tag:blogger.com,1999:blog-5303307482158922565Fri, 27 Nov 2015 15:10:24 +0000linksreviewcarnivalsalonteachingmythswcydwtinternationalmy sonpoemscienceanthologybase eightgender issuesimaginary numbersmath edmsrioctalproblem-solvingstoryMath Mama Writes...http://mathmamawrites.blogspot.com/noreply@blogger.com (Sue VanHattum)Blogger536125tag:blogger.com,1999:blog-5303307482158922565.post-8994491538821129314Sun, 25 Oct 2015 19:37:00 +00002015-10-25T15:11:12.666-07:00Math Teachers at Play, #91<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-QWruwGBIqGg/Vi0SUnWqbeI/AAAAAAAABm4/mdwWFZG3TJ8/s1600/91.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://1.bp.blogspot.com/-QWruwGBIqGg/Vi0SUnWqbeI/AAAAAAAABm4/mdwWFZG3TJ8/s200/91.jpg" width="200" /></a></div><span style="font-size: x-large;">Number 91</span> feels like we're closing in on 100. <a href="https://plus.maths.org/content/maths-minute-power-powers" target="_blank">The last time I hosted MT@P</a>, we were at #71 and I managed to include 71 posts. I wasn't quite that ambitious this time. (Old math posts don't go stale. You might enjoy browsing through a bunch of <a href="http://denisegaskins.com/mtap/" target="_blank">the old Math Teachers at Play blog carnivals</a>. And don't forget our partner carnival: the <a href="http://aperiodical.com/category/columns/carnival-of-mathematics/" target="_blank">Carnival of Mathematics</a>.) <br /><br />If there are 14 people in a group, and each shakes hands with each other, there will be 91 handshakes. (Can you see why?)<br /><br />91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13<br />(which makes it triangular)<br /><br />and<br /><br />91 = 7 * 13<br />(the middle and last numbers in the sum above)<br /><br />Will this always happen for triangular numbers?<br /><br /><br /><br /><br /><h3>Games & Puzzles</h3><ul><a href="http://4.bp.blogspot.com/-2d1H38gvef8/Vi0fgrMfN8I/AAAAAAAABnQ/q6qR6eKFUAk/s1600/number-tile-puzzles-primary.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-2d1H38gvef8/Vi0fgrMfN8I/AAAAAAAABnQ/q6qR6eKFUAk/s1600/number-tile-puzzles-primary.png" /></a><li><b>Shannon Duncan</b>, a 6th grade math & science teacher, shares <a href="http://blog.mindresearch.org/blog/game-based-learning-tips-from-math-educator" target="_blank">4 Reasons to Promote Math Success through Games</a> at the <i>MIND Research Institute</i> blog, illustrating her ideas with some of the games she has her students playing. I especially like the first point - making a mind-body connection.</li><li><b>John Golden</b> (@mathhombre) shares <a href="http://mathhombre.blogspot.com/2015/10/angle-of-coincidence.html" target="_blank">Angle of Coincidence</a> at his blog, <i>Math Hombre</i>, about an angle identification game he's developing. Ask your students to playtest it and give him feedback! John also wrote about the start of the semester, and included a game called <a href="http://mathhombre.blogspot.com/2015/09/a-sorted-beginning.html" target="_blank">In or Out?</a> that looks fun.</li><li><b>Jeff Trevaskis</b> shares a <a href="https://webmaths.wordpress.com/2015/10/18/multiplication-tic-tac-toe-in-3-acts/" target="_blank">Multiplication Tic-Tac-Toe Game</a> at his blog, <i>webmath<b>.</b></i><b> </b></li><li><b>Carole Fullerton</b> shares <a href="https://mindfull.wordpress.com/2015/10/17/number-tile-puzzles-primary-and-intermediate/" target="_blank">Number Tile Puzzles</a> at her blog, <i>Mathematical Thinking</i>.<b> </b></li><li><b>Gray Antonick</b> interviewed <a href="http://wordplay.blogs.nytimes.com/2015/06/01/salomon/" target="_blank">Paul Salomon in the New York Times Numberplay column</a>, about his Imbalance Puzzles, one of many puzzles and games featured in <i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i> (my book, published in April!).</li></ul><h3> </h3><h3>Arithmetic</h3><ul><li><b>Denise Gaskins</b> (@letsplaymath) shares an old favorite, <a href="http://denisegaskins.com/2008/09/22/things-to-do-hundred-chart/" target="_blank">30+ Things To Do with a Hundred Chart</a>, at her blog, <i>Let's Play Math</i>.</li><li><b>Brian Bushart</b> (@bstockus) shares <a href="https://bstockus.wordpress.com/2015/01/" target="_blank">Fraction Number Sense</a> at his blog, <i>Teaching To the Beat of a Different Drummer</i>. </li><li><b>Lior Pachter</b> shares <a href="https://liorpachter.wordpress.com/2015/09/20/unsolved-problems-with-the-common-core/" target="_blank">Unsolved math Problems and the Common Core</a> at his blog, <i>Bits of DNA</i>. (Lior writes about computational biology. I found this post thanks to Andrew Knauft, at <a href="http://blog.amathknauft.com/2015/10/share-from-repository-weekly_18.html" target="_blank"><i>LimSoup</i></a>.)</li></ul><h3><b> </b></h3><h3><b>Geometry </b></h3><ul><li><b>Stephen Cavadino </b>(@srcav) shares <a href="https://cavmaths.wordpress.com/2015/10/21/parallelograms/" target="_blank">Parallelograms</a> at his blog, <i>cavmaths</i>, on a student's creative way to find the area of a parallelogram.</li><li><b>Ioana I Pantiru</b> (@LThMathematics) shares <a href="https://lifethroughamathematicianseyes.wordpress.com/2015/10/17/playing-with-paper-folding/" target="_blank">Playing with Paper Folding</a> at her blog, <i>Life Through a Mathematician's Eyes</i>, showing the steps of an origami construction. In her post, <a href="https://lifethroughamathematicianseyes.wordpress.com/2015/10/15/maths-class-everywhere-project/" target="_blank">Maths Class Everywhere</a>, she asks readers to take her survey of math classes around the world. </li><li><b>Curmudgeon</b> shares <a href="http://matharguments180.blogspot.com/2015/10/498-circles-on-lattice.html" target="_blank">Circles on a Lattice</a>, at their blog, <i>Math Arguments 180</i>. I wonder if this would make a good problem for a math circle... </li><li><b>Greg Blonder</b>, a professor of manufacturing and product design, shares <a href="https://plus.maths.org/content/trisecting-angle-ruler" target="_blank">Trisecting the Angle With a Straightedge</a>, at <i>Plus Maths</i>.</li><li>There have been lots of posts in the past few months about classifications of pentagons (<a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile" target="_blank">here's one</a>), because a new (15th) type of pentagon that will tile the plane was recently found. Here's <a href="http://mathtourist.blogspot.com/2010/06/tiling-with-pentagons.html" target="_blank">a good background post</a>, from before the discovery, from the <i>Mathematical Tourist</i>.</li></ul><h3> </h3><br /><h3>It's All Connected</h3><ul><li><b>Miss D</b> shares <a href="http://www.missdtheteacher.blogspot.co.nz/2015/10/age-of-ultron.html" target="_blank">The Age of Ultron</a> at her blog, <i>Miss D the Teacher</i>, about teaching a unit on artificial intelligence in a way that gets at the deep ideas and really gets students thinking, partly through connecting math, science, and art. </li><li><b>Henri Picciotto</b> (@hpicciotto) posts about <a href="http://blog.mathedpage.org/2015/10/more-on-programming-in-education.html" target="_blank">Computer Programming and Math Education</a>. </li><li>What is the distance to Mars? It changes depending where the two planets are in their orbits. <b>John D. Cook</b> <a href="http://www.johndcook.com/blog/2015/10/24/distance-to-mars/" target="_blank">explains the math</a>. </li><li><b>Michelle</b> shares <a href="http://prairiecreek.typepad.com/herons/2015/10/making-time-for-the-serindipitous.html" target="_blank">Making Time for the Serendipitous</a> at <i>The Rookery</i>.</li> </ul><ul> </ul><h3>Ideas for Learning ...</h3><ul><li><b>Kate Snow</b> (@katesmathhelp) shares <a href="http://kateshomeschoolmath.com/how-to-teach-your-kids-to-read-math-and-be-more-independent-too/" target="_blank">How to Teach Your Kids to Read Math</a> at her blog, <i>Kate's Homeschool Math Help</i>. I'm still trying to teach my college students how to read math, with some of the same tips. </li><li><b>Manan</b> (@shalock) shares <a href="http://mathmisery.com/wp/2015/08/31/becoming-mathematically-fluent/" target="_blank">Becoming Mathematically Fluent</a> at his blog, <i>Math Misery.</i></li><li><b>Shecky</b> (@sheckyr) shares <a href="http://math-frolic.blogspot.com/2015/10/true-deep-beauty-comes-only-with.html" target="_blank">True Deep Beauty ...</a> at his blog, <i>Math-Frolic</i>, about the how our understanding of math deepens.</li><li><b>Chris Rime</b> is making <a href="https://partiallyderivative.wordpress.com/2015/09/30/october-2015-problem-calendars/" target="_blank">monthly math calendars</a> (Algebra I, II, and Geometry), available as doc or pdf at his blog, <i>Partially Derivative</i>.</li></ul><br /><h3>... And Teaching</h3><ul><li><b>Tom Bennison</b> (@DrBennison) shares <a href="http://blog.ifem.co.uk/how-to-enjoy-your-nqt-year/" target="_blank">How to enjoy your NQT Year</a> at his blog, <i>Mathematics and Coding</i>. [I had to look up NQT. It means newly qualified teacher, and in England and Wales, you are "inducted" in your NQT year, (generally) your first year of paid teaching.] I like his suggestion to make time for doing some math(s) yourself. </li></ul><br /><br /><br /><h3>Announcements</h3>I'm going to the <a href="http://jointmathematicsmeetings.org/jmm" target="_blank">Joint Mathematics Meetings</a> in January in Seattle. I'd love to connect with other bloggers who are going. There's a <b>math poetry reading</b> plus art exhibit on Thursday evening at 5:30. You can get all the details from <a href="http://poetrywithmathematics.blogspot.com/2015/10/jmm-seattle-1-7-16-poetrymathart.html" target="_blank">JoAnne Growney's Intersections blog</a>.<br /><br />http://mathmamawrites.blogspot.com/2015/10/math-teachers-at-play-91.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-4568363584153733410Fri, 18 Sep 2015 15:21:00 +00002015-09-19T11:35:29.397-07:00Joint Mathematics Meetings - Seattle in JanuaryI think I'd like to present. I've never done that at the JMM. I'd like your help. Here's (my second draft of) what I've written for my proposed abstract: <br /><blockquote class="tr_bq">Have you seen your students disengage from your calculus class in the first week as they struggle with the technical topic of limits? They don’t see the point, get mired in the algebra and can become alienated. I will share why I save limits for later and start out with an exciting and historical approach using slope and velocity. <br /><br />But perhaps your textbook, like mine, follows a traditional approach? I will also share how I used parts of two Open Education Resources (OER) by Matt Boelkins and Dale Hoffman, along with a few pages I created, to make a coursepack for my first unit. [Link to modifiable materials provided at talk, or by email.] Their materials gave my students the support they needed in our excursions off the traditional textbook’s beaten path. <br /><br />I’ll help you see why there’s a better order to the topics. (It’s not just the limits.) And I’ll show you one way to make Calculus fun for yourself and your students. <br /><br />You can use the experiences I share in my talk as inspiration to help you get started remixing OER to develop your own approach and materials. Using these materials in a coursepack alongside the required text may also be a way to show your reluctant department that they don’t need the $200-plus conventional textbooks. </blockquote><br /><ul><li> Have I said enough to make it clear what I have to offer?