tag:blogger.com,1999:blog-6272447367682852770Tue, 14 Apr 2015 14:37:29 +0000algebra101qsAsking the right questionDefining MathHAKASEJapanLocker ProblemMarian SmallMinds OnOpen QuestionsOrder of OperationsProportional ReasoningProportionsSolutionsSquare RootsWarm-upWhy are we learning this?Word Problemsabstractingabstractionassessment asassessment forassessment ofcontextconversionconvertingcreating relevanceengaging studentsequal signsheighthistogramimperiallinear patternsmath mistakesmetricpartial groceriesperceptionprime number generatorratesratiosstudent worktoothpickswhat students thinkYo: A Math Teacher's BlogIn search of some initial value and going from therehttp://ynaughtmath.blogspot.com/noreply@blogger.com (Nico Rowinsky)Blogger19125tag:blogger.com,1999:blog-6272447367682852770.post-4451215206057335410Wed, 23 Oct 2013 17:16:00 +00002013-10-23T10:16:12.467-07:00How My Students Impressed Me In The First WeekI started the year by prompting my students with this.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-hs9AV_KbDUQ/UlHJeEZIhYI/AAAAAAAABDc/b2Pazcp8y-k/s1600/20130906_084516.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-hs9AV_KbDUQ/UlHJeEZIhYI/AAAAAAAABDc/b2Pazcp8y-k/s320/20130906_084516.jpg" /></a></div><br />The idea was lifted from Justin Lanier, and this post is dedicated to him. <br /><br />I was impressed by their video creations. When given the chance (and the freedom to produce what they want) the results can be inspiring, touching, creative, and some were just funny. They had a blast making them and in the end the students and teacher learned some things along the way. This is a great way to start the year.<br /><br />Thank you Justin<br /><br />Enjoy.<br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/TNHicd58e78" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/MT4-vaJXm2A" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/oBdmRvTM9jE" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/ETXAk9E8tqY" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/aatg6MpKZEo" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/TL8zm2FE5YA" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/xgbe1NcOXj0" frameborder="0" allowfullscreen></iframe><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/jALPM2C7npo" frameborder="0" allowfullscreen></iframe><br /><br /><a href="http://www.youtube.com/watch?v=tgAjVo_eYmk">Video 9</a><br /><br /><iframe width="420" height="315" src="//www.youtube.com/embed/tgAjVo_eYmk" frameborder="0" allowfullscreen></iframe><br /><br /><br /><br />In other news, my blog was being attacked by someone who thought spamming in the comments will help his/her business.<br />I contacted the company based in India and I managed to get a response.<br />They apologized and removed me from their marketing strategy.<br />Sorry for the distraction.http://ynaughtmath.blogspot.com/2013/10/how-my-students-impressed-me-in-first.htmlnoreply@blogger.com (Nico Rowinsky)13tag:blogger.com,1999:blog-6272447367682852770.post-2806163101380281395Sat, 30 Mar 2013 21:51:00 +00002013-03-30T14:51:31.021-07:00Is this going to be on the test?File this one under 'SIGNS SOMETHING IS WRONG WITH MY CLASSROOM.'<br /><br />I feel like I have done something wrong when a student asks, "Is this going to be on the test?"<br /><br />The word THIS is the problem for me. To a student asking this question, THIS, means a piece of information that will be copied and reused very soon (and then likely never again).<br /><br />To me, THIS, might mean the act of thinking (or another complex variation on that same activity, but generally cognitive in nature.)<br /><br />I would like to answer the student like this:<br />student: "Is this going to be on the test?"<br />me: "Yes, thinking will be on the test."<br /><br />But more often then not, I answer, "This is the test!"<br /><br />The question, which is often a sincere plea, is the difference between a student that is aware that challenges happen (and you need persistence and some kind of search of prior knowledge to solve these challenges), and a student that is taking polaroids of moments in math class space and time.<br /><br /><b>Some initial questions that I need to consider:</b><br />How do I change this polaroid mentality?<br />How do I create a classroom where I am always encouraging persistence?<br />How can I avoid a student saying, "I never learned this," followed by pencil down and blank stare?<br />And it just occurred to me that these may also be related to the mother of all math class questions asked by students, "When are we ever going to need this?"<br /><br />But we've all been there and have our repertoire of answers for that one.<br /><br /><br /><br />http://ynaughtmath.blogspot.com/2013/03/is-this-going-to-be-on-test.htmlnoreply@blogger.com (Nico Rowinsky)20tag:blogger.com,1999:blog-6272447367682852770.post-887877168643075082Sun, 17 Mar 2013 16:34:00 +00002013-03-18T06:49:49.518-07:00algebraequal signsSome Equal Signs Are More Equal Than OthersHere's a challenge.<br /><br /><b>INTRODUCTION:</b><br />I'm introducing concepts of Algebra to my students. That is to say, if a student asked, "Mr.Rowinsky, what are we doing today in class?" I would likely answer, "Algebra." <br /><br />We're playing with the idea of equations. Balance. Left Side. Right Side. Equal sign in the middle. Some great discussion about this new mathematical animal takes place in the classroom. It's a challenge. It sometimes involves working backwards. It sometimes involves taking tiles and trying to match it with an algebraic abstraction. But mostly it's just abstract. There are moments of "I don't get it" and moments of "Oh ya, that makes sense!" It's frustrating in so many ways to some students. Likely more ways than I can imagine, and definitely more ways than I can teach. But this is the challenge.<br /><br /><b>MAIN:</b><br />A certain student has no difficulty solving x - 7 = 18. They will even momentarily suspend the trivial nature of the equation<sup><a href="#fn1" id="ref1">1</a></sup> and indulge in my suggestion that adding 7 to both sides might be helpful. x - 7 + 7 = 18 + 7 and yes, as you already knew x = 25.<br />However, when the question is flipped, 18 = x - 7, some interesting things happen.<br /><br />"Do I have to subtract 7 because it's reversed.<br />"How do I do this, it's all backwards?"<br />"I subtracted 18 from both sides but it didn't work."<br /><br />These are not the loudest voices in the class. Just a certain few. The few that I'm trying to highlight here because I believe I know why they are having difficulty with this equation. (Outside of the fact that <insert self-deprecating comment here.> <br /><br /><b>INITIAL THOUGHTS:</b><br />Students have been convinced to believe the this symbol <b>"="</b> actually means, "Put answer here".<br />That is math to them. <b>Find the answer!</b> And if they don't know the answer, they are not good at math. Or, they can grab a calculator and that solves everything.<br /><br /><b>FURTHER:</b><br />I am now collecting some like terms. Again, sometimes tiles are used. Some seemingly concrete ideas to explain the abstract. And, they are following and having fun with it.<br /><br />If I ask, what is 3x + 2x, they say 5x.<br />How about 7a - 3a, they say 4a.<br />And b + 8b - b...tough one, but yes, 8b works.<br /><br />But again, a certain student might step back a bit and then ask me, "But what's the answer?"<br />They are frustrated that there is no equal sign, and that they still don't know the value of x, or, a, or b?<br />"How is this helping me?" they'll ask.<br /><br />They want an equal sign. And they want a blank space for an answer.<br />They don't want to simplify.<br />They don't want to evaluate.<br />They don't want to factor.<br />They don't want to represent this equation with tiles.