tag:blogger.com,1999:blog-55446619683269100272020-02-12T01:09:35.627-08:00Three J's LearningJGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.comBlogger303125tag:blogger.com,1999:blog-5544661968326910027.post-15199783833704559602020-02-11T18:42:00.000-08:002020-02-11T18:42:32.457-08:00What is 8?I've had a chance to spend more time doing math with the kids again and am hoping to write up our activities more consistently. Let's see how this works out!<br /><br /><a href="https://twitter.com/gfletchy">Graham Fletcher</a> created a set of <a href="https://gfletchy.com/progression-videos/">Progressions</a> videos for various elementary school themes. J3 and I recently went back to his page and found he had a new(er than we knew) progression on early number and counting. Even for this simple topic, the video highlights some points we hadn't considered explicitly, for example distinguishing producers (of a number) and counters. Also, the cardinality point that smaller natural numbers are nested within larger numbers wasn't something we had talked about, but we soon realized it was part of many examples in how we understand numbers.<br /><br />With that as inspiration, J3 and I decided to search for a range of examples of a single number, we chose 8, in different forms. There is at least one obvious version we're missing.<br /><br /><b>Add a comment (with picture, if you can) to show other forms of the number 8!</b><br /><b><br /></b>Marking 8 on the 100 board, an easy place to start:<br /><b><br /></b><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-mXcU2c2FFlQ/XkNjUR70tZI/AAAAAAAACRI/Xs1gi5V0zaclLDlZDP0Ehr6_qxcwN4l5ACLcBGAsYHQ/s1600/IMG_0205.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-mXcU2c2FFlQ/XkNjUR70tZI/AAAAAAAACRI/Xs1gi5V0zaclLDlZDP0Ehr6_qxcwN4l5ACLcBGAsYHQ/s400/IMG_0205.jpeg" width="400" /></a></div><br /><br />8 beads on the abacus shows the relationships 3+5 = 8 and 10-2 = 8 (also 100- 92 = 8)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-9HiaCHhL2To/XkNjUmB_aTI/AAAAAAAACRM/p75uGU4cHD438CZ7hyepFssuFpw3qxGEgCEwYBhgL/s1600/IMG_0206.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-9HiaCHhL2To/XkNjUmB_aTI/AAAAAAAACRM/p75uGU4cHD438CZ7hyepFssuFpw3qxGEgCEwYBhgL/s400/IMG_0206.jpeg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><b><br /></b>8 can hide in plain sight. Without labeling the three lengths, it would have been hard to recognize the longer one as 8 cm and, for you at home, impossible to know without reference to show the scale.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-AtieNu3GxSc/XkNjToU5OFI/AAAAAAAACRU/1UL4u4nkJh4CkppCKlO8D3CJW9o24J5RACEwYBhgL/s1600/IMG_0204.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="400" src="https://1.bp.blogspot.com/-AtieNu3GxSc/XkNjToU5OFI/AAAAAAAACRU/1UL4u4nkJh4CkppCKlO8D3CJW9o24J5RACEwYBhgL/s400/IMG_0204.jpeg" width="300" /></a></div><b><br /></b>It happened that, within the precision of our scale, two chocolate wrapped chocolate bars were 8 oz (2x3.5 oz of chocolate + about half an ounce of wrapping for each):<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-3k0IR6Cf5DE/XkNjTi4c1EI/AAAAAAAACRg/JFY_vvRL27smb115n0q8_EvW0jc_7MwlQCEwYBhgL/s1600/IMG_0203.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-3k0IR6Cf5DE/XkNjTi4c1EI/AAAAAAAACRg/JFY_vvRL27smb115n0q8_EvW0jc_7MwlQCEwYBhgL/s400/IMG_0203.jpeg" width="400" /></a></div><br />8 cups of water ended up being a lot, so this version unintentionally revealed a relationship 4 + 2 + 2 = 8<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-JRFcRSr3QNs/XkNjTv7vcpI/AAAAAAAACRU/ob7aLkbwbmY9_m5rkNw85eycLcjGv8i8gCEwYBhgL/s1600/IMG_0202.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-JRFcRSr3QNs/XkNjTv7vcpI/AAAAAAAACRU/ob7aLkbwbmY9_m5rkNw85eycLcjGv8i8gCEwYBhgL/s400/IMG_0202.jpeg" width="400" /></a></div><br />Though I'm not sure I can articulate why or show supporting research, I feel it is very valuable to build experience with physical models of numbers to create familiarity and intuition about what they are/mean. In particular, I hope this helped J3 anchor the importance of <u><i>units of measure </i></u>and <i><u>scale </u></i>in the interpretation of numbers.<br /><br />Finally, this construction has nothing to do with the number 8 (or does it???)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-2xMdAONe3n8/XkNjU4cssLI/AAAAAAAACRg/Wy-LZq5pQ2EJs7dbbbz6q9gTAbTLeBBsQCEwYBhgL/s1600/IMG_0230.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="640" src="https://1.bp.blogspot.com/-2xMdAONe3n8/XkNjU4cssLI/AAAAAAAACRg/Wy-LZq5pQ2EJs7dbbbz6q9gTAbTLeBBsQCEwYBhgL/s640/IMG_0230.jpeg" width="480" /></a></div><br />JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-47986723279853331942019-05-30T15:24:00.001-07:002019-05-31T12:41:23.645-07:00Quick 2019 resources for parentsA quick list of resources for an elementary school parent<br /><br /><br /><div class="MsoNormal"><span style="font-family: inherit;">Resources depending on prep time:</span></div><ul style="margin-top: 0in;" type="disc"><li><span style="background-color: white; font-family: inherit; text-indent: 0in;">Grab-and-go</span></li><br /><ul><li>NRICH activities and games: <span style="background-color: white; font-family: inherit;"><a href="https://nrich.maths.org/primary">https://nrich.maths.org/primary</a></span></li><li>Peter Liljedahl numeracy tasks:<span style="background-color: white; font-family: inherit;"> <a href="http://www.peterliljedahl.com/teachers/numeracy-tasks">http://www.peterliljedahl.com/teachers/numeracy-tasks</a>. Despite the dull name, the activities are good for catalyzing interesting discussions. His card trick videos were really good with my sons: <a href="http://www.peterliljedahl.com/teachers/card-tricks">http://www.peterliljedahl.com/teachers/card-tricks</a></span></li><li>We’ve gotten a lot of mileage from the Beast Academy books: <span style="background-color: white; font-family: inherit;"><a href="https://beastacademy.com/books">https://beastacademy.com/books</a>. The problems are thoughtful and there’s usually one or two problems in each block that is a good inspiration for a conversation or deeper exploration.</span></li></ul><li>More prep time, in increasing order of advanced time required</li><ul><li><a href="http://mathpickle.com/" style="background-color: white; font-family: inherit;">http://mathpickle.com/</a><span style="background-color: white; font-family: inherit;">: the puzzles and games are very good. </span></li><li>Math Teachers at Play blog carnivals: <a href="https://denisegaskins.com/mtap/" style="background-color: white; font-family: inherit;">https://denisegaskins.com/mtap/</a><span style="background-color: white; font-family: inherit;">. Variable amounts of prep time, but usually there’s at least one activity that is ripe for exploration, may take a bit of reading through the carnival to find a suitable one.</span></li><li><u style="background-color: white; font-family: inherit;"><span style="color: #4472c4;"><a href="https://mikesmathpage.wordpress.com/"><span style="color: #4472c4;">Mike's Lawler's blog</span></a></span></u><span style="background-color: white; font-family: inherit;">: wonderful collection of (mostly) videos of his family working through problems, puzzles and mathematical explorations. Because his kids are older, it will take a little time to find something you think is suited for your son and then a bit to organize the activity.</span></li><li><span style="background-color: white;"><a href="https://www.georgiastandards.org/Georgia-Standards/Pages/Math.aspx">Georgia State math standards</a>: Despite the name "standards," these documents have a full curriculum with a collection of really great activities. As with any full curriculum, not everything is a complete winner, but there are enough gems. Also, this is probably the best resource for finding material to complement a kid's weaknesses.</span></li></ul></ul><div class="MsoNormal"><span style="font-family: inherit;">This page is still the most comprehensive list of our favorite resources: <a href="http://3jlearneng.blogspot.com/p/favorite-educational-resources.html">http://3jlearneng.blogspot.com/p/favorite-educational-resources.html</a><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;">Unfortunately, it is a little dated as I haven’t really been maintaining this blog in the last 2 years.</span></div><br />JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-20678091692843776302018-08-03T07:32:00.002-07:002018-08-03T07:32:58.449-07:00A context investigationNote: I drafted this a while ago and never finalized the post. Reading it again, it seems fine and maybe interesting without additional work, so I'm publishing it.<br /><br /><h2>Fake Math Models</h2><br />Robert Kaplinsky wrote a note recently discussing <a href="http://robertkaplinsky.com/beware-fake-math-modeling-problems/" target="_blank">fake math models and unnecessary context</a>. This prompted an activity with the kids.<br /><br />This issue seems to have come up a lot recently, so I've noticed a pattern: I really hate bad contexts.<br /><br />Robert wrote: "it looks like the context is completely unnecessary to do all of the problems."<br />I would go farther: this context is harmful. The context creates a conflict between the specific new material (rational vs irrational numbers) and other important concepts (measurement and measurement error). Subtly, we are discouraging students from<br />(a) forming connections across topics. For my taste, surprising connections has to be one of the most beautiful and delightful aspects of math.<br />(b) using all of their ideas and creativity to understand a challenge.<br />(c) putting new mathematical ideas into a broader mathematical context (maybe I'm just repeating point a?)<br /><br />I admit that the example only touches on these points lightly, but I suspect the accumulated weight over the course of a school math education is substantial.<br /><br />If I were full-time in a classroom with a textbook, I'd be tempted to use it as follows:<br />1. create censored versions of all problems and examples (as you did)<br />2. work through the questions with the kids<br />3. Ask them what context they think the publishers originally included and why<br />4. show the published version<br />5. discuss (does the published version relate to the math, does it help them understand, does it add confusion, does it conflict with something they know, etc)JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-41246513010359927892018-07-24T20:55:00.001-07:002018-07-24T20:55:33.580-07:00Playful Math Education Carnival #119<div style="background-color: white;"><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;">Welcome to <span style="color: #303030; font-family: "PT Sans", sans-serif;">Playful Math Education Carnival</span> #119! Just to be clear, that exclamation is to express excitement, not factorial. Fortunately, you will have a bit of time before there's any danger of confusing this post with the edition (119 factorial).</div><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;"><br /></div><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;">Anyway, only the very coolest folks get to handle a MTaP edition that can be written with a factorial. And I just realized how close (and yet how far) I was to such glory.</div><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;"><br /></div><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;"><b>119 Fun facts</b></div><br /><ul><li><span style="font-family: calibri, helvetica, sans-serif;">119 is the number to call for emergency services... in parts of Asia (<a href="https://en.wikipedia.org/wiki/119_(emergency_telephone_number)">wiki reference</a>).</span></li><li><span style="font-family: calibri, helvetica, sans-serif;">Of course, 119 backwards is 911 which is the US emergency services phone number</span></li><li><span style="font-family: calibri, helvetica, sans-serif;">119 is <i>aspiring, </i>the sequence formed by summing proper factors ends with a perfect number.</span></li><li><span style="font-family: calibri, helvetica, sans-serif;">119 isn't prime, but it almost feels like it</span></li><li><span style="font-family: calibri, helvetica, sans-serif;">119 = 7 x 17. I don't think products of consecutive primes ending in 7 has a name, but maybe it should?</span></li></ul><div><span style="font-family: calibri, helvetica, sans-serif;">Do you have other fun facts about 119? Please? Please?</span></div><div><br /></div><br /><h2 style="font-family: calibri, helvetica, sans-serif; font-size: 16px;">Dedication</h2></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">I'm saddened to note the passing of <span style="color: #444444; font-family: "arial" , sans-serif; font-size: 14px;"> Alexander Bogomolny this month, and I dedicate the edition of the carnival to him. The material he developed and made available on his site</span></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><a class="m_-5641341967216879287OWAAutoLink" data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.cut-the-knot.org/&source=gmail&ust=1532437263463000&usg=AFQjCNFv332vpFHxxFLkEVwPYI9kry1Fgw" href="https://www.cut-the-knot.org/" id="m_-5641341967216879287LPlnk332232" style="color: #1155cc;" target="_blank">https://www.cut-the-knot.org/</a> <wbr></wbr>is truly amazing and remains with us for our benefit.</div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /><br /></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><h2>Miscellaneous</h2>Aperiodical has an article from Benjamin Leis on the <a href="https://aperiodical.com/2018/07/one-fans-commentary-on-the-big-internet-math-off-so-far/">Big Internet Math-off</a>.</div><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;">Something James Propp wrote as part of the Big Internet Math-off: <a href="https://mathenchant.wordpress.com/2018/07/16/a-pair-of-shorts/">A pair of shorts</a>. </span><br /><br /><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><h2>Elementary</h2></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">I was reminded of the game of <a href="http://www.math.ucla.edu/~tom/Games/chomp">Chomp!</a> in <a href="https://mathtango.blogspot.com/">Shecky Riemann's linkfest</a> (most of which isn't elementary level, but worth investigating).<br /><br />Cathy O'Neil tells her <a href="https://www.bloomberg.com/view/articles/2018-07-21/how-dominoes-helped-make-me-a-mathematician">mathematician origin story</a>. I hope all our kids can have an empowering math experience like this.<br /><br />Discussion of a "square dancing" puzzle from Mike Lawler: <a href="https://mikesmathpage.wordpress.com/2018/07/21/sharing-a-problem-from-the-julia-robinson-math-festival-with-the-boys/">part 1</a> and <a href="https://mikesmathpage.wordpress.com/2018/07/22/the-square-problem-from-the-julia-robinson-math-festival-part-2/">part 2</a>. I think there is a lot more to explore here and hope some of you will write parts 3 and beyond... </div><br /><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">I always love game discussions. Set is a game you probably all know, but in case you don't here's an intro and a deeper analysis in the <a href="https://aperiodical.com/2018/07/the-mathematical-beauty-of-the-game-set/">Aperiodical</a>.</div><br style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;" /><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">Pat Ballew writes about <a href="http://pballew.blogspot.com/2018/07/why-divisible-by-eleven-rule-works-and.html">divisibility rules</a>. Pat also discusses a fun XKCD in <a href="http://pballew.blogspot.com/2018/07/prime-time-fun-revisited.html">prime time fun</a>.</div><div style="background-color: white;"><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;">I'm delighted at how this starts with something many take for granted (12 hour vs 24 hour time of day conventions) and then builds a fun exploration.</div><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;"><br /></div><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px;"><br /></div></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><h2>Middle school</h2>Have you been waiting for someone to write the perfect post giving you an introduction to tons of Desmos activities? Well, <a href="http://marybourassa.blogspot.com/2018/06/desmos-at-exeter-2018.html">Mary Bourrasa has done it for you</a>.<br /><br />Michael Pershan tweeted a pointer to a nice collection of logic puzzles on <a href="https://puzzling.stackexchange.com/questions/tagged/meta-knowledge+logical-deduction">puzzling stackexchange</a>.<br /><br /><br /></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /><div style="-webkit-text-stroke-width: 0px; background-color: white; color: black; font-family: calibri, helvetica, sans-serif; font-size: 16px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></div><br /><div style="-webkit-text-stroke-width: 0px; background-color: white; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><div style="font-family: calibri, helvetica, sans-serif; font-size: 16px; margin: 0px;">Denise Gaskins pointed out a past note about factor trees and some cute wordplay from<span> </span><a href="https://www.amazon.com/Math-Doesnt-Suck-Survive-Breaking/dp/0452289491">Danica McKellar's book</a>: <a href="https://denisegaskins.com/2010/01/14/prime-numbers-are-like-monkeys/">prime numbers are like monkeys</a>.</div><div style="margin: 0px;"><br /></div></div><br />This segues directly into a <a href="http://mymathclub.blogspot.com/2018/06/book-review-introduction-to-number.html">review of two number theory books </a>by Ben Leis (also the author of the Big Internet Math off post above) in which he discusses some other visualizations beyond factor trees: </div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><h2>High school</h2></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">Mr Honner: math around us and <a href="http://mrhonner.com/archives/19159">parallel lines cut congruent arcs on a circle in a nice picture</a>. </div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">Ben Orlin invents and illustrates a new adage that <a href="https://mathwithbaddrawings.com/2018/06/27/powers-great-and-small/">there are no puddles in mathematics, only oceans in disguise</a>. </div><br /><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><h2>More advanced</h2></div><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;">Mathematical theorems you had no idea existed because they are false: <a class="m_-5641341967216879287OWAAutoLink" data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.facebook.com/BestTheorems/&source=gmail&ust=1532437263464000&usg=AFQjCNGyOfpnrou_3PykRnm8cEeZ1MzLGg" href="https://www.facebook.com/BestTheorems/" id="m_-5641341967216879287LPlnk383216" style="color: #1155cc;" target="_blank">https://www.facebook.<wbr></wbr>com/BestTheorems/</a></div><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;">Have fun finding counterexamples. Also, link disproves the conjecture that there is nothing worthwhile on facebook.</span><br /><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;"><br /></span><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;">The Scientific American Blog has been running these columns on "my favorite theorem." Go back and take a look (I think this was their first one): <a href="https://blogs.scientificamerican.com/roots-of-unity/amie-wilkinsons-favorite-theorem/">Amie Wilkinson's favorite theorem</a>.