</li><li>What more should I say?</li><li>What should I change?</li><li>Would you come to my talk?</li></ul><br />(My deadline is in 4 days.) http://mathmamawrites.blogspot.com/2015/09/joint-mathematics-meeting-seattle-in.htmlnoreply@blogger.com (Sue VanHattum)4tag:blogger.com,1999:blog-5303307482158922565.post-7536002483892185123Sun, 23 Aug 2015 17:24:00 +00002015-08-23T10:24:26.972-07:00The algebra needed to read about climate change...<a href="http://www.occupy.com/article/kids-call-us-out-filing-lawsuits-science-based-climate-recovery" target="_blank">This article</a> (at <a href="http://occupy.com/">occupy.com</a>), on a lawsuit from a group of young people demanding that we do what it takes to recover from climate change, looks very interesting. One line seemed either wrong or surprising to me, though.<br /><br /><blockquote class="tr_bq">We must immediately commence carbon emissions reductions of 6% each year until the end of the century. Timing is crucial. If we wait until 2020 to begin emissions reductions the annual requirement is 15% per year. </blockquote>Starting only 5 years earlier, they are saying that we can do 2/5ths as much reducing each year, for 85 years instead of 80, and get the same result. It seems too dramatic. I want to think about how to analyze it. I don't yet know what assumptions I can make.<br /><br /><ul><li>Should I compare total emissions from now until 2100? (I think so.)</li><li>Should I assume emissions are <i>growing</i> exponentially from now until 2020 in the 2nd scenario? (I think so.)</li><li>What else would I need to know? (Are there other factors that make this more complicated?)</li></ul>This seems like a perfect question for pre-calculus. Too bad I'm not teaching it this semester.<br /><br />I think I got it. I think this assumes that we are currently increasing our carbon emissions at a rate of about 20% a year. We are not. It's more like a tenth of that - about 2.5%. (<a href="http://www.eia.gov/todayinenergy/detail.cfm?id=20872" target="_blank">Government source here</a>.)<br /><br />If you want to do some real math, think about what you would do before continuing. <br /><br />.<br /><br />.<br /><br />.<br /><br />.<br /><br />.<br /><br /><br />I figured it like this. I count this year's carbon emissions as 1. If we decrease 6% a year, that means we have 94% of the previous year's emissions. So the total emissions from now until 2100 is<br />S=1+.94+.94^2+...+.94^84. This simplifies to S = (1-.94^85)/(1-.94). Note that the .94^85 is so close to 0 that we can ignore it. We Get S=1/.06 = 16.666. So the article is saying that for the next 85 years, we can emit 16 times this year's emissions.<br /><br />If we increase until 2020, we would start with higher emissions, H. 15% decrease per year leaves 85% of the previous year's emissions. Our sum would be<br />S=H+.85H+.85^2H+...+.85^79H = H(1-.85^79)/(1-.85) = (almost) 1/.15 = 6.666.<br /><br />16.666 - 6.666 = 10. So somehow we get 10 times this year's emissions within the next 5 years. If our emissions are currently increasing so that our emissions next year is r, then<br />S = 1 + r + r^2 + ... +r^5 = (1-r^6) / (1-r) = 10. I asked <a href="http://wolframalpha.com/">wolframalpha.com</a> to solve this and got r = 1.2, for a 20% increase per year.<br /><br />I asked John Golden to check my work. <a href="http://mathhombre.tumblr.com/post/127403416469/climate-change-now-or-later-sue-van-hattum-got" target="_blank">He used a continuous increase model</a> and got close to 9%, mush lower. But still not low enough to match what's happening.<br /><br />So it seems that either the article has a typo, or my mathematical model is not including everything it should. Humanity seems to be at a tipping point. Can we change our ways of making decisions, from capitalism to something else, in time to save ourselves from our foolishness? I would like everyone to be able to do this sort of math.http://mathmamawrites.blogspot.com/2015/08/the-algebra-needed-to-read-about.htmlnoreply@blogger.com (Sue VanHattum)3tag:blogger.com,1999:blog-5303307482158922565.post-5117445434125085136Sat, 22 Aug 2015 16:54:00 +00002015-08-22T10:16:23.797-07:00Linear Algebra QuestionOn Thursday we arrived at Theorem 1 in David Lay's <i>Linear Algebra and Its Applications</i>:<br /><blockquote class="tr_bq">"<b>Uniqueness of the Reduced Echelon Form</b><br />Each matrix is row equivalent to one and only one reduced echelon matrix."</blockquote><br />The proof is in an appendix, which is a bummer, because this class feels like it could build from first principles nicely up to all its glory. The proof involves material from chapter 4, and I have to fight my way through it. Isn't he worried about being circular?<br /><br />I was thinking out loud in class. I said (more or less):<br /><blockquote class="tr_bq">If the system is consistent, it has a particular solution set. You can read the solution off from the reduced echelon form, so it can only give you one answer. [In class I wasn't thinking about free variables, and whether those could be different somehow. I was just thinking about problems with one unique solution.] We know it gives the right answer because <br />we've already shown that elementary row operations create row equivalent matrices, which have the same solution set.<br /><br />What about an inconsistent system? I'm not sure about that. If you can break his theorem, I'll give you extra credit. </blockquote><br />Well, I just broke his theorem, I think. (I hope none of my students are reading my blog yet.) Given the system<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-eJF8Ad0wYlI/VdiojIwWa7I/AAAAAAAABl8/sU-ZyelAoC8/s1600/matrices%2Bin%2Bproof.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-eJF8Ad0wYlI/VdiojIwWa7I/AAAAAAAABl8/sU-ZyelAoC8/s1600/matrices%2Bin%2Bproof.png" /></a></div><br />Have I broken his theorem? Should he have said this instead?<br /><blockquote class="tr_bq">"Each matrix representing a consistent system of equations is row equivalent to one and only one reduced echelon matrix."</blockquote>http://mathmamawrites.blogspot.com/2015/08/linear-algebra-question.htmlnoreply@blogger.com (Sue VanHattum)5tag:blogger.com,1999:blog-5303307482158922565.post-5830884857187811189Sat, 22 Aug 2015 05:35:00 +00002015-08-21T22:35:33.893-07:00Random Grouping Cards and SlipsI have just finished my first week of class.<br /><br />I have finally used <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxYllubzhQTHNQUTg/view?usp=sharing" target="_blank">Myra Snell's Random Grouping Cards</a>, to put students in groups. I've been wanting to do this for the past year, and finally got over my inertia problem. <a href="http://mathmamawrites.blogspot.com/2015/05/preparing-for-fall-semester-how-to-get.html" target="_blank">Research shows</a> that putting students in visibly random groups gets them participating more. (Visibly means they don't wonder if the teacher made it non-random.)<br /><br />Myra's cards work for a class of 32 students or (a bit) fewer. If you class is bigger or much smaller, you'll need something different. I couldn't figure out an easy way to get mine onto her format. So mine are Random Grouping Slips. I have sets for <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxT0QySUxaNkxDbEU/view?usp=sharing" target="_blank">16</a>, <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxb1dRSlk2S2QzTk0/view?usp=sharing" target="_blank">23</a>, <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxQlFSdm42MlZTY1E/view?usp=sharing" target="_blank">32</a>, and <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxVk5OUVd5UGZLYm8/view?usp=sharing" target="_blank">48</a> students. You cut off the first column, and then slice apart the rows.<br /><br />I was intrigued that I could not (easily) get 24 student slips. The last one would have put two people together in the last group who had been together before. The way I set it up was based on 16. There was no simple way to make it smaller.<br /><br />I ended up with classes with <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxYXNqMVpDQnByZDQ/view?usp=sharing" target="_blank">20</a>, <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxU3BZbU9SZjlVTTA/view?usp=sharing" target="_blank">40</a>, and <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxU3YxRnk4YkZtbXc/view?usp=sharing" target="_blank">28</a> students, so I've made those too now. They're organized a bit differently. I don't like the time it takes to cut them on the paper cutter. Hmm... <br /><br />Some of the students complain, but I think I am already seeing more of a community forming among the whole class. I'll be watching for ways in which this changes classroom dynamics.<br /><br />I have also finally begun to implement the Gallery Walk I learned about at the CAP (California Acceleration Project) conference from Myra. I hope to write about that soon.<br /><br />All three of my classes seem to be going well.http://mathmamawrites.blogspot.com/2015/08/random-grouping-cards-and-slips.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1749321542446669045Sat, 08 Aug 2015 20:29:00 +00002015-08-08T13:29:55.077-07:00Links on Saturday (lots for First Day)<b>First Day </b><br /><ul><li><a href="https://docs.google.com/document/d/1MoFqVB95zDA0meNMWLHbZD2H8BAw1xCmb4J8y1giP0M/edit" target="_blank">Julie Ruelbach wants help</a> getting more specific with her great first day plans. They already look fabulous to me. I will try to use some of her great ideas. </li><li>One of her commenters suggested <a href="http://www.scientificamerican.com/article/the-interleaving-effect-mixing-it-up-boosts-learning/" target="_blank">this Scientific American article on interleaving</a>. This may make me change my practices. And this <a href="https://www.psychologytoday.com/blog/make-it-stick/201406/make-it-stick-six-tips-students" target="_blank">Psychology Today article with 6 study tips</a> looks helpful for students.</li><li>I like this video <a href="https://www.youtube.com/watch?v=t5mGeR4AQdM" target="_blank">(the dot</a>), but I wouldn't be sure what the take-away was if I hadn't also saved Kristen Beck's post on <a href="http://teachtekbeck.blogspot.com/2015/07/finding-my-center.html" target="_blank">why she uses this in class.</a> </li><li><a href="http://maamathedmatters.blogspot.com/2015/01/setting-stage.html" target="_blank">Setting the stage, some questions to ask students in groups on day one. </a> </li><li> Math Plus has a lot of great articles. They've made some into <a href="https://plus.maths.org/content/put-plus-your-wall" target="_blank">posters for your classroom walls</a>. And they're free! (But I want to find a color printer to do justice to the one I picked.)