tag:blogger.com,1999:blog-4227811469912372962Sat, 26 May 2018 00:13:57 +0000geometrydigressionbrainstormingmath club ideasgeneral philosophyPythagorean Theoremalgebramanagementpuzzles ciphersbook reviewgamesnumber theorypi daypurple cometvirtual math clubworksheet homeworkCombinatoricscontestknights of pitriangle numbersdistributive propertymoems15-75-90amcartdecodingdivisibilitydreamboxfractionsgraph theorymath club math circlemath countsolympiadpythagorean tripleKaprekar's operationamc 8arthur benjamincarnival of mathchesscontinuing fractionsegyptian fractionseuler characcteristiceuler characteristicexponentsfactorizationfibonaccifive trianglesfold and cutfractalsherons formulainversionsjulia robinsonlecturemagic squaremath nightmiddle schoolmodular arithmeticmomathnotice wonderpair and sharepascal's triangleplatonic solidsproblemquarticrecruitingresourcessangakuseriessierpinskisoftware reviewsquare rootstatisticssymmetrytilingtopologytwitterulam'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)Blogger216125tag:blogger.com,1999:blog-4227811469912372962.post-53374022602213209Sat, 26 May 2018 00:13:00 +00002018-05-25T17:13:57.896-07:00I'm going to a Conference!<div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Dear Benjamin Leis,<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">On behalf of the 2018 Northwest Mathematics Conference Program Committee, we are pleased to inform you that your session, “Middle School and Math Circles”, has been accepted! An email will be sent with your specific presentation date & time by mid-June. Edits to your title and/or description may be made by the Program Committee.<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">We received an overwhelming number of proposals — more than twice as many proposals as we were able to accommodate. Please confirm your acceptance by <span style="color: red;"><span class="aBn" data-term="goog_1210437953" style="border-bottom: 1px dashed rgb(204, 204, 204); position: relative; top: -2px; z-index: 0;" tabindex="0"><span class="aQJ" style="position: relative; top: 2px; z-index: -1;">June 30, 2018</span></span></span>. If you have not confirmed by this date, we will begin to accept speakers from our lengthy waitlist.<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">As a lead speaker, your registration is complimentary and will be completed for you upon your confirmation of acceptance. If you have a co-speaker/s, they will need to register through the conference website, unless they are a lead speaker for a different session. The breakfast keynote <span class="aBn" data-term="goog_1210437954" style="border-bottom: 1px dashed rgb(204, 204, 204); position: relative; top: -2px; z-index: 0;" tabindex="0"><span class="aQJ" style="position: relative; top: 2px; z-index: -1;">on Saturday</span></span> morning with Annie Fetter is not included in your complimentary speaker registration. If you indicate on the confirmation page that you are interested in attending this keynote, information will be emailed to you at a later date.<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Thank-you for your willingness to share your ideas and contribute to the richness of the program for the 2018 Northwest Mathematics Conference hosted by the BCAMT.<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Chris Hunter & Janice Novakowski<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Program Committee<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">2018 Northwest Mathematics Conference<u></u><u></u></div><div class="MsoNormal" style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://www.bcamt.ca/nw2018/&source=gmail&ust=1527294184363000&usg=AFQjCNFRlvTkWoKCky7wNFbwbOkQOWA5NA" href="http://www.bcamt.ca/nw2018/" style="color: #1155cc;" target="_blank"><span lang="EN-US" style="color: windowtext; text-decoration-line: none;"><img alt="https://lh6.googleusercontent.com/cls7OA9lb5KHd6if7BXJvy0n2qHiavEf_fMuKVWqQNgnxT7YB-LmqqariSB8fbS_jDPP7kRvWvbFbmBDkcSy3D3Vnmyxoc2UwXku_BJ9pF7yrnHJ_66QI16vwMHzANFpCoQGGPdy" border="0" class="CToWUd" height="187" id="m_477837590861821498Picture_x0020_1" src="https://mail.google.com/mail/u/0/?ui=2&ik=fa8e727725&view=fimg&th=163949da52eeb2ff&attid=0.1&disp=emb&attbid=ANGjdJ9MRbUKDtt4GTfBV4AiFLcXfqdYvxXa6farPtiXsXacz2i26G-eVlv8WslhOXvIOxokf3iqm6qpsoMnsg40OXkA7r_Bfv6chM4wCaKpl5gwFLgJeu0GSUdo_Pc&sz=w400-h374&ats=1527207784360&rm=163949da52eeb2ff&zw&atsh=1" style="height: 1.9479in; width: 2.0833in;" width="200" /></span></a></div>http://mymathclub.blogspot.com/2018/05/im-going-to-conference.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-7229500802356189634Sun, 20 May 2018 23:17:00 +00002018-05-20T16:26:11.652-07:00algebrasymmetryTangles and SymmetryThis is a description of <a href="https://www.math.nmsu.edu/~davidp/">Dr. David Pengelley's</a> talk "All Tangled up and Searching for the Beauty of Symmetry" which I just attended. This makes an excellent Math Circle topic and I think I might use it next year.<br /><br /><h4> Materials</h4><div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both;"><a href="https://3.bp.blogspot.com/-pSz5XBnDJSc/WwH-VkKkbhI/AAAAAAAANVI/vtAcXpBj4Oc8yWyVXwuXMfDpL8CRrip4QCKgBGAs/s1600/20180520_144322.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://3.bp.blogspot.com/-pSz5XBnDJSc/WwH-VkKkbhI/AAAAAAAANVI/vtAcXpBj4Oc8yWyVXwuXMfDpL8CRrip4QCKgBGAs/s320/20180520_144322.jpg" width="180" /></a><a href="https://4.bp.blogspot.com/-HO4WaBxJVT0/WwH-LlZx8mI/AAAAAAAANVA/53GInbxxGX4v9fOzybKUOD4GjKTZlc4VwCKgBGAs/s1600/20180520_144317.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://4.bp.blogspot.com/-HO4WaBxJVT0/WwH-LlZx8mI/AAAAAAAANVA/53GInbxxGX4v9fOzybKUOD4GjKTZlc4VwCKgBGAs/s320/20180520_144317.jpg" width="180" /></a><a href="https://2.bp.blogspot.com/-aFeFS6grm_0/WwH-Qq2kw8I/AAAAAAAANVE/pG431NZUfnoeIf-TlpmnP2SMCeTl4STbQCKgBGAs/s1600/20180520_144313.jpg"><img border="0" src="https://2.bp.blogspot.com/-aFeFS6grm_0/WwH-Qq2kw8I/AAAAAAAANVE/pG431NZUfnoeIf-TlpmnP2SMCeTl4STbQCKgBGAs/s320/20180520_144313.jpg" /></a></div><br /><br />1. 3 different colors of dental floss or some equivalent string cut into 3 foot lengths.<br />2. A 6-8 inch wooden dowel<br />3. A note card with 3 holes for the strings and with an X on one side and an O on the other.<br /><br />Its helpful to tape the string in bundles of 3 to keep them from tangling before handing them out.<br /><br /><h4>Notecard Explorations</h4>1. First hand out the note cards. Have the kids explore how many different states of the note card there are with same orientation. <br /><br />There are 4 (2 with the x showing on top and bottom and 2 with o showing on top and bottom.) Note: the rectangle must stay with the longer side vertical. <br /><br /><a href="https://2.bp.blogspot.com/-bK8NQkDtZBY/WwH-5C03BCI/AAAAAAAANVc/zsZXKbvmdAQpuusK1sMFNr_rYnitGnPWwCKgBGAs/s1600/20180520_130924.jpg"><img border="0" src="https://2.bp.blogspot.com/-bK8NQkDtZBY/WwH-5C03BCI/AAAAAAAANVc/zsZXKbvmdAQpuusK1sMFNr_rYnitGnPWwCKgBGAs/s320/20180520_130924.jpg" /></a><br /><br /><br /><br />2. Next explore how many ways there are to transition between states.<br /><br /><br /><ul><li>Flip, Rotate and Spin and most importantly None. </li></ul><br /><br /><div class="separator" style="clear: both;">Make a an operation chart to explore the transitions:</div><br /><a href="https://3.bp.blogspot.com/-KBGhlHU0kGc/WwH_V1nijfI/AAAAAAAANVk/EPB5Qe-e_LAE_I2CN0EnQigCI6E0awZ3wCKgBGAs/s1600/20180520_141359.jpg"><img border="0" src="https://3.bp.blogspot.com/-KBGhlHU0kGc/WwH_V1nijfI/AAAAAAAANVk/EPB5Qe-e_LAE_I2CN0EnQigCI6E0awZ3wCKgBGAs/s320/20180520_141359.jpg" /></a><br /><br /><br />Things to look for: closure, identity element, commutativity. i.e. this is an Abelian Group.<br /><br /><h4>Full Space (Quarternions)</h4>3. Next add the strings. Each string should be threaded through the prepunched holes and tied to the dowel.