</span><br /><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"></div><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;"><a href="https://mathpresso.wordpress.com/2018/07/10/the-fields-medal-should-return-to-its-roots/">A fascinating discussion of the Fields' Medal </a>and some ideas about what it should be supporting. </span><br /><span style="background-color: white; font-family: "calibri" , "helvetica" , sans-serif; font-size: 16px;"><br /></span><div style="background-color: white; font-family: Calibri, Helvetica, sans-serif; font-size: 16px;"><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-A3Va5i7KDwg/W1f0b3J9ZkI/AAAAAAAACEw/cR0gwe8XqpYQrK4WmFiCGlLNfUQHquFcgCLcBGAs/s1600/IMG_1100.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://3.bp.blogspot.com/-A3Va5i7KDwg/W1f0b3J9ZkI/AAAAAAAACEw/cR0gwe8XqpYQrK4WmFiCGlLNfUQHquFcgCLcBGAs/s320/IMG_1100.JPG" width="240" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">What was the score? Maybe the sum of scores was 119?</td></tr></tbody></table><br /></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-60895705144931570812018-05-30T06:47:00.000-07:002018-05-30T06:47:20.361-07:00Ambiguity in math classMath class is a special place. We've talked before about some of the special assumptions that are based into that context: teachers pose questions, students answer questions, all questions have answers, questions include all the necessary information, answers are usually "nice," problems can be answered with the tools students have (just been) taught, diagrams are indicative while the underlying true forms are perfect, etc.<br /><br />Of course, not all math classes make these assumptions or leave them implicit, or are constant about which ones are in force, etc.<br /><br />In this post, I want to pick up a thread related to the "one true answer" myth: problems that have multiple interpretations.<br /><br /><b>Example</b><br />You are driving from your house to a soccer tournament. The distance is 120 miles. For half of the trip, you drive 60 mph. For the other half, you drive 30 mph. What is your average speed over the whole drive?<br /><br /><b>Where's the ambiguity?</b><br />For the teacher who poses this problem, there is no confusion. Obviously, students are meant to calculate that it takes 1 hour to drive the first 60 miles and 2 hours for the second 60 miles. That means it took 3 hours for 120 miles, or 40 mph average speed.<br /><br />The catch: what does "half of the trip" mean? As an alternative, it could mean half the time of the drive. If that feels contrived, consider the following natural statements about travel measured in time instead of distance:<br /><br /><ul><li>"The drive took 3 hours; we stopped for a snack half-way." In this case, time and distance are equally natural in normal conversation.</li><li>"The flight took 6 hours; I read half the time and slept the rest." In this case, time is the more common metric, but it wouldn't be considered unusual for someone to talk about the distance they flew.</li><li>"We were gone for 2 weeks, half at the beach, half visiting our cousins." Here, time is the natural metric, while it would seem strange to focus on distance. However, a vacation spent hiking the Appalachian trail or cycling across country would shift the balance back to distance.</li></ul><br /><h3><b>Sources of ambiguity</b></h3>I came up with four potential sources of ambiguity in math questions:<br /><h4>Things that can be measured in multiple ways. </h4>This extends the idea from “half a trip” ambiguity about distance or time. J1 and I had a discussion a couple of weeks ago where we measured chocolate bars and cookies using three different metrics: mass, cost, utils. For example, which is more:<br /><br /><ul><li>100 grams of chocolate that costs $2.00 and you value at 100 utils</li><li>80 grams of fresh baked sugar cookie that costs $2.50 and you value at 90 utils</li></ul><br />In business, it is common to have to deal with the ambiguity of whether “stuff” is measured in physical amounts or monetary value.<br /><h4>Pronoun ambiguity</h4><div class="MsoPlainText">For example: Ellis had 10 strawberries. Ellis gave 4 to his father and he ate 2. How many does he have now?</div><div class="MsoPlainText"><br /></div><div class="MsoPlainText">Who is meant by each occurrence of "he"? In each case, it could mean either Ellis or the father which leads to 3 distinct answers: 6, 4, or 2.</div><div class="MsoPlainText"><br /></div><div class="MsoPlainText">I accept that this is an example of bad English, but we're in math class and never claimed to be masters of language (did we?)</div><h4>Tense ambiguity</h4><div class="MsoPlainText">In the prior story it could be that giving the strawberries away and eating them happened before the state where Ellis had 10. Let's add some extra story context to make this alternative more clear:</div><div class="MsoPlainText">Ellis <i>still </i>had 10 strawberries. <i>He bought a pack of 16,</i> <i>but </i>Ellis gave 4 to his father and he ate 2.</div><div class="MsoPlainText"><br /></div><div class="MsoPlainText">I think this alternative interpretation is more of a stretch, but I've seen cases where the uncertainty about when things were happening is more natural. </div><div class="MsoPlainText"><o:p></o:p></div><h4>Assumption of scalability</h4><div class="MsoPlainText"><o:p></o:p></div><div class="MsoPlainText">Joe can bake 2 cookies in 20 minutes. How long does it take him to bake 4 cookies? 400 cookies? 4 million cookies? 4 quadrillion cookies?<o:p></o:p></div><div class="MsoPlainText"><br /></div><div class="MsoPlainText">I saw one math class question that involved writing books, a task which is very unlikely to happen at a constant rate.</div><div class="MsoPlainText"><br /></div><div class="MsoPlainText"><o:p></o:p></div><h3>Your challenges</h3><div><ol><li>Find other sources of ambiguity that can infect (or add spice to) math class problems.</li><li>For N a positive integer, create a puzzle that has N distinct solutions based on (reasonable) alternative interpretations.</li></ol></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-8523697103615496252018-02-20T19:13:00.000-08:002018-02-20T19:13:18.685-08:00What is your function? More excuses to delay bedtime<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-BKbVyP-dD1E/WozaSRPq3fI/AAAAAAAACA8/3HQzKPdHW6Q8cwvfWcWs6T-v6nHYg-AjwCLcBGAs/s1600/CJFunction.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="305" data-original-width="536" height="182" src="https://4.bp.blogspot.com/-BKbVyP-dD1E/WozaSRPq3fI/AAAAAAAACA8/3HQzKPdHW6Q8cwvfWcWs6T-v6nHYg-AjwCLcBGAs/s320/CJFunction.png" width="320" /></a></div><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J1 (5th grader, looking for an excuse to stay up): What are you working on?</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J0: I'm writing a review of a book.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J1: The one we got from math circle (<a href="http://store.doverpublications.com/0486256375.html">Martin Gardner's Perplexing Puzzlers and Tantalizing Teasers</a>)? </span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J0: No, the one about Funvillians.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-P5TRSEYTTLM/Wozb7msSFTI/AAAAAAAACBI/k6c_6DVTG187U6mec8oprXECUNkXt8OIgCLcBGAs/s1600/FV%2Badventures.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="471" data-original-width="311" height="400" src="https://4.bp.blogspot.com/-P5TRSEYTTLM/Wozb7msSFTI/AAAAAAAACBI/k6c_6DVTG187U6mec8oprXECUNkXt8OIgCLcBGAs/s400/FV%2Badventures.png" width="263" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">From <a href="https://naturalmath.com/funville/">Natural Math</a>!</td></tr></tbody></table><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J1: Tell them that it was fun!</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J0: You really enjoyed reading it. I'll make sure to mention that. I was thinking that we should have used it as an inspiration to make our own adventures.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J1: You mean, like creating new characters with their own powers? We could have heroes who control fire and ice, some others that can go forward and backward in time.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J0: Is that how the Funvillian powers worked? I thought they needed to have inputs. For example Marge's power only works on two exactly identical objects.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J1: Sure, the current time is an input and the output is the time in 5 minutes. Or another one can do the reverse.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">[pause, maybe he's starting to go to sleep?]</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">J1: Or... maybe we could make up some adventures where the Funvillians from the story have to solve their own challenges. They could meet some villains... not Villians! (laughing)</span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J1: The one who can duplicate things ... what if that power could be used on people? After they were copied, would they all have to do the same things? For example, if I were copied and I raised my arm, would the other one have to raise his arm, too? Could they think different thoughts?</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J0: Well, when they copied two toys, they could play with the toys separately. The toys didn't have to do identical things.</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J1: Oh! But what if they were changed a tiny amount? Would they still be considered identical and could they get reduced down to one copy?</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J0: I don't know. Where do you think the powers come from?</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J1: maybe from living in their magical land. Probably when they have spent enough time there, a power develops.</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J0: There, so that's what I'm going to write about. Thanks!</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="background-color: white; font-size: 12.8px;">J1: Remember to tell them it was fun!</span></span><br /><br /><hr /><span style="color: #222222; font-family: arial, sans-serif;"><span style="background-color: white; font-size: 12.8px;">On Fridays for the last several months, my fifth grader and I have been spending 2 hours in the evening doing math together. By that time of the week, I'm not always feeling energetic enough to properly plan an activity or exploration. Looking to give myself a break, last week, I brought Sasha Fradkin's book <a href="https://www.amazon.com/Funville-Adventures-Fradkin/dp/1945899026/ref=sr_1_1?ie=UTF8&qid=1519170559&sr=8-1&keywords=funville+adventures">Funville Adventures</a> for J1 to read during the session.</span></span><br /><span style="color: #222222; font-family: arial, sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: arial, sans-serif;"><span style="background-color: white; font-size: 12.8px;">He was engrossed and finished it with some amount of time to spare. Maybe 90 minutes of reading, leaving us 30 minutes to discuss. He had read the addendum, so was already primed for talking about functions. In addition, he still remembered past conversations about "function machines" and programming functions. Using the characters as references, though, he found it much more intuitive to understand invertible and non-invertible functions. We talked about examples of arithmetic functions that were similar to different characters' powers and had fun giving examples of what would happen if different characters used their powers in succession.</span></span><br /><span style="color: #222222; font-family: arial, sans-serif;"><span style="background-color: white; font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: arial, sans-serif;"><span style="background-color: white; font-size: 12.8px;">The experience, so far, suggests that this is a helpful model for understanding functions, more human and vivid than what we'd previously done with function machines.</span></span><br /><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></span><span style="background-color: white; color: #222222; font-family: arial, sans-serif;"><span style="font-size: large;">And remember, it was fun!</span></span><br /><span style="background-color: white; color: #222222; font-family: arial, sans-serif;"><span style="font-size: xx-small;">(now go to bed!)</span></span>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-7337609466259782942017-10-24T05:41:00.004-07:002017-10-24T05:49:47.338-07:00Math Teachers at Play Blog Carnival #113Welcome everyone to the 113th Math Teachers at Play blog carnival! As usual, I'm lucky to have the best month to curate. 113 is the prime MTaP because:<br /><ul><li>113 is prime</li><li>all permutations of the digits are prime</li><li>all 2 digit subsets of the digits are prime</li><li>the product of the digits is prime</li><li>the sum of the digits is prime</li><li>113<sub>(4)</sub> (113 in base 4, the smallest base that is sensible) is also prime!</li><li>2<sup>113</sup> - 2 is divisible by 113. Wow! (see <a href="https://blogs.scientificamerican.com/roots-of-unity/jordan-ellenbergs-favorite-theorem/" target="_blank">Jordan Ellenberg's Favorite Theorem</a>)</li></ul><h2>October is for Play</h2>October is a perfect month to talk about mathy play because of:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://nakedmeeple.files.wordpress.com/2016/09/spiel2.jpg?w=604&h=270&crop=1" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="270" data-original-width="604" height="178" src="https://nakedmeeple.files.wordpress.com/2016/09/spiel2.jpg?w=604&h=270&crop=1" width="400" /></a></div>Later this week is the largest international games convention in Essen, Germany. I'm jealous of any of you who get to go. For the rest of us, I've collected a bunch of great math games and explorations to keep us happy.<br /><br />Before we move off Spiel, though, take a look at the logo above again.<br /><div style="text-align: center;"><span style="font-size: large;">What do you notice?</span></div><div style="text-align: center;"><span style="font-size: large;">What do you wonder?</span></div><div style="text-align: center;"><br /></div>Click this button to compare with another version of the logo:<br /><div id="spoiler2" style="display: none;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-T0rw1UYSlGc/WeSPswvS7MI/AAAAAAAAB-E/Br23pImII5sMVcfYkdgldMmTMozg8ejwQCLcBGAs/s1600/spiel%2Btangram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="155" data-original-width="435" height="114" src="https://2.bp.blogspot.com/-T0rw1UYSlGc/WeSPswvS7MI/AAAAAAAAB-E/Br23pImII5sMVcfYkdgldMmTMozg8ejwQCLcBGAs/s320/spiel%2Btangram.png" width="320" /></a></div></div><button onclick="if(document.getElementById('spoiler2') .style.display=='none') {document.getElementById('spoiler2') .style.display=''}else{document.getElementById('spoiler2') .style.display='none'}" title="Click to show alternative logo" type="button">Show alternative logo</button><br /><br /><h2>Elementary</h2><div style="background-color: white;"><div style="color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">In the spirit of Malke Rosenfeld's <a href="http://www.mathinyourfeet.com/" target="_blank">Math in Your Feet</a>, Mrs. Miracle's post about <a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://www.mrsmiraclesmusicroom.com/2017/10/beat-passing-games.html&source=gmail&ust=1508805784293000&usg=AFQjCNGPcw2QGwaJEk-8oSwv5laTjU2xWA" href="http://www.mrsmiraclesmusicroom.com/2017/10/beat-passing-games.html" style="color: #1155cc; font-size: 12.8px;" target="_blank">beat passing games</a> can inspire a whole-body exploration of patterns. I like expanding beyond visual patterns and the fact even very small children can create their own beat pattern.</div></div><div style="background-color: white;"><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;">For some reason, this old Christopher Danielson post resurfaced on my RSS reader. While it </span></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">is an old one, I hadn't seen it before, so maybe you missed it too or will appreciate reading it again: <a href="https://talkingmathwithkids.com/2013/08/17/armholes/">Armholes</a>. Maybe it is easier to be patient while waiting for the kids to get dressed if we are also exploring math at the same time?</span></div><div style="background-color: white;"><br /></div><div style="background-color: white;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://i.pinimg.com/736x/a3/5a/06/a35a060fcaac29ed8c05913fe329ee4f--loose-knit-sweaters-oversized-knit-sweater.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="797" data-original-width="583" height="320" src="https://i.pinimg.com/736x/a3/5a/06/a35a060fcaac29ed8c05913fe329ee4f--loose-knit-sweaters-oversized-knit-sweater.jpg" width="234" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">How many holes?</td></tr></tbody></table><br /><br /></div><div style="background-color: white;"><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;">An online math competition for elementary kids: <a href="https://in.bricsmath.com/">BRICS Math</a>. I like using math competition questions as a jumping off point for further conversations and explorations. Sometimes the questions have natural extensions (what if we changed this number?) and other times we just talk about what the kids found interesting about the question or what it made them think about.</span></span></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white;"><div style="background-color: white; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; widows: 2;"><div style="-webkit-text-stroke-width: 0px; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; margin: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">Iva Sallay (who has hosted the last edition of MTaP) makes a <a href="https://findthefactors.com/2017/10/15/913-haunted-ten-frame-house-for-ten-timid-ghosts/">Halloween 10 Frame</a> (just in time!)</div><div style="-webkit-text-stroke-width: 0px; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; margin: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"><br /></div><div style="-webkit-text-stroke-width: 0px; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; margin: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">Here are some wonderful images and gifs from Gábor Damásdi. They could be a good prompt for Notice & wonder for young kids and older ones:</div><div style="margin: 0px;"></div><ul><li><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;"><a href="http://szimmetria-airtemmizs.tumblr.com/post/166356553843/sketch-i-am-working-on-fountain-the-idea-came">Fountain</a></span></span></li><li><a href="http://szimmetria-airtemmizs.tumblr.com/post/166281228373/hpgross-mathhombre-szimmetria-airtemmizs">A tesselation pattern</a></li><li><a href="http://szimmetria-airtemmizs.tumblr.com/post/166224732088/islamic-pattern-2-i-have-extended-the-previous">An islamic pattern</a></li></ul><br /><div style="-webkit-text-stroke-width: 0px; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px;"><div style="font-size: 12.8px;">AO Fradkin talks about a tricky game that helps develop mathematical language: <a href="https://aofradkin.wordpress.com/2017/10/13/a-figure-with-pointy-things-and-a-line-a-line-and-a-line/">A figure with pointy things...</a><br />This question: why do we bother with defined terms and mathematical language fits nicely with Chasing Number Sense's exploration of the definition of a polygon: <a href="https://telannannalet.wordpress.com/2017/10/22/polygon-is-a-shape-that-is-really-big/">Polygon is a shape that is really big</a>.