</li><li>Talking Points, <a href="http://cheesemonkeysf.blogspot.com/2014/07/tmc14-gwwg-talking-points-activity.html" target="_blank">intro by cheesemonkey</a>, and <a href="https://drive.google.com/drive/folders/0B8XS5HkHe5eNfmNVSjYzXzRtTWVfUm1xWE9uRHdJbWZ6U05OdW9XLTc3ejV2OHdXYlQtSnM" target="_blank">lots of files</a>. I need to figure out how to use this!</li></ul><br /><b> First Week</b><br /><ul><li>Can you <a href="https://christopherdanielson.files.wordpress.com/2015/07/keynote-016.jpg" target="_blank">describe a graph</a> so your friend can draw it? (for calc in first week, or precalc toward the end)</li><li><a href="http://mathteachermambo.blogspot.com/2013/08/calculus-day-1.html" target="_blank">Average vs instantaneous velocity</a></li></ul><br /><br /><b>Other Good Stuff</b><br /><ul><li><a href="https://www.sciencenews.org/blog/context/science-heroic-tragic-statistical-flaw" target="_blank">The flaw in statistics that messes with the way science is done</a>.</li><li>Puzzle (statistics again): <a href="http://datagenetics.com/blog/june32015/index.html" target="_blank">A standard deviation puzzle</a></li><li><a href="http://blog.matthen.com/post/120471240676/visualising-numbers-100-243-and-12-by-splitting" target="_blank">Visualizing factoring: a GIF</a> </li><li>James Cleveland warms my heart with this wonderfully nerdy post on <a href="https://rootsoftheequation.wordpress.com/2015/07/17/how-to-pack-your-boardgames/" target="_blank">trying to create a formula for which games to pack</a> for his trip to TMC.</li><li>Maria Andersen thinks deeply about education. Her desire to figure out how institutions of learning can change faster led her on <a href="http://busynessgirl.com/the-road-back-to-higher-education/" target="_blank">a very interesting path</a>. She is a visionary.</li><li>Video: David Kung on <a href="https://www.youtube.com/watch?v=V03scHu_OJE" target="_blank">Diversifying the Mathematical Community</a> (At 24 min in, he talks about racism in housing and how it affects family wealth.) Fabulous talk. (Thanks to Cathy O'Neil.) It's been a long time since I've watched <a href="https://www.youtube.com/watch?v=WwslBPj8GgI" target="_blank">an hour-long video on teaching</a>. (Eric Mazur is definitely worth watching too. On peer instruction in a big lecture class, using clickers.)</li><li>Find a way to continue the sequence. There are <a href="http://letsplaymath.net/2015/08/03/math-with-many-right-answers/" target="_blank">many right answers</a>... (Denise Gaskins)</li><li><a href="http://blog.mathedpage.org/2015/07/handwritten-pythagoras.html" target="_blank">Proving the Pythagorean Theorem with drawings on graph paper.</a> (Henri Picciotto)</li><li><a href="http://www.intmath.com/blog/computers/newtons-method-accuracy-and-floating-point-numbers-10324" target="_blank">Computers, big numbers, rounding, and Newton's Method</a>. (Murray Bourne)</li><li>A mathematician (Steven Strogatz) talks about being slow at math, and other things he notices while learning about <a href="https://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1?utm_source=Discovering+the+Art+of+Mathematics&utm_campaign=6878e64cc3-july-2015-news&utm_medium=email&utm_term=0_ba010f6015-6878e64cc3-88453817" target="_blank">inquiry-based learning</a>. </li></ul><br /><br /><ul></ul>http://mathmamawrites.blogspot.com/2015/08/links-on-saturday-lots-for-first-day.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-4225903435669012233Sun, 12 Jul 2015 17:02:00 +00002015-07-12T10:02:14.662-07:00Playing with Math: Can you write a review?<a href="http://www.amazon.com/Playing-Math-Homeschoolers-Passionate-Teachers/dp/0977693937/ref=sr_1_1?ie=UTF8&qid=1436720299&sr=8-1&keywords=playing+with+math" target="_blank"><i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i></a> is on Amazon now! But we don't yet have any reviews. If you've gotten a copy of the book, can you write a review on Amazon? We would be so grateful. <br /><br />Warmly,<br />Suehttp://mathmamawrites.blogspot.com/2015/07/playing-with-math-can-you-write-review.htmlnoreply@blogger.com (Sue VanHattum)1tag:blogger.com,1999:blog-5303307482158922565.post-8952400340473749767Fri, 10 Jul 2015 21:12:00 +00002015-07-10T14:12:16.927-07:00Links on Friday<ul><li>What is the Golden Ratio? A boy thought a museum had it wrong, and got in the news for correcting them. Really, they used the less common version of the ratio, still right. Read about it at <a href="https://sensemadehere.wordpress.com/2015/07/09/ee-therai-ther-calling-the-whole-thing-off-at-the-science-museum/" target="_blank">Sense Made Here</a>.</li><li>Jonathan Halabi blogged about <a href="http://jd2718.org/2015/07/10/cc-algebra-conclusion-why-fewer-strong-scores/" target="_blank">how crazy the scores on the NY common core math tests are</a>. I wonder how other states report scores.</li><li>I've been wondering whether I can use the <a href="http://www.storytellingandvideoconferencing.com/16.html" target="_blank">principles of storytelling</a> to improve my teaching. </li><li>I wonder if I can modify any of <a href="http://mathforlove.com/2015/05/quick-physical-games-for-the-math-classroom/" target="_blank">these math movement games for kids</a>, so they'd work well with adults students.</li><li>How can we shift math education from memorizing to problem solving? How can we help students learn problem solving? (<a href="http://parenting.blogs.nytimes.com/2015/04/02/the-problem-with-math-problems-were-solving-them-wrong/?smid=fb-share&_r=0" target="_blank">NY Times article</a>)</li><li>I've figured this out before, and the answer is even somewhere on my blog maybe. But I am once again stuck. <a href="http://mathriddles.williams.edu/?p=77" target="_blank">Flipping coins to one side without looking... (on a Math Riddles blog)</a></li></ul><br /><br />I'll be leading a Math Jam for eight days just before Fall semester starts, helping students prepare to succeed in Beginning Algebra. My eight topics:<br /><ol style="text-align: center;"><li>Number Sense</li><li>Fractions</li><li>Negatives</li><li>Algebra</li><li>Percents</li><li>Graphing </li><li>Slopes</li><li>Problem-Solving </li></ol><br />For fractions, I plan to do a bit with Egyptian Fractions. <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html" target="_blank">Here's a site that looks good for that</a>. I looked at the <a href="https://www.beastacademy.com/resources/printables.php" target="_blank">Beast Academy site</a> to see if they had anything good. I found 5 things I liked: one game and two puzzles using the area meaning of multiplication, one puzzle on ordering of decimals, and one game like Taboo for communicating about shapes. <br /><br /><ol style="text-align: center;"></ol>http://mathmamawrites.blogspot.com/2015/07/links-on-friday.htmlnoreply@blogger.com (Sue VanHattum)2tag:blogger.com,1999:blog-5303307482158922565.post-3727457472068492505Thu, 02 Jul 2015 19:55:00 +00002015-07-02T12:59:04.219-07:00Playing with Math: Inspiring Online Conversations<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-DCKxF2QFiWc/VZWX1s9dGxI/AAAAAAAABks/dNKeYwP7NYc/s1600/front%2Bcover.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://3.bp.blogspot.com/-DCKxF2QFiWc/VZWX1s9dGxI/AAAAAAAABks/dNKeYwP7NYc/s200/front%2Bcover.png" width="139" /></a></div>First sighting of a comment on a mathematical blog post that was inspired by seeing the content in my book...<br /><br /><br /><br />Jonathan Halabi writes <a href="http://jd2718.org/" target="_blank">jd2718</a>. His post, <a href="http://jd2718.org/2009/11/28/puzzle-who-am-i/" target="_blank">Puzzle: Who am I?</a>, became one of the puzzles in <a href="http://naturalmath.com/playingwithmath/" target="_blank"><i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i></a>.<br /><br /><br /><br />Today Lara H replied to his post:<br /><blockquote class="tr_bq">I came across this puzzle in the book “Playing with Math.” I found a different solution based on a wrong assumption I made at the beginning of solving the puzzle. I was thinking that a number with 3 digits also has 2 digits so I made both of those statements true and came up with 4097, which works for all the other conditions.</blockquote><br />I responded with:<br /><blockquote class="tr_bq">I’d say ‘different interpretation’ instead of ‘wrong assumption’. I wonder how many solutions the puzzle has using your interpretation. (Pretty exciting to see my book has inspired new discussion on Jonathan’s blog post!) </blockquote><br />We are hoping that the book will inspire online conversations. This is the first drop of what we hope will eventually become a deluge. http://mathmamawrites.blogspot.com/2015/07/playing-with-math.htmlnoreply@blogger.com (Sue VanHattum)2tag:blogger.com,1999:blog-5303307482158922565.post-5420465107832640900Sat, 20 Jun 2015 17:24:00 +00002015-06-20T10:25:09.057-07:00Book Review: The Archimedes Codex<a href="http://2.bp.blogspot.com/-rLtAWr7wKLM/VYWgsxXvfKI/AAAAAAAABkM/thzpAXYBo9Y/s1600/The-Archimedes-Codex-9780306815805.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-rLtAWr7wKLM/VYWgsxXvfKI/AAAAAAAABkM/thzpAXYBo9Y/s320/The-Archimedes-Codex-9780306815805.jpg" width="211" /></a>I bought this book because I wanted to understand more about Archimedes' role in the ancient development of calculus ideas. When I got it, I was worried it would be another book I wouldn't want to wade through. I was so wrong!<br /><a href="http://www.betterworldbooks.com/the-archimedes-codex-how-a-medieval-prayer-book-is-revealing-the-true-genius-of-antiquity-s-id-9780306815805.aspx" target="_blank"><br /></a><a href="http://www.betterworldbooks.com/the-archimedes-codex-how-a-medieval-prayer-book-is-revealing-the-true-genius-of-antiquity-s-id-9780306815805.aspx" target="_blank"><i>The Archimedes Codex</i></a>, by Reviel Netz and William Noel, is fascinating. Like much <i>g</i>ood science writing these days, <i>The Archimedes Codex</i> reads like a detective story. It is gripping! Netz writes chapters about Archimedes, his math, and translation issues. Noel writes chapters about the travels of the manuscript, and the attempts to use modern technology to get better images of Archimedes' writing.<br /><br />In 1998 Christie's auctioned off this battered medieval manuscript which on its face was a prayer book, but also contained traces underneath of Archimedes work, which had been scraped off. It sold for two million dollars to an anonymous bidder. William Noel, of the Walters Art Museum in Boston, followed the story and emailed the agent of the buyer. The buyer agreed to work with the museum to attempt restoration of the manuscript. Most experts expected little from the work, since the manuscript was in such bad condition. But the project, which took years, brought to light previously unknown work by Archimedes.<br /><br />Archimedes had explored the idea of infinity more carefully than had ever been realized. He also did work in combinatorics, which no one had even suspected. The math is pretty easy to follow, and it's amazing. I've dogeared about a dozen pages, so I can read passages to my calculus students.<br /><br />This is perfect summer reading. Enjoy! <br /><br />http://mathmamawrites.blogspot.com/2015/06/book-review-archmedes-codex.