<br /><br /><br /><a href="https://3.bp.blogspot.com/-xJVAlLqK_10/WwH_uBUYWjI/AAAAAAAANVw/QXdAHBqGX-cH37GdVq2dtSVXPnyGVzz4wCKgBGAs/s1600/20180520_132641.jpg"><img border="0" src="https://3.bp.blogspot.com/-xJVAlLqK_10/WwH_uBUYWjI/AAAAAAAANVw/QXdAHBqGX-cH37GdVq2dtSVXPnyGVzz4wCKgBGAs/s320/20180520_132641.jpg" /></a><br /><br /><br />4. Next explore whether anything has changed. Key question are the strings the same after a flip, rotate etc? <br /><br />Introduce notion of clockwise/counterclockwise transitions. Also Full rotation = 2 in one direction. <br /><br />5. Can we get back to the original state w.r.t to the strings and card after 2 and 4 rotations? You are allowed to move the strings but not the dowel or card and the strings may only move around the card. Split group in half and have each piece work on part of the problem. Remember to pre-practice play with the transitions before hand so they are very familiar.<br /><br /><br /><a href="https://2.bp.blogspot.com/-Cpz-1hj3cRA/WwIAGtD747I/AAAAAAAANV4/mgtyiBAU3WErosDuhrZgIpJt2KV-KzTvQCKgBGAs/s1600/20180520_140140.jpg"><img border="0" src="https://2.bp.blogspot.com/-Cpz-1hj3cRA/WwIAGtD747I/AAAAAAAANV4/mgtyiBAU3WErosDuhrZgIpJt2KV-KzTvQCKgBGAs/s320/20180520_140140.jpg" /></a><br /><br />(Only 4 rotations works. Several ways to move the strings to prove it. The easiest is to take all of them and move them around the card.) This takes some time.<br /><br />6. What state does a flip / rotate end up in. (Either a forward or backwards flip.<br /><br />7. Build the new operations table:<br /><br /><br /><a href="https://4.bp.blogspot.com/-ZPR2CQdMSwE/WwIAr4jxYGI/AAAAAAAANWE/RyeOnTd8YHsHpuJQOj2IXw5mbJNQWAnMACKgBGAs/s1600/20180520_140339.jpg"><img border="0" src="https://4.bp.blogspot.com/-ZPR2CQdMSwE/WwIAr4jxYGI/AAAAAAAANWE/RyeOnTd8YHsHpuJQOj2IXw5mbJNQWAnMACKgBGAs/s320/20180520_140339.jpg" /></a><br /><br /><br /><br /><br />8 states: None, Full Turn, Rotate Clockwise, Rotate Counter Clockwise, Forward Flip, Backwards Flip, Clockwise Spin, Counter Clockwise Spin.<br /><br />Note new patterns: This is not commutative for example. How do both tables relate to addition and subtraction?<br /><br />Wrap up: Tie to <a href="https://en.wikipedia.org/wiki/Algebraic_group">Algebraic Group</a> and <a href="https://en.wikipedia.org/wiki/Quaternion">Quarternion</a> I like bringing in a bit of math history about Hamilton and the Broome Bridge. See: <a href="https://en.wikipedia.org/wiki/History_of_quaternions">https://en.wikipedia.org/wiki/History_of_quaternions</a><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><div><br /></div></div>http://mymathclub.blogspot.com/2018/05/tangles-and-symmetry.htmlnoreply@blogger.com (Benjamin Leis)1tag:blogger.com,1999:blog-4227811469912372962.post-7465537152613969580Wed, 16 May 2018 18:34:00 +00002018-05-16T11:34:09.784-07:00fractalsmodular arithmetic5/22 Chaos + Mod Arithmetic<div class="separator" style="clear: both; text-align: center;"></div><div style="margin-left: 1em; margin-right: 1em;">We started the day looking at the problem of the week (from @mpershan):</div><blockquote class="tr_bq">Given a triangle with side length A, B, and C </blockquote><blockquote class="tr_bq"><ul><li>If A/B = B/C = 1, then it's an equilateral triangle. </li><li>If A/B = B/C = 2, it can't be a triangle. </li></ul></blockquote><blockquote class="tr_bq">What the largest value of A/B = B/C it's possible for a triangle to have?</blockquote><br /><br />I tried a slightly different structure this time and instead of asking for solution demos I asked the room for what they noticed about the problem. This was actually fairly productive. We started with several statements about the triangle inequality. That wasn't generally known so I demoed it on the board. I like creating pictures where the two smaller sides are very short so its really clear they can't meet. We then had the idea presented to fix one of the sides to length 1 and see what happens. Eventually the kids experimented with concrete numbers for the ratio of two and found the results: 1,2,4 violated the triangle inequality. Finally, one student came up with the general inequality x^2 < x + 1. I had another one solve it via the quadratic formula and to cap everything off I asked if anyone recognized the result. I ended with a small speech on the golden ratio and how it shows over and over again in unexpected places. <b>[I may go down this path for a future theme.]</b><br /><br /><div><img align="middle" height="320" src="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Sierpinski_Chaos.gif/200px-Sierpinski_Chaos.gif" style="float: center;" width="320" /></div><br />Based on an interesting post by Matt Enlow: I inserted a group activity next: <a href="https://en.wikipedia.org/wiki/Chaos_game">https://en.wikipedia.org/wiki/Chaos_game</a>.<br /><br />In an nutshell:<br /><br /><ul><li>Start with an equilateral triangle.</li><li>Pick a random point within it.</li><li>Randomly select a vertex and find the next point half way between the last one and that vertex.</li><li>Repeat</li></ul><br /><br />After thinking about some previous feedback comments I chose to do the simulation on a very large communal piece of butcher block paper (3 ft x 3ft). If I had a larger sheet I would have gone even bigger. I drew an equilateral triangle using a standard intersection of 2 arcs methods with a tape measure which actually elicited some discussion. Then everyone crowded around the side of the paper and we took turns rolling the dice, measuring the next point and marking it. (This also let me keep a rough tab that the points were accurate.) At strategic points I had the kids make predictions about what they observed. We generated over 30 points which takes a while but gave everyone a chance to roll the dice at least twice. This is enough to see the beginning but not the full pattern. But the kids were already able to make a good conjecture about why you couldn't get back to the center. After this point I switched to an internet simulation:<br /><br /><a href="http://thewessens.net/ClassroomApps/Main/chaosgame.html?topic=geometry&id=15">http://thewessens.net/ClassroomApps/Main/chaosgame.html?topic=geometry&id=15</a><br /><br />This let me run thousands of random choices and show the emerging Sierpinski triangle. As I hoped this produced a lot of spontaneous "Wows" We had a bit more followup fractal discussion but if I repeated I would love to find a second half that thematically linked here.<br /><br />Finally I returned to my original plan to go over a bit of Modular arithmetic using a Math Circle structure from <a href="https://math.berkeley.edu/~jhicks/links/MathCircleBook.pdf"> https://math.berkeley.edu/~jhicks/links/MathCircleBook.pdf</a> This is a bit dry for Middle School although I like the starting magic trick and that did work well. What I ended up doing was printing enough packets for pairs to work on and doing the first few questions as a group with some structured lecturing on my part and then circulating to help groups work through the later parts of the set. Overall: I kept the room moving and I finally introduced the topic which was a goal but I think there is more room to grow this session in the future. I think focusing on multiplication and addition tables is one of the more natural pieces to use and perhaps dropping some of the formal linear congruence proofs. For almost all of these, I found I was suggesting the kids try concrete numbers first and observe patterns to get at what the packet was suggesting.http://mymathclub.blogspot.com/2018/05/522-chaos-mod-arithmetic.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-5829697127473297581Fri, 11 May 2018 00:06:00 +00002018-05-11T07:40:33.832-07:00square root4/9 Square Roots<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 /><br />Today I returned to a topic, square roots, I've done before with younger kids : <a href="http://mymathclub.blogspot.com/2016/05/53-square-roots.html">http://mymathclub.blogspot.com/2016/05/53-square-roots.html</a>. My thinking was that I had a cool video I wanted to show that mentioned approximating the numeric value of a square root and doing it ourselves would motivate that part of the video and provide some embedded practice calculating with decimals. <br /><br />But before we could start in we needed to go over the old problem of the week.