<br /><br />Denise Gaskins, the wonderful unifying force behind this blog carnival, reminds us of <a href="https://denisegaskins.com/2008/09/22/things-to-do-hundred-chart/">30+ things to do with a 100 chart</a>. I have one more to add: our family first learned to play<a href="http://3jlearneng.blogspot.com/2017/01/100-board-go.html"> Go on a 100 chart </a>with some small blue cubes and bananagram tiles, before we had a chance to buy our first dedicated board. Here's an old snap from the beginning of the year:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-vc2imBlqJDY/WHm6KUUyqrI/AAAAAAAABz4/absV2aGol5gjbFZI__sC55uCYGsjZeDGwCLcB/s320/IMG_0862.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="320" data-original-width="240" src="https://2.bp.blogspot.com/-vc2imBlqJDY/WHm6KUUyqrI/AAAAAAAABz4/absV2aGol5gjbFZI__sC55uCYGsjZeDGwCLcB/s320/IMG_0862.JPG" /></a></div><br /></div></div></div></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><h2>Middle School</h2>If you like chained fraction puzzles (we do!) and you like thinking about concrete manipulatives (we do!!) then you'll enjoy this post from Bridget Dunbar: <a href="https://elsdunbar.wordpress.com/2017/10/14/thinking-in-the-concretemanipulatives-in-math-class/">Thinking in th Concrete</a>.<br /><br />Another post from Iva Sallay uses candy to teach equation solving: <a href="https://findthefactors.com/2017/10/19/917-how-to-solve-for-x-with-candy/">Solve for X with candy</a>. With Iva's help, we're certainly ready for a mathy Halloween.<br /><br />Presh Talwalkar at Mind Your Decisions occasionally posts viral puzzles with some nice explanations. I enjoyed this one (<a href="https://mindyourdecisions.com/blog/2017/10/15/the-octagon-in-a-parallelogram-can-you-solve-this-8th-grade-geometry-problem-from-russia/">octagon in a paralellogram</a>) because it fit with a problem solving strategy we've been practicing recently: test a special case. Here, we tried starting with a square, then discussed whether that was really a "special case" or fit the general situation.</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br />Mike Lawler, as usual, has some fun posts, this time I picked out his videos talking about <a href="https://mikesmathpage.wordpress.com/2017/10/14/sharing-tim-gowerss-nontransitive-dice-talk-with-kids/">Tim Gowers's intransitive dice</a>.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Intransitive_dice_2.svg/360px-Intransitive_dice_2.svg.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="122" data-original-width="360" height="135" src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Intransitive_dice_2.svg/360px-Intransitive_dice_2.svg.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Different from Tim Gowers's dice?</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"></div><br />Huge jars of coins are wonderful, for so many reasons. Kristen (Mind of an April Fool) shares a fun 3-act lesson: <a href="https://themindofanaprilfool.com/2017/10/09/sassy-cents-a-3-act-lesson/">Sassy Cents</a>. Our family has gotten a lot of mileage out of doing notice and wonder at home with similar 3-act lessons.</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br />Jim Propp contributed to the excitement of <a href="https://gmw.globalmathproject.org/?gclid=CjwKCAjw3_HOBRBaEiwAvLBbosS0HMgjrY55AHiphcd8t0PCKT18g9-I560NTMLaOQoqvJnPBYKPmxoCqVwQAvD_BwE">Global Math Week </a>with a sort of <a href="https://mathenchant.wordpress.com/2017/10/13/the-global-roots-of-exploding-dots/">History of Exploding Dots</a>. I have an especially warm feeling for this story because he includes mention of the "minicomputer" idea created by Frederique Papy. These minicomputers figured prominently in my own elementary math education.</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><h2>High School/More advanced</h2></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><span style="font-size: 12.8px;">Continuing with the theme above around language, definitions, precision and math, Mr Orr gives us <a href="http://mrorr-isageek.com/3-new-desmos-activities-talkers-drawers/">3 Desmos Activities for Talkers & Drawers</a>.</span><br /><span style="font-size: 12.8px;"><br /></span><span style="font-size: 12.8px;">Curiousa Mathematica shares a Putnam exam question that is actually very accessible: </span><a href="http://curiosamathematica.tumblr.com/post/166254593472/eka-mark-theres-a-problem-i-really-like-on-the" style="font-size: 12.8px;">Spots on a ball</a><span style="font-size: 12.8px;">. Try to think about it before reading the solution.</span><br /><br />Patrick Honner talks about some of the math related to gerrymandering in <a href="http://mrhonner.com/archives/17946">Wasted Votes</a>. In his discussion, he describes a game that sounds very much like Mathpickle's <a href="http://mathpickle.com/project/a-little-bit-of-aggression/">A Little Bit of Aggression</a> (pdf <a href="http://mathpickle.com/wp-content/uploads/2016/01/TpT-A-little-bit-of-aggression-2016.pdf">here</a>.)<br /><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a href="https://solvemymaths.com/">SolveMyMaths has done a series this month on trig identities</a>. These build step-by-step pictures to help understand what is going on, for example, in the angle addition formulas. Take a look (they are easier to understand than this final step picture, but it is one of my favorites):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://solvemymaths.files.wordpress.com/2017/10/trigrep4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="446" data-original-width="542" height="526" src="https://solvemymaths.files.wordpress.com/2017/10/trigrep4.jpg" width="640" /></a></div><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><h2>Teaching Resources</h2><div style="color: black; font-family: "Times New Roman"; font-size: medium;"><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">For all of us who sometimes have to find a math curriculum for our kids, David Wees has created his checklist of necessary characteristics: <a href="http://davidwees.com/content/questions-about-curriculum/">Questions about Curriculum</a>.</span><br /><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"><br /></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;">What makes a good school? </span><a href="http://janemouse.ru/good-schools" style="font-family: arial, sans-serif; font-size: 12.8px;">Jane Mouse (in russian)</a><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.8px;"> explains that there is no "best" school. For those of us who teach our own kids, one interpretation is that we should try to expose the kids to a variety of modes and styles. Also, what is working now might change over time.</span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;"><br /></span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;">Sam Shah offers a number of hacks for making <a href="https://samjshah.com/2017/10/22/the-diy-math-curriculum-simple-tricks-to-make-creating-your-own-material-feel-less-onerous/">your own material</a>. My personal recommendation is for you to try this out with your kids: make problems together and discuss the process. What makes a good question? What makes a hard vs an easy question? Can you create problems with only one, more than one, or no answers?</span></span><br /><span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: 12.8px;"><br /></span></span></div>Resourceaholic (Jo) has, you guessed it, a presentation on resources: <a href="http://www.resourceaholic.com/2017/10/mathsresources.html">Power of Six presentation</a>. Be sure to take a look at Jo's <a href="http://www.resourceaholic.com/p/resource-library.html">Resource Library </a>for ideas when you need secondary school material.<br /><br /><h3><b>I'm sure there are a lot of other great posts with families and teachers sharing their math games and explorations. Please add comments to let me know about your favorites from the month (or older ones)!</b></h3></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-74776052283970183242017-07-28T11:51:00.003-07:002017-07-28T11:51:41.650-07:00math recommendations for a 3 year oldI was recently asked for suggestions by a parent of a 3 year old.<br /><br /><div class="MsoNormal">There are a lot of different resources I could suggest, but they really depend on the child and the parents. The main question for customization is about the parents: what are their starting assumptions about math/math learning and how much do they want to engage on selecting/planning activities?<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">For example, if a parent doesn't really get the growth mindset, I would advise a heavy dose of Jo Boaler. If the parent wants open explorations and can build their own specific tasks, maybe the Vi Hart videos are good inspiration.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">That aside, there are a few resources/products good enough that I’m willing to give blanket recommendations:<o:p></o:p></div><div class="MsoNormal"><br /></div><ol start="1" style="margin-top: 0in;" type="1"><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level1 lfo1;">Lots of tools for measuring. Playing with measuring has so many benefits, I can’t list them all, but some of the highlights are (a) seeing math and numbers all around us, (b) tactile engagement, (c) inherent process of comparison, and (d) natural connection with language as the kids and parents talk about what they are measuring/why. The links I've provided just show examples, I am not necessarily recommending them over other versions.<o:p></o:p></li><ol start="1" style="margin-top: 0in;" type="a"><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;"><a href="https://www.bedbathandbeyond.com/store/product/betty-crocker-5-piece-measuring-cup-set-in-white/1016809136?skuId=16809136&mcid=PS_googlepla_nonbrand_furniture_online&product_id=16809136&adtype=pla_multichannel&product_channel=online&adpos=1o1&creative=43742643829&device=c&matchtype=&network=g&gclid=Cj0KCQjwwevLBRCGARIsAKnAJvfa_3I5TtFLSppgoGRDoMBbjLYqZXiivtvHhMZw8QVPgf-bD43tsFYaAoL_EALw_wcB">Set of plastic measuring cups (imperial units and fractions)</a><o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;"><a href="https://www.walmart.com/ip/Baumgartens-Tape-Measure-5-Assorted-Colors/13432597?wmlspartner=wlpa&selectedSellerId=0&adid=22222222227000324720&wl0=&wl1=g&wl2=c&wl3=40941347672&wl4=pla-78879037352&wl5=9061324&wl6=&wl7=&wl8=&wl9=pla_with_promotion&wl10=8175035&wl11=online&wl12=13432597&wl13=&veh=sem#about-item">Tape measure</a> (we just used standard adult tape measures, but as a recommendation, you need to be careful about tape measures that have fast return springs for cutting or catching small fingers)<o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;"><a href="https://www.hogentogler.com/ohaus/80410-00-mechanical-balance.asp?gclid=Cj0KCQjwwevLBRCGARIsAKnAJvfa11vT6tmU8fPSYC-LGCowvFXTTp_l0-o9f2xMmA56elIUWVrMQ7kaAsUOEALw_wcB">Balance scale</a> and set of <a href="http://www.hand2mind.com/item/metric-weight-set-plastic-set-of-58/1320?gclid=Cj0KCQjwwevLBRCGARIsAKnAJvfKiCKfvuHsdyq7YIbudFIwUTC7dFcAvy28Cxy8cxxlGJ9TOIQtb9UaAjIXEALw_wcB&gclsrc=aw.ds">standard weights</a> (<a href="http://www.hand2mind.com/item/math-balance/1140?gclid=Cj0KCQjwwevLBRCGARIsAKnAJvdyCL68Z_l3bT8bkQtYaF0fw4KbYLqubdQexuF3zs4OAw91CQSRVrMaAmMFEALw_wcB&gclsrc=aw.ds">this math balance</a> is a good option and one we bought)<o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;">Timer (<a href="https://www.trymezone.com/index.php?route=product/product&product_id=582">we liked this one</a>)<o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;">For older kids, a step counter, GPS wrist-watch showing speed, thermometer, pH meter, electricity meter are all interesting additional measuring devices.<o:p></o:p></li></ol><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level1 lfo1;">Talking Math with your Kids: <o:p></o:p></li><ol start="1" style="margin-top: 0in;" type="a"><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;"><a href="https://www.amazon.com/Talking-Math-Your-Kids-ebook/dp/B00EYSZH8E/ref=sr_1_2?ie=UTF8&qid=1378483262&sr=8-2&keywords=talking+math+with+your+kids">E-book</a><o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;"><a href="https://talkingmathwithkids.com/2013/08/page/3/">Blog</a>. I recommend reading all the posts, I think they are a superset of the material in the e-book, so this is a better resource unless you want the “curated” highlights. <a href="https://talkingmathwithkids.com/tag/3-years-old/">This link goes directly to posts tagged 3 years old</a>.<o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level2 lfo1;">Tiling toys and shapes book in the <a href="http://talkingmathwithkids.squarespace.com/">TMWYK store</a>. I particularly like <b>Which on doesn’t belong? A better shapes book</b>.<o:p></o:p></li></ol><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level1 lfo1;"><a href="https://tabletopacademy.net/playful-math-books/">Denise Gaskin’s Playful Math books</a>: these talk about general habits and methods in an intro section, then specific activities (mostly games) in the rest of the book.<o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level1 lfo1;">I got a lot out of these storybooks (free to print) with my kids: <a href="http://stern.buffalostate.edu/CSMPProgram/Storybooks/bygrade.html">CSMP Math Storybooks</a>.<o:p></o:p></li><li class="MsoNormal" style="margin-left: 0in; mso-list: l0 level1 lfo1;">Standard gambling tools: playing cards and dice (I like <a href="https://www.target.com/p/pound-o-dice-assorted-game-dice-set/-/A-15702809?ref=tgt_adv_XS000000&AFID=google_pla_df&CPNG=PLA_Toys+Shopping&adgroup=SC_Toys&LID=700000001170770pgs&network=g&device=c&location=9061324&gclid=Cj0KCQjwwevLBRCGARIsAKnAJvdjLmC8YbzLiYH1GtARqfXZ18kxETo8YjHksYu7fSRosJnPzw01KJEaAra2EALw_wcB&gclsrc=aw.ds">pound-o-dice</a> for the assorted colors, sizes, shapes)<o:p></o:p></li></ol><div class="MsoNormal">There are some computer games/systems, a lot of board games, and mechanical puzzles, but the stuff above is where I think parents should start for young children.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>What do you think of my recommendations? Any additions you think are worth adding to make a top 10?</b></div><div class="MsoNormal"><o:p></o:p></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-86961158442958682582017-07-24T05:18:00.002-07:002017-07-24T05:18:43.838-07:00Math Teachers At Play Carnival #110 Summer Vacation Edition<br />Hello again math folks! I've been in the middle of a major transition, moving between Asia and North America, so haven't really had time to post recently. Putting together this month's carnival was a nice opportunity to see what everyone else has been writing about and get some new ideas!<br /><br />As you scan through the links I've highlighted, please don't get too fixated on the grade level splits. These are really approximate and I expect you will find worthwhile activities for all ages in every section.<br /><h2>In Memoriam: Maryam Mirzakhani</h2>On the 14th of July, <a href="https://en.wikipedia.org/wiki/Maryam_Mirzakhani" target="_blank">Maryam Mirzakhani</a> passed away. She was the first woman to win the Fields Medal. It would be wonderful if you could do some exploration in her honor this month. One of her areas of research was on pool tables. Here are some places to get an idea of the way mathematicians have been inspired by this game:<br /><br /><ul><li><a href="https://www.youtube.com/watch?v=4KHCuXN2F3I" target="_blank">Elliptical pool (from Numberphile)</a></li><li><a href="https://www.youtube.com/watch?v=xhj5er1k6GQ" target="_blank">The Illumination Problem (also Numberphile)</a></li><li><a href="https://docs.google.com/viewer?a=v&pid=sites&srcid=aXR5bS5vcmd8d3d3fGd4OmMyYzkzYmZjNDBkNDgyNQ" target="_blank">A snooker (problem 3 from the 2013 International Tournament of Young Mathematicians)</a></li><li><a href="http://mathworld.wolfram.com/AlhazensBilliardProblem.html" target="_blank">Wolfram MathWorld Discussion of Alhazen's Billiard Problem</a> that indicates the link with optics</li></ul>If you find other kid-friendly projects related to Mirzakhani's work, please tell me in the comments!<br /><h2>Some 110 facts</h2><div>This was the best number carnival to be able to host, because:</div><div><ul><li>110 = 10 * 11. That means it is <i>pronic</i>, the product of two consecutive integers.</li><li>110 looks suspiciously like a binary number. Binary 110 = decimal 6. Decimal 110 = Binary 1101110, which I like to read as 110 1 110</li><li>Because it has an odd number of 1s in its binary expansion, 110 is <i>odious</i></li><li>110 is a <i>Harshad</i> number because it is divisible by the sum of its digits</li><li>The element with atomic number 110 is Darmstadtium (Ds).</li><li>110 is the number of millions of dollars spent in <a href="https://www.nytimes.com/2017/05/26/arts/design/110-million-basquiat-painting-yusaku-maezawa.html" target="_blank">March for a Basquiat painting</a>, the highest amount paid at auction for a work by an American artist.</li></ul></div><h2>A number talks picture that caught my eye</h2>I'm not sure there is anything especially 110 about this picture, but there are a lot of mathematical questions to ask and things to observe here:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://assets.bwbx.io/images/users/iqjWHBFdfxIU/iSatgVt9wewk/v0/800x-1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="567" data-original-width="800" height="452" src="https://assets.bwbx.io/images/users/iqjWHBFdfxIU/iSatgVt9wewk/v0/800x-1.jpg" width="640" /></a></div><br />In a related vein, if you and your kids need some mesmerizing math gifs, take a look at <a href="http://szimmetria-airtemmizs.tumblr.com/" target="_blank">Symmetry</a>.<br /><h2>Elementary skills</h2>Denise Gaskins has written a lot to help parents engage playfully and mathematically with their kids. In this blog post, she has collected highlights that are great with young students and worth remembering for older ones, too: <a href="https://denisegaskins.com/2017/07/18/how-to-talk-math-with-your-kids/" target="_blank">How to Talk Math with Your Kids</a>.<br /><br />I love board games and think there is still tremendous value in the physical games that electronic versions miss. Here's an example from Sasha Fradkin, where <i>cleaning up after playing</i> gives us a chance to think about whether skip counting is just a chant or if the words mean something: <a href="https://aofradkin.wordpress.com/2017/06/25/skip-counting-or-word-skipping/" target="_blank">Skip counting or word skipping</a><br /><br />While she's at it, Sasha Fradkin also has a <a href="https://aofradkin.wordpress.com/2017/06/22/numicon-combining-geometry-and-arithmetic/" target="_blank">nice puzzle activity with Numicons</a>. I would think of this as a progression step toward tangram and other dissection puzzles.<br /><br />Which one doesn't belong is a math meme you should know already. If you don't, ask in the comments and I'll point you in the right direction. Christopher Danielson has recently introduced <a href="https://talkingmathwithkids.com/2017/07/06/which-poster-doesnt-belong/" target="_blank">Which Poster Doesn't Belong?</a> While you are visiting his blog, enjoy his story about <a href="https://talkingmathwithkids.com/2017/06/21/young-children-are-more-logical-than-you-think-they-are/" target="_blank">The Three Year Old Who is Not a Monster</a>.