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-6773212512299832421Mon, 01 Jun 2015 19:54:00 +00002015-06-01T12:54:57.464-07:00Imbalance Abundance Puzzles (We're in the New York Times!!) Paul Salomon posted some delightful puzzles a few years back, I got in touch with him about including them in the book, and now <a href="http://wordplay.blogs.nytimes.com//2015/06/01/salomon/" target="_blank">his puzzles are featured</a> in the New york Times' Numberplay column!<br /><br />I met Gary Antonick (who writes Numberplay) in person a month or two ago at a lovely meeting of math popularizers. We were both excited to meet each other*, and he asked if he could share some of the book's material in his column. Of course I said yes.<br /><br />I knew the column was coming today, but forgot to look until I saw Mike South's Facebook post mentioning it. Mike writes great math explanations on Living Math Forum, but doesn't blog. I wanted to include something of his in the book, but didn't manage it. (<a href="http://mathmamawrites.blogspot.com/2010/01/mike-south-on-meaning-of-zero.html" target="_blank">Here's Mike on thinking about zero</a>.)<br /><br />Gary included a great photo that goes so well with the puzzles, I want to make up a new puzzle to go with it. Hmm.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-PV4cjKfJ_Rs/VWy365AVJUI/AAAAAAAABjM/WaZ675fDlWw/s1600/mobile%2Bfor%2Bimbalance%2Babundance.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="http://4.bp.blogspot.com/-PV4cjKfJ_Rs/VWy365AVJUI/AAAAAAAABjM/WaZ675fDlWw/s400/mobile%2Bfor%2Bimbalance%2Babundance.jpg" width="400" /></a></div><br /><br /><br />If you don't already have your own copy of <a href="http://naturalmath.com/playingwithmath/" target="_blank"><b><i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i></b></a>, you can <a href="http://naturalmath.com/playingwithmath/" target="_blank">buy one here</a>.<br /><br /><br /><br /><br /><br />_____________<br /><span style="font-size: x-small;">*I finally got to meet the fabulous <a href="http://fawnnguyen.com/" target="_blank">Fawn Nguyen</a> in person, too! What an exciting day that was!</span>http://mathmamawrites.blogspot.com/2015/06/imbalance-abundance-puzzles-were-in-new.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-6965860764776970174Wed, 27 May 2015 18:48:00 +00002015-05-27T11:50:33.726-07:00Preparing for the Fall Semester: How to Get Students to Participate MoreLast summer, at a conference for the <a href="http://cap.3csn.org/" target="_blank">California Acceleration Project</a>, Myra Snell used a cool way to set up random groups of participants/students. I wanted to use it in my classes, but just didn't get around to it. It seems, after almost 30 years of teaching, that it has become hard to change the way I run my classroom.<br /><br />But I did change one thing this past semester. I noticed, while sitting in on a colleague's Calc III class, that I really appreciated the notices he wrote on the board at the beginning of each class. So I began to do it too. Maybe I could implement a few more good habits by watching other teachers during the second and third weeks of class.<br /><br />Coming back to those random groups... I recently read research that found two effective strategies for getting students to participate more. One is <a href="http://www.peterliljedahl.com/wp-content/uploads/Visibly-Random-Groups.pdf" target="_blank">visibly random groups</a>. 'Visibly' means that they can't suspect the teacher of manipulating the group memberships. Myra's method is clearly random, looks easy to implement, and allows for up to four different groupings per class day. You have a slip for each student, with a number, a letter, an animal, and a food on it (for example). Those slips are set up so that no one is with any of the same other people more than once. I've asked Myra for her slips, but last night I was eager to think about it, and <a href="https://docs.google.com/spreadsheets/d/1XDqgjDCAoRhXfqK3hgrJC9Rq45e0RYX_6yQbOMk3UGg/edit?usp=sharing" target="_blank">created my own</a>. I don't know if this is the best way to do it, but I think it will work. Myra's slips had the 4 terms in a square and mine will be all in a row. I don't think that's a problem.<br /><br />The second strategy which made a difference in student participation was student use of vertical whiteboards. The researcher(s?) compared paper and whiteboard, used vertically and horizontally. [Unfortunately, I can't find the research I originally read, which mentioned both the visibly random groups and the vertical whiteboards.] I'd like to try this out with the class I'll be teaching for the first time this fall, a compressed version of beginning algebra (first half of the semester) and intermediate algebra (second half of the semester). It's officially the same courses we've always taught, but I get to use a different curriculum, and will be using something project-based. I'm excited about implementing this.<br /><br /><br /><br />http://mathmamawrites.blogspot.com/2015/05/preparing-for-fall-semester-how-to-get.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-6866736071142130484Tue, 26 May 2015 07:08:00 +00002015-05-26T07:33:48.972-07:00Machinery, Lines and Circles<div class="separator" style="clear: both; text-align: center;"></div><div style="margin-left: 1em; margin-right: 1em;"><img alt="" class="aligncenter" src="http://cdn.wonderfulengineering.com/wp-content/uploads/2014/07/Sewing-Machine.gif" height="387" width="371" /></div><br /><br />On Facebook, someone posted an animation of how a sewing machine works. It wasn't enough to help me understand how the top thread manages to get around the bobbin mechanism. I searched on youtube, and nothing helped. <a href="http://www.ams.org/samplings/feature-column/fc-2015-05" target="_blank">This article on math and the sewing machine</a> made me think for a moment that I was getting it, but I still am not. How is that bobbin mechanism held in place in a way that allows the thread to get around it? (Do you see how the top thread moves past the whole back of the bobbin? How is that possible?) They say that the bobbin is held snugly inside its case, but how is the out part attached? <br /><br />I think I need a transparent sewing machine, so I can really see how this is working.<br /><br />On thing leads to another (especially online!), and I ended up at <a href="http://php.math.unifi.it/archimede/archimede_NEW_inglese/curve/geomeccan0.php?id=2" target="_blank">this site from a museum for mathematics</a>, called The Garden of Archimedes, in Florence, Italy, where I encountered this very simple statement about the difference between constructing a circle and a line - something I had never thought about before.<br /><br /><blockquote class="tr_bq"><span style="font-family: Arial,Helvetica,sans-serif;">The simplest curves are doubtless the line and the circle. To draw circles, one uses a compass. It's sufficient to keep a constant distance between the tracing point and the centre, and one obtains a near-perfect circle, even with a primitive compass. At first sight, one would think that tracing a segment is also a very simple operation: you just need to use a ruler or pull a string taut. In fact, things don't work exactly like that. In order to draw a good straight line with a ruler, one needs the ruler itself to have a "straight" side, but the value of a ruled line depends on the ruler that was used to make it. So, who made the first ruler? To apply the same method to the circle would mean, for example, to take a coin and trace its edge - the circular profile would be "intrinsic" to the instrument itself.</span><br /><span style="font-family: Arial,Helvetica,sans-serif;"><br /></span><span style="font-family: Arial,Helvetica,sans-serif;"> It would be better to apply to the straight line the principle used to draw the circle, rather than vice versa. </span></blockquote>Inatead of using a ruler or straightedge, can't you use the "pull a string taut" method, with something a bit less flexible than string? Maybe something that freezes into position? Hmm... Apparently that's not the avenue that was followed. You can find out the fascinating history of the solutions people found for this problem by going to the <a href="http://php.math.unifi.it/archimede/archimede_NEW_inglese/curve/geomeccan0.php?id=3" target="_blank">Garden of Archimedes site</a>.<br /><br />http://mathmamawrites.blogspot.com/2015/05/machinery-lines-and-circles.htmlnoreply@blogger.com (Sue VanHattum)2tag:blogger.com,1999:blog-5303307482158922565.post-2666961631460296769Tue, 19 May 2015 15:02:00 +00002015-05-19T08:02:06.762-07:00Teaching My Son (Post One of Many?)I started out really believing in unschooling. (Advice to self: Beliefs are dangerous.) My son has attended a free school, where he didn't have to attend classes (K-2), and then was homeschooled at the homes of friends (I'm a single parent), in groups of 2 to 8, with very little required of him. He has learned a lot over the years, but not in the conventional ways. If you're an advocate of unschooling, that may not sound like a problem at all. But for him it was. He thought he was 'behind' in reading, and felt bad about that. He totally avoided math because of how far behind he thought he was. He thinks he's dumb because he hasn't done the conventional academics.<br /><br />Now he wants to go to a 'normal' school. So I signed him up for 8th grade at a charter school his friend goes to. (I've heard great things about it, and it is supposed to be project-based.*) Part of going to a regular school means catching up on all the 'regular subjects.' So I've begun requiring him to do 'academics' daily. (He asks if he has to. I say yes. He then shows subtle signs of relief. He really wants me to make him do this. This blows my mind.)<br /><br />About a month ago, we started with 15 minutes of reading and 15 minutes of handwriting practice each day. I don't care about his handwriting. He does. He is so embarrassed about it that he resisted signing in for his trampoline class. A few weeks ago, I added spelling (his desire), geography (identifying the states), and math. This week we're adding science and an essay of the history of bikes. My opinion is that the only things he really needs to catch up on are math and writing (essays, stories, ...). It helps that we're doing this, because he also needs to become more aware of conventions - how to write dates, what schoolwork looks like.<br /><br />For math, we're using <i><a href="https://www.beastacademy.com/" target="_blank">Beast Academy</a></i>. We started with book 3A. Yes, the 3 means third grade. We don't mention it, but he knows this is "supposed to be" for younger kids. <i>Beast Academy</i> has challenging work, though, and if we make it though all eight of the levels (3A-D and 4A-D), I think he'll be pretty well-prepared to join a class of 8th graders. I will look over the 'standards' for 5th to 7th grade later this summer, and see what might be missing from what we're doing. I have made a math plan for the next 14 weeks, leaving out some of the topics in the <i>Beast Academy</i> books (perfect squares, variables, counting, logic, probability). I'm sure they are excellent, but my goal was to find a way to pare it down, so he gets as much as possible of the foundational skills he'll need, in the short time we have before he starts 8th grade.