tag:blogger.com,1999:blog-53033074821589225652016-06-22T13:05:42.393-07:00Math Mama Writes...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger546125tag:blogger.com,1999:blog-5303307482158922565.post-27326833840775531592016-04-13T18:25:00.001-07:002016-04-16T23:01:14.619-07:00Kahoot I heard about <a href="https://getkahoot.com/" target="_blank">Kahoot</a> from my colleague, who heard about it from his wife who teaches third grade. It's a game site with lots of content already available. I looked up logarithms yesterday, <a href="https://create.kahoot.it/?_ga=1.136995064.990966096.1460488149&deviceId=f761c75b-eb18-471e-b1e8-e0cb73ea629a#quiz/b188f973-7be4-4a29-868b-9a34074722ec" target="_blank">found a kahoot* I liked</a>, and played it with my pre-calculus class.<br /><br />[To find a kahoot you like, choose Public Kahoots in the black bar at the top, search on a term like logarithms, click on"<span class="filter-teachers pull-right">Only show Kahoots made by teachers?", and search the list. I've been looking for the ones with high counts on the favourites list, but I might find better criteria later. Once you find one you like, favorite it right away. There doesn't seem to be an easy mechanism to get back to it later.]</span><br /><br /><span class="filter-teachers pull-right">We are about 2/3 rds of the way through the semester. The energy is a bit low about now. This game livened things up and kept us focused on mathematical ideas. The students loved it. </span><br /><br />This evening, I made <a href="https://create.kahoot.it/?_ga=1.244598221.990966096.1460488149&deviceId=f761c75b-eb18-471e-b1e8-e0cb73ea629a#quiz/6048bced-6996-49e4-8d4d-e833d34c7dfe" target="_blank">a pretty simple kahoot</a> to go along with my <a href="http://mathmamawrites.blogspot.com/2010/04/murder-mystery-project-for-logarithms.html" target="_blank">murder mystery</a>, which we're starting in precalc right now. I'll use this kahoot next week, when we're farther along in the murder mystery.<br /><br /><br /><br />______<br />*A kahoot is a gamified quiz. Each question is set up with multiple answers. Students use a pin shown on the screen to sign in using their phones. They get points for right answers based on how quickly they answer.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-69803894222965197442016-03-07T15:14:00.001-08:002016-03-09T16:34:00.376-08:00A Huge Bunch of Lovely LinksI have so many tabs with cool math posts, I don't know if I can possibly get them all into this collection. (I never seem to have enough time to finish, and then more goodies accumulate.)<br /><br /><b><br /></b><b>Math & Teaching Ideas I might use</b><br /><ul><li>I want to show this to my pre-calc class as part of their intro to trig: <a href="https://www.facebook.com/worldofskipper/videos/241010499356393/" target="_blank">sailboat leans to get under bridge</a></li><li>Wild Maths has a lovely collection of questions with photos: <a href="http://wild.maths.org/tags/move-it-prove-it" target="_blank">Move it to prove it </a></li><li><a href="http://samjshah.com/2016/01/17/snug-angles/" target="_blank">Snug angles</a>, from Sam Shah. He's doing it with a geometry class. I'm wondering if it might be good at the beginning of the trig unit. </li><li>As I prepared for Math Jam, our 3-day pre-semester math boost, I found lots of cool ideas here: <a href="http://map.mathshell.org/lessons.php" target="_blank">Math Assessment Project</a> (assessment doesn't sound promising, but these activities have lots of open-ended questions)</li><li>I just learned that if you start with the harmonic series, which diverges, and <a href="https://collegemathteaching.wordpress.com/2016/01/14/trimming-a-divergent-series-into-a-convergent-one/" target="_blank">take out all the terms with 9s in the denominator</a>, <a href="http://blogs.ams.org/blogonmathblogs/2015/12/21/mind-blowing-math-reminiscence" target="_blank">you'll get a converging series</a>. Too weird. I don't understand it yet, but I sure want to. (for Calc II)</li><li><a href="https://makingmathpeople.wordpress.com/2016/01/17/2016-blogging-initiative-week-2" target="_blank">Geometric construction on sciencevsmagic.net.</a> (I knew about this site, but reading this post made me decide to use it in Math Jam to get them playing around.)</li><li><a href="http://www.whatdowedoallday.com/2016/01/mobius-strip-hearts.html" target="_blank">Mobius Hearts.</a> Too fun not to do. (I didn't use it, though. Too overwhelmed this past month to do anything new...)</li><li>Sam Shah made<a href="http://explore-math.weebly.com/" target="_blank"> </a><a href="http://explore-math.weebly.com/" target="_blank">this fabulous website</a>, Explore Math, that pulls together gobs of cool math resources from the web. He has his students pick one (or was it a few?) to play around with and report on. I believe I'm going to do this in pre-calc.</li><li>What is proof? <a href="http://blog.amathknauft.com/2016/01/martin-asked-on-twitter-whether.html" target="_blank">Here's a good conversation</a> about proving the Pythagorean Theorem with visuals. Includes the best video I've ever seen, on my favorite proof.</li><li><a href="http://www.appetite-for-instruction.com/my-favorite-trig-tale/" target="_blank">Trig Fairy Tales</a> (having students write them) </li><li><a href="https://plus.maths.org/content/ping-pong-balls-and-super-powers" target="_blank">Infinity is so weird!</a> (infinite ping pong balls in, infinite ping pong balls out, how many left in?)</li><li><a href="https://plus.maths.org/content/population-growth" target="_blank">Infinite sums and China's demographics</a> </li><li><a href="http://tube.geogebra.org/material/simple/id/134244" target="_blank">Algebra Aerobics Stick Figure in Geogebra</a></li></ul><b></b><br /><b></b><br /><b>Problem Solving</b><br /><ul><li><a href="https://plus.maths.org/content/dropping-eggs-solution" target="_blank">Finding out how far you can drop an egg without breaking it</a></li><li><a href="http://musingmathematically.blogspot.com/2016/02/candies-pennies-and-inequalities.html" target="_blank">Systems of equations, using a problem with no solution</a></li><li><a href="https://mikesmathpage.wordpress.com/2016/01/18/my-favorite-watching-problem-solving-ideas-develop/" target="_blank">On problem solving, with videos</a>. I might give the absolute value problem in precalc, as a challenge.