<br /><br /><a href="http://mathforum.org/blogs/pows/free-scenario-exploding-shapes-anyqs-wcydwt/" target="_blank">Exploding Shapes</a> is a catalyst for notice and wonder from The Math Forum. I really like this because here are many different directions to go and no single "right" answer. Also, let's give a cheer because it looks like this recent set of posts shows the math forum folks have returned to posting nice conversation starters.<br /><br /><a href="https://barefootmath.blog/2017/06/09/barefoot-math/" target="_blank">Swine on a Line</a> by Jim Propp is a nice game/puzzle that seems a great companion to James Tanton's <a href="http://gdaymath.com/courses/exploding-dots/" target="_blank">Exploding Dots</a>. Hmm, maybe July 4th inspired me to look for lots of explosions...?<br /><br /><h2>Middle school(ish)</h2>Rupesh Gesota starts with a nice puzzle and shows us how it was analyzed by several different students: <a href="http://rupeshgesota.blogspot.in/2017/06/one-puzzle-many-students-many-approaches.html" target="_blank">One Puzzle, Many Students, Many Approaches</a>. I particularly like how the introduction to the puzzle encourages us to think of different methods.<br /><br />Mike Lawler has done a huge number of really great explorations with his kids. Here are some recent projects with books from the Park City Mathematics Institute: <a href="https://mikesmathpage.wordpress.com/2017/07/13/playing-around-with-the-pcmi-books/" target="_blank">Playing Around</a>. If you haven't been following Mike and his kids, I really encourage you to go through his past posts.This blog is fantastic for great projects and connections with other resources.<br /><br />Curious Cheetah shows us several ways to <a href="http://curiouscheetah.com/BlogMath/three-and-a-half-methods-for-finding-square-roots/" target="_blank">calculate square roots</a>. I would say, like long division, the value isn't in memorizing the algorithms, but understanding how they work and using them to play with numbers.<br /><br />Manan Shah has a couple of nice summer explorations. The first is an excursion into the digits of prime numbers: <a href="http://mathmisery.com/wp/2017/06/25/summer-excursion-1-prime-numbers/" target="_blank">Prime Numbers</a>. The second is a coin flipping and gambling game to ponder during these warm vacation months: <a href="http://mathmisery.com/wp/2017/06/30/summer-excursion-2-would-you-play-this-coin-flip-game/" target="_blank">Summer Excursion Coin Flipping.</a><br /><br />There are other, problem-based, posts on Benjamin Leis's blog, but this one made me jealous of his recent purchase of the <a href="http://mymathclub.blogspot.com/2017/06/first-impressions-decade-of-berkeley.html" target="_blank">A Decade of the Berkeley Math Circle</a>.<br /><h2>High school/more advanced</h2><div><a href="http://seekecho.blogspot.co.uk/2017/07/thinking-inside-box.html" target="_blank">Thinking Inside the Box</a>, Simon Gregg takes a new look at a familiar shape, the cube. His comment about the exploration really nicely captures something that is beautiful about mathematical exploration: <b>"I came back to a familiar place from an unfamiliar starting place."</b><br /><br />A cute absolute value game now appears as a nicely animated game: <a href="http://simplifyingradicals2.blogspot.ca/2017/07/absolutely-valuable-is-digital.html" target="_blank">Absolute Value</a>. I think this is a nice simplification and implementation of the <a href="http://simplifyingradicals2.blogspot.com/2015/02/absolute-value-equation-game.html" target="_blank">original game</a>.<br /><br />There's an improv game where the players have to switch between movie genres. Film noire or "hard boiled detective" comes up every time. <a href="http://eatplaymath.blogspot.com/2017/06/the-case-of-missing-fractals.html?m=1" target="_blank">This TedEd video </a>could introduce fractals and this film genre at the same time.<br /><br />Michael Pershan puzzles over two measures of steepness in his trigonometry class: <a href="https://problemproblems.wordpress.com/2017/07/12/when-measures-of-steepness-disagree/" target="_blank">When Measures of Steepness Disagree</a>. I really like the questions he raises about how to use two different scales that measure the same concept, but are not linearly related.</div><div><br /></div><div>Also, Michael links to the <a href="http://www.avalanche.net.nz/education/Online-Avalanche-Course/--Angle.asp" target="_blank">New Zealand Avalanche Advisory</a>, with a nice graphic showing a case where the greatest danger of avalanche is in the middle of a slope range:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.avalanche.net.nz/Files/Dome-slope-angle2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://www.avalanche.net.nz/Files/Dome-slope-angle2.jpg" data-original-height="338" data-original-width="700" height="308" width="640" /></a></div><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div><br /><a href="https://divisbyzero.com/2017/07/04/the-math-behind-a-reflected-double-rainbow/" target="_blank">Dave Richeson breaks</a> down an impressive rainbow photo:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://divisbyzero.files.wordpress.com/2017/07/19679462_10155516697704100_9164282315828366468_o.jpg?w=1024" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="450" data-original-width="800" height="360" src="https://divisbyzero.files.wordpress.com/2017/07/19679462_10155516697704100_9164282315828366468_o.jpg?w=1024" width="640" /></a></div><br /><br />Fair sharing is a really interesting theme to motivate a lot of great math. Tanya Khovanova looks at a couple of fair sharing problems and strategies in <a href="http://blog.tanyakhovanova.com/2017/06/fair-share-sequences/" target="_blank">Fair share sequences</a>.<br /><br />Also, check out the sister carnival to this one: <a href="http://aperiodical.com/2017/07/carnival-of-mathematics-147/" target="_blank">The Carnival of Mathematics</a> over at <a href="http://aperiodical.com/" target="_blank">The Aperiodical</a>.<br /><h2>Techniques for teaching</h2>This post is an old classic, but I've been reminded of it because it is used in a workshop that I frequently attend: <a href="https://justmathness.wordpress.com/2014/09/30/building-student-motivation-using-reflections/" target="_blank">using student reflections</a>.<br /><br />Have you visited NRICH recently? No?!?! Go over now (<a href="http://nrich.maths.org/" target="_blank">here's the link</a>) and find a really cool activity to do with your kids. Seriously!<br /><br />Some tips on giving feedback: <a href="http://www.activelylearn.com/blog/2016/6/21/effective-feedback-for-deeper-learning" target="_blank">Effective feedback for deeper learning</a>.<br /><br />Using Desmos to check your work: <a href="https://saravanderwerf.com/2017/07/11/desmos-is-the-new-back-of-the-book-and-not-just-for-the-odd-problems/" target="_blank">Desmos is the new back of the book</a>.<br /><br />A thought piece on the modern role of teachers: <a href="http://www.defantri.com/2017/07/teachers-sow-thirst-for-learning.html" target="_blank">Teachers Sow Thirst for Learning</a>. If you can read Indonesian (which I can't) you may find some other interesting pieces here on math education.<br /><h2><b>A special announcement</b></h2><div><div class="separator" style="clear: both; text-align: left;">James Tanton is leading a project for a world-wide week of math this fall. Please take a look at the project page <a href="http://www.theglobalmathproject.org/gmw" target="_blank">Global Math Week</a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://www.theglobalmathproject.org/gmw" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="554" data-original-width="984" height="360" src="https://3.bp.blogspot.com/-GFVZrq-PNZI/WXXZ4cI4oxI/AAAAAAAAB78/Ts-_sLqIheIBACp8qAPsFihsoTJ3doGqwCLcBGAs/s640/GMP.png" width="640" /></a></div><b><br /></b></div><div class="separator" style="clear: both; text-align: center;"></div><div><b><br /></b></div><br /></div><div></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com2tag:blogger.com,1999:blog-5544661968326910027.post-13937240270117232542017-03-29T02:22:00.002-07:002017-03-29T02:22:17.196-07:00Some simple dice gamesJ3 and I played several simple games recently that I want to record. One of them, race to the top, is a variation of a more sophisticated game that can be used more generally.<br /><h1>Digits three in a row</h1><b>Materials:</b> a 100-board, 2d10, colored tiles <br /><b>Players</b>: 2<br /><b>Goal</b>: mark three spaces in a row<br /><b>Basic play:</b><br />Two players take turns rolling the two dice. On a player's turn, they form a 2 digit number using the two digits and then claim that space on the 100 board. If the space has previously been claimed, they lose their turn.<br /><br />The first player to claim three adjacent spaces in a line (horizontal, vertical, or diagonal) is the winner.<br /><br /><b>Variations:</b><br />(1) we used one dice marked 0-9 and another marked 00 - 90 (all multiples of 10) and then added the values to get our two digit numbers. This eliminated any player choice, but helped reinforce the idea that the value in the 10's digit is the number of tens.<br /><br />(2) the winning condition can be increased to require a line of 4 spaces<br /><br />(3) the winning condition can be changed to allow any three (or four) spaces that are colinear; these spaces would not have to be adjacent.<br /><br /><b>Notes:</b><br />This is a very simple game, especially the variation we played, but J3 found it fun. It was a useful exercise to practice locating the numbers on the 100 board.<br /><h1>Race to the top</h1><b>Materials:</b> An 11x 6 grid with one side labelled 2 to 12, 4d6, 3 tokens. Optional: 11 distinct tokens/small objects per player.<br /><b>Players</b>: 2 - 4<br /><b>Goal</b>: capture 3 columns.<br /><b>Basic play:</b><br />This game is fairly simple, but has resisted our attempts to succinctly summarize the rules. Here is an explanation as we play two turns.<br /><br />Here's our playing material, the three green squares are temporary markers:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-skaXVgd1T_8/WNt4uKmQ-iI/AAAAAAAAB4g/Dwez8IUE-Yw5n9mRN1Ckcw_9lDVned-fwCLcB/s1600/IMG_1038.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://3.bp.blogspot.com/-skaXVgd1T_8/WNt4uKmQ-iI/AAAAAAAAB4g/Dwez8IUE-Yw5n9mRN1Ckcw_9lDVned-fwCLcB/s400/IMG_1038.JPG" width="400" /></a></div><br />First player, J1 rolls two ones and two fours. With this roll, J1 could group them into two fives or a two and an eight:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-rX_Feep2LzI/WNt4tieup8I/AAAAAAAAB4Y/xBKW2OQjX7wmWQB0XvmyDJ97TdkjcZVDgCLcB/s1600/IMG_1039.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-rX_Feep2LzI/WNt4tieup8I/AAAAAAAAB4Y/xBKW2OQjX7wmWQB0XvmyDJ97TdkjcZVDgCLcB/s320/IMG_1039.JPG" width="240" /></a></div><br />J1 decides on two fives and puts a temporary marker on the second level of the 5's column. After you understand the rules consider whether this choice is better or worse than the 2 and 8.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-SibaPkB42Ao/WNt4tnfBIGI/AAAAAAAAB4c/KIMhkucUltgQawzrTMFZf24WlfX6kDXrgCLcB/s1600/IMG_1040.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://2.bp.blogspot.com/-SibaPkB42Ao/WNt4tnfBIGI/AAAAAAAAB4c/KIMhkucUltgQawzrTMFZf24WlfX6kDXrgCLcB/s400/IMG_1040.JPG" width="400" /></a></div><br />J1 chooses to continue rolling and gets 1, 1, 4, and 6. J1 groups these as a 5 and a 7, then moves the temporary marker in the 5's column up one level and adds a marker at the bottom of the 7's column.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-W2pafoqmY2k/WNt40Ju5EuI/AAAAAAAAB4k/f1XNKskrdzEqFUHW9MphMDn62TKlLv_PQCLcB/s1600/IMG_1041.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://1.bp.blogspot.com/-W2pafoqmY2k/WNt40Ju5EuI/AAAAAAAAB4k/f1XNKskrdzEqFUHW9MphMDn62TKlLv_PQCLcB/s400/IMG_1041.JPG" width="400" /></a></div><br />J1 rolls a third time, getting 1, 1, 2, and 6. The only option is to group these as 3 and 7, so J1 places a temporary marker in the 3's column and advances the marker in the 7's column.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-dLS8BWPqU9c/WNt41LU5CeI/AAAAAAAAB4o/IM6g6hSezOA845hpEyIKK2DGkleRdkxNACLcB/s1600/IMG_1042.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://2.bp.blogspot.com/-dLS8BWPqU9c/WNt41LU5CeI/AAAAAAAAB4o/IM6g6hSezOA845hpEyIKK2DGkleRdkxNACLcB/s400/IMG_1042.JPG" width="400" /></a></div><br />At this point, J1 ends his turn and marks his progress. On his next turn, if he gets a 5, for example, the temporary marker will start at the fourth level of the 5's column (building on his consolidated progress).<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-WdgenpHH4GU/WNt41tYAKQI/AAAAAAAAB4s/0KLQWqKWzqMGS5ytww-FgQHv1cc2J6r-wCLcB/s1600/IMG_1043.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://4.bp.blogspot.com/-WdgenpHH4GU/WNt41tYAKQI/AAAAAAAAB4s/0KLQWqKWzqMGS5ytww-FgQHv1cc2J6r-wCLcB/s400/IMG_1043.JPG" width="400" /></a></div><br />D has the next turn. He gets 1, 4, 4, and 5, which he chooses to group as 5 and 9:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-aOrdt812_J0/WNt46e_1soI/AAAAAAAAB4w/N95RyeN5p4kVqsCda4et97TyizfUbo9BgCLcB/s1600/IMG_1044.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://1.bp.blogspot.com/-aOrdt812_J0/WNt46e_1soI/AAAAAAAAB4w/N95RyeN5p4kVqsCda4et97TyizfUbo9BgCLcB/s400/IMG_1044.JPG" width="400" /></a></div><br />D chooses to roll again, getting 4, 5, 5, and 6. This has to be grouped as 9 (advancing in that column) and 11:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-VhTFdOatVgQ/WNt4794Qn8I/AAAAAAAAB40/Ar-q3RC3xbg6Oqha1jPwmokgJmVJc8WHACLcB/s1600/IMG_1045.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://4.bp.blogspot.com/-VhTFdOatVgQ/WNt4794Qn8I/AAAAAAAAB40/Ar-q3RC3xbg6Oqha1jPwmokgJmVJc8WHACLcB/s400/IMG_1045.JPG" width="400" /></a></div><br />D chooses to roll again and gets 2, 3, 5, and 6. This is a lucky roll that can be grouped as 5 and 11, allowing two tokens to advance:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-kkhrV1RhpOY/WNt48Q1f8iI/AAAAAAAAB44/FYBYT6z37JAWj9ZJONc02EnUMpF706ghgCLcB/s1600/IMG_1046.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://3.bp.blogspot.com/-kkhrV1RhpOY/WNt48Q1f8iI/AAAAAAAAB44/FYBYT6z37JAWj9ZJONc02EnUMpF706ghgCLcB/s400/IMG_1046.JPG" width="400" /></a></div><br />D presses his luck and rolls a fourth time, getting 1, 3, 3 and 5. The dice can't be paired to get a 5, 9 or 11 and there are no more temporary markers available to place, so D loses his progress. J1 will have the next turn.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-iTu7Km-bgrU/WNt4-PKWvaI/AAAAAAAAB48/FjE6SCtpPwE5Adb3_Oe9uTsqBZ7C73OGACLcB/s1600/IMG_1047.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://1.bp.blogspot.com/-iTu7Km-bgrU/WNt4-PKWvaI/AAAAAAAAB48/FjE6SCtpPwE5Adb3_Oe9uTsqBZ7C73OGACLcB/s400/IMG_1047.JPG" width="400" /></a></div><br />To be clear about the failure condition: the player must be able to place or advance a temporary token for both pairs of dice. For example, if D had rolled 1, 3, 3, and 6, he still would have lost his progress.<br /><br />Further rules:<br /><br /><ul><li>the first person to end their turn on the sixth level of a column "claims" that column.</li><li>the first person to claim 3 columns wins the game.</li><li>columns that have been claimed by any player are safe values for all players. Players do not need to allocate a temporary token to those columns.</li><li>Players can occupy a square that an other player has marked.</li></ul><br /><br /><div><b>Variations:</b><br />(1) Change the winning condition so that the first player to capture a column wins<br />(2) Change the height of the columns, either fewer than 6 for a faster game or more than 6 for a slower game<br />(3) Change the failure condition so that only one pair of dice needs to be playable and reduce the temporary tokens to 2.<br />(4) only allow each player to roll one time. This eliminates the "press-your-luck" aspect of the game and is much more basic.<br />(5) allow players to jump over a square that has been occupied by another player. This rule particularly fits well if you use objects to record your consolidated progress (which also makes the grid re-usable).<br /><br /></div><div><b>Notes:</b></div><div>I was originally taught this game by Mark Nowacki of Logic Mills.</div><div>J3 and I played the variation where each player only rolled one time on their turn.</div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-19735147103112607692017-03-06T06:19:00.002-08:002017-03-06T06:19:58.836-08:00Cryptarithmetic puzzles follow-upI was asked to write a bit about strategies and answers for the <a href="http://3jlearneng.blogspot.com/2017/02/cryptarithmetic-puzzles-for-grades-1-to.html" target="_blank">puzzles we gave two weeks ago</a>.<br /><br /><b>BIG + PIG = YUM</b><br />Because the digits in YUM are all distinct from BIG and PIG and there are only 7 letters in this puzzle, we should expect there to be many solutions.<br /><br />The easiest way to get a feel for the puzzle is to start trying values and see what develops. This was part of the idea of using this puzzle as the opening challenge.<br /><br />As we play with examples, the kids should notice these things that constrain our possible solutions:<br /><br /><ol><li>B, G, I, M, P, U, Y must all be distinct</li><li>We are adding two three digit numbers and the sum is a three digit number</li><li>B, P, and Y are all leading digits</li><li>The largest sum possible with two numbers 0 to 9 is 18.</li></ol><div>Some conclusions:</div><div>(a) G is not 0. If it was, then M would also be 0.</div><div>(b) B, P, and Y are all not 0. They are leading digits, the rules of our puzzles say they can't be zero.</div><div>(c) G + G is at most 18. It may contribute at most one ten to the calculation of U. That will only happen if G is 5 or larger.</div><div>(d) I + I is at most 18. Along with a potential ten from G+G, that means we have at most 19 coming from the tens. That will only happen if I is 5 or larger.</div><div>(e) B+G is at most 9. If there is an extra hundred coming from the tens digits, B+ G is at most 8.</div><div>(f) If I is 9, G must be less than 5. Can you see why?</div><div>(g) If G is less than 5, I cannot be 0</div><div><br /></div><div>After these observations, I'd suggest picking values of G, then seeing what values of I are allowed, then checking what remains for B and P. Because we aren't allowed to have duplicates, we quickly see that our choices are constrained.</div><div><br /></div><div>For example, if G is 1 or 2, then I is at least 3 and we get the following possible solutions (B and P can be interchanged):</div><div>431 + 531 = 962</div><div>341 + 641 = 982</div><div>351 + 451 = 802</div><div>371 + 571 = 942</div><div>381 + 581 = 962</div><div><br /></div><div>132 + 732 = 864</div><div>132 + 832 = 964</div><div>152 + 652 = 804</div><div>152 + 752 = 904</div><div>182 + 582 = 764</div><div>192 + 392 = 584</div><div>192 + 592 = 784</div><div><br /></div><div>There are some more advanced ideas that could come out of trying to count or list all of the solutions, so I'd encourage people to explore. Even this simple puzzle can be a lot of fun!</div><div><br /></div><div><b>CAT + HAT = BAD</b></div><div>The A in BAD is the key part of this puzzle. We can get two cases:</div><div>(a) A is 0 and T is 1, 2, 3 or 4</div><div>(b) A is 9 and T is 5, 6, 7 or 8.