<br /><br />They want an equal sign and they want the answer!<br /><br />And by they, I mean certain students. And that is my challenge.<br />Oh, I'm up for it, but it's challenging nonetheless.<br /><br /><b>HELP ME:</b><br />I'd love to hear what works for you. Please share what you've seen, and what you've heard in your classes when you first introduce Algebra.<br /><br /><sup id="fn1">1. [This alone is a point to consider. A triumph in my eyes.]<a href="#ref1" title="Jump back to footnote 1 in the text.">↩</a></sup><br /><br /><br /><br /><br /><br />http://ynaughtmath.blogspot.com/2013/03/some-equal-signs-are-more-equal-than.htmlnoreply@blogger.com (Nico Rowinsky)20tag:blogger.com,1999:blog-6272447367682852770.post-6664690263769957484Sun, 24 Feb 2013 17:47:00 +00002013-02-24T09:47:02.244-08:00Math Survey Says...Following up on my <a href="http://ynaughtmath.blogspot.ca/2013/02/what-makes-great-math-teacher.html">What Makes A Great Math Teacher</a> post, here are the results of the survey in wordle form. And as <a href="https://twitter.com/mpershan">Michael Pershan</a> puts it, Wordles result in much rejoicing.<br /><br />What Makes a Great Math Teacher:<br /><a href="http://2.bp.blogspot.com/-cuQ3LrF_6mA/USpOzkimKdI/AAAAAAAAArI/O5Mi8RK0LhE/s1600/Screen+Shot+2013-02-24+at+12.21.50+PM.png" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-cuQ3LrF_6mA/USpOzkimKdI/AAAAAAAAArI/O5Mi8RK0LhE/s320/Screen+Shot+2013-02-24+at+12.21.50+PM.png" /></a><br /><br />What Makes a Great Math Student:<br /><a href="http://2.bp.blogspot.com/-K5WyY_VzsXE/USpO8lzhqlI/AAAAAAAAArQ/uO0_GQj86RM/s1600/Screen+Shot+2013-02-24+at+12.25.44+PM.png" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-K5WyY_VzsXE/USpO8lzhqlI/AAAAAAAAArQ/uO0_GQj86RM/s320/Screen+Shot+2013-02-24+at+12.25.44+PM.png" /></a><br /><br />What Makes a Great Math Classroom:<br /><a href="http://2.bp.blogspot.com/-wbCBc7KvsiM/USpPFfPH3HI/AAAAAAAAArY/aLZH2wSj15A/s1600/Screen+Shot+2013-02-24+at+12.29.17+PM.png" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-wbCBc7KvsiM/USpPFfPH3HI/AAAAAAAAArY/aLZH2wSj15A/s320/Screen+Shot+2013-02-24+at+12.29.17+PM.png" /></a><br /><br />I thought it would be a good idea to see what students and teachers think about what makes Math Ed work. Does this solve anything? No. We haven't solved teaching. But I was glad to see that Mr. Pershan attempted <a href="http://rationalexpressions.blogspot.ca/2012/09/what-kids-hate-about-school.html">something similar</a> with his classes earlier in the school year. <br /><br />Thanks to <a href="https://twitter.com/fawnpnguyen">Fawn Nguyen</a> for tweeting and getting the word out about the survey. Supportive as ever.<br />I'm going to post these in my classroom. Maybe even splurge and use the colour printer.<br /><br />Enjoy.http://ynaughtmath.blogspot.com/2013/02/math-survey-says.htmlnoreply@blogger.com (Nico Rowinsky)0tag:blogger.com,1999:blog-6272447367682852770.post-299758937389934644Thu, 21 Feb 2013 12:05:00 +00002013-03-02T12:52:48.185-08:00Grade 7 Math Class...the novelI am absolutely thrilled (and terrified to be honest) to announce that this weekend will be the launch of my first novel.<br /><br /><b><a href="https://leanpub.com/Sally_Strange_Grade_7">Sally Strange: And How She Learned To Stop Worrying And Love Grade 7 Math</a></b><br /><br /><a href="http://1.bp.blogspot.com/-DsNqc2-nLkI/USYJld7ERvI/AAAAAAAAAqI/7PTztvIkiv8/s1600/title_page.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-DsNqc2-nLkI/USYJld7ERvI/AAAAAAAAAqI/7PTztvIkiv8/s320/title_page.jpg" /></a><br /><br />The novel has been a project of mine for a couple of years and has been featured twice in the <a href="http://www.oame.on.ca/main/files/gazettefiles/Gz50i3.pdf">OAME Gazette</a>.<br /><br />It will be published <a href="http://leanpub.com/Sally_Strange_Grade_7">here</a>, where you can read <b>ABOUT THE BOOK</b> and see an <b>AUTHOR BIO</b>, along with some other goodies.<br /><br />I'm very excited to share this news and I wanted to give you a chance to share in my excitement.<br /><br />For your consideration, here is the foreword by Marian Small, followed by the opening lines of the novel.<br /><br /><center><b>FOREWORD:</b></center><br /><b>Here’s the reality of it.</b> Most students know when the math teacher is trying to trick them.<br /><br />Nico sent me a version of the manuscript for Sally Strange a couple of years ago, in hopes of some early feedback. He did this because I have published in math and have a long experience in the field. He wondered what I thought of how he had integrated the math into his fictional work. He didn’t want it to feel like a trick.<br /><br />Well, he succeeded.<br /><br />I was impressed at how easy and enjoyable it was to read, even without thinking about the math. What is so engaging about the book is how Nico has Sally down cold; you can just see her and hear her as you read the words. But what is so particularly creative is how the math is woven in through the story, the math that both he and I enjoy so much.<br /><br />He clearly knows what young students think and feel about school and life. For students, math does not lie behind a secret door, or on the 3rd and a half floor. For students, math happens in school, along with all the other drama we call life.<br /><br />A student who reads this will love the story, but will also be introduced to interesting math problems that are provided along the way. It is an opportunity for students to see the pleasure some of us see in math. Parents who read it will get a better understanding of their children, and will also probably get hooked into solving the problems inside.<br /><br />The first time I met Nico was in a teacher session I conducted years ago. He stood out in the crowd as a teacher with a unique and creative way of thinking. As I met Nico on subsequent occasions, I could see how right that was.<br /><br />Hearing Sally Strange’s words make it obvious how much fun it would be to be in such a classroom- another great message for young students; a math teacher can make a math class a great place to be!<br /><br />Marian Small<br /><br />Dean and Professor Emerita, University of New Brunswick<br /><br /><br /><br /><hr><center><b>CHAPTER 1</b></center><center><b>I've Got Problems</b></center><br /><b>tuesday september 15th</b><br /><br />If I was given the choice between going to math class or going to the orthodontist for a tightening, I’d probably choose the orthodontist. But I’m only 11 and I don’t get to make those choices.<br /><br />Yesterday, I had the painful tightening. Today, I’m here. Math class.<br /><br /><hr><br /><br />A free sample of the book will be available <a href="https://leanpub.com/Sally_Strange_Grade_7">here</a> as well.<br />The purchase of the novel will include 3 formats: <b>e-readers</b>, <b>iPads and Tablets</b> and also <b>PDF</b> for computer use.<br /><br />All readers of this blog can use the code <b>ynaughtmathblog</b> for a 33.3% discount.<br />A sincere thank you for sharing in the news, and I hope you enjoy the math adventure.<br /><br />Nico Rowinsky<br /><br /><br /><a href="http://leanpub.com/Sally_Strange_Grade_7">leanpub.com/Sally_Strange_Grade_7</a><br /><br />UPDATE: The novel is now available on <a href="http://www.amazon.com/Sally-Strange-Learned-Worrying-ebook/dp/B00BL8IQVU">Amazon.</a><br /><br /><br /><br /><br /><br />http://ynaughtmath.blogspot.com/2013/02/grade-7-math-classthe-novel.htmlnoreply@blogger.com (Nico Rowinsky)3tag:blogger.com,1999:blog-6272447367682852770.post-4307164950852118995Mon, 18 Feb 2013 01:01:00 +00002013-02-17T17:01:17.247-08:00What Makes A Great Math Teacher?This is an invitation to have your students fill out this <a href="https://docs.google.com/forms/d/1lZxfRQu_COBfjb6Ko1T4zYdmVuZgHga6pIx1ImVPxkM/viewform?sid=3bf3a003666e100f&token=yCjL6jwBAAA._3bKe_EXSWu6knwccobiLA.L4jwlDoBQ9HE6y9MwvQEEw">survey</a>.