of-history-and.html">last time</a>. Here's a topic that is both historical and mathematically interesting. I was going to originally title this week Funny Fractions and do a unit on both Egyptian Fractions and Farey Sequences but on consideration I decided there was enough to deal with just focusing on the first idea. That was right decision to make based on actual time management. As I discovered also over the hour, these provide a great platform for practicing other more basic skills,<br /><br />To start off I had everyone guess when fractions were first documented as being used. I mentioned the late entry of decimals as a starting point. I was pleased someone remembered the Babylonian base 60 fractions from last week. I then did a quick read of the background of the Rhind papyrus with some information and a printout of the scroll from: <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html">http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html</a><br /><br />I then used a modified version of the series of questions and activities from here:<br /><a href="https://nzmaths.co.nz/resource/egyptian-fractions">https://nzmaths.co.nz/resource/egyptian-fractions</a> I particularly focused on finding ways to break Egyptian fractions apart into sums of other Egyptian fractions and discovering algorithms to find an Egyptian fraction sum for a regular fraction. Once kids started to brainstorm on the whiteboard I started feeding further problems as different groups progressed:<br /><br />Further Problems:<br /><br />1. <b>The Mullah's horse</b>: The former Grand Wizier, Mullah Nasrudin was approached by three men with 19 horses. The men asked him to adjudicate the will of their recently dead father which required that his horses be divided among his three sons so that the oldest son receives 1/2, the middle son gets 1/3, and the youngest son would get 1/7. With little hesitation Nasrudin added his own horse to the herd and said, "What is half of 20, 1/4 of 20, and 1/5 of 20" After some time the men replied, "10, 5, and 4". The eldest son then took 10 of the horses, the middle son took 5 of the horses, and the youngest son took 4 of the horses. The Mullah Nasrudin, then took the remaining horse and rode home. Can you explain what occured?<br /><br /><br />2. Find all the solutions (there are less than 10) to the problem (n-1)/n = 1/a + 1/b + 1/c, where a < b,<br />b < c, a, b, and c are positive integers with least common multiple n. Note. a = 2, b = 4, c = 6, and n = 12 gives one solution.<br /><br /><br />3. How many different egyptian fractions can be used to describe 2/3? Two of them are 1/2 + 1/3 + 1/6 and 1/3 + 1/10 + 1/15.<br /><br /><br />4. <b>Want to solve an unsolved problem?</b> One of the most famous problems on Egyptian Fractions asks, "Can every proper fraction of the form 4/q be expressed with an egyptian fraction with less than 4 terms?" Can every proper fraction of the form 5/q be expressed with an egyptian fraction with less than 4 terms?<br /><br /><br />5. <b>The sailor, coconut, and monkey problem</b>: Five sailors were abandoned on an island. To provide food, they collected all the coconuts they could find. During the night one of the sailors awoke and decided to take his share of the coconuts. He divided the nuts into five equal piles and discovered that one was left over, so he threw the extra coconut to the monkies. He then hid his share and went back to sleep. A little later a second sailor awoke and had the same idea as the first. He divided the remainder of the nuts into five equal piles, discovered also that one was left over, and through it to the monkies before hiding his share. In turn each of the other three sailors did the same - dividing the observable amount into five equal piles, hiding one, throwing one left over to the monkies. The next morning the sailors, looking innocent, divided the remaining nuts into five piles with none left over. Find the smallest number of nuts in the original pile.<br /><br />6. find 1/a + 1/b + 1/c + 1/d + 1/e = 1<br /><br />7. 355/113 approximates<img src="https://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Africa%2Fpie.gif&container=blogger&gadget=a&rewriteMime=image%2F*" /> to 6 places. (355/113) - 3 = 16/113. Find an egyptian fraction whose sum is 16/113<br />I also printed out a fun geometric puzzle for tired students to relax with when they needed a break.