<br /><br />The first day that we did math, he was sitting next to me, saying his answers, waiting for me to confirm before he'd write them down. I did. (What he needs, as he takes on this huge emotional challenge, is support. Once he feels more secure, I'll be able to say things like "How can you decide whether that answer is right or not?")<br /><br />On the second and third days, I noticed that his wrong answers were usually one off. To me, that meant he wasn't noticing things I notice about even and odd numbers. I printed out <a href="http://nrich.maths.org/4308" target="_blank">something from the nrich site</a> that looked good. We haven't tried it yet.<br /><br />It's very fun for me to be planning out his math curriculum. But this is very stressful for him, so our work time can be full of conflict. Once he buckles down and gets started, I get to quietly support him. Mostly I just confirm his answers. He is already seeing progress, and feeling good about it. I am trying to use <a href="http://letsplaymath.net/2009/04/06/buddy-math/" target="_blank">Denise's technique of buddy math</a>, offering to do every other problem myself, and then talking my way through it. He seems to prefer doing the problems himself most of the time, but let me do one problem last night.<br /><br />The lessons he's working on are about finding perimeters. It has been a great way for him to work on adding numbers, with something extra thrown in. Most of the shapes have more than six sides. While we were working last night, I told him I noticed that he picked numbers that add to ten, which is a good strategy. I said that some people call those ten-bonds. He said he didn't know them all. I asked him for numbers that add to ten, and he got a bunch. The ones he hadn't mentioned, I asked him about: "Eight and ...?" He was surprised that there were only 5 pairs, I think. (At least two different stories in <a href="http://naturalmath.com/playingwithmath/" target="_blank"><i>Playing with Math</i></a> address this issue - Prison Math Circle and The Math Haters Come Around.) When he was unsure of 5 plus 8, I told him that I sometimes forget that one myself, and one way to figure it out is to move 2 from the 5 to the 8, so you get 3 and 10.<br /><br />I am exploring the balance between telling (ten bonds) and helping him to discover (hopefully we'll do that with odds and evens). I am so happy to be doing this, and marveling at how hard it was for me to see that he actually wanted me to make him do it.<br /><br /><br /><br /><br /><br />_____<br /><span style="font-size: x-small;">*Yes, I agree that charter schools are being used to mess up the regular public schools. Difficult situation all around.</span>http://mathmamawrites.blogspot.com/2015/05/teaching-my-son-post-one-of-many.htmlnoreply@blogger.com (Sue VanHattum)4tag:blogger.com,1999:blog-5303307482158922565.post-7879929957872002406Fri, 15 May 2015 06:59:00 +00002015-07-21T18:00:49.266-07:00Moebius Noodles is Delightful<a href="http://naturalmath.com/TheBook" target="_blank"><i>Moebius Noodles</i></a> is headed into its second printing soon. For the past few days I've been reading it over carefully to offer suggested edits. What a delightful task I gave myself! It has been so fun to remind myself of all the activities for young children Maria Droujkova and Yelena McManaman have put together.<br /><br />Their suggestion for creating an iconic times table got me dreaming. How can I get my son (who "hates" math, unfortunately) inspired to take photos for a times table collection? I was dreaming of a website that would show the whole table on one page, with each photo pretty small. And when you hover over a photo, that one would show up big. I don't know how to do that, though...<br /><br />Here's a photo (from <a href="http://lernertandsander.com/cubes">lernertandsander.com/cubes</a>) that feels like it belongs in the Grid section of <i>Moebius Noodles</i>, except that there's no pattern to the pieces. Well, the rows and columns are a bit wonky too. Hmm... <br /><br /><a href="http://function-of-time.blogspot.com/" target="_blank">Kate Nowak</a> posted this on Facebook. The question that came to her mind (among other less mathy questions) was ... How do <i>you</i> count these?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-V3wiv2uhcWo/VVWUFCsKHnI/AAAAAAAABgk/osYQhCA7XXk/s1600/food%2Bgrid.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="505" src="http://3.bp.blogspot.com/-V3wiv2uhcWo/VVWUFCsKHnI/AAAAAAAABgk/osYQhCA7XXk/s640/food%2Bgrid.jpg" width="640" /></a></div><br />[Edited to add: In the comments, Joshua described a very cool pattern he saw, and suggested that it's like 9 plus 4 is 13, which looks like my diagram below.]<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-5mPi4LObf8k/Va7q9hUKG8I/AAAAAAAABlM/sSlDYYLVK68/s1600/nine%2Bplus%2Bfour.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-5mPi4LObf8k/Va7q9hUKG8I/AAAAAAAABlM/sSlDYYLVK68/s1600/nine%2Bplus%2Bfour.png" /></a></div><br /><br /><br /><i>Moebius Noodles</i> has four sections: Symmetry, Number, Function, and Grid.<br /><br />The mirror book introduced in the symmetry section is so simple, and so cool to play with. Just get two small rectangular mirrors (at a dollar store), tape them together along one side, and use with photos or drawings, to see lots of symmetrical designs.<br /><br />My favorite game in the function section is Silly Robot. The grownup plays the robot, and follows orders exactly (while always trying to find a way to mess up the intention of the orders).<br /><br />If you know anyone with a child from one to eight who'd like to find ways to play around with mathematical ideas, <a href="http://naturalmath.com/TheBook" target="_blank"><i>Moebius Noodles</i></a> is a great resource. <br /><br />And my book, <a href="http://naturalmath.com/playingwithmath/" target="_blank"><i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i></a>, is delighting readers across the U.S. (and hopefully around the world). Here are a few photos of happy readers. Send me a photo of you with the book, and I'll add it to my collection (especially if you live far from me!).<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-dHD2CBXCn7w/VVWXKKoHTHI/AAAAAAAABhE/Wjw09pkDvLo/s1600/Dor%2BAbrahamson.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-dHD2CBXCn7w/VVWXKKoHTHI/AAAAAAAABhE/Wjw09pkDvLo/s320/Dor%2BAbrahamson.jpg" width="240" /></a><a href="http://2.bp.blogspot.com/-kMvSr1NMoVs/VVWXM9DlvcI/AAAAAAAABhM/IAKiUXtY3pY/s1600/Glenn%2BWaddell.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-kMvSr1NMoVs/VVWXM9DlvcI/AAAAAAAABhM/IAKiUXtY3pY/s320/Glenn%2BWaddell.jpg" width="180" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-EjBGv08aYoQ/VVWXNYW4ZqI/AAAAAAAABhQ/TZTDnIlZniw/s1600/joAnne.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-EjBGv08aYoQ/VVWXNYW4ZqI/AAAAAAAABhQ/TZTDnIlZniw/s1600/joAnne.jpg" /></a><a href="http://3.bp.blogspot.com/-9lstatPevtk/VVWXRvR2llI/AAAAAAAABhs/lf3zu6hsotw/s1600/rochelle%2B.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="158" src="http://3.bp.blogspot.com/-9lstatPevtk/VVWXRvR2llI/AAAAAAAABhs/lf3zu6hsotw/s320/rochelle%2B.png" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-rFquoTPGq8c/VVWXPjR69lI/AAAAAAAABhk/XON9oAc6Nxc/s1600/pei%2Band%2Bpaul.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-rFquoTPGq8c/VVWXPjR69lI/AAAAAAAABhk/XON9oAc6Nxc/s320/pei%2Band%2Bpaul.jpg" width="180" /></a><a href="http://2.bp.blogspot.com/-lN5hE9YKEo8/VVWXOurgqbI/AAAAAAAABhc/0bND7xxtACQ/s1600/megan%2Bschmidt.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-lN5hE9YKEo8/VVWXOurgqbI/AAAAAAAABhc/0bND7xxtACQ/s320/megan%2Bschmidt.jpg" width="180" /></a></div><br /><br />This is shaping up to be a very fun summer... <br /><br />http://mathmamawrites.blogspot.com/2015/05/moebius-noodles-is-delightful.htmlnoreply@blogger.com (Sue VanHattum)5tag:blogger.com,1999:blog-5303307482158922565.post-2083560167851424288Wed, 06 May 2015 03:43:00 +00002015-05-05T20:43:56.376-07:00Aunty Math<center> <span class="ambig"><b>Welcome to the world of</b></span><br /> <img alt="Aunty Math" height="135" src="http://web.archive.org/web/20081026000728im_/http://www.dupagechildrensmuseum.org/images/aunty_logo.gif" width="182" /> <br /> <span class="ambig"><b>Math Challenges for K-5 Learners</b></span> </center><br /><br /><br />One of the reasons I put together a book was my fear that good online writing often just disappears. One of the sites I had really liked - and thought of including somehow in the book - was a site with stories from Aunty Math (Aunt Mathilda). It disappeared before I could contact the author. And for years, I thought it was just plain gone.<br /><br />This evening I searched for Aunty Math, and found that someone had managed to get to this site through the <a href="http://en.wikipedia.org/wiki/Wayback_Machine" target="_blank">Wayback Machine</a>. It is now<a href="http://web.archive.org/web/20081026000728/http://www.dupagechildrensmuseum.org/aunty/index.html" target="_blank"> available as an archive</a>. Check out all eleven <a href="http://web.archive.org/web/20081026111109/http://www.dupagechildrensmuseum.org/aunty/chmain.html" target="_blank">past challenges</a>. I think you'll enjoy them.<br /><br />I would love to be in touch with the author, Angela G. Andrews. I googled her, but I don't see an email address. I'll just thank her here for her lovely stories. Thanks, Angela!<br /><br />My book, <i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i>, won't disappear. If you want<a href="http://naturalmath.com/playingwithmath/" target="_blank"> a copy to appear in your mailbox, order one now. </a><br />http://mathmamawrites.blogspot.com/2015/05/aunty-math.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-7135686202469414035Sat, 18 Apr 2015 22:34:00 +00002015-04-18T17:34:09.844-07:00The Book is Beginning to Arrive!!Dylan Kane (@math8_teacher) just posted this photo on <a href="https://twitter.com/math8_teacher/status/589531642571714560/photo/1" target="_blank">twitter</a> a few hours ago. It's the first sighting of <i>Playing with Math</i> in a crowdfunder's hand! <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ZfPkl2eLmv4/VTLagm8ZcDI/AAAAAAAABf8/qx5aQfZKPq4/s1600/book%2Bhas%2Barrived%2Bdylan%2Bkane.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-ZfPkl2eLmv4/VTLagm8ZcDI/AAAAAAAABf8/qx5aQfZKPq4/s1600/book%2Bhas%2Barrived%2Bdylan%2Bkane.jpg" /></a></div><br />We got a message from one more crowdfunder a few minutes later that her copy had arrived. The books are coming!<br /><br />My living room has stacks of books along one wall, sent to me by the publisher. I signed and repacked about twenty of them this morning, to send out to our $100 and over contributors.