</div><div><br /></div><div>Again, while there are a lot of solutions (and counting them would be a fun challenge) they are easiest to build up by choosing A (either 0 or 9), then T, then seeing what flexibility is left for C and H. Here are some examples:</div><div><br /></div><div>301 + 401 = 702</div><div>301 + 501 = 802</div><div>301 + 601 = 902</div><div>302 + 502 = 804</div><div>302 + 602 = 904</div><div>103 + 403 = 506</div><div>395 + 495 = 890</div><div><br /></div><div><b>SAD + MAD + DAD = SORRY</b></div><div>This was a puzzle without a solution. In this case, it isn't too hard to see that SORRY has too many digits. The best explanation was given by one student:</div><div><ul><li>The largest three digit number is 999. </li><li>If we add three of them, we will at most get 2997. </li><li>SORRY has to be bigger than 10,000.</li><li>This isn't possible</li></ul><div><b>CURRY + RICE = LUNCH</b></div></div><div>Unfortunately, this also doesn't have a solution, but the reasoning is more subtle than the previous puzzle.</div><div><br /></div><div>Here, we can reason as follows:</div><div><ul><li>R cannot be 0 because it is the leading digit in RICE</li><li>Because the tens digit of RICE and LUNCH are both C, R must be 9 and we must have Y + E > 10.</li><li>This also means R + C + 1 = 10 + C.</li><li>That will mean the 100s digit of RICE must be the same as the 100s digit of the sum.</li><li>However, the 100s digit of RICE and LUNCH are different.</li></ul><div>Too bad, it was such a cute puzzle!</div></div><div><br /></div><div><b>ALAS + LASS + NO + MORE = CASH</b></div><div>This is the most challenging puzzle from this set.</div><div><br /></div><div>Some things we notice:</div><div><ol><li>There are ten letters (A C E H L M N O R S) and they must all be distinct.</li><li>We are adding three 4-digit numbers and a two digit number to produce another 4 digit number.</li><li>A, L, N, M and C are leading digits, so they can't be zeros.</li><li>The tens and hundreds digits of CASH (S and A) are also involved in the sums for those digits.</li></ol><div>Point 4 has a subtle implication, which I'll illustrate with the hundreds digits. Since L + O must be more than 0, but A is the hundreds digit of the sum, we must have some number of thousands carried over. Because A, L and M are all distinct and larger than 0, the smallest their sum can be is 1+2+3. Putting these two observations together, C must be at least 7.</div></div><div><br /></div><div>In this case, I find it helpful to put together a table showing possibilities that we have eliminated:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Ui8BxZMCIHI/WLuNrcMOB0I/AAAAAAAAB3s/jOeEFeo4EREai32kvI5LqNxKbwnRraJ7QCLcB/s1600/ALASLASSNOMORECASJ.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="204" src="https://3.bp.blogspot.com/-Ui8BxZMCIHI/WLuNrcMOB0I/AAAAAAAAB3s/jOeEFeo4EREai32kvI5LqNxKbwnRraJ7QCLcB/s640/ALASLASSNOMORECASJ.png" width="640" /></a></div><div>We can see some more restrictions from the fact that A + L + M must be less than 9. That means we have only the following possible triplets (ignoring order):</div><div style="text-align: center;">{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}</div><div><br /></div><div>One thing we notice is that 1 is in all of these triplets, so either A, L or M must be 1 and none of the other letters can be 1. Another thing we notice is that we don't yet have any way of differentiating A, L, or M, so any ordering of our triplets is possible. That would mean we have 24 cases to consider.<br /><br />Let's see how we would work through the cases, starting with A = 1, L = 2, M = 3, the first on our list. Now this, happens to be a stroke of luck, as we'll see.<br /><br />Starting from the thousands digit, we see that this would make C = 7, if there is a single carry from the hundreds. Indeed, we can see that this must be the value (in the case we are testing), as the carry from there could only come from L + O (plus any carry from the tens digit). Since L is at most 5, L + O is at most 14 and any carry from the tens digit must be less than 6.<br /><br />Now, in the hundreds digit, we have 2 + O + carry from the tens = 10, so O = 8 - carry from tens.<br />We know there must be at least one carry from the tens, so O is at most 7. Since 7 is already used by C, let's try 6. That means we need to get 2 hundreds carried over from the tens, so we need<br />A + N + R + carry from ones = 20, or N + R + carry from ones = 19. Since we have already used 6 and 7, the only way this is possible is if N and R are 8 and 9 (in either order) and we are carrying 2 from the ones.<br /><br />At this point, the case we've worked through has:<br />121S + 21SS + 86 + 369E = 71SH<br /><br />We still have to allocate digits 0, 4, and 5. and we know that S + S + 6 + E = 20 + H. Given our remaining digits, the biggest the left hand can be is if S is 5 and E is 4, making 20. The smallest the right hand can be is if H is 0. Fortunately, this makes the equality hold, so we get our final answer:<br /><br />1255 + 2155 + 86 + 3694 = 7150<br /><br />Through the process of checking this case, we learned more about how the carry from lower digits is restricted and it would be faster for us to check through remaining cases.<br />Let me know how many other solutions you find!<br /><br /></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b><complete id="goog_1412240271"></complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL + LOL = ROFL</complete></b></div><div><b><complete><br /></complete></b></div><div><complete>There are 71 LOLs, so this is 71 x LOL = ROFL. While this looks daunting, there are some ideas which take us a long way to the solution.</complete></div><div><complete><br /></complete></div><div><complete>First, ROFL has 4 digits. If L were 2, 71 x LOL would be more than 14,000, so L must be 1. In fact, ROFL is less than 9861, so LOL is smaller than 9871 / 71 which is 139. We can quickly check</complete></div><div><complete>101, 121, and 131 and see that 131 works.</complete></div><div><complete><br /></complete></div><div><complete>71 x 131 = 9301</complete></div><div><complete></complete></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-55965637414730859812017-02-23T00:02:00.003-08:002017-02-23T00:02:29.212-08:00A bit of 3D(ish) geometryNote: this post started out focused on two recent geometry projects. However, the Desmos Function Carnival, which I originally just included as a miscellaneous item, is also worth your time.<br /><h1>Nets and solids</h1>The most ambitious project recently was led by P (the mom, of course). She found nets for 3-d shapes and supervised J1 and J2 as they created a nice display to take to school.<br /><br />Here are the nets: <a href="http://www.senteacher.org/worksheet/12/NetsPolyhedra.html" target="_blank">SenTeacher Polyhedral nets</a>. They have a collection of other printables, but this collection of nets seems to be the most interesting. Take a look and let me know if you see anything else worthwhile.<br /><br />Here's the completed board:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-676ULIQxdTs/WKzqVJy-8xI/AAAAAAAAB24/HoSERzvhoEwmLjaD1m-CcwalP90Y3QLBgCLcB/s1600/IMG_0935.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-676ULIQxdTs/WKzqVJy-8xI/AAAAAAAAB24/HoSERzvhoEwmLjaD1m-CcwalP90Y3QLBgCLcB/s400/IMG_0935.JPG" width="300" /></a></div><br />A profile picture to show that these are really 3-d:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-d8DUT_BrYxA/WKzqZPiB3uI/AAAAAAAAB28/eZ_u2gu_AwgvAHGawH2H0hyr4j9V9IeSQCLcB/s1600/IMG_0936.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-d8DUT_BrYxA/WKzqZPiB3uI/AAAAAAAAB28/eZ_u2gu_AwgvAHGawH2H0hyr4j9V9IeSQCLcB/s400/IMG_0936.JPG" width="300" /></a></div><br /><a href="http://gwydir.demon.co.uk/jo/solid/index.htm" target="_blank">Solid Shapes and Their Nets</a> (I think Jo Edkins is the author) has a nice discussion of nets and a little puzzle game to distinguish nets that fold into the platonic solids and which don't. Feel free to try to do this for the icosahedron or dodecahedron!<br /><br /><h1>A two dimensional challenge?</h1>J2 asked me about a triangle with two right angles. Of course, we all know that a triangle can't have two 90 degree angles, right? Well, this fits in my list of math lies from this old post: <a href="http://3jlearneng.blogspot.com/2015/01/23-isnt-prime-short-bedtime-story.html" target="_blank">23 isn't prime</a>.<br /><br />We talked briefly about triangles on a plane and agreed that two 90 degree angles wouldn't work. If we try by starting with a side and building two right angles on that side, we just get parallel lines. Ok, that's standard.<br /><br />However, what about this:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-JH3GiGySiCA/WKzqTOt8YRI/AAAAAAAAB20/S9NJ8hAfkT427RHUv6nZUlWTh8ck6u3AQCLcB/s1600/IMG_0930.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-JH3GiGySiCA/WKzqTOt8YRI/AAAAAAAAB20/S9NJ8hAfkT427RHUv6nZUlWTh8ck6u3AQCLcB/s400/IMG_0930.JPG" width="300" /></a></div><br />We discussed spherical geometry for a bit. Not shown is our first attempt on the other side of the tennis ball that did have two right angles, but the third wasn't. He wanted to see an equiangular triangle on a sphere. This led to further discussion of life on a sphere:<br /><br /><ul><li>what is the largest interior angle sum for a triangle?</li><li>what is the smallest interior angle sum?</li><li>are there any parallel lines?</li><li>are there any squares? are there even any rectangles?</li><li>is π (the ratio of the circumference to diameter of a circle) a constant?</li><li>do we have to think like this in real life because the earth is close to a sphere?</li></ul><div>Picking up the point about π, it occurs to me that this is another math lie. This point got a nice treatment recently by <a href="http://smbc-comics.com/comic/pi" target="_blank">SMBC</a>:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img height="320" src="https://www.smbc-comics.com/comics/1481553833-20161212.png" style="margin-left: auto; margin-right: auto;" width="132" /></td></tr><tr><td class="tr-caption" style="text-align: center;">I made this pic small so you will go to the <a href="http://smbc-comics.com/comic/pi" target="_blank">site and look at Zach Weinersmith's </a>other awesome work</td></tr></tbody></table><div class="separator" style="clear: both; text-align: left;"><b>Some open follow-ups:</b></div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>what is an equiangular quadrilateral on the sphere?</li><li>what about hyperbolic geometry? I think no triangles with two right angles. If I recall correctly, triangles there have angle sums smaller than 180 degrees (or π radians, ha!)</li></ul><h1>Other Misc Math</h1>Now for my usual grab bag of other things we've been doing. Some of these are really great activities, so don't skimp on this section!<br /><b><br /></b><b>Desmos Function Carnival</b><br />We learned about the <a href="https://student.desmos.com/carnival/student-welcome/5489163c20d9898c7e223aea" target="_blank">Desmos Function Carnival activity</a> from a post by <a href="http://www.kenthaines.com/blog/2017/2/13/functions-are-finally-clicking" target="_blank">Kent Haines</a> which, in turn, we learned about via <a href="https://problemproblems.wordpress.com/2017/02/16/functions-rules-formulas/" target="_blank">Michael Pershan's post</a>. They both were writing about teaching functions to their students and have outlined a really nice sequencing of lessons, if you're into that kind of thing.<br /><br />In case you are, you might like to know that there is another flavor of Function Carnival available through the <a href="https://teacher.desmos.com/carnival/" target="_blank">Desmos teacher site</a> with 2 other activities. Each version is worth checking out because both of worthwhile activities that aren't in the other. Also, Kent links to <a href="http://map.mathshell.org/lessons.php?unit=8225&collection=8&redir=1" target="_blank">this nice graphing activity from the Shell Center</a> which also worked well with J1 and J2.<br /><br />For our purposes, I was more interested in the mathematical physics (plots of position vs time or velocity vs time) and the fun of creating wacky animations with impossible plots.<br /><br />If I can figure out how to post the animations, I will update this page, since the animations really enhance the experience. For now, let me give you some screen caps of some of their proposed graphs from the Cannon Man (height) activity:<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-Fj4cAyqkPUQ/WK6VRYq-AbI/AAAAAAAAB3c/4el-dWbcnoE7Q_D5B_OreVgf11DNf_F2gCLcB/s1600/cannon%2Bman%2Bnames.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="245" src="https://2.bp.blogspot.com/-Fj4cAyqkPUQ/WK6VRYq-AbI/AAAAAAAAB3c/4el-dWbcnoE7Q_D5B_OreVgf11DNf_F2gCLcB/s400/cannon%2Bman%2Bnames.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Have a free-form drawing tool? Make your name!</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-xBWoWYHFnfg/WK6VRGD-_xI/AAAAAAAAB3Y/kKqthAkb2tc2kTTQkvD7glwNWBNnJgr2QCLcB/s1600/cannon%2Bman%2Bunion%2Bjack.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://1.bp.blogspot.com/-xBWoWYHFnfg/WK6VRGD-_xI/AAAAAAAAB3Y/kKqthAkb2tc2kTTQkvD7glwNWBNnJgr2QCLcB/s400/cannon%2Bman%2Bunion%2Bjack.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Note: the vertical stripe doesn't really work with this animation, but the patriotic spirit is there!</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-TGj7OCFkmPA/WK6VQi9FbdI/AAAAAAAAB3U/b_DW7lmGrwo4oZYX7lclYFkpYnH3xPZ_gCLcB/s1600/particle%2Bdiagram.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="291" src="https://1.bp.blogspot.com/-TGj7OCFkmPA/WK6VQi9FbdI/AAAAAAAAB3U/b_DW7lmGrwo4oZYX7lclYFkpYnH3xPZ_gCLcB/s400/particle%2Bdiagram.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Our first attempt at <a href="https://en.wikipedia.org/wiki/Feynman_diagram" target="_blank">Quantum Field Theory</a></td></tr></tbody></table><br /><br />Here is J2 playing with the function carnival, but J3 (4 years old) enjoyed it just as much:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-7iWPM9_6h5w/WKzqSzmzvzI/AAAAAAAAB2w/_tUpXUM10Qofio6h08xVcjLmcJxDLMbbQCLcB/s1600/IMG_0929.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-7iWPM9_6h5w/WKzqSzmzvzI/AAAAAAAAB2w/_tUpXUM10Qofio6h08xVcjLmcJxDLMbbQCLcB/s320/IMG_0929.JPG" width="240" /></a></div><br /><b>Different representations</b><br />J3 is working on place value and playing with different representations of numbers. Here, she's got the 100 board, an abacus, and foam decimal models. Sometimes I challenge her to make a number, sometimes she challenges me.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/--1evcdriQJU/WKzqKfZqDxI/AAAAAAAAB2s/VpNhelRGF9o40Kcf6_Ywnq7yV97DD-SxwCLcB/s1600/IMG_0928.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/--1evcdriQJU/WKzqKfZqDxI/AAAAAAAAB2s/VpNhelRGF9o40Kcf6_Ywnq7yV97DD-SxwCLcB/s320/IMG_0928.JPG" width="240" /></a></div><br />JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-42503566588879727182017-02-21T19:37:00.000-08:002017-02-21T19:37:37.466-08:00RSM International Math ContestSometime the first week of February, the RSM offers an on-line math competition. For the second year, I had J1 and J2 work through the problems at grade 3 and grade 4 level. This post is about our thoughts on competition, but I end with a problem from the contest that had us debating.<br /><h1>Compete and win!</h1>To get it out of the way, let me explain my interest in the competition and why I had the kids enter. Above all, I was curious about the problems and expected they would be interesting challenges. We found the puzzles last year interesting, so I was pretty sure this year's collection would also be nice.<br />While I know that there are tons of excellent activities, I couldn't resist making use of this resource when someone thoughtful had put together a convenient collection in one place.<br /><br />Second, I am curious about levels and assessments. What types of questions does RSM think 3rd and 4th graders should be able to answer, but find challenging? How difficult would our two little ones find them? Since we are currently operating outside a standard curriculum framework, it is hard for me to judge where they are or what we should expect of them. Admittedly, there are other ways I could form this assessment, but they take more effort.<br /><br />Notice that I don't really care about how well they perform on the test. It is interesting information for me, but I don't <em>need</em> them to do well.<br /><br />P has a different view. She was an olympiad kid and sees competitions as the easiest way for our kids to distinguish themselves. You know the anxiety: if they don't win competitions, they won't get into Harvard, they'll end up on the street somewhere.<br /><br />Perhaps unsurprisingly, the kids are getting a mixed message about the importance and reason for participating in competitions. This year, J1 was particularly sensitive. He was very resistant to doing the RSM test. I spent a lot of time talking with him. Mainly, I wasn't hoping to convince him to do the test, but I wanted to explore other issues related to the expectations he feels on himself, his relationship with J2, and his mindset about his own learning.<br /><br />I am trying to communicate: <br /><br /><ol><li>effort and progress are important, starting point and base ability aren't. I know this is debatable, but I'm talking about the differences between my kids where I'm on pretty firm ground.</li><li>there are many aspects to math and mathematical ability. Calculating and answering questions quickly are facets, logical reasoning, strategic thinking, spatial reasoning, asking good questions, gathering data, exploring connections, etc, etc are all components, too.</li><li>time isn't important. One issue with the RSM test is that it is timed. This goes in the face of our repeated efforts to emphasize that the time it takes to solve something is not a key consideration. To help with this, I screen capped the questions so he could work on them after the official time had expired. Note, however, his recorded performance on the test was based on work within the contest rules. </li><li>Math is not about right answers.</li></ol><div>The next section might help illustrate that last point.</div><br /><h1>Which area do we want?</h1>As mentioned above, one of the questions from grade 4 cause us to argue among ourselves. I'm paraphrasing slightly:<br /><blockquote class="tr_bq">Two tennis ball machines stand on opposite ends of a 25 meter by 10 meter court. The yellow machine shoots yellow balls that stop on the court 2 meters to 16 meters from the yellow machine's side. The green machine shoots green balls that stop on the court 5 meters to 20 meters from the green machine's side. <b>Find the area of the court that has balls of either color on it.</b></blockquote>Here are some ideas from our discussion:<br /><br /><ul><li>Option A: we want to find the area of overlap, because that's the only place we could find balls of either color.</li><li>Option B: we want to find the combined areas, because that's where we could find tennis balls, either color.</li><li>Option 1: the area with green balls is a rectangle, the area with yellow balls is a rectangle.</li><li>Option 2: think of the ball machines as single points, shooting balls at various angles and various distances. the target area for each machine is an intersection of of a circular ring and a rectangle. This idea came from J2 when we were looking over the grade 4 questions after time had expired.</li></ul><div>To answer the question, choose either A xor B and choose either 1 xor 2. So, what is the right answer?