<br /><a href="https://blogs.adelaide.edu.au/maths-learning/2016/10/19/panda-squares/">https://blogs.adelaide.edu.au/maths-learning/2016/10/19/panda-squares/</a><br /><br />Overall, this went well but I'd improve several things if repeating:<br /><br /><ul><li>This time too many kids went over to the puzzle a little too quickly. I have to think of way to keep everyone on task longer. </li><li>I also wanted to do some notice/wonder activities around patterns in the puzzle but that was not possible while focusing on the main activity. </li><li>I needed one or two more problems in the set to fully round things out. Several were of the type that kids could become stuck on. So a few more easier warm ups would work well. I'd have kids work out a variety of easy equivalent fractions next time i.e. find 3/4, 2/7 etc.</li><li>I had one student who out of character just wanted to read and not work on math today. Given the other needs of the kids I let her do that but I want to make sure next week she's engaged.</li></ul><div><br /></div><div>Also during the time I noticed a lot of fluency issues while the kids worked on the math. </div><div><ul><li>Adding fractions like 1/4 + 1/5.</li><li>Long division.</li><li>Mental math for fairly easy computations like 84 divided by 4.</li></ul><div>In each of these cases I ended up doing mini walk-throughs and I think the session acted as a way to review rusty skills. But overall, I'm toying with the idea of finding other activities that also stress these again. </div></div><div><br /></div><div><br /></div><div>P.O.T.W</div><div><a href="https://drive.google.com/open?id=1HjGYSNp8ngQet1lsXf2jxA6jEYoQhpIIxlMb2_Ye2TM">https://drive.google.com/open?id=1HjGYSNp8ngQet1lsXf2jxA6jEYoQhpIIxlMb2_Ye2TM</a></div><br /><br /><br /><br /><div><br /></div>http://mymathclub.blogspot.com/2017/11/1128-egyptian-fractions.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-8965766742481322478Wed, 22 Nov 2017 19:51:00 +00002017-11-29T16:47:35.688-08:0011/21 The intersection of History and MathematicsMy goal for this week in Math Club was to do something low key after AMC 8. Originally, I had been thinking about some tangram or panda block puzzles. I also had seen a recent Infinite Series video with a interesting triangle puzzle embedded within it that I thought looked promising.<br /><br />But as often occurs, I ended up going in a different direction. Earlier last week I was thinking about what educators mean by "humanizing math". The main claim is that mathematics is cold and sterile which I don't find totally convincing especially in the context of math circles. But some of the of the ideas bundled in with this subject are really interesting. In particular I liked this paper <a href="https://www.jstor.org/stable/27968440?seq=1#page_scan_tab_contents">https://www.jstor.org/stable/27968440?seq=1#page_scan_tab_contents</a> about using Math History to humanize a classroom. This dovetails with my worry that kids don't really understand the trajectory of their Mathematics education in the same way as other subjects and tend to view Mathematics as a complete set of knowledge rather than a developing field that people still work within. Even I as a student, couldn't imagine what Mathematics research really looked like.<br /><br />So I started researching Mathematics History videos that might be a good fit for a session. I found several candidates. My initial pick was "The Story of Maths" a BBC documentary that looked promising. But after picking out the clips when I went to prepare I found that they had all be taken down from youtube due to copyright issues. It turns out I can borrow the DVD version of this from the library which I will remember for the future. So instead I went with the following lecture given by Dr. John Dersch:<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/YsEcpS-hyXw/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/YsEcpS-hyXw?feature=player_embedded" width="320"></iframe></div><br /><br />What I like about this talk is that it covers a lot of ground and gives a good historical framework. Conversely, it lacks flashy visuals and does assume a college level background. So I prepped by starting with a talk with the kids where I had them guess when various mathematical discoveries occurred ranging from addition, to algebra to geometry to calculus. That set the stage for video. I also liberally stopped the video and talked about various topics. This was especially true when I thought the subject was new i.e. logarithms or derivatives. This led to several tangents that might be fun to do a whole session on:<br /><br /><ul><li>How did 17th century mathematics calculate square roots or logarithms?