<br /><br />It is so exciting to know the book is finally in people's hands, after 6 1/2 years of work.<br /><br />Want to have the book in <i>your</i> hands? <a href="http://www.playingwithmath.org/" target="_blank">Order a copy now. </a><br /><br /><br />http://mathmamawrites.blogspot.com/2015/04/the-book-is-beginning-to-arrive.htmlnoreply@blogger.com (Sue VanHattum)4tag:blogger.com,1999:blog-5303307482158922565.post-4311819606910958077Wed, 01 Apr 2015 02:29:00 +00002015-03-31T19:29:43.920-07:00A New Site for Critical Thinking (wodb.ca)<b>Which One Doesn't Belong?</b> Many of us have played with puzzles like that since we were very young. Most of those puzzles had one right answer. <a href="http://talkingmathwithkids.com/2015/01/07/building-a-better-shapes-book/" target="_blank">Christopher Danielson</a> has been championing versions of this where every item could be the right answer. He's created <a href="http://talkingmathwithkids.com/2015/01/07/building-a-better-shapes-book/" target="_blank">a 16-page shapes book</a> for young children, built on this principle. And he recently <a href="https://christopherdanielson.wordpress.com/2015/02/08/the-twin-cities-shapes-tour/" target="_blank">took it out to classrooms around Minneapolis</a>, learning much about kids' understandings of shape.<br /><br />Christopher's enthusiasm has engendered enthusiasm across the MTBOS (math twitter blog o sphere), and tonight I was able to attend a Big Marker online event discussing a new website dedicated to these puzzles: <a href="http://wodb.ca/">wodb.ca</a><br /><br />What fun!<br /><br />And so one more nifty tool is added to our techno toolbox for math class. (I have been loving <a href="http://desmos.com/">desmos.com</a> for a few years now, and use <a href="http://visualpatterns.org/">visualpatterns.org</a> and <a href="http://estimation180.com/">estimation180.com</a> whenever I get a chance.)http://mathmamawrites.blogspot.com/2015/03/a-new-site-for-critical-thinking-wodbca.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-984176555522905727Sat, 21 Mar 2015 16:57:00 +00002015-03-21T11:22:24.010-07:00Algebra Skills Needed for CalculusSam Shah posted his list <a href="http://samjshah.com/2012/06/01/algebra-bootcamp-in-calculus/" target="_blank">here</a>. I loved his list, but wanted to rewrite it a bit for myself. (Also, Sam finds it more effective to review the algebra ahead of time, while I think it's more effective to review once we see the need in our exploration of calculus.) I am posting this now, so it's available as an answer to <a href="http://matheducators.stackexchange.com/questions/7654/what-basic-algebra-skills-and-techniques-are-most-important-for-calculus-student" target="_blank">this question on math educators stack exchange</a>.<br /><br />I teach my calculus course in an order that I think will help students learn. I have four units:<br /><ul><li>Unit 1 includes history, graphing functions, slopes of tangent lines by approximation, algebraically finding the derivative using the limit (which we do not carefully define yet), seeing the similarities between velocity, rate of change, and slope, average versus instantaneous velocity, derivative from a graph, (estimated) derivative from a table of values.</li><li>Unit 2 includes derivative properties needed for polynomials, graphing, limits and continuity, trig derivatives, and optimization.</li><li>Unit 3 includes chain rule, derivatives of exponential functions, implicit differentiation, derivatives of inverse functions (ln x, tan<sup>-1</sup>x), and related rates.</li><li>Unit 4 includes integration (finding area under the curve), anti-derivatives, fundamental theorem of calculus, and substitution method. If there is time we include volumes of rotation (which I think is a perfect ending for the course).</li></ul><br /><br /><b>Algebra Skills needed for Unit 1 </b><br /><br /><b>Algebra </b><br /><ul><li>Determine the equation of a line given two points, or a point and a slope, or a graph of a line, </li><li>Find the average rate of change over an interval given a function or its graph, </li><li>Clearly express what is happening to an object given a position versus time graph, </li><li>Evaluate f(x+h) for any given function f(x), </li><li>Rationalize the numerator (to find the derivative of the square root function) , </li><li>Simplify complex fractions (to find the derivative of the 1/x function). </li></ul><br /><b>Algebra with Calculus Concepts </b><br /><ul><li>Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or its graph, </li><li>Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change, </li><li>Sketch a velocity versus time graph given a position versus time graph, </li><li>Construct the formal definition of the derivative by modifying the definition of slope, </li><li>Apply the formal definition of the derivative to simple polynomials and to simple square root functions.</li></ul><br /><br /><b>Algebra Skills needed for Unit 2</b><br /><br /><b>Algebra</b><br /><ul><li>Multiply out the expression (x+h)<sup>n</sup> (necessary to understand the proof for the derivative of y=x<sup>n</sup>),</li><li>Identify the holes, vertical asymptotes, x- and y-intercepts, horizontal or slant asymptote, and domain of any rational function,</li><li>Sketch the basic shape of a rational function,</li><li>Identify an equation for a rational function given a sketch of the function,</li><li>Explain clearly what a hole and an asymptote are,</li><li>Construct the equation of a piecewise function given its graph,</li><li>Sketch the graph of a piecewise function given its equation,</li><li>Work with inequalities,</li><li>Give both triangle and circle definitions of sin x, cos x, and tan x, and explain how they’re related,</li><li>Evaluate sin x, cos x, and tan x at all multiples of π/6 and π/4, without a calculator,</li><li>Understand trigonometry identities, including and sin(<i>x</i>+<i>h</i>)=sin <i>x</i> cos <i>h</i> + sin <i>h</i> cos <i>x</i>,</li><li>Accurately graph y = sin x and y = cos x.</li></ul><br /><b>Algebra with Calculus Concepts</b><br /><ul><li>Graph a polynomial or rational function, showing its maximums, minimums, and inflection points,</li><li>Follow complicated logic (in the definition of limit).</li></ul><br /><br /><b>Algebra Skills needed for Unit 3</b> <br /><br /><b>Algebra</b><br /><ul><li>Understand composition of functions,</li><li>Use logarithm properties to “break apart” a single logarithmic expression into simple logarithms,</li><li>Understand properties of exponents,</li><li>Be able to graph exponential and logarithmic functions.</li></ul><br /><b>Algebra with Calculus Concepts</b><br /><ul><li>Think in terms of composition of functions to determine outer and inner functions, in order to use the chain rule.</li></ul><br /><br /><b>Algebra Skills needed for Unit 4</b><br /><b><br /></b><b>Algebra</b><br /><ul><li>Work with summations. </li></ul>http://mathmamawrites.blogspot.com/2015/03/algebra-skills-needed-for-calculus.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-4417516451629367157Sat, 14 Mar 2015 00:20:00 +00002015-03-13T19:57:29.984-07:00Copy Number One of Playing with Math At 3:30 this afternoon, UPS knocked on the door and delivered copy number one of Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers!<br /><br />It is beautiful!<br /><br />Now we put in the full order. Books coming soon...<br /><br /><br />If you haven't ordered your copy yet, <a href="http://www.moebiusnoodles.com/playingwithmath/" target="_blank">you still can</a>. <br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-UiAwfNLVP2U/VQN-nXuYBnI/AAAAAAAABfA/aHWS3zBl6FI/s1600/me%2Bwith%2Bbook.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-UiAwfNLVP2U/VQN-nXuYBnI/AAAAAAAABfA/aHWS3zBl6FI/s1600/me%2Bwith%2Bbook.jpg" height="266" width="400" /></a></div>http://mathmamawrites.blogspot.com/2015/03/copy-number-one-of-playing-with-math.htmlnoreply@blogger.com (Sue VanHattum)5tag:blogger.com,1999:blog-5303307482158922565.post-7841318941683305336Fri, 06 Feb 2015 22:09:00 +00002015-02-06T14:09:19.564-08:00Linkfest for Friday, February 6Before I share all the delicious goodies I've stumbled on, news of the book is in order:<br /><br /><a href="http://www.playingwithmath.org/" target="_blank"><i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i></a> is just about done with page layout - and it's looking so beautiful! I am sending in the last proofreading corrections today, and will do the last fixes to page number mentions as soon as I've seen the final copy. Then it's off to the printers, then all the copies get shipped to the publisher, and finally get sent to the hundreds of people who ordered copies during the crowd-funding last summer. If you're eager for your own copy and weren't around for the crowd-funding, you can <a href="http://www.moebiusnoodles.com/playingwithmath/" target="_blank">order now</a>. (You know I'd be tickled if we sell out our first printing quickly!)<br /><br /><br /><br /><b>The Links </b><br /><ul><li><a href="http://tube.geogebra.org/student/mpFrPrV8D" target="_blank">Here's a nice multiplication model</a>, using number lines, that makes multiplication by negatives, and by numbers like <i>e</i> and π, make as much sense as 2<span style="font-family: "Trebuchet MS",sans-serif;">x</span>3.</li><li>I may have linked this before. But just in case I haven't, here's <a href="http://fawnnguyen.com/hotel-snap/" target="_blank">Fawn Nguyen's Hotel Snap activity.</a> I want to play!</li><li>Christopher Danielson has made a shapes book, <a href="http://talkingmathwithkids.com/2015/01/07/building-a-better-shapes-book/" target="_blank"><b><i>Which One Doesn't Belong?</i></b></a>, that will tickle your logic funny bone. At first glance it's like the other books for very young kids, but this one will be fun for any age. This might be my new go-to present for 2-year-olds.</li><li><a href="http://ncase.me/polygons/" target="_blank">Using visuals and math to explain the process of segregation</a> - a powerful combination, especially in the hands of Vi Hart.</li><li>I always like Mr. Honner's photos. <a href="http://mrhonner.com/archives/14528" target="_blank">This one of Riemann Shadows</a> looks good to share with my calculus class.</li><li><a href="http://mathmisery.com/wp/2014/10/06/factoring-quadratics-misery/" target="_blank">Good post on the problem with factoring</a> (it's not so useful in real life), and how much more useful completing the square is. Another post on Math Misery about <a href="http://mathmisery.com/wp/2014/11/16/how-real-functions-can-fool-you/" target="_blank">two anti-derivative problems that look similar, use very different techniques, but <i>could</i> be brought closer together</a>.</li><li><a href="http://www.moebiusnoodles.com/2015/01/fundamental-ideas-of-calculus/" target="_blank">Maria Droujkova has a question for you:</a> What's one central idea of calculus you'd want everyone in the world to understand? </li><li><a href="http://rationalexpressions.blogspot.com/2015/01/proofs-of-pythagorean-theorem-what-am-i.html" target="_blank">Michael Pershan asks:</a> What bigger ideas are proofs of the Pythagorean theorem connected to?