</li><li><a href="http://musingmathematically.blogspot.com/2016/02/my-favourite-surface-area-question.html" target="_blank">Doubling surface area, a good question </a></li><li><a href="http://considerlearning.com/2016/02/08/5-minute-problems-to-five-year-problems/" target="_blank">What's the longest time you've ever spent solving a problem?</a></li><li><a href="http://aperiodical.com/2016/02/open-season-pancake-flipping/" target="_blank">Flipping pancakes</a> </li></ul><br /><b><br /></b><b> </b><br /><b>Using Desmos</b><br /><ul><li><a href="http://mrhonner.com/archives/15951" target="_blank">An introduction to desmos</a> </li><li><a href="http://blog.amathknauft.com/2016/01/designing-and-assessing-desmos-calculus.html" target="_blank">Linearization in Calculus</a>, an amazingly detailed lesson using desmos, with commentary about how students did with it</li><li>I do a unit in trig called Days Of Our Lives, using minutes of daylight on each day of the year as data, and getting students to construct an equation for it. This <a href="https://student.desmos.com/activitybuilder/student/56ad0a34dd023fde0ba35760" target="_blank">Moon Illumination project</a> someone made on desmos using the activity builder looks like something I could imitate. (Where did they get their data? Who made this?)</li><li><a href="http://musingmathematically.blogspot.com/2016/01/desmos-art-project.html" target="_blank">Desmos art project </a></li></ul><br /><br /><b>On Teaching</b><br /><ul><li><a href="https://problemproblems.wordpress.com/2015/12/27/the-problems-of-writing/" target="_blank">Michael Pershan, on writing about teaching</a><b> </b></li><li><a href="http://blog.peerinstruction.net/2016/01/08/how-to-help-people-remember-what-they-learn/" target="_blank">How to help people remember what they learn (using retrieval practice)</a></li><li>How do you respond to wrong answers? <a href="http://profteacher.com/2016/01/16/explanatory-power-of-the-hierarchy-of-student-needs/" target="_blank">This post helps me think about that.</a></li><li>A good summary of <a href="https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/" target="_blank">Dweck's Mindset research</a> </li><li><a href="https://researchinpractice.wordpress.com/2016/01/28/lessons-from-bowen-and-darryl" target="_blank">Ben Blum-Smith on the strategies used at PCMI</a>. "when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They <i>never</i> ask for hands, and they never cold-call. <i>This means they already know more or less what the students are going to say." </i>And then <a href="http://cheesemonkeysf.blogspot.com/2016/02/lessons-from-lessons-from-bowen-and.html" target="_blank">Elizabeth responded</a>. I loved her katamari.<i><br /></i></li><li><a href="http://prairiecreek.typepad.com/herons/2016/02/lets-talk-about-it.html" target="_blank">Using sentence starters for math conversations</a> with 4th and 5th grade students </li><li><a href="http://www.fractiontalks.com/p/how-to.html" target="_blank">Fraction talks </a></li><li><a href="http://learn-always.com/2016/01/20/my-favourite-getting-students-talking-to-each-other-about-math-mtbos/" target="_blank">Getting students talking to each other</a> </li><li><a href="http://education.lms.ac.uk/2016/02/ronnie-brown-answer-to-a-students/" target="_blank">Getting students not to fear confusion</a> </li><li><a href="http://www.cbc.ca/news/health/physical-activity-class-lessons-1.3460346" target="_blank">Physical activity during lessons improves learning</a> (research with elementary students, but I imagine it would help my college students too. Yikes! I don't like this perspective: "the researchers found no differences on reading scores. They think activity works better for subjects with a lot of memorization and repetition." Math should not have lots of memorization!)</li><li><a href="https://www.washingtonpost.com/local/education/teachers-are-using-theater-and-dance-to-teach-math--and-its-working/2016/02/22/61f8dc0c-d68b-11e5-b195-2e29a4e13425_story.html" target="_blank">More movement and math</a>...</li><li>If I were a high school teacher, I'd seriously consider this. <a href="http://blog.amathknauft.com/2016/02/notes-and-homework-structures-calc-bc.html" target="_blank">Metacognition and homework</a></li><li><a href="http://www.tandfonline.com/doi/full/10.1080/10511970.2015.1027837" target="_blank">On Metacognition</a> (download pdf, interesting part for me is sections 3 and 4) </li></ul><a href="http://blog.peerinstruction.net/2016/01/08/how-to-help-people-remember-what-they-learn/" target="_blank"><br /></a><b><br /></b><b>Science</b><br /><ul><li><a href="http://ncase.me/emoji-prototype/?remote=-K71iWhftjtOIfx6b2Fo" target="_blank">Simulation, mathematically modelling</a> how chemistry and growth work together</li></ul><br /><b>Statistics</b><br /><ul><li><a href="http://drhagen.com/blog/the-missing-11th-of-the-month/" target="_blank">The missing 11th of the month</a></li><li><a href="http://markkreie.blogspot.com/2016/01/my-favorite-linear-regression-movies.html" target="_blank">Linear Regression and Movies</a></li><li><a href="http://www.johndcook.com/blog/2016/02/20/the-empty-middle-no-one-is-average/" target="_blank">Why no one is average</a> </li></ul><br /><br /><b>Estimation & Elementary</b><br /><ul><li><a href="http://gfletchy.com/the-apple/" target="_blank">How many blocks will equal an apple?</a> (3-act lessons, with video) </li><li><a href="https://aerecord.wordpress.com/2016/01/18/my-favorite-activity-number-talks/" target="_blank">Number Talks</a></li><li>Pre-algebra: <a href="http://www.mathedpage.org/manipulatives/slides/lg-2d-arithmetic/lab-gear-2d-arithmetic/assets/player/KeynoteDHTMLPlayer.html#15" target="_blank">Working with signed numbers</a></li></ul><br /><br /><b>Math for Parents</b><br /><ul><li><a href="http://education.lms.ac.uk/2015/08/parents-math-anxiety-can-undermine-childrens-math-achievement/" target="_blank">Parents’ Math Anxiety Can Undermine Children’s Math Achievement</a></li><li>Fractions may be elementary (previous topic), but the idea of fractions is also the first math concept that messes a lot of people up. Here's <a href="http://gdaymath.com/courses/fractions-are-hard/" target="_blank">James Tanton's new collection on fractions</a>. </li><li><a href="http://education.lms.ac.uk/2016/02/when-did-you-stop/" target="_blank">When did you stop</a> playing around with mathy ideas?