</li><li>Why can't you solve a 5th degree polynomial in a general fashion?</li><li>Fermat's Last Theorem.</li><li>Egyptian Fractions</li></ul><div><br /></div><div>Overall, I thought this went really well. If I repeat this topic, I do hope to find a better video resource or perhaps develop a slide deck of my own. I also wonder if I could thread various historical discoveries in during a year i.e. a talk on Babylonian tables, or Napier's bones.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="about:invalid#zClosurez" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Image result for clock image" border="0" src="data:image/png;base64,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" style="cursor: move;" /></a></div><div><br /></div><div>To round things out before this started I actually went to back to clock problems. This was strategic since I had noticed some of the kids were really interested in when the clock hands coincided already and had been looking up tables of the values online. (If I repeat and this wasn't already the case I would ask kid to observe during school beforehand.) The day really started with me drawing clocks on the whiteboard and having a few kids talk about what they already knew. I was hoping they had noticed a regular pattern but since that wasn't the case we worked on that in club. One focus I asked a few questions to point out was that here is a point of coincidence at every hour except 11. We then developed the basic equation to discover the actual values. </div><div><br /></div><div>m = h * 5 + m / 12 </div><div><br /></div><div>I was surprised that this seemed fairly new to everyone and the basic process of solving was not as smooth as expected especially developing the minute to hash mark ratio. I plan to return to ratios at some point. On the bright side kids quickly found the method for find when the hand form a straight line by the times we were done.</div><div><br /></div><div>P.O.T.W</div><div>This time I gave out a sample MOEMS test to prep for the first one of the series. I'm probably going to do it in two weeks which means I'll have to continue to slide the other ones around in order to balance activities out. I'm actually very curious to see how the kids do on it.</div><div><br /></div><div><br /></div><div><br /></div><br /><br />http://mymathclub.blogspot.com/2017/11/the-intersection-of-history-and.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-2825491761797802081Wed, 15 Nov 2017 16:56:00 +00002017-11-15T09:01:58.094-08:00amcdigressiongeometry11/14 2017 AMC 8 and a digression<script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); </script><script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> <br /><div class="separator" style="clear: both; text-align: left;">Math Club was super easy for me today. I paced outside the classroom while everyone took AMC8. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Qsu4tZ9pXok/WgxvVwwpCXI/AAAAAAAAKa0/sZYdRfg1uhknNCXCzGL3v2tlXzLSUfHXACKgBGAs/s1600/trollface.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="821" data-original-width="1460" height="223" src="https://2.bp.blogspot.com/-Qsu4tZ9pXok/WgxvVwwpCXI/AAAAAAAAKa0/sZYdRfg1uhknNCXCzGL3v2tlXzLSUfHXACKgBGAs/s400/trollface.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I was happy that the kids all were very focused. Hopefully we'll get good results and it was a positive experience for everyone. The problems are released in a few weeks. I'll come back to them if anything interesting appears.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So to fill the week here's the problem I looked at last night before bed. Its interesting to see the vast difference in approaches between mine and another online. Once again this is why I love geometry.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><img height="269" src="https://pbs.twimg.com/media/DOZBNH5UIAAaDaN.jpg" width="400" /></div><div class="separator" style="clear: both; text-align: left;">Thought Process:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">(Unusually this was a fairly linear process where each observation led farther forward.)</div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>I immediately noticed the right triangle and thought about the Pythagorean Theorem.