<br /><br />"<span style="font-family: "arial"; font-size: 11pt; white-space: pre-wrap;">In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure"</span><br /><br /><br />This one lent itself really well to whiteboard demos and I had kids present two different approaches. One was combinatoric, the other just brute force listed the cases. Since there is a max of 168 total in weight and only 9 weights either way is very approachable. However, none of the kids thought about putting the weights on both sides which presented me with a dilemma. I decided to approach it as follows "All the ideas you've mentioned are correct with some assumptions, lets make those explicit and double check if we can relax any of them." That way I could honor the thinking already done and point out the further avenue to explore.<br /><br />Next we started on the square root investigation. I framed the problem as follows:<br /><br />"How does the square root key work on the calculator? What do you think its doing? I'm going to put up some simple square roots \(\sqrt{2}, \sqrt{5}, \sqrt{7}\) can you find an approach to calculate value of these to a few decimal places (without a calculator)?"<br /><br />What followed was a fairly useful exercise on two levels. There was a ton of calculation practice and several variants on the bound and search for a better fit algorithm emerged. At the end I also raised the question "For what other calculator keys do you wonder about how they work and are there any themes in the implementations?" Logarithms were probably the best candidate for a future session. One possibility is stressing the use of series and iterative algorithms.<br /><br />From there I tried to fit in an old problem on the same theme I've been saving:<br /><br />Find the cube root of \(x^6 - 9x^5 + 33x^4 - 63x^3 + 66x^2 - 36x + 8 \). This we ended up doing as a group discussion. The kids eventually found the first and last coefficients of the root Ax^2 + Bx + C but were stuck on the middle one B. I really wanted to carefully go through the distributive law work and finish the solution but I was short on time and had to call this to a close. Moral: This is more than a 10 minute problem (and maybe more problems requiring polynomial mult. are called for)<br /><br />In the midst of all of this I had audio issues with the projector and had to rush around to find another room we could use with a working system. So we filed over next door and spent the last 25 minutes watching this really fascinating 3Blue1Brown video which I alluded to originally:<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/b7FxPsqfkOY/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/b7FxPsqfkOY?feature=player_embedded" width="320"></iframe></div><br />Right at the beginning the square root approximation problem is discussed and hopefully it had extra resonance after trying it ourselves.<br /><br /><br />P.O.T.W.<br /><br />A fun triangle inequality / golden ratio problem from @mpershan:<br /><br />https://drive.google.com/open?id=1y-LAjQmdOuyTYd7NlNXfayi-AX73IF4CzkujMZta3PE<br /><br /><br />I've now seen an excellent numberphile video on Phi that I could easily integrate with this. So maybe we'll do a golden ratio day. Although I also want to fit in a 3-D weaving art project as well in the next few weeks.<br /><br /><br /><br />http://mymathclub.blogspot.com/2018/05/49-square-roots.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-983153730869641367Fri, 04 May 2018 18:14:00 +00002018-05-07T08:23:05.544-07:004/2 Following my passionThis week I ended up switching my focus on the fly. I had been originally been planning on doing a graph theory math circle activity centered around chicken pecking hierarchies. (Yes really) But I became so excited about thinking about polynomial deltas that I ended up asking myself the question "Why not go with what you're excited about?" All the kids have enough background with polynomials so we could just jump in which was an added bonus.<br /><br />I stuck with one part of my planning process. I had already decided I wanted to start with some group white boarding focusing on some of the geometry puzzles from @solvemymaths.<br /><br />So I picked 3 of them included the now infamous pink triangle. In each the goal is to figure out which fraction of the shape is shaded pink and usually any polygons are regular.<br /><br /><img src="https://pbs.twimg.com/media/DbfbxhlW0AAKEQk.jpg" /><br /><img src="https://pbs.twimg.com/media/DMbfvU4WAAIBlqk.jpg" /><br /><br /><br /><img src="https://pbs.twimg.com/media/DNAiWsOXkAEPKRW.jpg" /><br />I put each of these up on different sections of the whiteboards and let the kids circulate among them forming organic groups. (Occasionally I'll nudge kids to work together) They then spent about 15-20 minutes attempting to find solutions while I circulated and interacted with individual groupings. My particular focus this time was to emphasize thinking about the problems and coming up with ideas. I used the "What do you notice/wonder?" prompt quite a bit. There was a lot of good thinking but I definitely still see room for encouraging more experimentation. At the end of the process I had everyone regroup for a discussion of what various people had found. Interestingly, the first pink triangle solution was analytic i.e. the student setup to equations for the lines and found the intersection. I think this reflects the emphasis the curriculum places on these type approaches over pure synthetic reasoning. Students don't see similar triangles quite as quickly. As an aside I had to explain the expression "broke the internet" to one of the boys as in "this puzzle just broke the internet this week."<br /><br /><br />For the second half, I switched over to looking at the patterns within polynomial deltas. See: <a href="http://mymathclub.blogspot.com/2018/05/polynomial-deltas.html">http://mymathclub.blogspot.com/2018/05/polynomial-deltas.html</a> for my motivation.<br /><br />Example:<br /><br />x^2 + x + 1<br /><br /><br />1 3<br /> 4<br />2 7 2<br /> 6 0<br />3 13 2<br /> 8<br />4 21<br /><br /><br />The way I structured this section was to demonstrate calculating deltas on a sample polynomial and have the kids then come with up with their own polynomials and look for patterns on what was occurring.<br /><br /><ol><li>How many levels of deltas before you hit you and why?</li><li>Is there some pattern to what the second to last value was etc?</li><li>What does this mean? I had one kid talk about velocity/acceleration and I think I would draw this out more if I repeated as well.</li></ol><div><br /></div><div>When they started coming up with enough ideas, I then suggesting trying to investigate general classes of polynomial like Ax^2 + Bx + C. This generated lots of good distributive law practice on the whiteboards.</div><div><br /></div><div>Finally for my 3rd prompt I asked can you go backwards as well as forwards and why do you think it does or doesn't work?</div><div><br /></div><div>I also put up the original problem from the last post as an extension at the end for a few kids who worked quickly enough to get there.</div><div><br /></div><div>This structure worked pretty well the one element I would improve on was to add in graphing. I only have a single computer to use but it occurred to me after the fact that I could have done some group desmos activities in the beginning with various polynomials and graphically shown the deltas.<br /><br /><br />[Another interesting angle to pursue would be function inverses. What's the behavior of the deltas when a function has an inverse vs. when it doesn't can we make a test?]</div><div><br /></div><div>To close, I mentioned this kind of investigation is closely related to another branch of mathematics and had everyone take guesses at what it was. As I expected no one even came close to saying Calculus which shows how mysterious it is . </div>http://mymathclub.blogspot.com/2018/05/42-following-my-passion.