</li><li> A video about <a href="http://gfletchy.com/2016/03/04/the-progression-of-addition-and-subtraction/" target="_blank">what kids learn in the early grades about addition and subtraction</a> (Please let me know what you think!)</li><li><a href="http://ww2.kqed.org/mindshift/2013/10/01/finding-the-beauty-in-math/" target="_blank">Finding the Beauty in Math</a> </li></ul><br /><b>Social Justice</b><br /><ul><li><a href="https://sonatamathematique.wordpress.com/2016/02/21/an-evolution-of-my-reaction/#comment-723" target="_blank">On responding to people's surprise that I'm a math teacher</a></li><li><a href="http://considerlearning.com/2016/01/27/an-actual-response-to-chief-justice-roberts/" target="_blank">How affirmative action makes for a better physics education</a></li><li>"<a href="http://hiphopchess.blogspot.com/2016/02/why-dont-black-kids-like-math-and.html" target="_blank">Why Black kids don't like math...</a>" </li><li><a href="http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169" target="_blank">The master's tools...</a> (Dr. Danny Martin's talk at NCTM conference)</li></ul><br /><br /><br /><b>Playing with Math</b><br /><ul><li>As usual, <a href="http://gameaboutsquares.com/" target="_blank">this game (called this game is about squares)</a> is more about logic than about math. What I'm finding interesting is how impossible it seems, and then when I (and others) go away and come back, it can suddenly seem so easy.</li><li> Tracy Zager wrote a great post on <a href="https://tjzager.wordpress.com/2016/01/05/my-criteria-for-fact-based-apps" target="_blank">evaluating math fact apps</a>. Lots of good ones are mentioned in the comments. [My comment: I would really love to be able to find this app online so I can recommend it. I have this game on my phone. It seems to be called 1 Whole. There are rectangular shapes that fill with liquid. You push one toward another and they go together if the sum is less than or equal to one. You watch the liquid rise. If it’s 1, it goes away and you get points. You keep going until the screen is full of things that won’t combine (sum > 1). There is no time pressure, the conceptual basis seems strong to me, and mistakes aren’t allowed. No penalties, no bad sounds, it just won’t work. I think it’s pretty good. I wish I could find it online. Cna anyone help me?]</li><li>Kids like doing the simple math involved in thinking about the Collatz Conjecture. [Start with any number (whole, >1). If odd, triple it and add 1. If even, cut in half. Repeat. Does this always end up at 1? Conjecture is 'yes'.] Mathematicians don't know the answer, but they like to explore the question in sophisticated ways. Here's a <a href="http://gottwurfelt.com/2016/01/10/logarithmic-approximations-for-collatz/" target="_blank">post on what sorts of functions come close</a> to modeling the number of steps it takes to get to 1 from each number.</li><li>This game would have made it into my book, I think. <a href="https://mindfull.wordpress.com/2016/01/10/cross-over-a-game-for-practicing-addition-and-subtraction/" target="_blank">Cross Over</a> looks like it has enough strategy to entertain us jaded adults, and it's for addition and subtraction practice. Coolo.</li><li>Not math. Go. <a href="http://aperiodical.com/2016/01/learning-to-play-go/" target="_blank">Learning to play go</a>. </li><li>New game for iphone (really, it's logic not math), <a href="http://blog.tanyakhovanova.com/2016/02/ringiana/" target="_blank">Ringiana </a></li><li>I love <a href="https://mikesmathpage.wordpress.com/2016/02/04/i-think-you-can-share-the-surreal-numbers-with-kids/comment-page-1/#comment-1992" target="_blank">surreal numbers</a>. I need to come back and read this more carefully when I have more time to play with it. </li><li>A silly little game. Totally violates Tracy's criteria (nothing timed). But mathy folk may like it. <a href="http://isthisprime.com/game/" target="_blank">How many primes can you identify in a minute</a> (with no mistakes)? (Use y and n for y and no.)</li></ul><br /><br /><b>Books</b><br /><ul><li>Here's a great list of <a href="http://aperiodical.com/2016/01/books-a-14-year-old-whos-good-at-maths-might-enjoy/" target="_blank">fun math books</a>, compiled with a 14-year-old in mind, but almost all good for adult mathophiles too. I think <a href="https://mikesmathpage.wordpress.com/2016/01/18/some-book-suggestions-for-a-14-year-old-who-loves-math/" target="_blank">this list</a> came from the same question and has a different set of books.</li><li>My publisher is having a <a href="http://naturalmath.com/goods/" target="_blank">sale</a>. All 5 books published by <a href="http://naturalmath.com/goods/" target="_blank">Natural Math</a> for $50 total. What a great way to expand your playful math collection. </li></ul><ul></ul>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-6139186327488816372016-01-16T22:14:00.000-08:002016-01-16T22:14:23.621-08:00My Favorite Course (to teach): Calculus<b><span style="color: #cc0000;"><span style="font-size: large;"><span style="color: #3d85c6;">Why is calculus my favorite?</span> Let me count the ways ...</span></span></b><br /><ol><li>It tells a story.</li><li>It has cool historical connections,</li><li>... and great connections to science.</li><li>It's a good time to help students start to see what proof means.</li><li>I keep learning more.</li></ol><br /><br /><b>Calculus Tells a Story...</b><br />...if we let it. And the conventional textbooks don't. So I used two different creative commons texts (Boelkins and Hoffman), some of my own materials, and a few things from some of my favorite bloggers, and I made a coursepack to use for the first three weeks. I gave a talk about it at the Joint Mathematics Meeting a week ago. As part of my preparation for that, I made a new blog page. Click 'calculus' above, and you'll see all of my materials, including the slides from my talk, links to the creative commons texts I used, and lots more.