</li><li>Then it occurred to me that D was the incenter and it would be interesting to draw in all the altitudes from it and to connect it to C.</li><li>That also meant CD would bisect angle C into 2 45 degree angles. </li><li>At about this point I noticed the square that formed.</li><li>I then started to think about the line AE and how it bisected the triangle and could be used with the angle bisector theorem.</li><li>I thought this was almost enough and I actually used 3 variables at this point to see how much I could combine. That didn't quite work so I actually plugged a sample number in just to watch how it played out.</li><li>At that point I went back to the picture and angle chased to find the similar triangles. That gave me a way to only use 2 variables and I was sure I was almost there.</li><li>I did some algebraic simplification and at this point I wasn't sure if I needed another equation/invariant. </li><li>But I lucked out since I was looking for the sum of the 2 variables, everything was in place.</li></ul><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-atuAVF0DTWs/WgvSFjQE9rI/AAAAAAAAKZo/Ws-ZtHRx8a4ffoBaZAgcL-hpeXaB_B4_gCLcBGAs/s1600/eylem.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="370" data-original-width="809" height="292" src="https://1.bp.blogspot.com/-atuAVF0DTWs/WgvSFjQE9rI/AAAAAAAAKZo/Ws-ZtHRx8a4ffoBaZAgcL-hpeXaB_B4_gCLcBGAs/s640/eylem.PNG" width="640" /></a></div><br /><br />Setup: Note O is the incenter since its the intersection of the angle bisectors. So drop another one from point C. This forms the 45-45-90 triangle CHO, let r = CH = HO = GO = the inradius, let x = DH . We want to find r + x.<br /><br />1. After angle tracing triangle AGO is similar to DHO. so \(\frac{AG}{GO}=\frac{HO}{DH}\)<br /> \(AG=\frac{HO \cdot GO}{DH} = \frac{r^2}{x}\)<br /><br />2. From the angle bisector theorem: \(\frac{AC}{CD}=\frac{AB}{BD}\)<br /> \( \frac{\frac{r^2 }{x} + r}{r +x}=\frac{\frac{r^2}{x} + x + 3}{3} \) which simplifies to: \(3r = r^2 + x^2 + 3x\)<br /><br />3. We also know from the Pythagorean theorem on triangle BHO that \(4^2 = r^2 + (x+3)^2\) which simplifies to \(r^2 + x^2 = 7 - 6x\)<br /><br />4. Substitute r^2+x^2 from the 2nd into the 1st equation: \(3r = (7 - 6x) + 3x = 7 - 3x\)<br /> \(3(r+x)= 7\) or <b>\(r +x = \frac{7}{3}\) </b>and we're done.<br /><br /><br />The Trig Approach:<br /><br />Another user @mathforpyp put this soln up. Notice how completely differently this works. I like to think of trig as a bulldozer for these type problems but applying it is actually a bit tricky. The key observation here which I didn't use above was the relationship between the angles A and B.<br /><br /><br /><img height="289" src="https://pbs.twimg.com/media/DOcMTmrXkAECYBZ.jpg" width="640" /><br /><br />http://mymathclub.blogspot.com/2017/11/1114-2017-amc-8-and-digression.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-2766315643709490732Thu, 09 Nov 2017 22:55:00 +00002017-12-13T14:31:05.789-08:0011/7 DecodingI decided to do a second smaller sampler of AMC 8 problems for Math Club this week. Unlike last week (see: <a href="http://mymathclub.blogspot.com/2017/10/1031-put-bird-on-it.html">http://mymathclub.blogspot.com/2017/10/1031-put-bird-on-it.html</a>) this time I wanted to approach them as a group and only do 5-6 max but concentrate on the hard ones and have the group demo solutions.<br /><br />So I picked the last 6 problems from AMC 2014: <a href="https://drive.google.com/open?id=1ZoLgBb3wTQBrqsqPcBymgGED-wG-ffTtI5ffm0w1FFc">link to partial set</a> and had the kids divide up and work them in groups. My goal was to only spend 15 minutes but because the work looked productive we ended using about half the time again. <br /><br />Overall:<br /><br /><br /><ul><li>focus was less good today especially during the demos. I'm going to need to work on improving the classroom norms here or be more mindful to limit this to fewer problems.</li><li>There was one really interesting argument about the solution to the 2nd problem in one group. One girl had a general solution and her partner didn't understand how it worked. I intervened to try to get the two students to slow down and listen to each more carefully. </li><li>I noticed a general hole in modular arithmetic that would make a good topic for one of the upcoming sessions.</li><li>One other student has a weakness for linear systems. I really like his thinking but he almost always tries to setup a system regardless of the problem. My personal goal here is to work to get him to expand his tool set.