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-8301094354584218797Tue, 01 May 2018 21:33:00 +00002018-05-01T14:33:56.067-07:00algebraPolynomial Deltas<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">Find the polynomial </span><a href="https://www.codecogs.com/eqnedit.php?latex=Ax%5E5%20%2B%20Bx%5E4%20%2B%20Cx%5E3%20%2B%20Dx%5E2%20%2B%20Ex%20%2B%20F%0" style="text-decoration-line: none;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"><img height="16" src="https://lh5.googleusercontent.com/FwkyRZrLsHZv-ZS1PhoQWhCg4zIOLm8YfPtNvC2LjLw1j5JeRiav2VBIiVoSONNH2-nliS5k1dc-XtOdYiwo4yLzFyuHfQltiwzYB43q-qE6E_53551VNseExF5A5zCA6bDOTUfv" style="border: none; transform: rotate(0rad);" width="255" /></span></a><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> given:</span></div><span id="docs-internal-guid-49ed9999-1d8f-a54e-21ff-9bac8ba769b8" style="font-weight: normal;"><br /></span><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">f(1) = 1 f(4) = 3</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">f(2) = 1 f(5) = 5</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">f(3) = 2 f(6) = 8</span></div><span style="font-weight: normal;"><br /></span><span style="font-weight: normal;">In the past I would have viewed this as a 6 variable linear system and given the complexity gone to linear algebra and matrices probably with a computer assist. </span><br /><span style="font-weight: normal;"><br /></span><span style="font-weight: normal;">But I saw a new way to approach this via @eylem_99 that relies on analyzing the deltas. The idea comes from a tool I've only ever seen used for making sense of polynomials. Calculate the deltas between values of a func for instance f(2)-f(1) = 0 and then calculate the second level deltas etc. Eventually after n iterations where n is the same as the degree of the equation you'll reach a constant value. You can prove this via algebra but essentially these deltas are approximations of the various derivatives. When you reach the nth level the derivative becomes constant and the approximation is completely accurate.</span><br /><br />So its a fun exercise to do this on various polynomials and see what falls out. What had never occurred to me was this also can be used to reverse additional values for the function by generating the deltas in reverse!<br /><br />So for example from the above values we can regenerate f(<span style="font-family: "arial"; font-size: 11pt; white-space: pre-wrap;">0), f(-1), and f(-2) starting with constant delta -3 on the right:</span><br /><span style="font-weight: normal;"><br /></span><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">-2 110 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> -74 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">-1 36 46 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> -28 -25 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">0 8 21 11 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> -7 -14 -3</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">1 1 7 8 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> 0 -6 -3</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">2 1 1 5</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> 1 -1 -3</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">3 2 0 2</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> 1 1 -3</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">4 3 1 -1</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> 2 0</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">5 5 1</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> 3</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">6 8</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">The final delta is actually </span><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><a href="https://www.codecogs.com/eqnedit.php?latex=%20f%27%27%27%27%27(x)%20%0" style="text-decoration-line: none;"><img height="16" src="https://lh6.googleusercontent.com/RldpHPq3Zbxf-hEGrYbLjVrae_FyJjgwYau-xJrT--97unA2DocFmGY_nX_RTGphVeNzilOzu_Fov-UlgkaItKFpvwxTAZrsfKKv6V7KbZKsglGWgZvtzlG7vxZ0HsGzMLhA3d1m" style="border: none; transform: rotate(0rad);" width="44" /></a> and note that in general this would 5! * A</span><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"> so </span><a href="https://www.codecogs.com/eqnedit.php?latex=A%20%3D%20-%5Cfrac%7B3%7D%7B5!%7D%20%3D%20-%5Cfrac%7B1%7D%7B40%7D%0" style="text-decoration-line: none;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;"><img height="33" src="https://lh3.googleusercontent.com/3WVlNMYVfBLdfwaOtsiyWFS2Bx0Sau_m-48N5S-EOKwFo_MKYj8YWHM0N3bt_zYKoU-kgHE4tN-l4eBkDQp2LLiskwDDi_w9j3uFQQWIxa2OkH94xAY1UeuN3N2K1Aqb4k53sv0C" style="border: none; transform: rotate(0rad);" width="115" /></span></a></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">f(0) = 8 so F = 8</span></div><span style="font-weight: normal;"><br /></span><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">Now adding and subtracting complements f(1) and f(-1) and f(2) and f(-2) generates simpler</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">linear equations for finding B,C,D, and E.</span></div><span style="font-weight: normal;"><br /></span><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">2B + 2D + 16 = 37 and -1/20 + 2C + 2E = -35 </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre;">32B + 16D + 16 = 111 -32/20 + 16C + 4E = -109</span></div><span style="font-weight: normal;"><br /></span><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">From there its not to difficult to get the rest of the constants and certainly simpler than the matrix math.</span></span></div>http://mymathclub.blogspot.com/2018/05/polynomial-deltas.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-7792526389960814431Wed, 25 Apr 2018 19:21:00 +00002018-05-02T20:36:25.940-07:004/24 Tic Tac Toe (Naughts and Crosses)<div class="separator" style="clear: both; text-align: left;">Today started with some administrative tasks. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>I had the kids who went to the Quiz bowl talk about their experience and whether it was worth repeating. The consensus is that its fun (although not particularly math related) So I'd like to figure out a way to do this more next year without sacrificing any club time. The best option would be to spin off another ASB club but that requires another sponsor and quite a bit of leg work.</li><li>I also asked about the Purple Comet Meet. Again the kids also really enjoyed it so we will do it again next year.</li><li>A quick review of last week's Problem of the Week.</li></ul><div><br /></div><div>I had two main goals for the day day: go over a few problems from the Purple Comet meet so kids could see solutions to problems they hadn't figured out or alternatively have a chance to present a solution they found to the group. I figured that 3 questions would be about the right number from a focus perspective. </div><div><br /></div><div>I projected the problem set on the board and the kids chose numbers 4,9 and 20 to look at as a group. The best discussion was on the first problem where we had 3 different solutions presented. On the last I really wanted to draft a usually shy kid to draw the diagram on the board but he was very adamant about not wanting to do so. I didn't insist and had another kid do it instead. I'm going to think more about the best way to do this is. I<i>s it enough to have kids participate in groups with each other or is it important to strongly encourage them to also talk in front of everyone? </i>As a first step I intend to privately chat and see if I can get him to volunteer in future weeks.</div><div><br /></div><div><br /></div><div>Then for contrast I wanted some game or puzzle to do for the remainder of the time which I assumed we would be around 20-30 minutes. My problem was I didn't have any solid ideas in mind and on a lark I mentioned that on twitter. In a very affirming moment I received a bunch of interesting suggestions:</div><br /><blockquote class="twitter-tweet" data-lang="en"><div dir="ltr" lang="en">In which the twittersphere was incredibly helpful. Thanks <a href="https://twitter.com/mpershan?ref_src=twsrc%5Etfw">@mpershan</a> <a href="https://twitter.com/KentHaines?ref_src=twsrc%5Etfw">@KentHaines</a> <a href="https://twitter.com/DavidKButlerUoA?ref_src=twsrc%5Etfw">@DavidKButlerUoA</a> <a href="https://twitter.com/MrCorleyMath?ref_src=twsrc%5Etfw">@MrCorleyMath</a> etal. <a href="https://t.co/9fiCCTIE5N">https://t.co/9fiCCTIE5N</a></div>— Benjamin Leis (@benjamin_leis) <a href="https://twitter.com/benjamin_leis/status/988636694504030208?ref_src=twsrc%5Etfw">April 24, 2018</a></blockquote>A theme emerged around variants of tic-tac-toe that I thought would work very well.