<br /><br />What stories does calculus tell? It takes one of the central concepts from algebra, that of slope, and twists it so it will work for curves. To do that, we need to consider two points that are "infinitely close together," whatever that means. So we have to delve into the weirdness of "infinitely close." Once we get good at all that, we can find out where things reach their maximum and minimum values, and use that to graph all sorts of curves. We also use that to optimize, to get the most volume with the least surface area (when building boxes), for instance. And then we play with finding areas of strange shapes, and how that's connected to slopes. <br /><br /><b><br /></b><b><br /></b><b>Calculus has cool historical connections, and great connections to science.</b><br />Archimedes figured out all sorts of things that are really a part of calculus (call it proto-calculus), and used the 'method of exhaustion' which is a foundation for what we now do with limits. Newton and Leibniz are credited with inventing calculus, even though lots of what we do in Calculus I had already been figured out. The main thing they discovered was what we call the Fundamental Theorem of Calculus, which says that areas and rates of change are inverse functions. It makes sense that two different people invented calculus because it was needed at the time for the science questions that were being considered: lenses and light, paths of planets, gravity, angle to shoot a cannon, volume of the Earth. And then it took 150 years to get that limit thing just right, and another 150 years (in 1960 Abraham Robinson invented non-standard analysis) to prove that Newton's original conception (of fluxions) wasn't so far off.<br /><a href="http://2.bp.blogspot.com/-AYDhD3yloxg/Vpsp8ICsNzI/AAAAAAAABpA/rubX2d-kln0/s1600/220px-CircleArea.svg.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://2.bp.blogspot.com/-AYDhD3yloxg/Vpsp8ICsNzI/AAAAAAAABpA/rubX2d-kln0/s200/220px-CircleArea.svg.png" width="184" /></a><br /><br /><br /><b>It's a good time to help students start to see what proof means.</b><br />Did you realize that the two 'formulas' we all know for circles are very different sorts of creatures? The first, C=2*pi*r, is really just a restatement of a definition. pi is <i>defined</i> to be C(ircumference) over D(iameter), so it takes 2 or 3 algebraic steps to get to C=2*pi*r. But the other, A = pi*r<sup>2</sup>, should be proved. The simplest almost-proof comes from cutting the circle up and rearranging it.<br /><br /><br /><br /><b>I keep learning more.</b><br />I learned two cool things while preparing for that talk: <a href="http://mathmamawrites.blogspot.com/2016/01/newton-and-notion-of-limit-he-knew-more.html" target="_blank">Newton had a clearer conception of limits than we usually think</a>, and <a href="http://mathmamawrites.blogspot.com/2015/12/the-roots-of-calculus-archimedes.html" target="_blank">Archimedes' calculation of an approximation for pi</a> was easier to follow than I would have imagined, and really simple and beautiful (in our modern notation).<br /><br />And to make this post a fun one for all you MTBOS folks, here's the worksheet I designed to share with my calculus class (.<a href="https://drive.google.com/file/d/0B4Lou9CsLnQxQTR1X0RzU1VCZWc/view?usp=sharing" target="_blank">doc</a> and .<a href="https://drive.google.com/file/d/0B4Lou9CsLnQxWV9UMVVza0Nldlk/view?usp=sharing" target="_blank">pdf</a>), leading them through Archimedes' first few steps as he worked toward the 96-gon to approximate pi. Go ahead, try it and put your answer for the 96-gon in the comments. (I couldn't find it anywhere else online!)<br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-i10vclQgGmw/VpsvZavb-hI/AAAAAAAABpU/sbwLX3foai0/s1600/pi%2Bworksheet%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="457" src="http://2.bp.blogspot.com/-i10vclQgGmw/VpsvZavb-hI/AAAAAAAABpU/sbwLX3foai0/s640/pi%2Bworksheet%2Bimage.png" width="640" /></a></div><br /><br /><br /><br />_____<br /><span style="font-size: x-small;">*(There's a better way to show word docs, right? Someone tell me. I should know that after all these years of blogging!)</span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-11228907619096759132016-01-02T09:07:00.000-08:002016-01-02T09:07:00.814-08:00Newton and the Notion of Limit (he knew more than I thought he did)Preparing to give a math talk has been very educational for me. I posted about ten days ago about finally figuring out <a href="http://mathmamawrites.blogspot.com/2015/12/the-roots-of-calculus-archimedes.html" target="_blank">how Archimedes calculated pi with his 96-gon</a>.<br /><br />Now I just found out that <a href="http://www.sciencedirect.com/science/article/pii/S0315086000923012" target="_blank">Newton wrote more about limits than we're usually led to believe</a>. In 1687, Newton wrote:<br /><br /><blockquote class="tr_bq"><span style="font-size: small;"><span style="font-family: "Times";">"Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...." </span></span></blockquote><br />This quote comes from Bruce Porciau's paper, <a href="http://www.sciencedirect.com/science/article/pii/S0315086000923012" target="_blank">Newton and the Notion of Limit</a>, in Historia Mathematica. He gives much more evidence that Newton understood the limit concept pretty well.<br /><br />I guess I can still say that it took the best minds in all the world 150 years to come up with a precise definition of limit. But Bishop Berkeley's complaint ...<br /><blockquote class="tr_bq">"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"</blockquote>... now seems to me more the product of a small mind and less the careful quest for precision of a mathematician. Now I lean more toward thinking Newton (and Leibniz?) got it, but it took 150 years for a mathematician to create a precise definition that would convince all the other mathematicians. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-37513673826811674512016-01-02T08:01:00.001-08:002016-01-02T08:01:12.059-08:00Joint Mathematics Meetings in Seattle this coming weekI leave on Wednesday for the <a href="http://jointmathematicsmeetings.org/jmm" target="_blank">Joint Mathematics Meetings</a> in Seattle. I'm giving a talk there on using creative commons textbooks in calculus. Friday, 1:20pm, room 620. I'd like to meet online friends there!Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-77443934460629170002015-12-27T17:51:00.002-08:002015-12-27T17:51:58.773-08:00Does your kid hate math? Try a new angle.Long before I became a parent, in my teaching (of community college students), a number of them told me how bad they were at math even though their mom or dad taught it. I figured the parents pushed too much or something. (Blame the parents much, do we?) I ‘knew’ I wouldn’t do that.<br /><br />Well, I don’t think I pushed. But my son hates math, and is consequently way behind his peers. (He unschooled for years and there was no ‘behind’. But he chose to go to a regular middle school this year, where the other kids have mostly had the standard schooling.) So when two people I respect got into a meaty conversation about this, my antennae popped up. They’ve allowed me to share this conversation, which occurred in a closed group on Facebook called <a href="https://www.facebook.com/groups/1001mathcircles/" target="_blank">1001 Math Circles</a>. (Ask to join if you’d like - group description: A place to share and discuss your #mathcircles plus learn more about the Natural Math principles! Run by Shelley Nash and Maria Droujkova of NaturalMath.com.)<br /><br /><br /><br /><br /><i>Lhianna</i>: Hi. I'm a homeschool mom of daughters 7 and 13. I absolutely love math and creative problem-solving and my oldest daughter hates it. My failure to transfer my love of math to her drove me to find better ways of teaching and sharing the beauty and excitement that I see. I found out about Math Circles and have done <a href="http://themathcircle.org/" target="_blank">the summer training camp with Bob and Ellen Kaplan</a> for several years now. I run Math Circles around Philadelphia as time and opportunity allow. I love getting inspired by all the great ideas of a wonderful math community like this one. Thanks for letting me join!<br /><br /><br /><i>Maria</i>: Lhianna, welcome! The Kaplans’ community is wonderful. Maybe we can have a live chat sometime about your circles? When someone hates math, there is usually what I call a grief story. Even with homeschooling, our children can get enough grief "second-hand" from us, or from the society... When I ask people who hate math what happened to them, they usually do know, and tell their stories. Do you know what happened to your 13-year-old? And what does your 7 year-old like to do? It's such interesting age for girls!<br /><br /><br /><i>Lhianna</i>: My 7 year-old loves logic problems. (The island of knights and knaves kind. I have a special fondness for all of Raymond Smullyan's books!) She likes unit origami (especially the sonobe units). And she seems fascinated by anything to do with parity. Also building with geometric shapes of all kinds.<br /><br />I think my 13 year-old has a deep fear of getting things wrong in any subject and in general in life. In other subjects she finds ways around it. But it is especially devastating for mathematical exploration. You really have to try many different avenues and be able to look at your failures and analyze them to arrive at a solution in math. Math is about exploring what is unknown to you and she can't stand that. She prefers the familiar.<br /><br />It has been an interesting journey for me. I started thinking how lucky she is to get an exploratory background in math. I then realized my own shortcomings that, while I loved to explore math, I hadn't been able to communicate that idea to my child. Which led me on a wonderful journey of discovering Math Circles and many more amazing people and sources full of creative ideas about learning math.<br /><br />But as my daughter continued to hate it (and trying to do math with other people too, not just me), I also learned that math is not for everyone like I originally thought. It's ok now that she doesn't like math! That is a homeschooling journey to learn and accept this. (When she does do some math she is perfectly able to learn and understand the concepts. She just has zero interest and will not voluntarily spend any time on math study).<br /><br />I am currently dragging her through "The Art of Problem Solving" book series so she can have enough math to go on to higher education. (And it's a pretty decent series for a textbook!) I am very much an amateur. I am constantly learning and open to new ideas. Any suggestions would be greatly helpful.<br /><br /><br /><i>Maria</i>: Lhianna, thank you for sharing. Yes, I am with you - love of math for its own sake isn't for everyone (just like any other area); but I do feel that everyone can feel good doing some math-rich activities in their own ways. I see a pattern in your interaction with math and with your 13 year-old. Do most of your math activities center on problem-solving?<br /><br />In contrast, have you ever tried math activities that don't involve problems, solutions, answers, or unknowns? There are activities where you: (1) only work with what you know, and (2) don't seek any answers or solutions. When I say that now, can you picture 4-5 examples of activities that I am talking about?<br /><br /><br /><i>Lhianna</i>: Not off the top of my head. What kinds of activities are you thinking about?<br /><br /><br /><i>Maria</i>: Logic is so lovely! Smullyan's books made a difference for many people. <a href="http://naturalmath.com/camplogic/" target="_blank"><i><b>Camp Logic</b></i></a>, which we published this year, is one of our most popular books, too. I just sent three big boxes of it to groups. Next year, "Bright, Brave, Open Minds" will be out, by Julia Brodsky - there are very lovely logic activities in there, too. Here are a few things to try from that book:<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-uPUmmRWhwfE/VoCVCbiH4eI/AAAAAAAABn0/4cg41Z8ccU0/s1600/brodsky%2Bi%2Bam%2Blying.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://3.bp.blogspot.com/-uPUmmRWhwfE/VoCVCbiH4eI/AAAAAAAABn0/4cg41Z8ccU0/s640/brodsky%2Bi%2Bam%2Blying.jpg" width="414" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-5WNZSdiRlfE/VoCVDhUDAUI/AAAAAAAABn8/hCgHa2ovlb0/s1600/brodsky%2Bdinosaurs.