</li></ul><div><br /></div><br />At this point we switched to my main focus, encoding problems which ran better than the first half. I've done these in the past but this time I switched my sequence of problems up a bit which I think worked really well.<br /><br />Encoding Problems:<br /><br />First I started with a general open-middle type problem: Using the digits 1-9 form a valid addition<br />equation with the form below:<br /><br /> _ _ _<br />+ _ _ _<br />=======<br /> _ _ _<br /><br /><br />This was great for a low barrier to entry and due to the many/many possible solutions. After the group had found 3 or so we move on to a class decoding problem.<br /><br />Each letter stands for a distinct digit<br /><br /> DOG<br />+GOD<br />======<br /> DEEP<br /><br />Interestingly, my best solver in the first part also cracked this one first.<br /><br />Finally we finished with this multiplication problem which was not solved before time ran out:<br /><br /><br />A B C D E F A B C<br />x 6 and + D E F<br />----------------- ---------<br /> D E F A B C 9 9 9<br /><br /><br />Spare problem we didn't reach:<br /><br /> _ 5 3 <br /> -----------------<br />_ _ 9 | 6 _ 8 _ _ _<br /> _ _ _ _<br /><br /> ======<br /> _ 9 _ _<br /> _ _ 4 _<br /> =======<br /> _ _ 4 _<br /> _ _ _ _<br /><br /><br />Overall, I would have preferred to have only done my main activities but I think again for AMC 8 it was worth one more session of prep. I also had a few issue with a few students rough housing today that I'm working hard to nip in the bud. I'm going to go over behavioral standards and pull one student aside before we start next time. Looking forward, I'm super excited to see the kids take the test next week. I have a small scheduling issue with MOEMS which is on the same date. I really don't want 2 contest in a row so my plan is to to slide the MOEMS tests around fairly aggressively to free up time for more focused math circle sessions.http://mymathclub.blogspot.com/2017/11/11-7-decoding.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-4096608619089375228Wed, 01 Nov 2017 04:47:00 +00002017-10-31T21:47:54.433-07:0010/31 Put a bird on it.<blockquote class="twitter-tweet" data-lang="en"><div dir="ltr" lang="en">Its the math worksheet version of "Just put a bird on it." <a href="https://t.co/lNP3fs0wA2">https://t.co/lNP3fs0wA2</a></div>— Benjamin Leis (@benjamin_leis) <a href="https://twitter.com/benjamin_leis/status/925378626190581760?ref_src=twsrc%5Etfw">October 31, 2017</a></blockquote><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-_tj_BmTdac8/WflMXENvHaI/AAAAAAAAKUA/kO5VuP9hClAiziEbxIxKMLOBNgs732BUACLcBGAs/s1600/hall1.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="417" data-original-width="854" height="156" src="https://1.bp.blogspot.com/-_tj_BmTdac8/WflMXENvHaI/AAAAAAAAKUA/kO5VuP9hClAiziEbxIxKMLOBNgs732BUACLcBGAs/s320/hall1.PNG" width="320" /></a></div><span style="font-size: x-small;"><br /></span><span style="font-size: x-small;">My surprisingly well read retweet from today.</span><br /><br /><br />I joke (or recycle Ed's joke) above but that's essentially what I did today in Math Club. It's about 2 weeks until AMC8 and I really felt the need to have the kids do one sample test before hand so they are familiar with the format. At the same time, I also know from experience handing kids a multi-page set of problems doesn't work that well and the group tends to get unfocused. So I went through several options beforehand:<br /><br /><br /><ul><li>Select 3-5 representative questions. </li><li>Do a few problems over several weeks.</li><li>Create some kind of competitive relay.</li><li>Slog through a larger selection and just work really hard at keeping everyone focused.</li></ul><div>There are drawbacks to all of these ideas. What I finally went with was a 30 minute practice. I printed out the entire 2011 test from <a href="https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8">https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8</a> </div><div>but<u> stripped away</u> the multiple choice answers. I also slapped a Halloween pumpkin on the sheet. </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: Arial; font-size: 11pt; margin-left: 1em; margin-right: 1em; vertical-align: baseline; white-space: pre-wrap;"><img height="274" src="https://lh6.googleusercontent.com/uTWQdwIre79PZfm5suHTGTcUlLr6wwPpQN79aKQyM2dcXQnl1lfTsvrrmvqSMc53ez9OcyjDXCZS0-cdaFHbmy8p58ey_dEgJ2SbcHYrrQk3709zVn78I-KPwz3L_aQB1fjBLC3p" style="border: none; transform: rotate(0rad);" width="279" /></span></div><div><span id="docs-internal-guid-db4e8ff6-75da-4e56-f776-4936984e9d50"></span></div><div><br /></div><div><br /></div><div>I then had all the kids put their names on the whiteboard and had them work through every third problem individually. After each problem was done I told everyone to put the problem number on the board under their name and find someone else who had finished the same problem to compare answers with.