<br /><br />I ended up with 3 versions to have the kids try:<br /><br /><br /><ol><li>Tic-Tac-Toe on a torus <a href="http://mathforum.org/library/drmath/view/55291.html">http://mathforum.org/library/drmath/view/55291.html</a></li><li>Ben Orlin's ultimate Tic Tac Toe. <a href="https://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/">https://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/</a> which is done with a grid of 3x3 tic tac toe games. This was probably the favorite version for the room.</li></ol><br /><br /><br /><br /><script async="" charset="utf-8" src="https://platform.twitter.com/widgets.js"></script> <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-IN0kqK0PyZk/WuDPvU5rcII/AAAAAAAANLk/liElSuOyS8AwXFltF2x08v7pQQxIY6MNgCKgBGAs/s1600/20180424_165518.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/-IN0kqK0PyZk/WuDPvU5rcII/AAAAAAAANLk/liElSuOyS8AwXFltF2x08v7pQQxIY6MNgCKgBGAs/s320/20180424_165518.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-gVbWbttJTNk/WuDPvSK_3UI/AAAAAAAANLk/O2efmNxnJOAaFY3yJQB65ELZIkjlyDrqgCKgBGAs/s1600/20180424_165458.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/-gVbWbttJTNk/WuDPvSK_3UI/AAAAAAAANLk/O2efmNxnJOAaFY3yJQB65ELZIkjlyDrqgCKgBGAs/s320/20180424_165458.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-pYLICrESZ6g/WuDPvYmakUI/AAAAAAAANLk/rVUPP90rMYsIQDI36ncO5mEW-qq_nQ7LwCKgBGAs/s1600/20180424_165510.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/-pYLICrESZ6g/WuDPvYmakUI/AAAAAAAANLk/rVUPP90rMYsIQDI36ncO5mEW-qq_nQ7LwCKgBGAs/s320/20180424_165510.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">3. David Butler's 4D Tic-Tac-Toe <a href="https://t.co/ryATOQOZiK">https://t.co/ryATOQOZiK</a> I had mostly run out of time so we just barely described the rules which are a bit complex.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Overall these were a big hit. Middle schoolers still find the game fun and I had a few kids I literally had to shoo out of the room past our closing time.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><b><br /></b></div><div class="separator" style="clear: both; text-align: left;"><b>Problem of the Week:</b></div>I'm still in a purple comet meet mood so I took a mid-level problem from a previous year that I liked:<br /><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="https://drive.google.com/open?id=1fY3zq6jk1330-X9rxAQI7GOivdMvoi62bTgpK7jY-SE">https://drive.google.com/open?id=1fY3zq6jk1330-X9rxAQI7GOivdMvoi62bTgpK7jY-SE</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;"><br /></div><br />http://mymathclub.blogspot.com/2018/04/424-tic-tac-toe-naughts-and-crosses.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-7884359565291987753Fri, 20 Apr 2018 21:43:00 +00002018-04-28T13:57:04.857-07:00purple comet4/16 Purple Comet Online Contest<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> Three years ago I discovered the Purple Comet contest @ <a href="http://purplecomet.org/">purplecomet.org</a>. It has close links to the AwesomeMath and I really liked the problems in the old tests. So I tried it out with my fifth graders at the time. That <a href="http://mymathclub.blogspot.com/2015/04/421-purple-comet-online-meet.html">2015 Experience</a> discouraged me from doing it again. Despite the kids theoretically having up to Math7 knowledge the contest was too hard and I needed material that was better levelled for them to be most productive.<br /><br />Cut to this year when I have actual 6th and 8th graders and I decided to participate again. My current motivation was less the problems themselves than the timing. We don't have any real contests to participate in during the Spring and I wanted to do one meaningful one for the kids who like doing them. Since you have a testing window when you can administer the contest and it just needs a few computers, the overall experience is very low barrier (much easier than an AMC test).<br /><br />Overall this year went much better. I split into two teams and we worked in the library. Both groups found answers to over half the problems and most kids stayed on target. I had few at the last 10 minutes who had reached their limit for the day which is not unexpected.<br /><br />I'm once again trying to decide which problems to review as a group next week. I'm thinking we probably should only do 3 maximum so perhaps I'll bring a set and let the kids vote on which ones they want to see the most.<br /><br />In the meantime and in no particular order here are my observations about a few problems I noticed and thought about while proctoring, now that its ok to discuss them:<br /><br /><b>Problem 17</b><br /><br />Let a, b, c, and d be real numbers such that \( a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018. \). Evaluate 3b + 8c + 24d + 37a.<br /><br />This one is phrased in such a way as to suggest non integer answers but just looking at it the temptation is to say what if the two expression are completely identical and 3a = a^2, 8b = b^2 etc? And if you compute the squares sure enough \(3^2 + 8^2 + 24^2 + 37^2\) does equal 2018.<br /><br />That's fairly easy for a problem at the end and its not a very satisfying method. So what I think they might have been aiming at was to subtract the two expressions and complete the square to get<br /><br />\( (a-\frac{3}{2})^2 + (b-4)^2 + (c - 12)^2 + (d-\frac{37}{2})^2 = \frac{1009}{2} \) The right hand side is 1/4 of 2018 so you can plug the original equations back in to get<br /><br />\( 4( (a-\frac{3}{2})^2 + (b-4)^2 + (c - 12)^2 + (d-\frac{37}{2})^2 ) = a^2 + b^2 + c^2 + d^2 \)<br /><br />Then subtracting the right hand side again you get:<br /><br />\( 4(a-\frac{3}{2})^2 - a^2 + 4(b-4)^2 - b^2 + 4(c - 12)^2 - c^2+ 4(d-\frac{37}{2})^2 - d^2 = 0 \)<br /><br />Each one of those pieces is a difference of square for example the first one is \( (2a - 3 + a )(2a - 3 - a) = (3a - 3)(a - 3) \) and if we set a to either 1 or 3 will zero out etc. You can do a similar operation with the other original expression and see that a = 3 for instance is the overlapping solution to both new equations. That gets to the original observation once you test the values in the original function. But it doesn't rule out alternate solutions where the pieces balance each other out in various ways.<br /><br /><b><br /></b><b>Problem 16 </b><br /><br />On \( \triangle{ABC} \) let D be a point on side AB, F be a point on side AC, and E be a point inside the triangle so that DE is parallel to AC and EF is parallel AB. Given that AF = 6, AC = 33, AD = 7, AB = 26, and the area of quadrilateral ADEF is 14, find the area of \( \triangle{ABC} \)<br /><br />A significant part of the difficult with problems like these is getting an accurate drawing from the description. That's something we can definitely practice as a group.<br /><br />Assuming you arrived at the following:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-gLnwqASTEF4/WtpVykuwDpI/AAAAAAAANH0/yPGCnKDxueAeUpBcVGu9Sx8Kt02wzln-wCLcBGAs/s1600/p16.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="273" data-original-width="651" height="268" src="https://1.bp.blogspot.com/-gLnwqASTEF4/WtpVykuwDpI/AAAAAAAANH0/yPGCnKDxueAeUpBcVGu9Sx8Kt02wzln-wCLcBGAs/s640/p16.PNG" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br />You can deal with the easier part of the problem which is all about triangle area ratios. First lets remove the unneeded lines and split the parallelogram in half.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Trw-3V8NP6U/WtpWTyNQiWI/AAAAAAAANH8/tAzuGxyOzC8R2RsqWZnmBvl924ktfqmLwCLcBGAs/s1600/p16.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="255" data-original-width="729" height="222" src="https://3.bp.blogspot.com/-Trw-3V8NP6U/WtpWTyNQiWI/AAAAAAAANH8/tAzuGxyOzC8R2RsqWZnmBvl924ktfqmLwCLcBGAs/s640/p16.PNG" width="640" /></a></div><br /><br /><ul><li> First note ADF is half of the parallelogram and has an area of 7.