<br /><br />Fawn Nguyen recently tweeted this from a conference in Las Vegas:<br /><br /><a href="http://2.bp.blogspot.com/-FTdFaWODhgQ/USF5wxQtN-I/AAAAAAAAAow/tSmpDAoMPvU/s1600/Screen+Shot+2013-02-17+at+7.44.56+PM.png" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-FTdFaWODhgQ/USF5wxQtN-I/AAAAAAAAAow/tSmpDAoMPvU/s320/Screen+Shot+2013-02-17+at+7.44.56+PM.png" /></a><br /><br />I proposed that I would ask my students on Tuesday (my next scheduled class) the question 'What makes a great Math teacher?' With 130+ students I should be able to get a good <a href="http://www.wordle.net/">wordle</a> out of it. But 130+ seems a bit small considering the resources we have and the online math community. So I'm going to UP my proposal and offer all your classes this <a href="https://docs.google.com/forms/d/1lZxfRQu_COBfjb6Ko1T4zYdmVuZgHga6pIx1ImVPxkM/viewform?sid=3bf3a003666e100f&token=yCjL6jwBAAA._3bKe_EXSWu6knwccobiLA.L4jwlDoBQ9HE6y9MwvQEEw">survey</a> for this week.<br /><br />The survey has 3 questions and looks like this:<br /><br /><br /><u>A Great Math Class</u><br /><br />Trying to define what makes a great math class<br /><br /><b>What makes a great math teacher?</b><br />Think about what you'd like in a math teacher, or think about a math teacher in the past and what made them great.<br /><br /><b>What makes a great math student?</b><br />Think about a quality you have that makes you a good student, or a quality you would like to have.<br /><br /><b>What makes a great math classroom?</b><br />Think about things you want to see in the classroom. More than one answer is okay on this one.<br /><br /><br />Try it out with your math classes this week and I'll post the responses next weekend.<br /><br />If the link for the survey above doesn't quite work try:<br />https://docs.google.com/forms/d/1lZxfRQu_COBfjb6Ko1T4zYdmVuZgHga6pIx1ImVPxkM/viewform?sid=3bf3a003666e100f&token=yCjL6jwBAAA._3bKe_EXSWu6knwccobiLA.L4jwlDoBQ9HE6y9MwvQEEw<br /><br />Thank you <br />http://ynaughtmath.blogspot.com/2013/02/what-makes-great-math-teacher.htmlnoreply@blogger.com (Nico Rowinsky)1tag:blogger.com,1999:blog-6272447367682852770.post-7522506312267277628Sat, 09 Feb 2013 20:44:00 +00002013-02-09T12:55:22.483-08:00engaging studentsperceptionwhat students thinkWhat Do Students Say About Math To Other Teachers: A Humble Brag<b>Brag:</b><br />A fellow teacher came up to me this week and gave me a beautiful compliment. She retold a story of what a student had said about my math class. The student had mentioned that I challenged her into going beyond just the math question. "Taking it to the next level," and "Not just the same old questions." The student in question was one that I consider to have difficulties in my class, but I felt good that I've been able to give her the right challenge. I thanked my colleague for the great story. I walked a little taller that day, chest out, ego that much more inflated.<br /><br /><b>Humbled:</b><br />Another fellow teacher came up to me later that same week and presented the following anecdote. She retold a story of what another student had said about my math class. The student had mentioned that I didn't really challenge them and that I taught her nothing. "We just have to answer questions," and "He doesn't really help us." The student in question was one that I consider to have difficulties in my class, but I felt bad that I have not been able to engage her. I thanked my colleague for the feedback. I walked around a bit slower that day, slouched, ego that much more deflated.<br /><br />But it's the second story that is going to drive me to make changes. Make the lesson better. Make the class better.<br />http://ynaughtmath.blogspot.com/2013/02/what-students-say-about-math-to-other.htmlnoreply@blogger.com (Nico Rowinsky)5tag:blogger.com,1999:blog-6272447367682852770.post-1451189127919310394Tue, 22 Jan 2013 00:31:00 +00002013-05-18T14:36:00.289-07:00101qscontextMinds Onpartial groceriesProportional ReasoningProportionsratesratiosProportional Reasoning And Other Boring WordsLast week I did some proportion questions with my students. I found that some methods for solving these questions were preferred to others. I'll highlight these methods plus a few other things I noticed but first, here's a quick question I wanted to test out with some readers.<br /><br />How would you solve this question:<br /><br /><b>Reader Question #1.</b><br />To make 10 pancakes, a recipe calls for 16 Tbsp of sugar. <br />How much sugar is used for 17 pancakes?<br /><br />Take your time. Write down the solution that comes to your mind. Go ahead...<br />Good. Got it? Save your solution until end.<br /><br />Here's the lesson I did with students and some observations.<br /><br /><center><u><b>Proportional Reasoning Lesson</b></u><sup><a href="#fn1" id="ref1">1</a></sup></center><br />This is the pre-activity from <a href="http://map.mathshell.org/materials/download.php?fileid=1306">MARS</a>, thanks to <a href="http://fawnnguyen.com/2012/11/11/my-gratitude-and-one-share.aspx?ref=rss">Fawn Ngyuen</a> for citing this resource.<br /><br />First:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-EJ95wiJeQ0Q/UPi5P7Vci2I/AAAAAAAAAlo/MKla8gkXbWg/s1600/Screen%2BShot%2B2013-01-17%2Bat%2B9.45.00%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="169" width="400" src="http://2.bp.blogspot.com/-EJ95wiJeQ0Q/UPi5P7Vci2I/AAAAAAAAAlo/MKla8gkXbWg/s400/Screen%2BShot%2B2013-01-17%2Bat%2B9.45.00%2BPM.png" /></a></div>Results:<br /><ul><li>Often students found the price of one pancake (Unit Price) and then x 10.</li><li>Less frequently students did 10 ÷ 4 (Scale Factor) then x 6.</li><li>Doubling and halving was quite popular.</li><li>One student did cross multiplication "because that's the way I learned it."</il><br /></ul>Then: <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-_axD4T3rjGs/UPi5VaqNaNI/AAAAAAAAAl0/nVS7OlS3H3E/s1600/2Screen%2BShot%2B2013-01-17%2Bat%2B9.52.45%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="322" width="400" src="http://1.bp.blogspot.com/-_axD4T3rjGs/UPi5VaqNaNI/AAAAAAAAAl0/nVS7OlS3H3E/s400/2Screen%2BShot%2B2013-01-17%2Bat%2B9.52.45%2BPM.png" /></a></div>Results: <ul><li>the increase in size was easier than the decrease.</li><li>doubling and halving was used for some of the .75 and the 2.5</li><li>No cross multiplication was used.</li></ul>And finally: <div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-gshgKfbYW2E/UPi5ixuUx-I/AAAAAAAAAmA/gGkIGHdIJKw/s1600/3Screen%2BShot%2B2013-01-17%2Bat%2B9.54.26%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="241" width="400" src="http://2.bp.blogspot.com/-gshgKfbYW2E/UPi5ixuUx-I/AAAAAAAAAmA/gGkIGHdIJKw/s400/3Screen%2BShot%2B2013-01-17%2Bat%2B9.54.26%2BPM.png" /></a></div>Results: <ul><li>some adding of 15 to get 31 was used.<sup><a href="#fn2" id="ref2">2</a></sup></li><li>this question, more so than any other, was left blank.</li></ul>Solutions were shared in small groups and each group presented the solution they felt was most successful. As a class we sorted, critiqued and supported the solutions presented. They fell into 4 categories. <ul><li>Scale Factor</li><li>Unit Price</li><li>Doubling and Halving</li><li>And the rarely used Cross Multiplication</li></ul>Sharing (as a whole class) the pros and cons of each method was the most rewarding. My biggest issue was actually reading the MARS timelines for each phase of the lesson. I was way off! The ideal situation as advised by the MARS document calls for, <ul><li>15 min. working on the sheet individually. (took me 35)</li><li>15 min. producing small group solutions (took me 40+)</li><li>20 min. Class discussion (took 40)</li></ul>I also have to mention that I put this on the screen as a Minds On: <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-w-Gh2wRSPh4/UPi_tz2jKQI/AAAAAAAAAmc/5kOUHpsFNzo/s1600/4Screen%2BShot%2B2013-01-17%2Bat%2B9.57.13%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="230" width="400" src="http://1.bp.blogspot.com/-w-Gh2wRSPh4/UPi_tz2jKQI/AAAAAAAAAmc/5kOUHpsFNzo/s400/4Screen%2BShot%2B2013-01-17%2Bat%2B9.57.13%2BPM.png" /></a></div>Kudos to the brilliant 101qs for this <a href="http://www.101qs.com/68-partial-products">perfect</a> 'Minds On' task. One of my favourite 101qs. Questions were (in order from popular to least) <ol><li>How much for five cans?