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-NGUhCb2f56U/WflODFV-FRI/AAAAAAAAKUM/OfQ-G5j9jV8bZ-U56LXjku3rXehLavLeQCKgBGAs/s1600/20171031_164636.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://3.bp.blogspot.com/-NGUhCb2f56U/WflODFV-FRI/AAAAAAAAKUM/OfQ-G5j9jV8bZ-U56LXjku3rXehLavLeQCKgBGAs/s320/20171031_164636.jpg" width="320" /></a></div><div><br /></div><div>This structure worked out surprisingly well. A lot of the kids got into running up to the board to write down what they finished but it didn't feel super competitive. At the same time, this farmed out the answer checking so I didn't need to act like a living answer key while there was still feedback for everyone. I was also able to keep the process going with a bit of nudging over multiple problems and circulate to help out with questions. Further by skip counting questions the kids were able to try the range of problems from the easier ones in the beginning to the more difficult ones at the end. I didn't really need it but by not condensing the questions there were enough to not worry about running out.</div><div><br /></div><div>To keep things interesting, I interrupted several times during this process to tell some corny Halloween Jokes:</div><div><br /></div><div>"Why do mathematicians mix up Halloween and Christmas. Because Oct. 31 = Dec. 25" and to take a field trip to my brand new Math Club Poster Board:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-fCrk4Uj5cnU/WflP2HrJS7I/AAAAAAAAKUc/P-Kq1kmLj8AMp5jbkJN-cCEZh7GlZDcogCKgBGAs/s1600/20171031_170326.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://4.bp.blogspot.com/-fCrk4Uj5cnU/WflP2HrJS7I/AAAAAAAAKUc/P-Kq1kmLj8AMp5jbkJN-cCEZh7GlZDcogCKgBGAs/s400/20171031_170326.jpg" width="400" /></a></div><div><br /></div><div><br /></div><div>After the 30 minute mark I did a cool down / party. First I handed out candy to celebrate Halloween. This was strategically delayed until after the serious practice was done and then I found a lovely set of Halloween themed logic puzzles from here <a href="http://geekfamilies.co/halloween-math-and-logic-puzzles-for-kids/">http://geekfamilies.co/halloween-math-and-logic-puzzles-for-kids/</a> Today confirms these are still irresistible even to older students.</div><div><br /></div><div><br /></div><div>P.O.T.W.</div><div>A final holiday themed geometry problem, I found from @five_triangles:</div><div><a href="https://drive.google.com/open?id=1gJmBkiPcJtq-XYBct6tDCHR6oP7b7D5Q4-mMoA1whaA">https://drive.google.com/open?id=1gJmBkiPcJtq-XYBct6tDCHR6oP7b7D5Q4-mMoA1whaA</a></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><br /><br /><br /><br /><br />http://mymathclub.blogspot.com/2017/10/1031-put-bird-on-it.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-2211664267882265370Wed, 25 Oct 2017 01:56:00 +00002017-10-24T19:02:15.493-07:00number theory10/25 Strange BasesThis was a funny week. All the eighth graders were out on the class trip so Math Club skewed younger. That played a part in my planning. I aimed a bit less complex this time and hoped to draw all the sixth graders in.<br /><br />My main inspiration was an article by @RobJLow on <a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjriP_90IrXAhVD7mMKHScaBEkQFggoMAA&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBalanced_ternary&usg=AOvVaw1Rd8YQQo3VgiLVFZC9vBg3">balanced ternary</a> number systems. This got me thinking about the <a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiQwpCQ0YrXAhVJyWMKHaDRCnsQFggoMAA&url=https%3A%2F%2Fwww.theglobalmathproject.org%2F&usg=AOvVaw2xoden72X7Sf5ptRM1l7bK">Global Math Project</a> and the exploding dots work James Tanton has been doing. I've seen a lot of folks participating and it looked like it might be a fit.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-tfPjlCTtVt0/We4IOL29gxI/AAAAAAAAKQo/dbur0RJAp-E7d6qIX1G6U6HPHXDbaZ_5gCPcBGAYYCw/s1600/20171023_081548.