</li><li>Then the area ratio of ADF : ACD is 6:33 </li><li>You then repeat this process: the area ratio of ACD : ABC is 7:26</li></ul><div><br /></div><div><br /></div><br /><br /><b>Problem 13 </b><br /><br />Suppose x and y are nonzero real numbers simultaneously satisfying the equations \( x + \frac{2018}{y} = 1000 \) and \( \frac{9}{x} + y = 1. \) Find the maximum possible value of x + 1000y.<br /><br />My first instinct in these problems is to always remove the fractions to get:<br /><br />$xy + 2018 = 1000y$ and $9 + xy = x$ Then just on inspection we have the 2 parts of the expression we want to simplify \( x + 1000y = 2xy + 2027 \)<br /><br />And we also have an way to isolate either x and y, I picked y, to plug them back in and get<br /><br />$y = \frac{9}{1-x}$ => $x + 1000 \cdot \frac{9}{1-x} = 2x \cdot \frac {9}{1-x} + 2027$<br /><br />That cleans up quickly to: \( 1000x^2 - 3009x + 2018 = 0\) and while the numbers are high the factorization is still not too hard (1000x - 1019)(x - 2). So despite the phrasing which suggests an optimization problem there are only 2 solutions for (x,y) and you just have to plug them in and compare to get the final result.<br /><br /><div id="prob15"><b>Problem 15</b><br /><b><br /></b>There are integers \(a_1, a_2, a_3, . . . , a_{240}\) such that \( x(x + 1)(x + 2)(x + 3)· · ·(x + 239) = \sum_{n=1}^{240} a_n x^n \). Find the number of integers k with 1 ≤ k ≤ 240 such that \(a_k\) is a multiple of 3.<br /><br />At first glance this is a slog of a counting problem. Multiplying all 240 binomials together is impractical without a computer program. So the first step I took was to look for patterns by doing the first few terms,<br /><br />Ignoring x which doesn't change the coefficients:<br /><br />$$(x+1)(x+2) = x^2 + 3x + 2$$<br />$$(x+1)(x+2)(x+3) = x^3 + 6x^2 + 11x + 6$$<br /><br />I noticed a few things from this:<br /><br /><ul><li>all the x+3n terms always added 1 more multiple of 3 coefficient than the last term. This makes sense the x term just shifts the previous result and then we add a multiple of 3 to it which doesn't change the value modulus 3 and finally we get one new multiple of 3 constant term at the end.</li><li>So grouping the terms looked interesting. Note: there are 79 terms with a multiple of 3 in them. So that's 79 coefficients with a 3 at the end as a minimum. </li><li>Next: modular arithmetic seems useful here. We can simplify everything to mod 3 and not change the result so now we have x(x+1)(x+2)(x+3)(x+1)(x+2)(x+3) .... I chose 3 rather than zero deliberately so we didn't lose any terms.</li></ul><div>I also started simplifying the calculations letting x be an implicit place value system and never carrying so</div><div><br /></div><div>for example squaring the first term (x+1)^2 becomes (1 1)^2 = 1 2 1 </div><div><br /></div><div>Somewhere around this point I also converted to balanced ternary and used \( ( x+3)^{79} \cdot (x+1)^{80} \cdot (x-1)^{80} \)</div><div><br /></div><div>That's convenient because we can mostly easily expand the two final terms using the binomial theorem mod 3. At this point you're left with multiplying two highly symmetrical 81 digit terms. </div><div><br /></div><div>As far as I can see there's no escaping working out this product at least to a few terms before the pattern is visible but its a fairly simple process and you quickly see the result looks like (1 0 1 0 1 ... 0 1) with 161 terms and therefore 80 more zeros which are really coefficients that are multiples of three. That gives the baseline number of coefficients. So we add the two parts together to get 80 + 79 = 159. </div><br /></div><br /><br />P.O.T.W:<br /><br />I went with another UWaterloo problem set this week: <a href="http://www.cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWC-17-NN-PA-26-P.pdf">http://www.cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWC-17-NN-PA-26-P.pdf</a> with a fairly approachable number theory/lcm type problem.<br /><br /><br />http://mymathclub.blogspot.com/2018/04/416-purple-comet-online-contest.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-4518754668035094229Wed, 11 Apr 2018 02:07:00 +00002018-04-10T19:07:38.136-07:00Tricky Geometry ProblemIn this walkthrough we start with one that didn't look too hard at first:<br /><br /><br /><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> <img src="https://pbs.twimg.com/media/DYaOXVpXcAER5nv.jpg" /><br />Goal: Find angle \(\angle \alpha \) [Ignacio Larrosa]<br /><br />It quickly became apparent though that was going to be tricky.<br /><br />Avenues of investigation<br />1. Angle chase. The isosceles trinagle ADM is the starting point .At this point things were going well. All the angles can be described in terms of alpha and many are similar.<br /><br />2. Somewhere at around this point I decided to model in Geogebra to see what the goal really was. <br />This gave the target 18 degrees which was helpful. It also showed that ABC was a right triangle. So one path of investigation was to look into whether we could find expressions for the side lengths and show they satisfied the Pythagorean Equation. But it was hard to get them in a common variable.<br /><br />3. I started looking for similar triangles - BM reflected across BH creates another isosceleses for instance in an effort to build up side lengths but couldn't derive the 3rd side without another variable.<br /><br />4. I also added a few other auxiliary lines to see if they aided including splitting DAM in half, the perpendicular bisector of AC. I extended AB to create a larger outer isosceles triangle.<br /><br />5. Played with Stewarts Theorem and various similar triangles trying to derive BM's length and looked for ways to show BM = AM. That would prove B was on the circle curmscribing AC.<br /><br />6. I then became interested in subdividing BAM and the cyclic quads there. This was the breakthrough I found enough similar triangles in this division to create two expressions for a subsegment. Even this idea took a bit of playing before it shook out.<br /><br /><br />Here's what I came up with in the end:<br /><br />1. Angle chase and a lot of similar triangles and cyclic quadrilaterals fall out.<br />2. I subdivided angle \(\angle DBA\) on the right so I had more combinations of angle alpha to play with. It also seemed useful to add the perpendicular bisector of AB.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-XWqsMuds-g0/WrHDX-HZfuI/AAAAAAAAM00/7DDb3T4uR3sOBxF8IGVo5pMj_dsXg8A2QCLcBGAs/s1600/diag1.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="449" data-original-width="818" height="350" src="https://3.bp.blogspot.com/-XWqsMuds-g0/WrHDX-HZfuI/AAAAAAAAM00/7DDb3T4uR3sOBxF8IGVo5pMj_dsXg8A2QCLcBGAs/s640/diag1.PNG" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br /><br />Note: green angles are \(\angle \alpha\) and can be combined to make \(2\alpha\) or \(3\alpha\) red angles are \(90 - 3\alpha\). and the blue angle is \(2 \alpha\)<br /><br />In particular: \( \triangle BIJ \simeq \triangle EHI \) forming a cyclic quad and therefore angle \( \angle EHB \) is a right angle.<br /><br />1. So let BC = AC = BG = m and let CJ = a and BJ = m - a. Then since \(\triangle EHI \) is some scaled version of \(\triangle IBJ\) lets assign k equal to the scale factor. Based on that EH = \( k \cdot BJ = k\cdot (m -a)\) Likewise EI is k scaled version of BI.<br /><br />2. Next note \( \triangle BEI \simeq \triangle ACE \) . Since EI is a k scaled version of BI, CE is a k scaled version of AC or \(k \cdot m\) and CH = CE - EH = \(k \cdot a \)<br /><br />3. Also \( \triangle BIJ \simeq \triangle BCH \simeq \triangle CEJ \) all \(\alpha\) right triangles and from this we get<br /><br />\( \frac{CH}{BC} = \frac{CJ}{CE}\) or \(\frac{k \cdot a}{m} = \frac{a}{k \cdot m} \)<br /><br />But after simplification this implies k = 1 and \( \triangle BIJ \) is not just similar but congruent to<br />\( \triangle EHI \)<br /><br />4. So EI = BI and \(\angle IBE = \angle BEI \) This means \(2 \alpha = 90 - 3 \alpha \) and \(\alpha = 18 ^{\circ} \)<br /><br /><br /><br />http://mymathclub.blogspot.com/2018/04/tricky-geometry-problem.