</div><div><br /></div><div>I think the right answer is to have this discussion, to encourage multiple interpretations (with justification), to see how the answers compare, to think about what other things we could do to make the intended interpretation more clear (a diagram seems the most obvious), to recognize that this is part of the richness of math and it can't be represented by a single numerical answer on a timed test.</div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-50855163986311816432017-02-21T02:06:00.000-08:002017-02-21T02:06:29.794-08:00Cryptarithmetic Puzzles for Grades 1 to 4Inspired by a series of puzzles from <a href="http://mathmisery.com/" target="_blank">Manan Shah</a>, I decided to have the kids play with cryptarithmetic puzzles today. In addition to borrowing some of Manan's puzzles, I also used some from this puzzle page: <a href="http://www.brain-fun.com/Cryptic-Math-Puzzles/" target="_blank">Brain Fun</a>. I've included some more comments below about the Brain Fun puzzles.<br /><br />My main concern was whether the puzzles were at the right level. In particular, I was afraid that the puzzles would be too hard. In fact, I tried solving a bunch of them yesterday and actually found myself struggling. I'll ascribe some of that to being tired and sick. However, my intuition was to make some simpler puzzles of my own. In particular, I added:<br /><ul><li>puzzles that have many solutions: I figured that many solutions would make it easy to find at least one.</li><li>a puzzle that "obviously" has no solution. Now, obviously, the word "obviously" is a sneaky one in math, but I was pretty sure the kids could see the problem with this structure.</li></ul><div><h1>Grades 1 and 2</h1></div><div>For the younger kids, I started with a shape substitution puzzle. This is one our family explored almost 2 years ago: <a href="http://3jlearneng.blogspot.com/2015/05/twist-on-old-puzzle-and-our-number.html" target="_blank">Shape Substitution</a>. I don't recall the original source.</div><div><br /></div><div>Two reasons why I started with this. First, it has a lot of solutions, but there is an important insight that unlocks those solutions. Second, by using shapes, we can write possible number solutions inside them as we solve or guess-and-check the puzzle. This made it easier for the kids to see the connection that all squares have the same value, etc.</div><div><br /></div><br /><div>The second puzzle: BIG + PIG = YUM</div><div>Really just a warm-up practicing the rules and doing a little bit of checking that we haven't duplicated any numbers.</div><br /><div><br /></div><div>The third puzzle: CAT + HAT = BAD</div><div>Again, lots of solutions, but noticing leads to a good insight.</div><div><br /></div><div>Fourth puzzle: SAD + MAD + DAD = SORRY</div><div>This is a trick puzzle. The kids know that I like to tease them, so they are aware they need to look out for things like this. We discussed this in class and I suggested they give this puzzle to their parents.</div><div><br /></div><div>Fifth puzzle: CURRY + RICE = LUNCH</div><div>When I translated this to Thai, all the kids laughed. I was sneaking a little bit of English practice into the lesson and then they realized that it was worth trying to read all the puzzles, not just solve them.</div><div><br /></div><div><i style="font-weight: bold;">Sources: </i>I think I made up all of these puzzles (original authors, please correct me if I'm wrong).</div><div><h1>Grades 3 and 4</h1></div><div>The older kids already had experience with these puzzles. We did refresh their memory a bit with BIG + PIG = YUM</div><div><br /></div><div>I asked them to give me the rules and explain why those rules made sense. As with most games, I want to communicate that we're doing things for a reason, but those reasons can be challenged. If they think it makes sense to do it a particular way, we're open to their ideas.</div><div><br /></div><div>Second puzzle: SAD + MAD + DAD = SORRY</div><div>Same discussion as for the younger kids. When prompted, this was pretty easy for them to spot, but they weren't naturally attuned to think about whether a puzzle had solutions or how many. This led me to take a vote on all the puzzles at the end to see who thought the puzzles would have 0, 1 or many solutions.</div><div><br /></div><div><div>Third puzzle: ALAS + LASS + NO + MORE = CASH</div><div>A puzzle from Brain Fun. I think this is one of the easier ones on that page. Again, a bit of English practice.</div></div><div><br /></div><div><div><div>Fourth puzzle: LOL + LOL + LOL + .... + LOL = ROFL (71 LOLs)</div><div>This was from Manan. I think it is one of the easier ones in his collection, but it looks daunting. Turns out none of the kids in the class were familiar with (English) texting short-hand, so my attempt to be <i style="font-weight: bold;">cool</i> fell flat.</div></div></div><div><br /></div><div>Fifth puzzle: CURRY + RICE = LUNCH</div><div>Again, everyone was delighted when I translated this one. We're in Thailand, after all, so at least one puzzle had to be about food.</div><div><br /></div><div><h1>The key exercise</h1></div><div>The final assignment everyone (all four grades) was given was to make up a puzzle for me to solve. I was thinking it would be nice to have one in Thai, but we decided to keep it in English as further language practice.<br /><br />Manan wrote a nice post about having kids design their own puzzles. If it goes well, this is actually the activity that ties a lot of the learning messages together: they think about structure, they think about what allows multiple or single solutions, they apply their own aesthetic judgment, they use their knowledge of the operations, they are empowered with an open-ended task that cannot be "wrong."<br /><br />We'll see how it goes. At the very least, I expect a lot of work for myself when their puzzles come in!</div><div><br /></div><div><b>An extra sweetener</b></div><div>Two kids asked if we could use other operations than addition. That prompted me to put this on the table (also from Brain Fun):</div><div><br /></div><div>DOS x DOS = CUATRO</div><div><br /></div><h1>Brain Fun Problems</h1>The first time I'd seen the Brain Fun problems, I added them to a list and called them "basic" (see <a href="http://3jlearneng.blogspot.com/2016/11/puzzling-puzzlers.html" target="_blank">this page</a>.) When I actually went to solve them, though, they didn't seem so easy.<br /><br /><b>Big confession time: </b>I actually looked at some of the solutions. However, I was disturbed to see that the solutions involved extra information that wasn't included as part of the problem statement! For example, in THREE + THREE + FIVE = ELEVEN, the solution assumes that ELEVEN is divisible by 11. This seems to be the case for several of the puzzles involving written out arithmetic:<br /><br />TWO + TWENTY = TWELVE + TEN (assume 20 divides TWENTY and 12 divides TWELVE, I wasn't clear about whether any divisibility was assumed for TWO and TEN)<br /><br />I'm not sure if similar assumptions are allowed/required for any of the others.<br /><br />Maybe I shouldn't complain, since this assumption creates an additional constraint without which there could be further solutions. Perhaps part of the reason it doesn't sit well is aesthetic. In the 3 + 3 + 5 = 11 puzzle, 3 doesn't divide THREE and 5 doesn't divide FIVE.<br /><br />Lastly, there is a typo in the final puzzle of the Brain Fun page. That puzzle should be<br />TEN x TEN = FIFTY + FIFTYJGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-55097574243410505082017-02-16T01:10:00.000-08:002017-02-16T01:10:08.736-08:00More Man Who Counted (gaps and notes)As previously mentioned, we have been reading <a href="https://www.amazon.com/Man-Who-Counted-Collection-Mathematical/dp/0393351475" target="_blank">The Man Who Counted</a>. While the story is good and there are nice math puzzles, we've found some of our best conversations have come from errors or weaknesses in the book. Here are three examples:<br /><h1>How old was Diophantus?</h1>In chapter 24, we encounter a puzzle to figure out how old Diophantus was when he died. In summary, the clues are:<br /><br /><ol><li>he was a child for 1/6 of his life</li><li>he was an adolescent for 1/12 of his life. (J1: "what's that?" J0: "a teenager")</li><li>childless marriage for 1/7 of his life</li><li>Five more years passed, then had a child</li><li>The child got to half its father's age, then died.</li><li>Diophantus lived for four more years</li></ol>Perhaps we are wrong about our interpretation of the clues, but we noticed two things:<br />(a) the answer is not a whole number of years.<br />(b) the answer given in the book doesn't fit the clues.<br /><br />For the first part, it seems a natural assumption of these types of puzzles that we are only working with whole number years. Sometimes, this is an interesting assumption to directly challenge.<br />Here, since the clues involve a second person (Diophantus's child) we felt whole numbers were a strong assumption. Also, the name <a href="https://en.wikipedia.org/wiki/Diophantine_equation" target="_blank">Diophantus</a>, you know?<br /><br />Each clue required some discussion for us to agree on the interpretation. The one that seems most open is the fifth clue. In particular, did the child live until its age was half of the age of its father at the time of birth or to the point that, contemporaneously, it was half its father's age?<br /><br />For completeness, I'd note that neither interpretation matches the book's answer. The first interpretation does allow a whole number answer, but it doesn't give whole numbers for all the listed segments of Diophantus's life.<br /><br />Just so you can check for yourself, the solution given in the book is 84 years old.<br /><br /><b>How do you fix it?</b><br />We discussed several possible fixes:<br /><br /><ul><li>accept answers that aren't whole numbers or require whole number segments for each clue. This allows us to take the alternative interpretation of the fifth clue (though that still isn't satisfying) or to accept the clues and just take a new answer. This isn't satisfactory because... Diophantus.</li><li>Change clue 4 or clue 5 to match the book's answer. This approach seemed to fix the puzzle without distorting it or changing the mathematics required to analyze it.</li><li>Change clue 1, 2, or 3. While possible, these seemed to open the possibility of changing the character of the puzzle. Also, these fractions were plausible based on our own experience of human life spans.</li></ul><div>Of course, an even more satisfying answer would be to introduce a further variable and make the puzzle into one that makes heavy(ier) use of the integer restriction.</div><h1>Clever Suitors</h1>In chapter 31, Beremiz is confronted by a nice logic puzzle. Three suitors are put to a test, each is blindfolded and has disc strapped to his back. The background of the discs: other than color, the discs are all identical, there are five to choose from, 2 black and 3 white. <br /><br />The first suitor is allowed to see the colors of the discs on the backs of his two competitors, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.<br /><br />The second suitor is allowed to see the disc on the back of the third suitor, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.<br /><br />Finally, the third suitor is required to identify the color of his own disc and explain his reasoning. He succeeds.<br /><br /><b>Weakness 1</b><br />As a logic puzzle, we enjoyed this. Our problems came from the context in the story. This challenge was set to the three suitors as a way of fairly judging between them by finding the most clever suitor. However, this process was clearly unfair. In fact, it is inherent in the solution that it was impossible for the first and second suitors to determine the color of their own discs.<br /><br />This led to a nice discussion about who really held the power in this process: the person who structured the problem by deciding what color disc should be on which suitor and what order they would be allowed to give their answers.<br /><br />Extensions:<br /><ul><li>consider all arrangements of discs. Are there any arrangements where none of the suitors can answer correctly?</li><li>What is the winning fraction for each suitor? If you were a suitor, would you prefer to answer first, second, or third?</li></ul><div><br /></div><div><b>Weakness 2</b></div><div>Our second objection was non-mathematical, but again related to the story context. The fundamental problem wasn't how to choose a suitor. The fundamental problem was how the king could remain peacefully friendly toward all the suitors' home nations through this process.</div><div><br /></div><div>For this discussion, we went back to the story of Helen of Sparta, which we'd read a long time ago in the <a href="https://www.amazon.com/DAulaires-Greek-Myths-Ingri-dAulaire/dp/0440406943" target="_blank">D'Aulaire's Book of Greek Myths</a>. Of course, that also led to discussion of the division of the golden apples, another puzzle we all felt surely could have been solved more effectively with some mathematical reasoning...</div><h1>The Last Matter of Love</h1>The last puzzle of the book is in chapter 33. It is another logic puzzle, again intended to test the merit of a suitor in marriage. The test:<br /><br /><ul><li>there are five people</li><li>two have black eyes and always tell the truth</li><li>three have blue eyes and always lie</li><li>the suitor is permitted to ask three of them, in turn, a "simple" question each.</li><li>the suitor must determine the eye color of all five people</li></ul><div>As a logic puzzle, we readers get some extra information:</div><div><ol><li>The first person is asked: "what are the color of your eyes?" The answer is unintelligible.</li><li>The second is asked: "What did the first person say?" The answer is "blue eyes."</li><li>The third is asked: "What are the eye colors of the first and second people?" The answers are "the first has black eyes and the second has blue eyes."</li></ol><div><b>Simple questions</b></div></div><div>Our first objection was the part about asking "simple" questions. Having developed our taste for these types of puzzles through the knights and knaves examples of <a href="https://en.wikipedia.org/wiki/Raymond_Smullyan" target="_blank">Raymond Smullyan</a> (RIP, we loved your work!!!), the third question really bothered us. If you're going to go that far, why not ask the third person for the color eyes of all five people?</div><div><br /></div><div>Personally, I would prefer that the puzzle require us to ask each person a single yes/no question.</div><div><br /></div><div>As an extension: can you solve the puzzle with that restriction? </div><div><br /></div><div><b>Getting lucky</b></div><div>Again, we felt that this puzzle didn't meet the requirements of the context: to prove the worthiness of the suitor. Putting aside the question of whether this is really an appropriate way to decide whether two people should be allowed to marry, the hero here got lucky.</div><div><br /></div><div>Extension: what eye color for the third person would have caused the suitor to fail?</div><div>Extension: what answer from the third person would have caused the suitor to fail?</div><div>Extension: for what arrangement of eye colors would the questions asked by the suitor guarantee success?</div><div>Extension: what was the suitors' probability of success, given those were the three questions asked?</div><div><br /></div><div><b>Waste</b></div><div>Our final objection was the simple waste in the first question. From a narrative perspective, this is justified and even seems made to serve the purposes of the suitor. However, it opens another idea:</div><div>can you solve the puzzle, regardless of eye color arrangement, with only two questions?</div><div><br /></div><div>Feel free to test this with yes/no questions only or your own suitable definition of a "simple" question.</div><div><br /></div><h1>The power of...</h1>As a final thought, let me say that I think errors and ambiguity in a text are a feature, not a bug. It is another great opportunity for us to emphasize that mathematics is about the power of reasoning, not the power of authority.JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-8070710959439062112017-02-14T00:30:00.002-08:002017-02-14T00:30:46.938-08:00Good games and badRecently, we have been playing the following games:<br /><br /><ol><li>Go (baduk, weiqi, หมากล้อม). For now, we are playing on small boards, usually 5x5 or smaller.</li><li>Hanabi</li><li>Cribbage</li><li>Qwirkle (not regulation play, a form of War invented by J3 and grandma)</li><li>Munchkin</li><li>UNO</li><li>Vanguard</li></ol><div>I've ordered these by my own preference. In fact, I would be delighted playing just the first two exclusively and am happy to play cribbage or Qwirkle when asked.</div><div><br /></div><div>For the other three, I find myself biting my tongue a bit and grudgingly agreeing to be part of the game. I'm in the mood for strategic depth and a moderate (but not large) amount of pure chance. Part of my feeling was echoed in a recent My Little Poppies post: <a href="http://my-little-poppies.com/gateway-games/" target="_blank">Gateway Games</a>.</div><div><br /></div><div>However, as in the MLP post, I recognize that my enjoyment of the game is only a part of the reason for the activity. I guess the kids' enjoyment counts, too. </div><div><br /></div><div>Beyond that, even the games with limited depth are helping to build habits and skills:</div><div><ul><li>executive control: assessing the situation, understanding what behavior is appropriate, understanding options and making choices.</li><li>general gaming etiquette: taking turns, use of the game materials</li><li>meta-gaming: helping and encouraging each other, making sure that the littler ones have fun, too</li><li>numeracy and literacy: every time a number or calculation comes up or when something needs to be read, they are reinforcing their observation that math and reading are all around them.</li><li>meta-meta gaming: game choice, consensus building, finding options that interest and are suitable for all the players, knowing when it is time to play and when it isn't.</li></ul><div>As a family, and a little team, they are also building a shared set of experiences and jargon as they absorb ideas from each of the games.</div></div><div><br /></div><div>All of these are, of course, enough reason to make the effort to be open minded and follow their gaming lead.</div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-77774018593666749962017-02-13T23:10:00.003-08:002017-02-13T23:10:59.629-08:00NRICH 5 Steps to 50A quick note about the game we played in first grade today: <a href="http://nrich.maths.org/10586" target="_blank">5 Steps to 50</a>.<br /><br />This is an <a href="http://nrich.maths.org/frontpage" target="_blank">NRICH</a> activity that I've had on my radar for a while. I even made a <a href="http://jgplay.pencilcode.net/home/jumpGameTree2" target="_blank">pencilcode program</a> to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.<br /><br /><b>Our lesson outline</b><br />I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.<br /><br />I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:<br /><br /><ol><li>the only operations allowed are +1, -1, +10, -10</li><li>we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you <i>can</i> then make 5 jumps")</li><li>we are allowed to do the operations in any order</li><li>we can mix addition and subtraction operations</li></ol>After everyone had been through 3 rounds, we regrouped to summarize our findings:<br /><br /><ul><li>Which starting numbers can jump to 50?