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="497" src="http://1.bp.blogspot.com/-5WNZSdiRlfE/VoCVDhUDAUI/AAAAAAAABn8/hCgHa2ovlb0/s640/brodsky%2Bdinosaurs.jpg" width="640" /></a></div><br /><br /><br /><i>Lhianna</i>: I see my 13 year-old use math in other activities (she really likes to cook and make up her own recipes which involves experimentation and therefore doubling and tripling many measurements as well as analyzing the ratios of one ingredient to another). Is this what you are talking about? Or math games? She likes to play SET.<br /><br /><br /><i>Maria</i>: Lhianna, so the goal is to find math-rich activities that: (1) are not problem-solving, and (2) center on what you already know, and yet (3) are open and can be made uniquely yours. Let’s see if we can find a fresh angle on what your daughter can try…<br /><ul><li>Storytelling. You tell what you know; you make the story interesting, fun, pretty, and may invent details, but you know your story (and math therein). Vi Hart videos are like that. Or storybooks like The Cat in Numberland.</li><li>Illustrations. Take something you know. Illustrate it with a picture, comic, video, toys, interpretive dance smile emoticon Basically, represent it by some medium you like. A lot of math comics are illustrations of math jokes, for example.</li><li>Programming. Take a formula or pattern you know and use, and make your computer (spreadsheet, solver, etc.) do it for you.</li><li>Scavenger hunt. Find some math idea you know (e.g. ratio) in what you like (e.g. Star Wars, your favorite park, or your room). Or find a lot of math ideas in one book, movie, room... Make a curated collection. There are a lot of those online. Have you tried that sort of approach? How did it go?</li></ul>SET is a very good game too. To use this as an example of doing what you like and know... We do this activity where we make our own set of SET cards from scratch, using our own shapes and themes. On the one hand, it's something you know. On the other, the amount of delicious a-ha moments you have along the way is just incredible!<br /><br /><br /><i>Lhianna</i>: Great idea! Thanks. And thanks for the advice. I will start looking for activities and examples that follow along the lines of familiar but open. I appreciate the new perspective.<br /><br /><br /><i>Maria</i>: I would love to hear what else you find, because you have such a thoughtful approach to the whole thing! Moving the focus to, "Love SET, like Vi Hart videos, like Tangram puzzles..." (from, "hate math").<br /><br /><br /><br />Do you have a kid who hates math? Do any of these ideas sound like something you might want to try out with them? <br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-73946664094019545982015-12-24T15:58:00.001-08:002015-12-24T15:58:43.484-08:00Question for my ReadersLately, when I'm trying to write a post, I often get shifted over to some sort of ad. Does that happen to any of you reading my posts? If it does, I may move my blog over to Wordpress.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com8tag:blogger.com,1999:blog-5303307482158922565.post-55768467239837351942015-12-23T21:31:00.000-08:002015-12-28T19:41:53.779-08:00The Roots of Calculus - ArchimedesArchimedes did a lot that nowadays looks like calculus...<br /><br />He determined the value of pi very precisely, by starting with a hexagon inscribed in a circle, then a 12-sided polygon, then he kept doubling the number of sides until he got to a 96-gon. A procedure like this is called the 'method of exhaustion', and it looks a lot like what we do nowadays with limits.<br /><br />I am embarrassed to admit that I couldn't figure out how he did it. (I think I was focusing on area, and that might be harder.) I just found <a href="https://www.youtube.com/watch?v=_rJdkhlWZVQ" target="_blank">a great video by David Chandler</a> (whose youtube channel is Math Without Borders).<br /><br />Here's a summary:<br />Start with a hexagon inscribed in a circle of radius 1 (giving diameter of 2). The perimeter of the hexagon will be 6. This gives a lower bound on pi, which is the ratio of circumference to diameter. We know the circumferences is bigger than this perimeter of 6, so pi is bigger than 6/2 = 3.<br /><br />If you cut one of the triangles that made the hexagon into two, you get a radius that crosses a side of the hexagon at right angles. You can use the Pythagorean Theorem (twice) to find the new side length. Repeat 3 times and you're at the 96-gon. Archimedes had none of our technology, and little or none of our algebraic symbolism, so the calculations were much harder for him. We can do all this on a spreadsheet, and up comes pi (if you have a column for the perimeter over the diameter). So satisfying!<br /><br />If this doesn't make sense, watch <a href="https://www.youtube.com/watch?v=_rJdkhlWZVQ" target="_blank">this lovely video</a>. Thank you, David!<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/_rJdkhlWZVQ/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/_rJdkhlWZVQ?feature=player_embedded" width="320"></iframe></div><br /><br />Archimedes did a lot more than find a value for pi! What's your favorite bit of calculus that started out with Archimedes?Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com9tag:blogger.com,1999:blog-5303307482158922565.post-12688593991900354262015-12-21T09:08:00.000-08:002015-12-21T18:18:34.784-08:00Fun Mathy BooksIs it too late to suggest good holiday gifts<i><b>? </b></i>Here are some books I think you might like.<br /><br /><br /><br /><i><b><a href="http://www.amazon.com/This-Not-Maths-Book-Activity/dp/1782402055" target="_blank">This is Not a Math Book</a></b></i>, by Anna Weltman<br /><br /><i><b><a href="http://www.abebooks.com/servlet/BookDetailsPL?bi=17662548102" target="_blank">Patterns of the Universe: A Coloring Adventure in Math and Beauty</a></b></i>, by Alex Bellos <br /><br /><a href="http://www.abebooks.com/servlet/BookDetailsPL?bi=17481782380" target="_blank"><b><i>Mathematical Mindsets</i></b></a>, by Jo Boaler<br /><i><br /></i><a href="https://www.bookbyte.com/textbooks/intentional-talk-how-to-structure-and/9781571109767-1571109765" target="_blank"><i><b>Intentional Talk: How to Structure and Lead Productive Mathematical Discussions</b></i></a>, by Elham Kazemi <i><br /></i><br /><br />Dan MacKinnon wrote a lovely review of a book I hadn't heard of before, at his blog, Math Recreation. Here's the beginning of it...