</li><li>How much is one can?</li><li>What happened to the missing can?</li><li>Where are they shopping?</li></ol>Almost exclusively they all calculated 1 can, then multiplied by 5. (Unit Price Method) One student calculated 1 can and subtracted that from 6. </br> </br> Ok. Now try:</br> <b>Reader Question #2.</b></br> A picture has dimensions 16 height x 10 width. A proportional enlargement is made. The width is now 17, what is the height? Write down a solution that comes to mind. My question to you is: <b>Did you solve #1 and #2 differently?</b> My students do. Does the pseudo-context used change the method of solution? <hr></br> <sup id="fn1">1. [I have to say the word 'Proportional' in combination with 'Reasoning' is a tough sell for students. We need a snazzier name. Rates and Ratios is only slightly better. Is Scale enough?]<a href="#ref1" title="Jump back to footnote 1 in the text.">↩</a></sup> </br> <sup id="fn1">2. [I should submit that for a <a href="http://mathmistakes.org/">math mistake</a>]<a href="#ref2" title="Jump back to footnote 2 in the text.">↩</a></sup> http://ynaughtmath.blogspot.com/2013/01/proportional-representations-and-other.htmlnoreply@blogger.com (Nico Rowinsky)4tag:blogger.com,1999:blog-6272447367682852770.post-8117547638134633491Sat, 10 Nov 2012 22:08:00 +00002012-11-13T06:48:01.899-08:00abstractingabstractionhistogramThe Abstraction of a HistogramBefore the students entered the class, I asked them to tell me how long it took them (to the nearest minute) to get to school this morning. I took a marker and sticky notes and each student told me their number, I wrote it on the sticky and told them to stick it on the board.<br /><br />Naturally, the students really wanted to round to the nearest 5 minutes. But I emphasized the "nearest minute" part, in an attempt to get more accuracy and less repetition.<br /><br />So the class is in the room and we're looking at a board filled with sticky notes. For the most part, as a class, we tried to organize, sort, structure, and then label our way into a graphical representation. The end product was a histogram of travel time intervals versus frequency.<br /><br />HOWEVER, during one of my classes, I tried something a little different. I took two students aside and worked with them as the rest of the class caught up on some work from the previous day.<br /><br /><b>My goal? <br />Document the abstraction of a histogram. </b><br /><br />It is noteworthy that the two students I chose had shown specific characteristics that I wanted.<br /><ul><li>They have shown in the past that they don't give up</li><li>They have shown in the past to struggle more than average with certain acts of abstraction</li></ul><br />So, staring at the white board with random sticky notes I wanted to guide the process and see where they took it.<br /><br /><b>Instruction #1 from the teacher:</b> <br />"Can you group these stickys in whatever way you want so they are more organized?"<br /><br />Here's the result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-eSW23TvjH_s/UJ67tJNknPI/AAAAAAAAAjw/uVYpM1jx_DM/s1600/1groups.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-eSW23TvjH_s/UJ67tJNknPI/AAAAAAAAAjw/uVYpM1jx_DM/s400/1groups.jpg" /></a></div><br /><b>Instruction #2: </b><br />"I can't really see some of the numbers, can you fix that?"<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-21XdV6dfTkA/UJ673uZB3yI/AAAAAAAAAj8/dwBc5d1mM2c/s1600/2groups.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-21XdV6dfTkA/UJ673uZB3yI/AAAAAAAAAj8/dwBc5d1mM2c/s400/2groups.jpg" /></a></div><br /><b>Instruction #3:</b><br />"Oh, I'm wondering why you chose those groups?" and "I'm getting a bit distracted by the circle, the line and the star shape, can we fix that?"<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-u6dCCeR26Fk/UJ6SBnEX_KI/AAAAAAAAAhU/tlSaLiJ5lJY/s1600/3refined.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://2.bp.blogspot.com/-u6dCCeR26Fk/UJ6SBnEX_KI/AAAAAAAAAhU/tlSaLiJ5lJY/s400/3refined.jpg" /></a></div><br /><b>Instruction #4:</b><br />"I noticed you changed some of the groups, why?" and "As I'm looking at these, the order is kind of from left to right, but then I have to go up to the next level and go left to right again. I wish it was just left to right without having to back track." Now, this instruction is MY personal preference because I need that left to right for the histogram, but both students were accepting, although I felt manipulative here.<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-fU3eK1RKHMY/UJ6TO_qWZjI/AAAAAAAAAhg/FTFh6t64Uw4/s1600/4horizontal_sort.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://2.bp.blogspot.com/-fU3eK1RKHMY/UJ6TO_qWZjI/AAAAAAAAAhg/FTFh6t64Uw4/s400/4horizontal_sort.jpg" /></a></div><br /><br />So now I'm staring at this new structure and I realize the impact of my previous instruction. Encouraging them to go left to right broke down their group stacks. But I rolled with it and gave them,<br /><br /><b>Instruction #5:</b><br />"Ok, can we add something here so I know what the groups are?"<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ugT26DDE2jU/UJ6UyZ2GzTI/AAAAAAAAAhs/Yxq8Fp5zVm0/s1600/5labelled.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-ugT26DDE2jU/UJ6UyZ2GzTI/AAAAAAAAAhs/Yxq8Fp5zVm0/s400/5labelled.jpg" /></a></div><br /><b>Instruction #6:</b><br />"Ok, now I see all your groups. It looks like you have 3 groups. Is that true?" They looked at it and changed some things because of my question. Note, they didn't answer me, they just went to work.<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-vemtdUjg-IM/UJ6VkCDkWzI/AAAAAAAAAh4/2NpiXrsn1K4/s1600/6label_refined.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-vemtdUjg-IM/UJ6VkCDkWzI/AAAAAAAAAh4/2NpiXrsn1K4/s400/6label_refined.jpg" /></a></div><br /><b>Instruction #7:</b><br />"I'm noticing you are having trouble reaching the top right of the board, both to reach the sticky and to even label your groups. Can we fix that?"<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-wXVziETaXSM/UJ6WHf-zMjI/AAAAAAAAAiE/_pWbzDQkuLo/s1600/7lowered.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-wXVziETaXSM/UJ6WHf-zMjI/AAAAAAAAAiE/_pWbzDQkuLo/s400/7lowered.jpg" /></a></div><br />Ok, here I didn't have a one liner. So during <br /><br /><b>Instruction #8:</b><br />I discussed that the numbers look like they're just in a straight line, AND, the labels don't seem to match where the group of numbers are, AND, is there a fix, AND, is this even a problem.<br /><br />They started stacking stickys on top of (covering) each other.<br />"Wait, I still want to see them all."<br /><br />Now they paused, thought about the 'problem' and eventually came up with the following result. It is noteworthy that this seemed to be the most challenging step. Perhaps the most abstracting (<a href="http://blog.mrmeyer.com/?p=14587&utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+dydan1+%28dy%2Fdan+posts+%2B+lessons%29&utm_content=Google+Reader">verb</a>) they had to do?<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Qho298YJEoM/UJ6Y9URuBoI/AAAAAAAAAic/Rn88PT8_9dM/s1600/8stacked.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-Qho298YJEoM/UJ6Y9URuBoI/AAAAAAAAAic/Rn88PT8_9dM/s400/8stacked.jpg" /></a></div><br />At this point I felt like I could bring in the X and Y axis. I had chart paper ready and asked them,<br /><br /><b>Instruction #9:</b><br />"Can you move your stacks onto the chart paper?"