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1165" height="320" src="https://3.bp.blogspot.com/-tfPjlCTtVt0/We4IOL29gxI/AAAAAAAAKQo/dbur0RJAp-E7d6qIX1G6U6HPHXDbaZ_5gCPcBGAYYCw/s320/20171023_081548.jpg" width="233" /></a></div><br /><br />Then last Friday, our official MathCounts team packet arrived with the lovely poster from my <a href="http://mymathclub.blogspot.com/2017/10/this-years-math-count-poster.html">last post</a>. My own son found the poster intriguing and I was fairly sure everyone else would too. So I started with a group whiteboard attempt at the problem.<br /><br />With closer to 9 kids, I could have everyone work in to two groups up at the board. I brought in some magnets and tacked the poster in the middle. We worked on the problem for about 10 minutes and I circulated making sure to keep stragglers at the board working. What was nice was the first group to solve was fairly collaborative so I could have the kids take turns explaining their thinking to the other half of the group. (This was a theme for today: making lots of opportunities for kids to demo to each other.)<br /><br /><br />This week I remembered to tackle the P.O.T.W early on. After 3 or 4 kids presented it was clear we didn't have a solution yet to the problem:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://blog.mrmeyer.com/wp-content/uploads/161027_3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://blog.mrmeyer.com/wp-content/uploads/161027_3.png" data-original-height="756" data-original-width="632" height="320" width="267" /></a></div><br /><br />So I made the executive decision to spend some more time engaging with it on the whiteboard. Once again, I had everyone come up. This time I seeded the groups with a few ideas after a few minutes like: try adding the origin of the circle and radii from there to all the points on the circumference.<br /><br />One of the topics I worked on a bit with several students was applying the Pythagorean Theorem to find hypotenuse side lengths. At the end, I lucked out and had two different solutions from two students that again I had each explain to the larger group.<br /><br />As a consequence, we were now half way into the time and just about to start working on my main focus: alternate bases. It was clear, we'd only have time for one of the two examples. So I told the group "We can work on base 3/2 or a balanced ternary base system, which would you like to explore?" The room voted for base 1.5, probably because it sounded less formal.<br /><br />In looking through the source material: I found Tanton's videos not quite to my taste and I wasn't sure if I'd have a projector. So I decided to focus on topic 9: <a href="http://gdaymath.com/lessons/explodingdots/9-1-welcome/">source link</a> but work the material directly. I started by asking who had played with alternate base system before. Most of the room had some experience. I also asked if anyone had seen exploding dots (nope). So I spent a few minutes explaining how the model worked on base 2 and then jumped right into base 3/2.<br /><br />First we started by counting up and recording the numbers. Here I enlisted one of the students as a scribe which freed me up to talk more. We then worked on what was the meaning of the place values, why were some values missing, and was there any pattern to the numbers. Just like the source material, I had the kids verify that the number worked and converted correctly back into the familiar decimal system. This all went fairly well. The back half of the exploration, we built multiplication and addition tables and figured out how to manipulate numbers in this system With a minute to go we started to look for divisibility rules: like the one for multiples of 3.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://gdaymath.com/wp-content/uploads/2017/04/ed4-4-300x82.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://gdaymath.com/wp-content/uploads/2017/04/ed4-4-300x82.png" data-original-height="82" data-original-width="300" /></a></div><br /><br />All told this material worked really well. Building the tables up was particularly satisfying. If I had another half an hour, I would have loved to have broken into the second system but I think not rushing and digesting the topics we already had was necessary today.<br /><br /><br />P.O.T.W<br />I took a page out of the MathCounts manual so the kids would have an initial exposure to some of these style problems<br /><br />http://mymathclub.blogspot.com/2017/10/1025-strange-bases.htmlnoreply@blogger.com (Benjamin Leis)0