<br /><blockquote class="tr_bq">In <i><b><a href="http://www.bookfinder.com/search/?ac=sl&st=sl&ref=bf_s2_a1_t1_1&qi=.Mzh9tzyG8iHT9ljpY0FCP6W5As_1450617514_1:33:543&bq=author%3Divan%2520moscovich%26title%3Dpuzzle%2520universe" target="_blank">The Puzzle Universe: A History of Mathematics</a></b></i>* <b>in 315 Puzzles </b>(TPU), <a href="http://yoz.com/wired/2.09/features/moscovich.html">Ivan Moscovich</a> stretches the concept of puzzles to encompass almost anything that combines curiosity and playfulness (<i>playthinks</i> is his preferred term for this more general category of puzzling items). No surprise - these playful curiosities are inherently mathematical. In an informal and accessible way, Moscovich details the development of these puzzles, revealing their surprising family resemblances and the deep mathematics behind their playful exterior. [<a href="http://www.mathrecreation.com/2015/12/a-universe-of-puzzles.html" target="_blank">read the rest at Dan's blog...</a>] </blockquote> <br />And of course, there's my book, <b><a href="http://naturalmath.com/playingwithmath/" target="_blank"><i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i></a></b><i>, </i>along with <a href="http://naturalmath.com/goods/" target="_blank">all the other cool books at Natural Math</a>.<br /><br /><br /><br /><br /><br /><span style="font-size: x-small;">_________</span><br /><span style="font-size: x-small;">*This link goes to bookfinder.com, which will point to other sites. It's the best way I know of to find the least expensive copy available. (My other links point to the sites that were cheapest at bookfinder on the day I wrote this.)</span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-46992092513026887362015-12-21T08:53:00.001-08:002015-12-21T09:03:56.799-08:00Lots of LinksFor months I've been saving cool things in tabs in my browser. I think I was up to over 80 tabs when I started cleaning up yesterday. Here are the goodies... <br /><br /><ul><li><a href="http://wild.maths.org/solve-very-old-problem" target="_blank">Trisect the Angle</a> (using origami axioms)</li><li><a href="https://rootsoftheequation.wordpress.com/2015/12/14/growth-vs-fixed-mindset-on-npr/" target="_blank">Growth Mindset</a> (a conversation with Neil deGrasse Tyson)</li><li><a href="http://denisegaskins.com/2015/12/15/understanding-math-area-of-a-rectangle/#comment-143037" target="_blank">Denise Gaskins on Understanding Math</a> (She references an article by Richard Skemp that differentiates between 'instrumental' and 'relational' understanding.)</li><li><a href="https://plus.maths.org/content/what-are-sigma-levels" target="_blank">What are sigma levels?</a> (statistics)</li><li><a href="http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem?utm_source=facebook.com&utm_medium=social&utm_campaign=npr&utm_term=nprnews&utm_content=20150814" target="_blank">A new pentagon for tiling the plane</a></li><li><a href="https://christopherdanielson.wordpress.com/2015/12/16/project-pentagon/" target="_blank">Project Pentagon</a> (Christopher Danielson is playing around, and thinking about the math.)</li><li><a href="http://www.intmath.com/integration/6b-fundamental-theorem-calculus-interactive.php" target="_blank">Fundamental Theorem of Calculus with proofs and an applet</a></li><li><a href="http://www.geogebra.org/student/m59882" target="_blank">What is a radian?</a> (geogebra applet)</li><li><a href="http://mathybeagle.com/2015/12/02/making-groups-work/" target="_blank">Teaching students to work well in groups</a></li><li><a href="http://mathwithbaddrawings.com/2015/10/28/the-differentiation-a-survivors-tale/" target="_blank">The Differentiation: A Survivor's Tale</a> </li><li><a href="http://www.intmath.com/blog/mathematics/wallis-pi-and-quantum-theory-10494" target="_blank">John Wallis, Pi, and Quantum Theory</a> (I need to read this again, and the next one)</li><li><a href="https://plus.maths.org/content/ramanujan" target="_blank">Ramanujan and Fermat's Last Theorem</a></li><li><a href="http://blogs.scientificamerican.com/roots-of-unity/teaching-the-controversy-is-5-3-five-3s-or-three-5s/" target="_blank">Is 5x3 Five Threes or Three Fives?</a> (Scientific American)</li><li>Steven Strogatz, in Scientific American, on <a href="http://www.newyorker.com/tech/elements/einsteins-first-proof-pythagorean-theorem" target="_blank">Einstein's First Proof</a> (my favorite proof of the Pythagorean Theorem, based on symmetry)</li><li><a href="https://plus.maths.org/content/secret-club-diverse-triangles-0" target="_blank">Using theater exercises to teach math</a> (Malke would like this!)</li><li><a href="http://drawingonmath.blogspot.com/2015/11/sine-and-cosine-waves-with-activity.html" target="_blank">Trig graphs on Desmos</a> (using their new <a href="https://teacher.desmos.com/activitybuilder/" target="_blank">activity builder</a>)</li><li><a href="http://www.theatlantic.com/education/archive/2015/11/math-showing-work/414924/" target="_blank">If you can't explain it, does that mean you don't understand it?</a></li><li><a href="http://untilnextstop.blogspot.com/2015/11/what-it-means-to-slow-down-problem.html" target="_blank">What it means to slow down a (calculus) problem</a></li><li><a href="http://youcubed.org/">youcubed.org</a> is Jo Boaler's new site (her new book is <i>Mathematical Mindsets</i>, which I hope to review soon)</li><li><a href="https://plus.maths.org/content/plus-advent-calendar-door-4-konigsberg-movie" target="_blank">A video on the Konigsberg Bridges Problem</a></li><li><a href="http://prairiecreek.typepad.com/herons/2015/10/turtle-triangles.html" target="_blank">Turtle Triangles</a> (on programming using turtle)</li><li><a href="http://blogush.edublogs.org/2015/11/01/it-takes-courage-to-play-in-a-world-that-does-not-play/" target="_blank">It takes courage to play in a world that does not play</a></li><li><a href="http://samjshah.com/2015/11/05/playing-with-blocks-three-dimensional-visual-sequences/" target="_blank">High school students playing with blocks</a> (3D visual sequences)</li><li><a href="http://www.epsilon-delta.org/2015/10/related-rates-related-to-you.html" target="_blank">Making related rates relevant by using students' names</a></li><li><a href="http://musingmathematically.blogspot.com/2015/10/wodb-polynomial-functions.html" target="_blank">Which one doesn't belong? (with polynomial functions)</a></li><li><a href="http://homeschoolerpost.com/16105/130218/a/the-emotional-connection-to-math" target="_blank">Pam Sorooshian on emotions and math</a></li><li><a href="http://blog.plover.com/aliens/dd/intro.html" target="_blank">A message to the aliens</a> </li><li><a href="http://mathpages.com/rr/s8-01/8-01.htm" target="_blank">Kepler, Napier, and the Third law</a> (I'm trying to learn more of the history of calculus, to help me teach calculus more effectively. This article is good.)