<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-tBNbmz0MZoA/UJ6fTzhxUDI/AAAAAAAAAi0/mb7AW0oBbO4/s1600/9axis.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-tBNbmz0MZoA/UJ6fTzhxUDI/AAAAAAAAAi0/mb7AW0oBbO4/s400/9axis.jpg" /></a></div><br /><b>Instruction #10:</b><br />"Do you mind if I label these for you? What would I put here?" (pointing to the left side of the x-axis) They told me to put 0-10...and then 10-20. The result created the next level of organization, including a category (interval) that was empty.<br /><br />Result:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-HT-jV2SAEGY/UJ6gEMnynMI/AAAAAAAAAjA/cIGMf_y3HWY/s1600/10finished.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://3.bp.blogspot.com/-HT-jV2SAEGY/UJ6gEMnynMI/AAAAAAAAAjA/cIGMf_y3HWY/s400/10finished.jpg" /></a></div><br />To wrap things up I added some vocabulary at the end.<br />Words like intervals, frequency, axis...also some details regarding spacing and the notable differences with respect to bar graphs.<br /><br /><b> Some Initial Thoughts:</b><br /><ol><li>It was interesting to note which instruction created the most difficulty #8</li><li>Although I tried to get out of the way, I felt I was imposing myself too much in certain spots #4</li><li>I don't know if getting out of the way is an intrinsic part of my goal (documenting the abstraction of a histogram)</li><li>The difficulties students experience while abstracting are very student specific. In other words, this would have taken less instruction for others, and perhaps more for others.</li></ol><br />Any thoughts are welcome.<br /><br />Bonus footage.<br />Here is a partially completed histogram superimposed onto a stem and leaf plot. These were built by a different class and resulted in a "Oh, that's cool" moment. <br /> <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-doe4ChllVqg/UJ6g1O1NF-I/AAAAAAAAAjM/asH-c8qaL6g/s1600/stemandleaf.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://2.bp.blogspot.com/-doe4ChllVqg/UJ6g1O1NF-I/AAAAAAAAAjM/asH-c8qaL6g/s400/stemandleaf.jpg" /></a></div><br /><br />http://ynaughtmath.blogspot.com/2012/11/the-abstraction-of-histogram.htmlnoreply@blogger.com (Nico Rowinsky)6tag:blogger.com,1999:blog-6272447367682852770.post-2908709791492897345Sat, 27 Oct 2012 23:32:00 +00002012-10-27T16:57:34.614-07:00algebraassessment asassessment forassessment ofcreating relevancestudent workStudent Work on the QuizAt the end of this first run at algebra, I used student work on the actual quiz. <br />Here's what it looked like:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-WAqs71XbRwU/UIxdN7dCI2I/AAAAAAAAAf8/VV7_QafY4vw/s1600/Screen%2BShot%2B2012-10-27%2Bat%2B5.51.18%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="233" width="400" src="http://4.bp.blogspot.com/-WAqs71XbRwU/UIxdN7dCI2I/AAAAAAAAAf8/VV7_QafY4vw/s400/Screen%2BShot%2B2012-10-27%2Bat%2B5.51.18%2BPM.png" /></a></div><br />I'm not sure what this means about creating assessments though. <br /><br />I think I'm looking to make it relevant for the students. But this seems like it's relevant in a very 'meta' way. <br /><br />In the build up, I asked students to create patterns with toothpicks, and they did. They also discovered the numeric growing patterns and created the algebraic expressions accordingly. The work was displayed around the room.<br /><br />And when it came to the assessment of their learning, they did seem somewhat excited to see their classmate's work on the quiz. I know I was excited because I've never done this before, so maybe my excitement rubbed off on them.<br /><br /><b>Some Initial Questions and a Comment:</b><br /><ol><li>Does this "somewhat excited" justify the use of actual student work on the quiz?</li><li>In the past I've used student work as an assessment <b>AS</b> learning, but is it just as valid as an assessment <b>OF</b> learning? </li><li>Put another way, is this just a novelty/semi-interesting use of technology or does it give me some return on investment?<br /><li>It was a relatively small investment in time and effort to take the pictures and throw it on the quiz.<br /></li><br /><br /></ol>Now I'm wondering what you think.<br />http://ynaughtmath.blogspot.com/2012/10/student-work-on-quiz.htmlnoreply@blogger.com (Nico Rowinsky)0tag:blogger.com,1999:blog-6272447367682852770.post-8720082042588757292Fri, 19 Oct 2012 21:51:00 +00002012-10-19T14:51:57.870-07:00algebralinear patternstoothpicksLet the toothpicks fall where they may.I wanted growing patterns.<br />Actually, I wanted patterns that grow by the same amount every time.<br />We might call these linear patterns. At least that's what I was hoping for.<br />And for the most part that's what I got.<br /><br />See...<br /><table> <tr> <td><br /> <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-XCDkCM59Dpw/UIG-4-lefmI/AAAAAAAAAZI/dkSqWSVl-xY/s1600/20121017_141934.jpg" imageanchor="1" style="clear:left; float:left;margin-right:1em; margin-bottom:1em"><img border="0" height="320" width="240" src="http://1.bp.blogspot.com/-XCDkCM59Dpw/UIG-4-lefmI/AAAAAAAAAZI/dkSqWSVl-xY/s320/20121017_141934.jpg" /></a></div> </td> <td><br /> <div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-SS1sqX-lb-8/UIG_CNSwrXI/AAAAAAAAAZU/cz_WmYT9J0E/s1600/20121017_135211.jpg" imageanchor="1" style="clear:left; float:left;margin-right:1em; margin-bottom:1em"><img border="0" height="320" width="240" src="http://4.bp.blogspot.com/-SS1sqX-lb-8/UIG_CNSwrXI/AAAAAAAAAZU/cz_WmYT9J0E/s320/20121017_135211.jpg" /></a></div><br /> </td> </tr></table>But then something like this happened:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-uW3Y2yBZHhc/UIHAJWGpglI/AAAAAAAAAZg/zTFBODeeGwQ/s1600/20121017_115345.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://2.bp.blogspot.com/-uW3Y2yBZHhc/UIHAJWGpglI/AAAAAAAAAZg/zTFBODeeGwQ/s320/20121017_115345.jpg" /></a></div>And this:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-jR2JbEuf1To/UIHFI5ugJ4I/AAAAAAAAAaY/Ly5n-VKqO4Y/s1600/20121019_120459.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://3.bp.blogspot.com/-jR2JbEuf1To/UIHFI5ugJ4I/AAAAAAAAAaY/Ly5n-VKqO4Y/s320/20121019_120459.jpg" /></a></div><br />Q: "Hey Mr. Rowinsky, we couldn't seem to find the algebraic expression for this one."<br /><br />A: "Umm, ya, those aren't, ummm linear. They don't grow by the same amount. What you've discovered is a pattern that grows by an amount that grows. That's advanced."<br /><br /><i>Cheers</i><br /><br />Q: "But do these have an algebra expression we can use?"<br /><br />A: "Good question!" as I think to myself, 'not one that I know off-hand.'<br /><br /><i>Awkward smiles and blank stares</i><br /><br />I rush to Wolfram Alpha to find the solution. And Wolfram Alpha came through. <br />See...<br /><table> <tr> <td><br /> <div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-sRNdX92I-VM/UIHHyiUA1QI/AAAAAAAAAaw/IJNXaDflzLA/s1600/Screen%2BShot%2B2012-10-19%2Bat%2B5.34.11%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="61" width="200" src="http://2.bp.blogspot.com/-sRNdX92I-VM/UIHHyiUA1QI/AAAAAAAAAaw/IJNXaDflzLA/s200/Screen%2BShot%2B2012-10-19%2Bat%2B5.34.11%2BPM.png" /></a></div> </td> <td><br /> <div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JNHTxm1zVB4/UIHH27i4hLI/AAAAAAAAAa8/EhkXgXTjPPk/s1600/Screen%2BShot%2B2012-10-19%2Bat%2B5.34.24%2BPM.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="73" width="200" src="http://3.bp.blogspot.com/-JNHTxm1zVB4/UIHH27i4hLI/AAAAAAAAAa8/EhkXgXTjPPk/s200/Screen%2BShot%2B2012-10-19%2Bat%2B5.34.24%2BPM.png" /></a></div> </td> </tr></table>You can check it out <a href="http://www.wolframalpha.com/input/?i=3%2C+9%2C+18%2C+30">here</a> and <a href="http://www.wolframalpha.com/input/?i=6%2C+16%2C+30%2C+48">here</a>.