</li><li><a href="https://www.youtube.com/watch?v=v678Em6qyzk&feature=youtu.be" target="_blank">A video of Donald Knuth</a> (author of the lovely little book, <b><i>Surreal Numbers</i></b>), talking about learning from his mistakes.</li><li>Kate Nowak pulls together what looks like a great (5-part) <a href="http://function-of-time.blogspot.com/2015/01/ssa-asa-and-all-rest.html" target="_blank">lesson on triangles</a> - what you can figure out from what you are given (different combinations of sides and angles). I wonder if I can do anything with this in just one day with my pre-calc class... Shireen's <a href="http://mathteachermambo.blogspot.com/2015/01/inverse-trig-graphing.html" target="_blank">lesson for inverse trig functions</a> might be helpful too...</li><li><a href="http://wordplay.blogs.nytimes.com/2015/01/12/finkel/?_r=1" target="_blank">Numberplay</a> appears in the New York Times (at least online) every Monday. Last month Daniel Finkel (of <a href="http://mathforlove.com/" target="_blank">Math for Love</a>) provided this new take on a favorite puzzle type: <br /><blockquote class="tr_bq"><div class="story-body-text" itemprop="articleBody"><em>There are two bags of coins. One contains genuine silver dollars, and the other contains a mix of two types of counterfeits, the first of which is too heavy by 0.01 ounce, and the second of which is too light by 0.01 ounce.</em></div><div class="story-body-text" itemprop="articleBody"><em>Using a balance, you weigh the two bags and find that they both weigh exactly the same amount. How many additional weighings will it take to determine which is the bag of real silver dollars if there are 32 coins in each bag?</em></div></blockquote></li></ul><blockquote></blockquote><ul><li><a href="http://mathteachermambo.blogspot.com/2015/01/two-truths-and-lie.html" target="_blank">Two Truths and a Lie</a>: Get calculus students to make up stories from their lives, using the idea of rate of change, and matching given graphs. Brilliant, Shireen!</li><li>I like this for a first day activity! (I just figured out how to link to this on my google calendar to remember to look at it in August!) <a href="http://maamathedmatters.blogspot.com/2015/01/setting-stage.html" target="_blank">Getting the students involved in discussing</a> what education should be, and what productive failure might look like.</li><li><a href="http://setosa.io/ev/" target="_blank">Explained Visually</a> has animated graphics for trig functions, exponential growth, statistical processes, and more. Fun.</li><li><a href="https://tjzager.wordpress.com/2014/09/17/you-just-listened-so-then-i-could-figure-it-out/" target="_blank">Beautiful teacher story</a>. “You just listened, so then I could figure it out.” </li><li>This post asks: <a href="https://mathexchanges.wordpress.com/2014/12/16/is-there-room-for-math-that-isnt-hard/#comment-810" target="_blank">Is there room for math that isn't hard?</a> The post and comments are both interesting reading, and I'd enjoy seeing more comments. The blog is called Math Exchanges, and their more recent post, <a href="https://mathexchanges.wordpress.com/2015/01/21/over-or-under-a-fraction-number-sense-routine/" target="_blank">Over or Under</a>, is great too.</li><li>About half a year ago, I joined in the crowd-funding for the math game <a href="http://www.amazon.com/Math-for-Love-Prime-Climb/dp/B00PG9590G" target="_blank">Prime Climb</a>. It arrived in early December (or was it in Novemeber?) and we played it at my holiday party. People definitely enjoyed it. Now I've heard about another game being crowd-funded. <a href="https://www.indiegogo.com/projects/three-sticks-taking-geometry-to-the-next-level" target="_blank">Three Sticks</a> is a geometric game, developed in India. It looks fun. For a $35 contribution, you get the full set (and escape the very high shipping charges). </li><li>The math in the solutions may be too hard to follow, but <a href="http://blogs.ams.org/visualinsight/2015/01/15/hammersley-sofa/" target="_blank">this problem is charmingly simple</a>: Your hallway is one meter wide, and turns a corner. What is the greatest base area of an object that can be carried flat through the corner?</li><li>I'm not so good at making things (origami, etc), but <a href="https://onegoodthingteach.wordpress.com/2015/01/09/pretty-mathematical-sculptures/" target="_blank">these pretty mathematical sculptures</a> do look fun. </li><li>Every textbook I've seen that includes conic sections shows the conic, and then shows another definition, and never connects the two. <a href="http://plus.maths.org/content/conic-sections" target="_blank">This blog post</a> makes some of the necessary connections. (Anything on Dandelin's spheres catches my eye.) </li><li><a href="http://matharguments180.blogspot.com/2015/01/368-number-puzzle.html" target="_blank">Tricky puzzle</a>. (Do you like that sort of thing?) The 7 at the bottom is NOT a typo.</li><li>I'm always happy to hear about new math circles. <a href="http://blogs.kqed.org/mindshift/2015/02/playing-with-math-how-math-circles-bring-learners-together-for-fun/" target="_blank">Here's one in Santa Cruz, in the news.</a></li><li>Estimation questions are a great way to build number sense. And Andrew Stadel has <a href="https://twitter.com/Estimation180" target="_blank">a twitter feed</a> just for that. This week included a few questions about these Lego Lions: How many legos? How long to build? How many legos tall?<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-LnCH4KT36iU/VNTkznVRHyI/AAAAAAAABeI/8jZ2xr8g-B4/s1600/lego%2Blions.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-LnCH4KT36iU/VNTkznVRHyI/AAAAAAAABeI/8jZ2xr8g-B4/s1600/lego%2Blions.jpg" height="320" width="240" /></a></div></li></ul><br /><br /><b>A Question</b><br />I'm teaching Linear Algebra, and I find it a bit odd that linear transformations by definition don't include lines like y = mx+b (with b not 0). A student asked the significance of the word linear (she thought it was a silly question, and I assured her it definitely was not silly), so I started searching online. I noticed <a href="http://www.psychstat.missouristate.edu/introbook/sbk15.htm" target="_blank">this site</a>, which defines a linear transformation for statistics - differently from the linear algebra definition. It looks like the two definitions contradict one another. Any ideas about how standard this statistics definition is, or pointers to discussions of this difference in definition?<br /><br /><br /><br />[Oops! I lost a few weeks on the #YourEduStory challenge. Maybe I can get back to it. My pre-calc class is going better than usual. My calculus students loved having all those handouts in a coursepack. And I love thinking about all the connections in linear algebra. This week's topic: Define "learning" in 100 words or less.]http://mathmamawrites.blogspot.com/2015/02/linkfest-for-friday-february-6.htmlnoreply@blogger.com (Sue VanHattum)2tag:blogger.com,1999:blog-5303307482158922565.post-4492080188621427548Sun, 18 Jan 2015 20:19:00 +00002015-01-18T12:38:12.865-08:00My Favorite Teachers and MeThe <a href="https://sites.google.com/site/shareyouredustory/" target="_blank">#YourEduStory blogging challenge</a> question of the week:<br /><blockquote class="tr_bq">How are you, or is your approach, different than your favorite teacher?</blockquote><br />I don't have just one favorite teacher. I have lots. Long, long ago, before I started teaching, I made a list of my favorite teachers:<br /><blockquote class="tr_bq">Mr. West, high school biology, and then anatomy and physiology<br />Ms. Purvins, high school Shakespeare teacher<br />Mr. A, high school poetry teacher <br />Mr. X, UM philosophy prof<br />Ms. Y, UM history of feminism prof<br />Gisela Ahlbrandt, EMU math prof</blockquote> There were probably more on the list at the time. These are the ones I still remember. (And I'm losing the names. Yikes!) When I made the list, I noticed something interesting. There were about equal numbers of men and women on the list, but they were very different sorts of teachers. The men were good performers, and the women were good facilitators. A few did both well (the poetry guy and Gisela). I wanted to do both well. I thought about taking some drama courses to improve my performance skills. I did that while teaching in Muskegon, and realized I needed a different sort of course. Performing in a play is a lot different than performing as a teacher. Improv might be good for me. Hmm... I also learned a lot about facilitation over the years.<br /><br />I know now that the best performers make students happy to come to class, but that's not enough. We need to get students actively engaging with the material for them to learn much. (Mr. West did that in lab, even though I remember his great lectures.) If you don't know the research done by Eric Mazur on this, check it out. (<a href="https://www.youtube.com/watch?v=rvw68sLlfF8" target="_blank">This video</a> might include the best parts of the hour-long video I watched a few years ago.)<br /><br />How is my approach different than theirs? I think it's only in the combination that I'm different. I try to pull in all my students (like my Shakespeare and history of feminism profs did). I ask them multiple times each class to show me with thumbs up, down, or sideways how well they understand what I've just explained. I call on students randomly. (Because teachers tend to call on male students more.) I come in as excited as my bouncy philosophy prof. I suggest my students try strange experiments, like my poetry prof did (he had us write at a cemetery and a mall). I try to be as accepting and as challenging as my best teachers were. http://mathmamawrites.blogspot.com/2015/01/my-favorite-teachers-and-me.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-2718060294792438815Sun, 18 Jan 2015 19:54:00 +00002015-01-18T11:54:52.610-08:00Math Circles at Nueva School<a href="http://www.nuevaschool.org/outreach/math-circles" target="_blank">Nueva School</a>, in Hillsborough, south of San Francisco, puts on a math night three times a year, with multiple math circles, along with a puzzle and game room. Nancy Blachman invited me to lead two math circles last night, one for 2nd and 3rd graders and another for 4th and 5th graders.<br /><br /><br /><b>2nd and 3rd grade Circle</b><br />This circle met for just 30 minutes. I know that the Collatz conjecture is dependably fun for kids this age, so that was our main activity. I asked the kids what they thought mathematicians do, and got a reasonable answer, but saw that there wouldn't be time for useful discussion. So I said a bit about math being like a game for mathematicians, and how fun it was to come up with a new puzzle.<br /><br />In 1937 (I just said it was about a hundred years ago), Lothar Collatz came up with this puzzle/game:<br /><ul><li>Pick a number.</li><li>If it's even, cut it in half. Write your new number.</li><li>If it's odd, triple it and add one. Write your new number.</li><li>(We drew an arrow from each number to the next.)