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-4179305204371958127Wed, 04 Apr 2018 15:52:00 +00002018-04-04T08:52:05.429-07:00purple comet4/3 Spring is Sprung<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>The last few weeks I've been concentrating on arranging the topics up to the last MOEMS Olympiad. This week I looked forward enough to realize that first Spring Break was coming up and that immediately after it was our only window to do the Purple Comet Math Meet. So I tabled my original idea for a new game in favor of doing a practice problem day. To start things off, I had one new student join us so we went through introductions again. As usual I had everyone say their name, math class and favorite activity from the last few months (except for the new boy who I asked to say why he joined) There were a couple of trends in what was mentioned. Pi day was apparently quite memorable and a group favorite as well was the Math Counts competition itself. But my favorite comment from someone a bit unexpected was to the effect "I really like all the problems, they're not like math class at all." </div><div><br /></div><div><div>Before we started I had the kids go over the problem of the week (from @solvemymaths) on the board:</div></div><div><br /></div><div><span id="docs-internal-guid-1ac5fa44-9143-341e-b118-24b422f59c9f"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><img height="425" src="https://lh6.googleusercontent.com/GE-spzJUjA23TH7cSGqNif84kK5ZvM5-7RwGE4fqdMrxWmwAuDx8owIGKc_AcNJ7utVRD4KIwUZ08MB50YkgZz9h6JoMuFsJe1ko8wMwDUakE6U8s2nzcdC2AWYCrr3PhuAuk2lc" style="border: none; transform: rotate(0rad);" width="602" /></span></span></div>I was happy to get both compute the inner area and compute the outer area demonstrated as approaches by two different students. (As I type this write-up I just notice the slanted 90 degree angle and wish we had talked about it.)<br /><div><br /></div><div>Since Purple Comet is a team event that you run in your own room I asked whether the kids wanted me to randomize the groupings or if they wanted to form them on their own. There was a very strong strong preference to choose teammates. As no one was left out I let them do so although every time this happens I wonder a bit about random groupings and if I should occasionally impose them on the kids. So far my feeling is that as a voluntary club its ok to honor their wishes and that regular class will cover some of this ground. </div><div><br /></div><div>Once everyone was divided I used a structure that has worked well in the past. I wrote the problem numbers up on the whiteboard and had the teams record their answers there. If they noticed a difference, they then went and discussed their solutions between teams to come to a consensus (and I also could focus on the groups at that point too.)</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-aPTfPy4SxXE/WsTwAq-ielI/AAAAAAAANAs/31Dp3gwTAQ0YU6Oy6yBk2gIPG9PS5z6pACKgBGAs/s1600/20180403_162932.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://2.bp.blogspot.com/-aPTfPy4SxXE/WsTwAq-ielI/AAAAAAAANAs/31Dp3gwTAQ0YU6Oy6yBk2gIPG9PS5z6pACKgBGAs/s640/20180403_162932.jpg" width="640" /></a></div><div><br /></div><div><br /></div><div>Even though its not really competitive this is just enough structure to hook a lot of the kids in and keep them going. I also brought some slant puzzles to hand out if I saw kids "burning out."</div><div><br /></div><div><a href="https://mathequalslove.blogspot.com/2017/03/slants-puzzles-from-brainbasherscom.html">https://mathequalslove.blogspot.com/2017/03/slants-puzzles-from-brainbasherscom.html</a> but I really only used them with 2 kids at the very end. Engagement overall maintained itself.</div><div><br /></div><div>I had the kids go over last year's contest: <a href="https://purplecomet.org/resource/problems/1136/2016MS_Problems.pdf">2017 MS Contest</a> Based on yesterday it looks like everyone should be able to do at least 6-10 problems on the real event which for me means the levelling is pretty good this time.</div><div><br /></div><div>During this whole process I then circulated and helped out with individual groups. Interestingly, the one problem I ended up focusing on the most with all the groups the was the tower of 7's.</div><div><br /></div><div><span style="font-size: large;">Find the remainder when \( 7^{7^7} \) is divided by 1000. </span></div><div><br /></div><div>There's not a lot of number theory or modular arithmetic exposure in school so this isn't so surprising.<br />I emphasized a few ideas:<br /><br /><br /><ol><li>Find the pattern/cycle in the last 3 digits of the powers of 7 i.e. make a table and see what happens.</li><li>You don't need all the digits to keep calculating only the last 3 (and why?)</li><li>Once you know the cycle length you just need to find where you are in the cycle i.e. what's the remainder of 7^7 divided by 20.</li><li>That can also be done by the just looking at the remainder (mod 20) after each multiplication of 7 and there is a simple pattern there as well.</li></ol><div>I'm almost completely decided we will do something with modular arithmetic before the end of the year based on this experience. </div></div><div><div><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><br /></span></div></div>http://mymathclub.blogspot.com/2018/04/43-spring-is-sprung.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-9118612297266805938Mon, 02 Apr 2018 18:43:00 +00002018-04-02T15:03:25.851-07:00algebradigressionIn 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 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.<br /><br /><br /><b>Example 3:</b><br /><b><br /></b>Solve:<br />$$x^2 - xy + y^2 = 13$$ $$x -xy + y = -5$$<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-4yC5FB6f5gY/WsE4C05qiLI/AAAAAAAAM_Q/-vcc733XiBU17K7-uc1P8YAh-e7qGcLXACLcBGAs/s1600/inverted.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="423" data-original-width="355" height="400" src="https://3.bp.blogspot.com/-4yC5FB6f5gY/WsE4C05qiLI/AAAAAAAAM_Q/-vcc733XiBU17K7-uc1P8YAh-e7qGcLXACLcBGAs/s400/inverted.PNG" width="335" /></a></div><br />Note: this is the symmetric intersection of a tilted ellipse and hyperbola. There's a clever substitution that can simplify this which I'll mention at the end but in the general case this system is actually a quartic in disguise.<br /><br />If directly attacking the problem, the starting point is the second equation with the simpler degree terms that allow us to easily isolate x or y.<br /><br />\( x + 5 = xy - y = y (x -1)\) or after confirming x - 1 => x = 1 is not a solution and can be safely divided \( y = \frac{x + 5}{x -1}\)<br /><br />We can substitute that back into the first equation to get:<br /><br />$$x^2 + x \cdot \frac{x+5}{x-1} + (\frac{x+5}{x-1})^2 = 13$$<br /><br />Now multiply everything by \((x-1)^2\) to arrive at:<br /><br />$$x^4 - 2x^3 + x^2 - x^3 - 4x^2 + 5x + x^2+ 10x + 25 = 13x^2 - 26x + 13$$ which cleans up to the following quartic:<br /><br />$$x^4 -3x^3 - 15x^2 + 41x + 12 = 0$$<br />Again using the rational roots theorem you only have to test 1,2,3,4,6, and 12. I also check from the bottom up since its simpler and you don't need to recheck again if you find a factor and divide to a simpler polynomial. Going up from one, first 3 and then 4 test successfully as roots leading to the following factorization once you do the polynomial division:<br /><br />$$(x-3)(x-4)(x^2 + 4x + 1) = 0$$ and from here the four (symmetric) solutions fall out.<br /><br />As promised the clever factorization in problem like this where there are only xy and x + y terms is to consolidate and substitute:<br /><br />First rearrange the original equations: \((x+y)^2 - 3xy = 13\) and \((x + y) + 5 = xy\)<br /><br />Substitute for xy to arrive at \( (x+y)^2 -3(x+y) -28 = 0\)<br /><br />Then substitute again w = x + y and solve the quadratic \(w^2 -3w - 28 = 0\) Once done replug the solutions x + y = 7 and x + y = -4 back into the original equations to solve a second set of quadratics. Note: this is actually probably just as much work as directly attacking the quartic since in effect you have to handle 3 quadratics.