<br /><br />Some initial thoughts:<br />1) Sometimes students take it to the next level without any help<br />2) Sometimes I need help with the next level<br />3) We live in a world where we shouldn't fear 1 or 2.<br />http://ynaughtmath.blogspot.com/2012/10/let-toothpicks-fall-where-they-may.htmlnoreply@blogger.com (Nico Rowinsky)1tag:blogger.com,1999:blog-6272447367682852770.post-4381670633134931771Mon, 15 Oct 2012 14:23:00 +00002012-10-15T09:06:58.154-07:00Square RootsWhat happens when you've never even heard of a square root?On the board, <blockquote>Using the chart paper, <b>draw a square</b> with an area of area 40 units<sup>2</sup></blockquote>I circulated around the class and recorded the conversations. I didn't answer any questions but I did repeat, 'it has to be a square' a few times. <br /><br />I hadn't even asked a question yet and this is what I got in return.<br /><br />What I heard and what I saw:<br /><br /><b><u>Class A</u></b><br />Wait, how big is that? <br />Can we use a calculator? <br />Does it have to be a square…can it be a rectangle? <br />I know what it is! ~ (…what does he mean by it?)<br />But 40 is not….<br />Is 40 a perfect square!<br />Points…we have to use points… point 5<br />6 x 6 = 36.<br />40 is not a perfect square.<br />It must be between 6 and 7<br />What's 40 divided by…<br />(and again)<br />What’s 40 divided by… (not sure how to complete this sentence)<br />Does it have to be a perfect square?<br />What can be divided by 40 so that it makes a perfect square?<br />What number by what number will get us 40…but it has to be the same number.<br /><br /><b><u>Class B</u></b><br />Does it have to be a square?<br />No rectangles?…is a rectangle a square?<br />It has to be times by the same number<br /><i>(found written on a page)<br />2x20 <br />6.6 x 6.6<br />4x10</i><br /><i>7.5 x 7.5<br />6.7<br />6 x 7 =42</i><br /><br />What’s 40 divided by…<paused to think><br />Can we do point 5’s<br />HAS TO BE A SQUARE RIGHT?<br />Factors of 40<br />40 is NOT A PERFECT SQUARE!<br />6 is closest to 36.<br /><br /><b><u>Class C</u></b><br />Guys, it’s 40 divided by 15<br />2.66<br />Wait, 40 isn’t a square.<br />20 x 20 that’s it...no wait.<br />20 x 20 is not equal to 40...it’s like...400<br />Let's find something that = 40<br />Something that multiplies to give 40<br />What times itself = 40, let’s start with that....<br />Has to be lower than 6.5<br />Try multiplying 6.5, 6.8, 6.7<br />More than 6, because 6 x 6 is 36.<br />6.3 is too low.<br /><br /><b><u>Class D</u></b><br />A rectangle is a square<br />When you say a square what do you mean?<br />Area is 40 units (2)<br />Rectangle is a square or a square a rectanlge?<br />2 of the same...numbers<br />Something times something....= 40<br />It needs to be squared<br />Can we go to decimals<br />Definitely not 7, no wait, definitely not 6<br />6 x 6 = 36<br />6.5 x 6.5<br />What’s on these sides? (points to the sides of the square)<br />It’s a decimal point!<br />8 x 5 = 40<br />6.3 x 6.3<br /><br /><b><u>Some End Results</u></b><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-b47za7zFxqQ/UHs6_LHED7I/AAAAAAAAAWA/jMMhpsIL-6Y/s1600/63.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://2.bp.blogspot.com/-b47za7zFxqQ/UHs6_LHED7I/AAAAAAAAAWA/jMMhpsIL-6Y/s400/63.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-KyVIeLWxupw/UHs7Hc01z_I/AAAAAAAAAWM/LDbYAq1vXA4/s1600/64.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-KyVIeLWxupw/UHs7Hc01z_I/AAAAAAAAAWM/LDbYAq1vXA4/s400/64.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-aRxPxfYnO1I/UHs7Qsj9PqI/AAAAAAAAAWY/sFq4a1y_Ol0/s1600/633.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-aRxPxfYnO1I/UHs7Qsj9PqI/AAAAAAAAAWY/sFq4a1y_Ol0/s400/633.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-MvkSroYO8y8/UHs76y4i7pI/AAAAAAAAAWk/PBghV-SAB68/s1600/6325.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://3.bp.blogspot.com/-MvkSroYO8y8/UHs76y4i7pI/AAAAAAAAAWk/PBghV-SAB68/s400/6325.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-iSKx60N7mu8/UHs8EcC2WWI/AAAAAAAAAWw/-QiKMDV6H0Q/s1600/64writing.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-iSKx60N7mu8/UHs8EcC2WWI/AAAAAAAAAWw/-QiKMDV6H0Q/s400/64writing.jpg" /></a></div><br />I also noticed a surprising amount of reluctance at not being exactly at 40 units<sup>2</sup>. So much so that some groups were paralyzed and did not show any numbers, like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-QsdNP-jEZfk/UHs8c1-bNtI/AAAAAAAAAW8/Bl1_r0TSCGc/s1600/64blank.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-QsdNP-jEZfk/UHs8c1-bNtI/AAAAAAAAAW8/Bl1_r0TSCGc/s400/64blank.jpg" /></a></div><br />And, to be honest, I even saw some of these (despite work being done somewhere on the side.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-HUJhfc836ik/UHs8wvbs5sI/AAAAAAAAAXI/Irj7mqV264E/s1600/reallyBlank.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-HUJhfc836ik/UHs8wvbs5sI/AAAAAAAAAXI/Irj7mqV264E/s400/reallyBlank.jpg" /></a></div><br />And then, after all that, I mentioned square roots.<br />I feel like my return on investment is very high here.<br />Which of these student comments stands out for you?<br />Are there some comments worth noting more than others?<br /><br /><br /><br />http://ynaughtmath.blogspot.com/2012/10/what-happens-when-youve-never-even.htmlnoreply@blogger.com (Nico Rowinsky)10tag:blogger.com,1999:blog-6272447367682852770.post-2272349447144276631Fri, 05 Oct 2012 18:52:00 +00002012-10-05T11:56:33.871-07:00Teaching an Old Word Problem New Tricks: PART IIIf you missed <a href="http://ynaughtmath.blogspot.ca/2012/10/teaching-old-word-problem-new-tricks.html">PART 1</a>, I emphasized how impressed I was by the number and variety of different solutions provided by my students.<br /><br />And then <a href="http://www.youtube.com/watch?v=rFve12E58s4&feature=g-hist">this</a> happened!<br /><br />Keep in mind, this student is just in grade 8 and is still learning the terms, multiples, factors, primes, composites and perfect squares.<br /><br />By the way, that's the sound of my jaw dropping and my teaching life changing all at once. Who knew it would sound so good with a little Shayne Ward playing in the background.<br /><br /><b>Initial Conclusions:</b><br /><ol><li>The kids need more ways to express their solutions (and I might not even know what those ways are)</li><li>Always review student solutions in front of the whole class, it can change where you go with the lesson</li><li>Grade 8 students are stepping up their game</li><li>I hear ya kids...time to step up my game</li><li>Fading to <b>TO Mr. Rowinsky</b> is a crowd pleaser</li>And it got to me...gonna need a moment. </ol><br />What initial conclusions can you add for me? <br />Help me out, I have to take advantage of this going forward.<br />http://ynaughtmath.blogspot.com/2012/10/teaching-old-word-problem-new-tricks_5.htmlnoreply@blogger.com (Nico Rowinsky)0tag:blogger.com,1999:blog-6272447367682852770.post-3147263044723285318Thu, 04 Oct 2012 02:01:00 +00002012-10-03T19:48:24.654-07:00Asking the right questionLocker ProblemSolutionsWord ProblemsTeaching an Old Word Problem New TricksToday, I used the fairly well known 'Locker Problem' with my students. If you haven't heard of it, here's my version:<br /><br /><blockquote>Imagine 50 closed lockers in a hallway.<br />One student goes by and opens every one.<br />Another students goes to every second locker and closes it.<br />A third student goes to every third locker and if it is open, closes it, and if it is closed, opens it.<br />A fourth student then does the same but for every fourth locker.<br />A fifth student does the same but for every fifth.<br />And so on.