</li><li>Which starting numbers cannot jump to 50?</li></ul><div>We helped the kids resolve disagreements and then posed the following:</div><div><ul><li>What is the smallest number that can jump to 50?</li><li>What is the largest number that can jump to 50?</li></ul></div>For those to challenges, we kept the restriction that the numbers must be possible to generate from 2d6.<br /><br /><b>Basic level</b><br />To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.<br /><br />For these kids, it was helpful to ask a couple of prompting questions:<br /><br /><ul><li>What do you notice? This is a standard that never gets old!</li><li>If this path doesn't get to 50, does that mean there is no path to 50?</li></ul><div>This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.</div><div><br /></div><div><b>Getting more advanced</b></div><div>The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....</div><div><br /></div><div>The next major step is thinking about a way to systematically write down the paths.</div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-14781773949297313772017-02-01T22:16:00.004-08:002017-02-01T22:16:34.556-08:00Perfect Play for My closest neighborJoe Schwartz at Exit10a wrote a <a href="http://exit10a.blogspot.com/2017/02/ball-dont-lie.html" target="_blank">fraction comparison post</a> that prompted me to write up more of my experience and thoughts on this game.<br /><br /><b>Let's find perfect play</b><br />This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:<br /><br /><b><i>If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?</i></b><br /><br />My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?<br /><br /><b>How did it go?</b><br />In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:<br /><br /><ol><li>common denominators</li><li>common numerators</li><li>distance to 1</li><li>relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark. </li></ol><br />For visualization, drawing on a number line seemed to work best.<br /><br />I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.<br /><br /><b>What have I learned?</b><br />This game is really effective at distinguishing levels of understanding:<br />(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.<br /><br />(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.<br /><br />(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.<br /><br />(3) proficient: have at least one consistent strategy they can work through to make a comparison<br /><br />(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.<br /><br /><b>What would I do differently?</b><br />Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions <b><i>mean</i></b>. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.<br /><br />More locally, for this game in a class of mixed levels, I would<br /><br /><ul><li>lean toward doing this more as a cooperative puzzle</li><li>re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)</li><li>I also would consider allowing equivalent fractions to the target as winning plays</li></ul>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com2tag:blogger.com,1999:blog-5544661968326910027.post-77217319334891125012017-02-01T19:45:00.000-08:002017-02-01T19:45:13.370-08:00Impassable Din Daeng (BKK intersections 3)It has been a while since I've done one of these. For your topological and civil engineering pleasure, I present Din Daeng intersection (แยกดินแดง):<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-m1bhvPTMb14/WJKpDDEj4BI/AAAAAAAAB2E/jMwn3ulIW7kUjJdWTrHv1Wcd9e7Ac9rTACLcB/s1600/dindaeng.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="226" src="https://2.bp.blogspot.com/-m1bhvPTMb14/WJKpDDEj4BI/AAAAAAAAB2E/jMwn3ulIW7kUjJdWTrHv1Wcd9e7Ac9rTACLcB/s400/dindaeng.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Notice how the expressway obscures key details?</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-k0UG93Mml_0/WJKpC-FzLWI/AAAAAAAAB2A/cN43U3f2csMXHTPVJaB43ZcCr5UhkWtFgCLcB/s1600/dindaeng2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="213" src="https://2.bp.blogspot.com/-k0UG93Mml_0/WJKpC-FzLWI/AAAAAAAAB2A/cN43U3f2csMXHTPVJaB43ZcCr5UhkWtFgCLcB/s400/dindaeng2.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Magnified view. We now see a tunnel, but where does it emerge?</td></tr></tbody></table><br />This intersection has a special place in my heart. Last month, J1 and J2 played in a squash tournament at the Thai-Japan Youth Centre. It isn't visible on either map I've included, but is just to the northeast of the intersection. Since we live on the west of the intersection, we needed to cross somehow.<br /><br />After three failed attempts (following the driving directions on google maps) to get through from west to east, I ended up parking our car in one of the small side streets and we just walked. The walk took about 30 minutes...JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-45303886108245631792017-02-01T06:30:00.001-08:002017-02-01T06:30:16.723-08:00ideas for upcoming classes<h4>warm-ups for all</h4><div>WODB: (1) shapes book (2) <a href="http://wodb.ca/">wodb.ca</a></div><div><b style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">any:</b><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"> Traffic lights/inverse tic tac toe/faces game</span></div><div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><span style="font-size: 12.8px;">good options here, mostly grades 1/2:</span><span style="font-size: 12.8px;"> </span><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://3jlearneng.blogspot.com/2014/08/adding-and-subtracting-games.html&source=gmail&ust=1486000547474000&usg=AFQjCNG9Y4HlaLS74fGPccPv3J7wc0L1oA" href="http://3jlearneng.blogspot.com/2014/08/adding-and-subtracting-games.html" style="color: #1155cc; font-size: 12.8px;" target="_blank">some games</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">dots & boxes (maybe with an arithmetic component)</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">loop-de-loops</div></div><div><br /></div><h4><b>Grades 1 and 2</b></h4><div><a href="https://denisegaskins.com/2015/04/21/math-game-thirty-one/" target="_blank">31 Game</a>: could be used by any grade<br /><a href="https://denisegaskins.com/2008/05/29/hit-me-math-game/" target="_blank">Integer addition game </a><br /><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://nrich.maths.org/10586&source=gmail&ust=1486000547474000&usg=AFQjCNEX_NgqPntRyIxELfBLg5e6sUqToQ" href="http://nrich.maths.org/10586" style="color: #1155cc; font-family: arial, sans-serif; font-size: 12.8px;" target="_blank">jump to 50</a><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">,</span><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"> </span><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://nrich.maths.org/221&source=gmail&ust=1486000547474000&usg=AFQjCNGsQnJZwEl1F-gBPcDF5ISnr3CLnA" href="http://nrich.maths.org/221" style="color: #1155cc; font-family: arial, sans-serif; font-size: 12.8px;" target="_blank">chain of changes puzzles</a></div><div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><div style="font-size: 12.8px;">close to 100 game: </div><div style="font-size: 12.8px;"><span style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px; line-height: 18.2px;">Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.</span><br style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;" /><div style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;"><span style="line-height: 18.2px;">Procedure: </span></div></div><blockquote style="border: none; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;"><span style="line-height: 18.2px;">- Deal out 6 cards to each player</span></div></div><div><div style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;"><span style="line-height: 18.2px;">- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers. The goal is to get the sum of the two numbers as close to 100 as possible but </span><span style="line-height: 18.2px;">cannot exceed 100.</span></div></div><div><span style="line-height: 18.2px;"><br /></span></div></blockquote></div><div 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=https://mikesmathpage.wordpress.com/2014/09/03/fawn-nguyen-shares-a-really-neat-math-forum-problem/&source=gmail&ust=1486000547474000&usg=AFQjCNHKtW8n0GByoPJhaSC3souihiOAKw" href="https://mikesmathpage.wordpress.com/2014/09/03/fawn-nguyen-shares-a-really-neat-math-forum-problem/" style="color: #1155cc;" target="_blank">3 chips puzzle</a></div></div><div><br /></div><h4><b>Grades 3 and 4</b></h4><a href="https://denisegaskins.com/2015/05/08/math-games-with-factors-multiples-and-prime-numbers/" target="_blank">Factor finding game</a> (maybe warm-up?)<br /><a href="http://nrich.maths.org/5468" target="_blank">Factors and Multiples game</a><br /><a href="http://www.mathwire.com/games/contig.pdf" target="_blank">Contig for 3 and 4</a> (<a href="https://denisegaskins.com/2008/11/24/contig-game-master-your-math-facts/" target="_blank">explanation</a>).<br /><a href="https://denisegaskins.com/2011/03/16/game-times-tac-toe/" target="_blank">Times tic-tac-toe</a>: review for Grades 3 and 4<br /><a href="https://denisegaskins.com/2006/12/29/the-game-that-is-worth-1000-worksheets/" target="_blank">Fraction war for grade 4</a> (smallest card is numerator)<br /><a href="https://letsplaymath.files.wordpress.com/3012/07/multiplication-matching-cards.pdf" target="_blank">Multiplicaton models</a>: worth making for grades 3-4 for solidifying concepts? <a href="https://denisegaskins.com/2013/12/17/multiplication-models-card-game/" target="_blank">Associated games</a><br />d 2<br /><br /><br /><br /><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div 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://3jlearneng.blogspot.com/2014/11/subtraction-games-math-class-2.html&source=gmail&ust=1486000547474000&usg=AFQjCNGDjFkVyLK6ht_MHoHpka2M93A5SQ" href="http://3jlearneng.blogspot.com/2014/11/subtraction-games-math-class-2.html" style="color: #1155cc;" target="_blank">card subtraction</a></div><div 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://mathsolutions.com/documents/978-1-935099-02-4_NL36_L1.pdf&source=gmail&ust=1486000547474000&usg=AFQjCNH4GGZhUx8SRRDLdJo8hrXhhaElwg" href="http://mathsolutions.com/documents/978-1-935099-02-4_NL36_L1.pdf" style="color: #1155cc;" target="_blank">factor game</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">card on head game</div><div 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://3jlearneng.blogspot.com/2014/12/math-games-class-5.html&source=gmail&ust=1486000547474000&usg=AFQjCNEW2vXfy7gBqZx3YKIgQfxCGRx4hQ" href="http://3jlearneng.blogspot.com/2014/12/math-games-class-5.html" style="color: #1155cc; font-size: 12.8px;" target="_blank">Pattern revealed</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div 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://3jlearneng.blogspot.com/2015/01/calendar-tricks-and-break-bank-math.html&source=gmail&ust=1486000547474000&usg=AFQjCNFP6znksS3ily1H84D43kNYIFabXQ" href="http://3jlearneng.blogspot.com/2015/01/calendar-tricks-and-break-bank-math.html" style="color: #1155cc;" target="_blank">Calendar tricks</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Pico Fermi Bagel</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Magic triangle puzzles</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">damult dice</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div 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://nrich.maths.org/179&source=gmail&ust=1486000547474000&usg=AFQjCNHAx1Lbj-3MbHcO1hgfLxG5Vx-IGw" href="http://nrich.maths.org/179" style="color: #1155cc;" target="_blank">4 dominoes puzzle</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div 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://nrich.maths.org/2908&source=gmail&ust=1486000547474000&usg=AFQjCNHshPqexGG-MIivlsoVYoTdXZFJhA" href="http://nrich.maths.org/2908" style="color: #1155cc;" target="_blank">tables and chairs</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><div style="font-size: 12.8px;"><b><span style="font-size: medium;">(1) Dice <span class="il">game</span> perudo</span></b></div><div style="font-size: 12.8px;"><b>Equipment</b></div><div style="font-size: 12.8px;">- multi-player, 2-5</div><div style="font-size: 12.8px;">- Everyone gets the same number of 6 sided dice (full <span class="il">game</span> they get 5, I would start with 3)</div><div style="font-size: 12.8px;">- Everyone has a cup to shake and conceal their dice</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><b>Basic Play</b></div><div style="font-size: 12.8px;">- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.</div><div style="font-size: 12.8px;">- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s. </div><div style="font-size: 12.8px;">This bid signifies that the player has 2 (or more dice) showing the value 3.</div><div style="font-size: 12.8px;">- the next player has two choices: </div><div style="font-size: 12.8px;"><ul><li style="margin-left: 15px;">call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.</li><li style="margin-left: 15px;">raise, either the number of dice or the value or both get increased </li></ul><div><b>Advanced rules</b></div></div><div style="font-size: 12.8px;">- Ones are wild, they count as any number toward the target bid</div><div style="font-size: 12.8px;">- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round</div><div style="font-size: 12.8px;">- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><b>remainder jump</b></div><div style="font-size: 12.8px;"><span style="font-size: 12.8px;">we played this </span><span class="il" style="font-size: 12.8px;">game</span><span style="font-size: 12.8px;"> before, but we could give them blank boards and let them create. See the last page here: </span><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf&source=gmail&ust=1486019989770000&usg=AFQjCNFUmyIglOcPjrSLQ_l_cwz1zSy6kw" href="http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf" style="color: #1155cc; font-size: 12.8px;" target="_blank">http://ba-cdn.<wbr></wbr>beastacademy.com/store/<wbr></wbr>products/3C/printables/<wbr></wbr>RemainderJump.pdf</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.beastacademy.com/store/products/5B/printables/GCF_LCM_Webs.pdf&source=gmail&ust=1486019989775000&usg=AFQjCNGWSNN0TEuiIoz-Es7mvfVZyCrjhg" href="https://www.beastacademy.com/store/products/5B/printables/GCF_LCM_Webs.pdf" style="color: #1155cc; font-size: 12.8px;" target="_blank">https://www.<wbr></wbr>beastacademy.com/store/<wbr></wbr>products/5B/printables/GCF_<wbr></wbr>LCM_Webs.pdf</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><span style="font-size: 12.8px;">another good </span><span class="il" style="font-size: 12.8px;">game</span><span style="font-size: 12.8px;">:</span><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://saravanderwerf.com/2015/12/13/5x5-most-amazing-just-for-fun-game/&source=gmail&ust=1486019989777000&usg=AFQjCNEu9-BperhhonKQ-2G3XkoJGHqfxA" href="https://saravanderwerf.com/2015/12/13/5x5-most-amazing-just-for-fun-game/" style="color: #1155cc;" target="_blank">https://saravanderwerf.com/<wbr></wbr>2015/12/13/5x5-most-amazing-<wbr></wbr>just-for-fun-<span class="il">game</span>/</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">We should still use the pyramid puzzle sometime:</div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/pyramid-puzzles/&source=gmail&ust=1486019989778000&usg=AFQjCNES2-o5XkMjJTB9AmidWA7OuPa-Kg" href="http://mathforlove.com/lesson/pyramid-puzzles/" style="color: #1155cc;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>pyramid-puzzles/</a></div></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><span style="font-size: 12.8px;">(1) double digit and double dollar:</span><div style="font-size: 12.8px;">We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.</div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/double-digit-and-dollar-digit/&source=gmail&ust=1486019990897000&usg=AFQjCNFOB3X-FUmvlg5nmkRzue2tNBgCdw" href="http://mathforlove.com/lesson/double-digit-and-dollar-digit/" style="color: #1155cc;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>double-digit-and-dollar-digit/</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here <a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/wp-content/uploads/2016/02/MFL_Rectangle-Worksheet-2_Q.pdf&source=gmail&ust=1486019990897000&usg=AFQjCNEcNELxKyJ7tl7ER2jvWu28J5BMHA" href="http://mathforlove.com/wp-content/uploads/2016/02/MFL_Rectangle-Worksheet-2_Q.pdf" style="color: #1155cc;" target="_blank">are 5 questions</a>).</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/1009569184/&source=gmail&ust=1486019990897000&usg=AFQjCNFmqnY1m7W--feMeJjewVqxzeM60g" href="http://mathforlove.com/lesson/1009569184/" style="color: #1155cc;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>1009569184/</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">(3) some of these <span class="il">games</span> are promising:</div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://danburf.files.wordpress.com/2014/11/tapson-games.pdf&source=gmail&ust=1486019990897000&usg=AFQjCNHo_EVEYa_QDFKti_GgvDcPyeIsCQ" href="https://danburf.files.wordpress.com/2014/11/tapson-games.pdf" style="color: #1155cc;" target="_blank">https://danburf.files.<wbr></wbr>wordpress.com/2014/11/tapson-<wbr></wbr><span class="il">games</span>.pdf</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/pyramid-puzzles/&source=gmail&ust=1486019990899000&usg=AFQjCNGR_HzFaVOiKV0gRMRvEVjR0H_Vcw" href="http://mathforlove.com/lesson/pyramid-puzzles/" style="color: #1155cc; font-size: 12.8px;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>pyramid-puzzles/</a></div><div style="font-size: 12.8px;"><br /></div></div></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com4tag:blogger.com,1999:blog-5544661968326910027.post-69872954780258196522017-01-31T06:27:00.000-08:002017-01-31T06:27:12.662-08:00Quadratic Friends (The Man Who Counted)J1, J2, and I are currently reading <a href="https://www.amazon.com/Man-Who-Counted-Collection-Mathematical/dp/0393351475" target="_blank">The Man Who Counted</a>. Here are some quick thoughts:<br /><br /><b>Quadratic Friends</b><br />The book is a great entry point for mathematical discussions. In fact, it makes it questionable as bedtime reading, since I have to be careful to find a more narrative section to close the evening. Otherwise, we would just continue talking and they'd never get to sleep.<br /><br />Fortunately, the J's are willing to extend some of these conversations over to the next day, so we're not obligated to wrap up everything in one evening.<br /><br />Here is an example discussion: in one of the early chapters, the protagonist Beremiz talks about the special relationship between 13 and 16. Namely:<br /><blockquote class="tr_bq">13 * 13 = 169<br />1 + 6 + 9 = 16<br />16 * 16 = 256<br />2 + 5 + 6 = 13</blockquote><b>Finding more</b><br />We wondered: what other pairs of numbers share this property?<br /><br />Our first instinct was to gather data, so we started calculating some examples. We began with 0 and worked up, squaring, adding the digits, repeating. We found a couple of cases that flowed into the 13-16 relationship, for example 7. This gives a feeling that 7 is very fond of 13, but 13 only has eyes for 16. Not the usual way people think about numbers, I guess.