</li><li><a href="https://bookzoompa.wordpress.com/2015/05/03/the-animated-equation-book/" target="_blank">Function flip books </a>(I thought I was done, but twitter is dangerously good!)</li></ul><br /><br /><b>Games, Puzzles, & Problems</b><br /><ul><li><a href="http://gamedesign.jp/flash/chatnoir/chatnoir.html" target="_blank">Chat Noir</a> (Can you corral the cat? I did it once. Can't do it again.)</li><li><a href="http://wild.maths.org/drips" target="_blank">Drips</a> (a nim game)</li><li><a href="http://mrhonner.com/archives/15551" target="_blank">How many sides of a pentagon can you see?</a></li><li><a href="http://nautil.us/issue/30/identity/how-to-solve-the-hardest-logic-puzzle-ever" target="_blank">A very hard truth and lies logic puzzle</a></li><li><a href="http://matharguments180.blogspot.com/2015/10/497-factor-grids.html" target="_blank">Factor Grid</a> (I wonder if I could make up my own versions of this) </li><li><a href="http://cemc.uwaterloo.ca/resources/potw.php" target="_blank">Some good problems of the week</a> (this site changes each week)</li><li><a href="http://mrhonner.com/archives/7717" target="_blank">A simple trig challenge</a> (I need to save this for my precalc class)</li><li><a href="http://matharguments180.blogspot.com/2015/12/512-factory-ratios-3.html" target="_blank">Factory Ratios</a><a href="http://www.ohiorc.org/for/math/stella/setintro/problem.aspx?id=415#" target="_blank">Speed of sound</a></li><li><a href="http://samjshah.com/2015/09/03/blermions-cyclic-quadrilaterals-and-cross-chords/#comment-98592" target="_blank">Blermions (an approach to some geometry questions)</a></li><li><a href="http://dailydesmos.com/2015/11/04/parabola-of-lines-1-advanced/" target="_blank">Can you make this graph?</a></li></ul><br /><br />I have to admit that I skip the Intermediate Value Theorem when I teach Calc I (please tell me if you think I'm short-changing my students), but here are two great posts about it. <a href="http://blogs.scientificamerican.com/roots-of-unity/math-on-the-run/" target="_blank">If you ran a race at an average pace of 3:07 per kilometer, did you run any single kilometer in exactly 3:07?</a> (from Scientific American) and <a href="https://christopherdanielson.wordpress.com/2015/12/08/a-new-calculus-activity-builder-activity/" target="_blank">an activity using Desmos</a> (from Christopher Danielson).<br /><br /> <br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-89944915388211293142015-10-25T12:37:00.001-07:002015-10-25T15:11:12.666-07:00Math 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 />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-45683635841537334102015-09-18T08:21:00.004-07:002015-09-19T11:35:29.397-07:00Joint 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.) Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-75360024838921851232015-08-23T10:24:00.001-07:002015-08-23T10:24:26.972-07:00The 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.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-51174454341250851362015-08-22T09:54:00.000-07:002015-08-22T10:16:23.797-07:00Linear 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>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com5tag:blogger.com,1999:blog-5303307482158922565.post-58308848571878111892015-08-21T22:35:00.003-07:002015-08-21T22:35:33.893-07:00Random 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.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-17493215424466690452015-08-08T13:29:00.000-07:002015-08-08T13:29:55.077-07:00Links 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>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-42259034356690122332015-07-12T10:02:00.001-07:002015-07-12T10:02:14.662-07:00Playing 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 />SueSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com1tag:blogger.com,1999:blog-5303307482158922565.post-89524003404737497672015-07-10T14:12:00.000-07:002015-07-10T14:12:16.927-07:00Links 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>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-37274574720684925052015-07-02T12:55:00.001-07:002015-07-02T12:59:04.219-07:00Playing 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. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-54204651078326409002015-06-20T10:24:00.000-07:002015-06-20T10:25:09.057-07:00Book 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 />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-67732125122998324212015-06-01T12:54:00.004-07:002015-06-01T12:54:57.464-07:00Imbalance 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>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-69658607647769701742015-05-27T11:48:00.001-07:002015-05-27T11:50:33.726-07:00Preparing 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 />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-68667360711421304842015-05-26T00:08:00.002-07:002015-05-26T07:33:48.972-07:00Machinery, 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 />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-26669616314602967692015-05-19T08:02:00.000-07:002016-04-17T08:22:10.554-07:00Teaching 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 on 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>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-78799299578720024062015-05-14T23:59:00.000-07:002015-07-21T18:00:49.266-07:00Moebius 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 />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com5