<br />At the end, which lockers will be open and which will be closed? Why these numbers? </blockquote><br />I think there is merit in analyzing the best way to deliver the problem. Case in point, the students did question the words 'And so on.' "Do you mean forever?", "How far should we go?", "When do we stop?" were fairly common.<br /><br />But here, I want to emphasize something that totally surprised me. About how the students solved the problem, and what they chose as their method. The options were wide open. I'm defining wide open here as 7 different ways of solving this. <br /><br />1) Pen and paper (immediately popular when they were reading the problem)<br />2) Chart paper (the response was, muted)<br />3) A long roll of mural paper (the rolling, intriguing, but not much commitment from the crowd) <br />4) The white board (notable enthusiasm)<br />5) Cube Links (crickets)<br />6) 2-sided plastic counters (aha moments for most groups, interest was building)<br />7) Sticky Notes (game over...close the door...the kids are getting rowdy)<br /><br />But I didn't close the door. Instead, I offered two groups of students (per class) to take the problem outside, into the hallways, with their sticky notes. Inside the classroom, 2-sided counters were popular, followed by some independent pen and paper enthusiasts.<br /><br />Having given some time for the groups outside to get going while I watched students inside, I wandered out the door to check on the progress. And there was the surprise. I thought, Sticky Notes outside the class, was one option. My misconception was that I could imagine how students outside were using those Sticky Notes. It wasn't one option. And I had no idea the students were going to do all of this:<br /><br /><blockquote>Lay numbered Stickys on the Floor</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-SaRQiyRC1TU/UGziF6MGijI/AAAAAAAAAT8/JjGYMGpxnrY/s1600/numbersline.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="300" src="http://4.bp.blogspot.com/-SaRQiyRC1TU/UGziF6MGijI/AAAAAAAAAT8/JjGYMGpxnrY/s400/numbersline.jpg" /></a></div><br /><br /><blockquote>Using O's and C's</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-aiZG6a95-Gg/UGziaBO1xwI/AAAAAAAAAUI/ECTjwxvcmFo/s1600/OsCs2.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-aiZG6a95-Gg/UGziaBO1xwI/AAAAAAAAAUI/ECTjwxvcmFo/s400/OsCs2.jpg" /></a></div><br /><blockquote>Only OPEN Stickys without using Close Stickys</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-GWw0mlbNXVI/UGziqlgfeVI/AAAAAAAAAUY/St1V1157nA8/s1600/setting_up.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="300" src="http://4.bp.blogspot.com/-GWw0mlbNXVI/UGziqlgfeVI/AAAAAAAAAUY/St1V1157nA8/s400/setting_up.jpg" /></a></div><br /><blockquote>Choosing to skip lockers instead of all being labelled.</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-kfFaszQnRg8/UGzi4gtTYUI/AAAAAAAAAUg/ycnXyGRRdMg/s1600/skipping3.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-kfFaszQnRg8/UGzi4gtTYUI/AAAAAAAAAUg/ycnXyGRRdMg/s400/skipping3.jpg" /></a></div><br /><blockquote>Tally Marks on each Sticky as needed.</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-KZC459kFA5s/UGzjC3qYnTI/AAAAAAAAAUs/-EvEwnuvAjY/s1600/tally_sticky.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://2.bp.blogspot.com/-KZC459kFA5s/UGzjC3qYnTI/AAAAAAAAAUs/-EvEwnuvAjY/s400/tally_sticky.jpg" /></a></div><br /><blockquote>Moving Stickys up and down, for Open and Closed</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-eGTUUsavo40/UGzjpTntY0I/AAAAAAAAAU4/FmSqm5aq_S8/s1600/TopOpen.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-eGTUUsavo40/UGzjpTntY0I/AAAAAAAAAU4/FmSqm5aq_S8/s400/TopOpen.jpg" /></a></div><br /><blockquote>How long will 50 remain OPEN?</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-XLe36KoTpLs/UGzkG80ntcI/AAAAAAAAAVE/-sZLeGvLzAs/s1600/Open50.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://1.bp.blogspot.com/-XLe36KoTpLs/UGzkG80ntcI/AAAAAAAAAVE/-sZLeGvLzAs/s400/Open50.jpg" /></a></div><br /><blockquote>24 has been through some changes:</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-698h_ssS9wo/UGzkQy3vKvI/AAAAAAAAAVQ/NcWaNt66xQQ/s1600/factors24.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-698h_ssS9wo/UGzkQy3vKvI/AAAAAAAAAVQ/NcWaNt66xQQ/s400/factors24.jpg" /></a></div><br />Some inside highlights can be seen here:<br /><br /><blockquote>Here come the counters,</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-topzu_ogNlU/UGzmU704hbI/AAAAAAAAAVc/Igeq-s-WOII/s1600/counters.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-topzu_ogNlU/UGzmU704hbI/AAAAAAAAAVc/Igeq-s-WOII/s400/counters.jpg" /></a></div><br /><blockquote>Some planning with Paper and Pencil.</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-G9CtyqgIWVk/UGzmhxE8ZhI/AAAAAAAAAVs/3bHOV3AZT5s/s1600/multiples.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-G9CtyqgIWVk/UGzmhxE8ZhI/AAAAAAAAAVs/3bHOV3AZT5s/s400/multiples.jpg" /></a></div><br />I thought I knew what I was going to see. Today was a pleasant surprise. This is good. <br /><br />At the end of the day, I still have these questions:<br /><br />How could I have widened their options (choices) before starting?<br />Logistics and timing were an issue. How can I manage this better?<br />What prompting questions, if any, are needed here?<br />How can I better word the question, "Which lockers will be open and which will be closed"?<br />An attempt at answering that last one: Can I just say, "What will happen?"<br /><br />http://ynaughtmath.blogspot.com/2012/10/teaching-old-word-problem-new-tricks.htmlnoreply@blogger.com (Nico Rowinsky)3tag:blogger.com,1999:blog-6272447367682852770.post-4791817299500995341Sat, 29 Sep 2012 22:41:00 +00002012-09-30T10:52:29.522-07:00Marian SmallOpen QuestionsOrder of OperationsWarm-upJust Warming Up, BackwardsThis past week I posted a warm-up for the students as they came into class. It read:<br /><blockquote>This is the second line of an 'order of operations' question.<br />What might the first line be?</blockquote><center><b>=26 - 84 ÷ 3</b></center><br />The students had reached a certain comfort in their Order of Operations ability and had seen most questions that I had already planned on presenting to them. But for this warm-up I thought I might try something that Marian Small often suggests. Give them the answer (or in this case the second line) and not the question.<br /><br />Examples of this technique are scattered throughout her book,<br /><u><a href="http://www.amazon.ca/Good-Questions-Differentiate-Mathematics-Instruction/dp/0807753130"><center>Good Questions: Great Ways to Differentiate Mathematics Instruction.</center></a></u><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-zDrHEYXAECo/UGdwM1q0E4I/AAAAAAAAATo/_Nz6z-n5nGw/s1600/goodQuesCover.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="320" width="220" src="http://1.bp.blogspot.com/-zDrHEYXAECo/UGdwM1q0E4I/AAAAAAAAATo/_Nz6z-n5nGw/s320/goodQuesCover.png" /></a></div><br />I have a copy on my desk for quick inspiration when I'm looking to start building something new.<br /><br />This was not the first "work backwards" warm-up I've given my students. But some still looked at it for a minute before diving in. But then answers (or questions rather) were being offered with some ease. Very few students (about one per class) actually misjudged the warm-up and simply solved the next two lines.<br /><br />Things I liked about this warm up:<br /><ul><li>it was open and offered lots of options (thanks Marian)</li><li>it gave them a new look at and old question. After all, they saw bedmas last year didn't they? (pemdas for those in the US)</li><li>they were impressed by the number of solutions and quickly understood that there were infinite solutions</li><li>it didn't take much time and provided lots of discussion including some valuable mistakes</li></ul><br />Are there any other points that make this warm-up useful?