</li><li>Repeat until you get back to a number you've already written.</li></ul><br />Collatz conjectured (guessed) that the sequence would end up at 1, no mater what number you started with, but he couldn't prove his conjecture. Mathematicians have tried to prove this for over 75 years, and it is still an open question. (It is very likely to be true. Using computers, people have tested every number up to and past <a href="http://en.wikipedia.org/wiki/Collatz_conjecture" target="_blank">5 quintillion</a>.)<br /><br />As I expected, the kids loved it. At the end, I showed them a "mind reading" trick.<br /><ul><li>Pick a number from 1 to 31. Don't say it, just keep it in your brain.</li><li>(I pretend I'm sucking their thoughts over to my own head.)</li><li>Now show me which of these five cards it's on.</li><li>(I barely glance at the cards.)</li><li>Your number is ___.</li></ul>After we did it a few times, I had the parents cover their ears and told the kids how it worked. I had <a href="http://www.mathmaniacs.org/lessons/01-binary/Magic_Trick/" target="_blank">the five cards</a> on the board, and half-size index cards for them to make their own cards. They loved it.<br /><b><br /></b><b><br /></b><b>4th and 5th grade Circle</b><br />This circle met for an hour and a half. My plan was to analyze <a href="http://mathmamawrites.blogspot.com/2012/01/math-adventures-thinking-about-spot-it.html" target="_blank">Spot It</a> with them. (I've written at least 4 posts on using Spot It for math circles. Search on Spot It to find them.) We started out playing the game for about 15 minutes, which they all enjoyed.<br /><br />The problem was, half of them had done this last year in their math class at Nueva! Luckily, one girl had come early and I had shown her the number trick. I asked her if she wanted to teach it to the others. She did.<br /><br />I split the group in two, and she showed her group the number trick, while my group started thinking about the game. I had one boy who answered every question very quickly, and asking him to slow down didn't help. So, after we had figured out that there would be 57 different pictures, I got out the half-size index cards and suggested they make their own decks, with 4 pictures per card. Or, if they weren't into drawing pictures, 4 numbers per card. They worked hard at trying to make a deck where each card matched every other card on exactly one picture. Towards the end, they wanted to play with the number trick too.<br /><br />About halfway through the girl who led the other group came over and said, "The number trick is done." So I joined their group for a bit, and asked, "Why does it work?" A few parents were there, thinking about it with their kids. I should have asked them to work with all the kids (about 6 of them), but didn't think to say it. A few kids wandered away, to the puzzle room, no doubt.<br /><br />The kids who stayed worked hard on the problems and had fun. I had a great time.http://mathmamawrites.blogspot.com/2015/01/math-circles-at-nueva-school.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-3655403413846221696Fri, 16 Jan 2015 16:42:00 +00002015-01-16T08:42:20.228-08:00Days Three and Four<b>8am</b><br />Calculus. Wednesday: Circle area. Archimedes. Zeno. Started Boelkins' Velocity of a Ball activity. On Thursday, we got through most of the Velocity of a Ball activity. The students did not recognize that (s(b) - s(a)) / (b-a) is a slope. So We are working through the parts they need to review. I am goign slower than in other semesters. I hope I'm not going too slowly.<br /><br /><br /><b>10am</b><br />Linear Algebra. Wednesday: Discussed differences between Echelon Form and Reduced Echelon Form. I started with: a matrix in Echelon Form, and got them to tell me the values of the variables. I explained that this way is quicker for computers. We talked about number of possible solutions, and drew examples in 2D and 3D. Quiz tomorrow. (Quiz made and copied.)<br /><br />Thursday: Most of them aced the quiz. The ones who didn't will be in my office to retake. We finished 1.2. (I hate referring to book sections, instead of math topics. Basically, we are working on row reducing matrices. We've started to think about parametric representation of solutions, where there are free variables.)<br /><br /><br /><b>11am</b><br />Pre-Calc. Wednesday: We practiced an arithmetic sequence (find the nth term) and a geometric sequence. We looked at a problem that used a recursive definition for a <sub>n</sub>. I mentioned the Fibonacci sequence, but didn't do much with it. Quiz tomorrow.<br /><br />Thursday: Only a few aced the quiz. It was harder than what we had done in class. I'll give a retake on Tuesday in class. We reviewed lines. I walked them through my proof that perpendicular lines have slopes that are negative reciprocals. (It's different from the text's proof.) In the process, I also walked them through the proof that the angles in a triangle add up to 180 degrees. I love how the result suddenly pops out of the picture. I asked them to show me with their thumbs (up, down, sideways) how cool it was. They all gave it a thumbs up and I said they were being too nice. The bigger proof (for perpendiculars) gets an 80% coolness rating from me.<br /><br /><b>1pm</b><br />Calc III. (I am sitting in on this class.) Wednesday: Ed showed us how to connect the tops first and use dotted lines for hidden lines. I noticed that it felt like we were seeing the xz-plane from the back. Thursday: Over an hour of lecture. Ed is a good lecturer, but that's too long for me. I fell asleep. I woke up for the quiz. It included drawing 3D surfaces. I <i>understand</i> all of this, but how well did I draw? I'm not satisfied yet.<br /><br /><br />http://mathmamawrites.blogspot.com/2015/01/days-three-and-four.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-2271206563798208273Tue, 13 Jan 2015 21:19:00 +00002015-01-13T18:20:36.336-08:00Day Two<b>8am</b><br />Calculus. I talked about what we had done yesterday with finding a line tangent to y=x<sup>2</sup> at x=3. In algebra, we find the slope when we are given two points. We know one point, (3,9), and there is no other point that we know. [Last semester, at least one person used the y-intercept of the tangent line they had graphed as their second point. I liked that, but forgot to mention it today.]<br /><br />I asked them to give their definitions of the word tangent.<br />First student definition of tangent: A line that touches the curve in one place only. <br />Sue's counter-example: I drew y=x<sup>3</sup> and drew at tangent line at about x=1. They agreed that I had drawn a tangent. Then I extended the curve and the line. They cross at x = -2. I suggested that we could add the word nearby, and maybe this would work.<br /><br />Second student definition of tangent: A line that touches the curve but doesn't cross it.<br />Sue's counter-example: I asked them what the tangent to y=x<sup>3</sup> at x=0 would look like. They told me it would be horizontal. I drew it in. Hmm. (I told them that later we'll talk about concavity, and showed it with my hand curved. I said that I think the only time the tangent line crosses the curve is when it's tangent at an inflection point. Is that true? I should try to prove it.)<br /><br />Third student definition of tangent: A line that determines the direction of the curve.<br />I think this one is about as good as we can get at this point, although it's hard to turn it into something precise. I talked about thinking of the curve as a road, and your point being a car driving along the curve. Its headlights make half the tangent line, and its taillights make the other half.<br /><br />Talked just a bit about history of calculus, and gravity. Got some volunteers who will drop a heavy and a light object, and see what happens.<br /><br />Then we started our circle activity. I had a picture of a circle of radius 10cm on the back of the handout. I asked for the radius, a rough estimate of the area, and a more careful estimate of the area. (I asked them to pretend they knew no formulas. Next I had them fold a round coffee filters in half through the middle over and over, then cut it into wedges, and play with them. Tomorrow we'll do the area formula from that. Today I gave the definition of pi (C/D), and talked about how C=2*pi*r comes easily from this definition. I got a few volunteers who will measure around a circle and across it, using string, and will bring in their string tomorrow. Area is different...<br /><br /><br /><b><br /></b><b>10am</b><br />Linear Algebra. I used a desk corner as the origin, drew the x and y axes with my finger along its edges, and the z axis coming up from the corner. I asked them to figure out (in groups of four) what the equation x+y+z=1 would look like. I heard someone say circle. It is not at all obvious to most of them yet that it will be a plane. But we got there.<br /><br />Was that before or after we worked on the definition of a linear equation? Yesterday I had asked for their definitions from their heads. I got four volunteers today (yay!) to give me their definitions to put on the board. They were all different, and none matched the official definition. So, after I went over the official definition from our textbook, I asked them to use that to prove or disprove each of the statements given by students. I think this will help them with proofs and with what a linear equation is.<br /><br />Next I continued with the problem we had done, algebra style (no matrix), yesterday. I talked about computers, and representing it with just the coefficients, and wrote the matrix. I showed them the matrix that would represent the solution, and said our steps will be similar to those we used yesterday, but our order will be different. We did our same problem matrix-style, and I identified the three elementary row operations as we used them. (We never used the swap rows operation, but I talked about when it would be needed, and how you'd never do that with the algebra-style method.)<br /><br />I finished up with one book problem.<br /><br /><br /><b>11am</b><br />Pre-Calc. Stamped their homework. Had them share with their group the list of 5 problems they couldn't do. Had them each pick a problem from their partner's list, that they would later explain to their partner. Some people working hard; others feeling unsure what to do. (Everyone willing to participate.)<br /><br />Showed them y=mx+b on desmos, but got caught up in another problem. We'll come back to this tomorrow.<br /><br />They worked on finding a<sub>n</sub>, with the hint that it might be good to find a<sub>100</sub> first, for the sequence 12, 17, 22, 27. (I got starting value and jump size from students. Good it was five - some people struggle with arithmetic.) We worked on that a while, and then did a problem from 12.1 (Stewart) that turned out to be geometric. It was good to see the similarities.<br /><br />I loved my day. Now I'm off to the chiropractor.<br /><br />http://mathmamawrites.blogspot.com/2015/01/day-two.htmlnoreply@blogger.com (Sue VanHattum)0