<br /><br />Along the way, we made some interesting observations about this iterative process. I won't spoil the surprise, but would encourage you to explore yourself.<br /><br />I'd note that J1 did the calculations up to 30 in his head, while I was a bit lazy and wrote a pencilcode program.<br /><br /><b>An extension</b><br />This conversation branched in an interesting way. Squaring is a natural thing to do with numbers, but summing the digits is a bit artificial. It depends on a choice of base. So, a natural follow-up question:<br />what quadratic friends exist in other bases? This is an exploration for another day.JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-47611632012325494232017-01-24T02:03:00.000-08:002017-01-24T02:03:00.564-08:00Delta Pack the uncreative way (euclidea series)Returning with a very brief installment of geometric constructions. Basically, I was just brute-forcing the constructions in this pack, so most of the cases where I managed the minimum move constructions were just because those were straightforward. Reviewing the pack in preparation for this post, I did find a couple of ways to shave down some of the constructions, but I don't feel like there was a major breakthrough.<div><br /></div><div>Here's a snapshot showing (almost) where the V stars are:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-HZScYHlsNj8/WIclZLbW-4I/AAAAAAAAB1w/VJz1hvctHC00DCNFLLaD3Jc4RhX7xlWYwCLcB/s1600/delta%2Bpack.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="458" src="https://4.bp.blogspot.com/-HZScYHlsNj8/WIclZLbW-4I/AAAAAAAAB1w/VJz1hvctHC00DCNFLLaD3Jc4RhX7xlWYwCLcB/s640/delta%2Bpack.png" width="640" /></a></div><div><br /></div><div><br /></div><div>If you do some simple counting, you'll see that there is one more V-star missing. Since this somehow escaped me the first time, I'll leave its location as an exercise for you, too.</div><div><br /></div><div><b>Favorites</b></div><div>I've mentioned the idea from 4.5 before and I still like that construction, even though it is very simple.</div><div><br /></div><div>Constructing the two equilateral triangles also seemed nice. Constructing the inscribed and circumscribing circles from the triangle seems more like the "usual" direction of construction, so these reverse constructions appealed to me.</div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-7901438494263867422017-01-23T00:13:00.003-08:002017-01-23T00:29:55.974-08:00Fractions and Farey AdditionBenjamin Leis (who posts at <a href="http://mymathclub.blogspot.com/" target="_blank">Running a Math Club</a>) flagged this video in response to our recent fractions work: <a href="https://www.youtube.com/watch?v=0hlvhQZIOQw" target="_blank">Funny Fractions and Ford Circles (Numberphile)</a>. <br /><br /><b>Ex ante discussion ideas</b><br />The video gave me several ideas for possibly interesting conversations with the kids:<br />(1) Some basic geometry, particularly for J3. Circles that are tangent, nesting pictures, pictures that have fractal qualities.<br />(2) Comparing Farey addition and regular addition<br />(3) Well-defined operations on fractions. I always like to discuss whether the operations gives us the same results regardless of the equivalent form we start with? Farey addition is a good example where the choice of representation is important (indeed, Prof Banahon is careful to keep reminding us that he wants the fractions in lowest terms.)<br />(4) why do we want the fractions in simplest terms? Possibly relate this to the Cat in Numberland (showing rationals are countable).<br />(5) what happens if we try Farey addition of three fractions in a row: e.g., (1/5) @ (1/3) @ (1/2)? This is one of the few "naturally occurring" non-associative operations I know.<br />(6) Since associativity doesn't work, surely distribution of multiplication over Farey addition must not work, right? What about commutativity?<br />(7) Linking back with our comparison game, if a<b, how do a@c and b@c compare? If a@c < b@c, what can we say about a and b?<br />(8) what if we allow negative numbers? How should we define Farey addition, then?<br /><br /><b>How the conversation actually went</b><br />J2 was especially taken with the picture of the Ford circles and immediately had two requests: he wanted to draw them and he wanted me to create a pencilcode program to draw them.<br /><br />The former was a great activity with a lot of figuring and fraction practice. Here he is, hard at work:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-wVwuI5NYIio/WITq_iticpI/AAAAAAAAB0w/ll20sdH26HwDQrEgMO5msGuZN5WfFTbWgCLcB/s1600/IMG_0876.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-wVwuI5NYIio/WITq_iticpI/AAAAAAAAB0w/ll20sdH26HwDQrEgMO5msGuZN5WfFTbWgCLcB/s320/IMG_0876.JPG" width="240" /></a></div><br />Along the way, there was lots of discussion about where to position each fraction on the number line (he scaled with 20 cm as the unit distance from 0 to 1), and how big to make each circle. Tangency condition was a nice check on his work. He would see right away when something was wrong (which did happen several times:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-kyN-wdw6Cgk/WITrILJ4DwI/AAAAAAAAB00/4HwSfgG11_MNIpb88Ms5biNHaWPOl1hCQCLcB/s1600/IMG_0877.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://1.bp.blogspot.com/-kyN-wdw6Cgk/WITrILJ4DwI/AAAAAAAAB00/4HwSfgG11_MNIpb88Ms5biNHaWPOl1hCQCLcB/s320/IMG_0877.JPG" width="320" /></a></div><br /><br />We did talk through some of the ideas on my pre-planned list: is Farey addition well-defined on fractions (no! point 3), does associativity work (no! point 5), could we extend to negative numbers (yes, make the numerator negative seems to work best, point 8). Other areas are still open for future discussion.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-2dtyESv7sVw/WIW-5WKe9VI/AAAAAAAAB1g/PikoP2npAigu6io2UMqaDt1Rz4ij8epnwCLcB/s1600/IMG_0885.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://3.bp.blogspot.com/-2dtyESv7sVw/WIW-5WKe9VI/AAAAAAAAB1g/PikoP2npAigu6io2UMqaDt1Rz4ij8epnwCLcB/s320/IMG_0885.JPG" width="320" /></a></div><br /><br /><b>Pencilcode result</b><br />I wrote a quick program here: <a href="http://jgplay.pencilcode.net/edit/Math/FareyFord" target="_blank">FareyFord</a>. You'll notice that it doesn't actually generate Farey sequences. Instead, it creates generations of fractions, starting from 0 and 1 as the original parents. For each new generation, it uses Farey addition to create a new fraction between each adjacent pair in the previous generation.<br /><br />Here's a picture of the associated Ford circles:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-LNiaQKn3m8s/WIW6YqS_cfI/AAAAAAAAB1U/n3ZhoFbxPIUfAnXqYSmItYF93ewKc8C0gCLcB/s1600/Ford%2BCircles.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="342" src="https://1.bp.blogspot.com/-LNiaQKn3m8s/WIW6YqS_cfI/AAAAAAAAB1U/n3ZhoFbxPIUfAnXqYSmItYF93ewKc8C0gCLcB/s640/Ford%2BCircles.png" width="640" /></a></div><br />This method raised an interesting question: what is the largest denominator in each generation? If you don't know, it is cute and worth considering.<br /><br /><b>More Go (miscellaneous)</b><br />Note: this part is unrelated to fractions or farey sequences.<br /><b><br /></b>J3 wasn't in the mood to play more capture go with me, but I had an idea. I noticed in one of Nick Sibicky's lectures that one of his students was a young girl, roughly around the age of our three kids. I showed that part of the video to J3 and she made the connection: "this is something girls like me do."<br /><br />We went and played some silly games on very small boards: 1x1, 2x2, 3x3. In the picture below, we set out a blue-green alternating boundary around a 3x3 board. Then, I asked J3 how many different moves were available. She pointed first to the center, then I asked if there were any other spaces that were the same as the center, if we moved the board around or tipped ourselves upside down.<br /><br />No, so we made the center red. What other moves? She then chose a side square and figured out that there were three other places that were equivalent. Those became yellow. Finally, we figured out that the four corners were also identical, so that gave us the final picture:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-uKerzA8MTDY/WITrIEbVGrI/AAAAAAAAB04/_XAo16JkBUYTmRRRT2Y3xUuDxeIuH6QAQCLcB/s1600/IMG_0880.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-uKerzA8MTDY/WITrIEbVGrI/AAAAAAAAB04/_XAo16JkBUYTmRRRT2Y3xUuDxeIuH6QAQCLcB/s320/IMG_0880.JPG" width="240" /></a></div><br />Later, I was playing 9x9 with J2. Instead of go stones, we used Banangram tiles for the white stones. At the end of the game, we tried to make words with the captured tiles from the game. Here was one case where we could (sort of?) make a complete scrabble chain with all the captures:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-0yxnMgjhVCY/WITr1i5NWLI/AAAAAAAAB1A/ofunAcwT2nsG6ZoezYD6QPZiDbOcAl8jgCLcB/s1600/IMG_0882.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-0yxnMgjhVCY/WITr1i5NWLI/AAAAAAAAB1A/ofunAcwT2nsG6ZoezYD6QPZiDbOcAl8jgCLcB/s320/IMG_0882.JPG" width="240" /></a></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-91628910203992331892017-01-18T22:38:00.000-08:002017-01-18T22:38:12.264-08:00Closest neighbor one-on-oneIn my last post, I wrote about playing Denise Gaskins' closest neighbor fraction game with our 4th grade class. Yesterday, I spent time with J2 and used the game as a semi-cooperative puzzle.<br /><div><br /></div><div>This activity worked really well and the experience gave me some additional ideas about how to use the core ideas again with the 4th grade class.</div><div><br /></div><div><b>Puzzle or game?</b></div><div>First, there were only two of us, one a kid and another an adult, so that background naturally makes the activity very different. As the key modification for play, we played all of our hands open and helped each other find the fraction in each of our hands that was closest to the target for that round. Then, we worked together to determine which of those two "champions" was closest overall.</div><div><br /></div><div>Some of the consequences:</div><div><ul><li>the activity was not really competitive (see below)</li><li>J2 had to do a lot more fraction work.</li></ul><div>Let me explain the second point here. Because we were looking for the best play, J2 had to consider all of the combinations in his hand (20 choices). Some of those can be rejected quickly with simple analytical strategies depending on the target. Even this is good number sense thinking. Also, some combinations are close competitors and need to be analyzed more carefully.</div></div><div><br /></div><div>If we were playing with closed hands, he could choose two cards, play a fraction based on them, and I wouldn't be able to say anything about whether those were his best options or not.</div><div><br /></div><div>Second, while I write that "we worked together," as a sneaky dad, that means that I pretended to do work, while actually getting J2 to analyze my hand as well as his. Really, the only thing I offered was an alternative comparison strategy, once he had already worked through his own approach.</div><div><br /></div><div><b>An example of some strategies</b></div><div>We found that some of the comparisons that arise naturally in this game are quite tricky, even for me. For example, quickly tell me which is closer to 1/3: 1/5 or 4/9?</div><div><br /></div><div>We found that placing the fractions on a number line was a really helpful strategy for many of the comparisons. We also made very heavy use of the two strategies involving common numerators or common denominators.</div><div><br /></div><div>Finally, you can see in this example that J2 is comfortable mixing decimals and fractions, for example converting to 1/2 to 3.5/7 to aid some comparison:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-dP_4HiV5qQE/WIBWhBnBiFI/AAAAAAAAB0U/PilxbBW3NTIn782SeBEnbDbXPB_kM4U9wCLcB/s1600/IMG_0871.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-dP_4HiV5qQE/WIBWhBnBiFI/AAAAAAAAB0U/PilxbBW3NTIn782SeBEnbDbXPB_kM4U9wCLcB/s400/IMG_0871.JPG" width="300" /></a></div><div><br /></div><div><br /></div><div><b>Our grid</b></div><div>Through our play, we filled out this grid, taking turns putting in our best results and congratulating each other when our hand was the ultimate champion for that round:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-wACjF58Nw4k/WIBWhDgkpVI/AAAAAAAAB0Q/2cMoXCUZVnUtipe--QingicVzOXc48mZQCEw/s1600/IMG_0872.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-wACjF58Nw4k/WIBWhDgkpVI/AAAAAAAAB0Q/2cMoXCUZVnUtipe--QingicVzOXc48mZQCEw/s400/IMG_0872.JPG" width="300" /></a></div><div><br /></div><div><br /></div><div><b>Competition and Strategic thinking</b></div><div>I was particularly pleased by one comment J2 made about this overall game: "this is mostly luck, how well we can play depends on the cards we get." This comment came after one round where he had several duplicate cards in his hand, reducing the number of distinct values he could play. We've discussed elsewhere my goals of helping the kids think about game structure, so I always love it when they bring those ideas up themselves.</div><div><br /></div><div>Some thoughts about competition. While we played this game non-competitively, I'm not opposed to competition nor do I think that this game always needs to be played non-competitively. Ultimately, my litmus test is how to play in a way that is the most fun. If I were a more serious educator, I suppose I would also consider which way is the most educational, too.</div><div><br /></div><div>It won't always be obvious what is the best way to play each game. In this case, I got to benefit from the prior experience with the class and my close knowledge of J2. Many times, I'll tell the kids that there are several ways to play and we'll try them out together, then review the experience.</div><div><br /></div><div>Among other things, this is why I love handicap games like Go. By adjusting the starting advantages, we can create scenarios where it is very competitive and very fun, even though the players have very different levels of experience and current strength in the game. And also, there are things we can do together when we want a non-competitive activity.</div><div><br /></div><div><b>Ideas for going back to class</b></div><div>From this time with J2, here are my ideas about taking the game back to the 4th grade class are:</div><div><ol><li>Spend a lot of time on fraction comparison strategies before we play</li><li>Reduce the number of cards dealt to each player</li><li>play as teams</li><li>convert to open hands with a lot of talk about why we chose particular plays</li></ol></div><h1>An actual puzzle</h1>As a reward for reading down this far, here's an actual puzzle related to the closest neighbors fraction game: <br /><br />During the round where the target is 1/2, Jay plays 6/6 = 1. Was that her best play? How do we know?<br /><br /><div id="spoiler1" style="display: none;">Remember, we are playing our game with a single deck of playing cards and each player is dealt five cards</div><button onclick="if(document.getElementById('spoiler1') .style.display=='none') {document.getElementById('spoiler1') .style.display=''}else{document.getElementById('spoiler1') .style.display='none'}" title="Click to show/hide construction" type="button">Hint 1</button><br /><div id="spoiler2" style="display: none;">The target for the next round will be 3/4</div><button onclick="if(document.getElementById('spoiler2') .style.display=='none') {document.getElementById('spoiler2') .style.display=''}else{document.getElementById('spoiler2') .style.display='none'}" title="Click to show/hide construction" type="button">Hint 2</button> <br /><div id="spoiler3" style="display: none;">Burning 2 identical values might allow Jay to increase the diversity of her hand, especially if she happened to have 3 or 4 sixes.<br />Hey, life doesn't promise that all puzzles will have solutions that can be wrapped up in a nice neat package, does it? </div><button onclick="if(document.getElementById('spoiler3') .style.display=='none') {document.getElementById('spoiler3') .style.display=''}else{document.getElementById('spoiler3') .style.display='none'}" title="Click to show/hide construction" type="button">Hint 3</button>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-37090240877032814382017-01-17T04:47:00.002-08:002017-01-17T04:47:25.625-08:00My Closest Neighbor Fraction game Denise Gaskins recently flagged a post about a good fraction game: <a href="https://denisegaskins.com/2014/08/06/fraction-game-my-closest-neighbor/">My Closest Neighbor</a>. I tried this out in class today.<br /><br /><b>A pre-test</b><br />First, I wasn't sure whether the level of the game would be right for the kids. I was considering it for the 3rd and 4th graders, but had some alternative activities planned in case. To start, I posed the following questions:<br /><ul><li>Which is closest to one-half: 1/3 or 2/5? The third graders really struggled with this, so I left it alone and went to my plan B games. The fourth graders were all confident on this one.</li><li>Which is closest to 3/4: 5/11 or 11/12? This was a challenge for the fourth graders, but I thought it would be ok to play the game.</li></ul><div>In our discussion of the second question, we explored two strategies:</div><div><ol><li>making a common denominator</li><li>comparing with reference numbers</li></ol><div>The common denominator is a bit of a pain, since 11 is prime, though at least we have the fact that 4 is a factor of 12. One student soldiered through this approach, but it was difficult for the other kids to follow.</div></div><div><br /></div><div>For the second strategy, we made use of some observations that were more elementary for the kids:</div><div>(a) 5/11 < 5/10 = 1/2</div><div>(b) 3/4 is halfway between 1/2 and 1</div><div>(c) 11/12 < 1</div><div><br /></div><div>Combining these, it was easy for us to draw a rough number line, place 5/11, 1/2, 3/4, 11/12, 1 and see that 11/12 must be closer.<br /><br /><b>The game</b><br />We played three rounds: target 0, target 1/3 and target 1/2. I think this game was very challenging for the kids. Everyone had to work to figure out the best play from their hand and didn't always make the right (local) choice. For example, whether to choose 5/8 or 5/9 for a 1/2 target.<br /><br />Once everyone had played, the challenge was still just starting. They had to figure out who was closest. I structured the discussion by helping them figure out which plays were lower than the target and which were higher. For the ones that were lower, they could put them in order and only needed to consider the highest. Then, we worked on the ones higher than the target and got the lowest of those.<br /><br />In the course of this discussion, we added a third strategy to the ones listed above:<br /><br /><ul><li>making a common numerator</li></ul><div><b>Summary thoughts</b></div><div>Fraction comparison like this was still too difficult for the kids to make an engaging game. If I were to do it again, I would change to make it more of a puzzling exercise, removing competition and any sense of time pressure.</div><div><br /></div><div>Once the kids gain a bit more experience, though, I think this game has some nice features. It is particularly good for practicing fraction sense, and the multiple rounds allow some scope for strategic play.</div></div>JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.com0