<br /><br />Further Digression:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-lXCbmWNpPu4/WsGtyPD5HPI/AAAAAAAAM_g/M4JY8AT0u5oqOuKM37cPLTK6Qna9LAWIACLcBGAs/s1600/rotated.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="430" data-original-width="537" height="256" src="https://4.bp.blogspot.com/-lXCbmWNpPu4/WsGtyPD5HPI/AAAAAAAAM_g/M4JY8AT0u5oqOuKM37cPLTK6Qna9LAWIACLcBGAs/s320/rotated.PNG" width="320" /></a></div><br />Conceptually I actually prefer the following approach. From either the graph or the symmetrical nature of both equations it looks interesting to rotate them 45 degrees back to non-tilted form. This is desirable because it will remove xy terms.<br /><br />So applying the standard rotation equations: $x' = x cos(\theta) + y sin(\theta)$ $y' = x sin(\theta) - y cos(\theta)$ to the two equations for \(\theta = -\frac{\pi}{8}\) we get:<br /><br />\(x^2 - xy + y^2 = 13\) becomes \((\frac{\sqrt{2}}{2}x - \frac{\sqrt{2}}{2}y)^2 - (\frac{\sqrt{2}}{2}x - \frac{\sqrt{2}}{2}y) (\frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y) + (\frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y)^2 = 13 \)<br /><br />That simplifies to \((x')^2 + 3(y')^2 = 26 \)<br /><br />Likewise \(x - xy + y = -5\) becomes \( (\frac{\sqrt{2}}{2}x - \frac{\sqrt{2}}{2}y) - (\frac{\sqrt{2}}{2}x - \frac{\sqrt{2}}{2}y) (\frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y) + (\frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y) = -5 \)<br /><br />Once again this simplifies nicely to \( (x' - \sqrt{2})^2 - (y')^2 = 12\) and these two equations are solved in a fairly standard fashion via elimination i.e.<br /><br />\( 3(x' - \sqrt{2})^2 - 3(y') ^2 = 36 \)<br />\(+ (x')^2 + 3(y')^2 = 26 \)<br />\( 4(x')^2 - 6\sqrt{2}x' + 6 = 62 \)<br /><br />Solving with quadratic equation and \(x' = \frac{7\sqrt{2}}{2}, y' = \pm \frac{\sqrt{2}}{2} \) or \( x' = -2\sqrt{2}, y' = \pm \sqrt{6} \)<br /><br />If you look back at the previous solutions at first this looks odd but remember these have to be tilted back 45 degrees to give the proper solutions and sure enough for instance \( (\frac{\sqrt{2}}{2} \cdot \frac{7\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \cdot \frac{7\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}) = (4,3) \)http://mymathclub.blogspot.com/2018/02/in-praise-of-rational-roots-theorem.htmlnoreply@blogger.com (Benjamin Leis)0tag:blogger.com,1999:blog-4227811469912372962.post-404629101421268933Fri, 30 Mar 2018 15:35:00 +00002018-03-30T11:05:29.983-07:003/27 Olympiad #5As is our routine, I started by going over the problem of the week:<br /><div style="text-indent: 50pt;"><br />Find six distinct natural numbers A,B,C,D,E,F such that<br /><br /><span style="text-indent: 50pt;">A + B + C = D + E + F</span><br /><br />And<br /><br /><a href="https://www.codecogs.com/eqnedit.php?latex=A%5E2%20%2B%20B%5E2%20%2B%20C%5E2%20%3D%20D%5E2%20%2B%20E%5E2%20%2B%20F%5E2%0"><img height="22" src="https://lh3.googleusercontent.com/dxY8c-CaWkp_GcRco9xUj-inMAYPC6nWMMA6LwN7JNqq3K_43neqFZcspvAQBVFGMwqzauWmf3OiGoMF0LNqekhY791HkO7ahjrIr_H0t_LN6zVLc1Q0CqZpk0x_6CmBLytpDYGu" width="320" /></a><br /><br /><br /></div>Once again, I had a girl write a python program to find the solution. I love all the computational computing that is occurring. I need a better way to harness this energy. This time I had her talk about the structure of the loops she used to search the problem space. In this case, it was a straightforward brute force attack loop over all 6 variables up to a limit and just check for each permutation if the two conditions were met.<br /><br />This is relatively slow and it produced lots of duplicate solutions as she pointed out that had to be manually disambiguated. I didn't go into it because not enough of the kids have a programming background but there are a few easy improvements that can be made on this approach:<br /><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;">limit = 100</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><br /></span><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;">for total in range (6,limit):</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> results = dict()</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> solns = 0</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> for a in range (1,total - 3):</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> for b in range (a+1, total - (3 + a)):</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> c = total - (a+b)</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> if c < b:</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> break</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> sum = a*a + b*b + c*c</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> if sum in results:</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> solns += 1</span><br /><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"> print "Found %d %d %d and %s total:%d" %(a, b, c, results[sum], tota</span><span style="font-family: "courier new" , "courier" , monospace;">l)</span></span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> break</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><br /></span><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> results[sum] = (a, b, c)</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><br /></span><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> ratio = float(solns) / total</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"> print "%d solutions for %d ratio=%f" % (solns, total, ratio)</span><br /><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><br /></span><br />Looping over the sum and then over the 3 digits in increasing order eliminates duplicates and cuts the number of comparisons down significantly. Note: the key observation is that you only have 2 degrees of a freedom once you've picked a sum for a + b + c.<br /><div><br /></div><div>After this point, we did the final MOEMS olympiad for the year. I delayed this round to fit better with the other activities I wanted to do. So technically we only had this week to finish and submit the scores. Two things stood out at me. There was a fraction question that dovetailed with the 2 weeks we've worked on Farey Sequences and Wilf-Calkin trees. Also yet again there was another combinatorics problem that most kids enumerated over rather than calculating a true combination. Overall, I think the kids scored the highest average of all the rounds. This afforded me the opportunity to have to cold call a few kids that usually are more reticent to demo on the board. One highlight for me was a girl proudly getting all the problems right and telling me that she thought this one was easy. I know she meant relative to the other ones for her but I still had to make a comment to avoid language like "this is easy." Nevertheless that was a huge victory. </div><div><br /></div><div>I chose a page from the "This is not a math book" for the early finishers with a fun rectangle dividing project. (I brought my box of crayons in for this.)</div><div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-sP6pZlPr8m8/Wr5YxHb1ZxI/AAAAAAAAM8Q/CgMNuAAYgNcNRuIKHBdzfei0CzcKjpmtwCKgBGAs/s1600/20180330_083239.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/-sP6pZlPr8m8/Wr5YxHb1ZxI/AAAAAAAAM8Q/CgMNuAAYgNcNRuIKHBdzfei0CzcKjpmtwCKgBGAs/s640/20180330_083239.jpg" width="640" /></a></div></div><div><br /></div><div><br />This was popular but didn't occupy as much time as I expected and I ended up giving out the P.O.T.W early to some kids as well as breaking out my game of 24 cards.</div><div><br /></div><div><br /></div><div>P.O.T.W:</div><div><br /></div><div>This one comes from Ed Southall and is a fraction talk activity:</div><div><a href="https://drive.google.com/open?id=1enhyxB4sBONd1AefGa8ieeuuWOrUggEi8xUq3ksdUsE">https://drive.google.com/open?id=1enhyxB4sBONd1AefGa8ieeuuWOrUggEi8xUq3ksdUsE</a></div><div><br /></div><div><br /></div>http://mymathclub.blogspot.com/2018/03/327-olympiad-5.htmlnoreply@blogger.com (Benjamin Leis)0tag: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" 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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-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)0