<br />What can I do to improve it? Or make them see bedmas without the same old question?<br /><br />Your comments (or a comment for that matter) are welcomed.<br />colon close bracket<br /><br />http://ynaughtmath.blogspot.com/2012/09/just-warming-up-backwards.htmlnoreply@blogger.com (Nico Rowinsky)2tag:blogger.com,1999:blog-6272447367682852770.post-5048775867418798391Wed, 26 Sep 2012 01:16:00 +00002012-09-25T18:16:09.909-07:00Go Low or Go HomeLast year I played a math game with all the students at my school. The rules were simple:<br /><br />1) Choose a positive integer.<br />2) If your number is chosen by someone else that number is out.<br />3) Lowest number wins.<br /><br />I gave out little tickets to all the teachers to distribute and collect once students filled them out.<br /><br />I love this game and first discovered it years ago on the rubiks cube website.<br />It is discussed <a href="http://blog.mrmeyer.com/?p=12354">here</a> and done <a href="http://mathjokes4mathyfolks.wordpress.com/2011/12/05/results-for-my-favorite-game/">here</a> by another teacher.<br /><br />Ideally, I'd like to write a program (or use google doc forms) and run this competition weekly and get automatic google doc spreadsheets created. However, last year I did it the old fashion way.<br /><br />The math team helped me place all the individual tickets down on a lengthy roll of yellow paper. This was quite fun as you might be able to tell in these pictures.<br /><br />And that is why I'm writing today. I recently got these pictures that had been misplaced due to some yearbook photo/camera logistic meltdown. I don't do the yearbook, but I did finally manage to get these pics.<br /><br />Let me know what you think.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-_kHX3ivmQRQ/UGJVz81_iLI/AAAAAAAAAR0/LAby5LbjRfU/s1600/numbergame5.JPG" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://3.bp.blogspot.com/-_kHX3ivmQRQ/UGJVz81_iLI/AAAAAAAAAR0/LAby5LbjRfU/s320/numbergame5.JPG" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-_jzetXLgqwk/UGJWAxdo_sI/AAAAAAAAASA/xOjGx7DwXmM/s1600/numbergame4.JPG" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://4.bp.blogspot.com/-_jzetXLgqwk/UGJWAxdo_sI/AAAAAAAAASA/xOjGx7DwXmM/s320/numbergame4.JPG" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-04lpgS7owCg/UGJWLfIR0zI/AAAAAAAAASM/iSzm-Rt-jMg/s1600/numbergame1.JPG" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://3.bp.blogspot.com/-04lpgS7owCg/UGJWLfIR0zI/AAAAAAAAASM/iSzm-Rt-jMg/s320/numbergame1.JPG" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-myibqNRLVZw/UGJWddObZtI/AAAAAAAAASY/s3YRgt91aew/s1600/numbergame2.JPG" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://1.bp.blogspot.com/-myibqNRLVZw/UGJWddObZtI/AAAAAAAAASY/s3YRgt91aew/s320/numbergame2.JPG" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-BPR4EwcNNcc/UGJWndERcDI/AAAAAAAAASk/Cb462sp29QY/s1600/numbergame3.JPG" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="240" width="320" src="http://3.bp.blogspot.com/-BPR4EwcNNcc/UGJWndERcDI/AAAAAAAAASk/Cb462sp29QY/s320/numbergame3.JPG" /></a></div><br />By the way, can you tell the winning number was 10?http://ynaughtmath.blogspot.com/2012/09/go-low-or-go-home.htmlnoreply@blogger.com (Nico Rowinsky)0tag:blogger.com,1999:blog-6272447367682852770.post-2923481033564856786Tue, 18 Sep 2012 20:52:00 +00002012-09-18T13:59:11.982-07:00HAKASEJapanprime number generatorWhy are we learning this?Faster, Higher and Stronger...for mathLast week my students compared methods of converting centimetres into feet and inches. At one point during the presentations a student made this statement,<br /><br /><i>"When would we ever use that method, since this way is much faster."</i><br /><br />This instinctive lean towards the more efficient method did not go unnoticed. I think it would be fair to say that students are always looking for the faster method. It might also be fair to say that students enjoy the 'easy way', as opposed to the 'hard way'. (But I'm not defining either of those terms.)<br /><br />They are looking for efficiency. This is a very natural mathematical idea within students. <br /><br />It is highlighted in this <a href="http://www.youtube.com/watch?v=K_sV1HNZIQk&feature=player_embedded">video</a> where a Japanese math teacher (at minute 4:23) brings the students back to the pneumonic HA-KA-SE. Fast, Easy, Accurate.<br /><br />It is the math teachers equivalent to the Olympic slogan, Faster, Higher, Stronger.<br /><br /><center><blockquote><b>Citius, Altius, Fortius</b></blockquote></center><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-auvlZDz4Hsk/UFZGHyPsqqI/AAAAAAAAARM/cMEVL8ordy4/s1600/Olympic-Rings.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="155" width="320" src="http://4.bp.blogspot.com/-auvlZDz4Hsk/UFZGHyPsqqI/AAAAAAAAARM/cMEVL8ordy4/s320/Olympic-Rings.png" /></a></div><br />This led me to question whether this HAKASE is a constant consideration in Japanese schooling. Is it standard? In watching more of the video you can hear the students suggesting that the given method might only be easy and accurate, and not necessarily fast.<br /><br />The lesson is analyzed here by <a href="http://blog.mrmeyer.com/?p=14656&utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+dydan1+%28dy%2Fdan+posts+%2B+lessons%29&utm_content=Google+Reader">Dan</a> in his ongoing discussion about what he calls the Ladder of Abstraction. <br /><br />Is this <i>the</i> math slogan we can always make reference to?<br />How often will I be able to refer back to these 3 words when a student asks, <i>"Why are we learning this?"</i><br /><br />And when a student does point out, <i>"When would we ever use that method, since this way is much faster?"</i> I can just agree, unless someone can point out a faster or easier method.<br /><br /><br /><br /><b>TOO COOL NOT TO SHARE</b> <br />Gorgeous <a href="http://www.numbersimulation.com/">prime number generator</a>.<br /><br />It took me a minute to realize what I was looking at but then...ahhhhhh...a beautiful design. Don't miss out on zooming in and out, and speeding it up.<br /><br /><br /><br />http://ynaughtmath.blogspot.com/2012/09/faster-higher-and-strongerfor-math.htmlnoreply@blogger.com (Nico Rowinsky)1tag:blogger.com,1999:blog-6272447367682852770.post-1015397443825059033Sun, 09 Sep 2012 15:50:00 +00002012-09-09T14:04:50.824-07:00conversionconvertingheightimperialmath mistakesmetricThe Empire Strikes BackLast week, my students took a few minutes to measure their heights. In partners, they found a couple spots around the classroom that had tape measures attached to the wall. Results were coming in fairly quickly. A few 170's, lots of 160's and several 150's. These are grade 8 students and the tallest of them I believe was 179cm.<br /><br />Simple. But were there #anyqs?<br />Actually just one question, but it came up too often to ignore.<br /><br /><i>Student: Mr. Rowinsky, I'm 163cm tall, how tall is that?</i><br /><br />For American readers this Canadian nuance might be new. There are certain things where the metric system is quite natural to us. And by natural I mean the first thing that comes to mind.<br /><br />Travel distances for example. (Toronto to Montreal is 550 km). <br />Another is speed limits. (100km/h is a common speed limit).<br />Track and Field events. (In school they run the 100m, 200m etc.)<br /><br />But there are two glaring exceptions. <br />Personal heights and personal weights.<br /><br />When it comes to answering the question, <b>How tall are you?</b>, centimetres are not the first choice. And for, <b>How much do you weigh?</b>, pounds will often trump the poor kilogram.<br /><br />Hence our lesson: <b>Converting from Metric to Imperial</b><br /><br /><br /><blockquote>You must unlearn, what you have learned.</blockquote><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-336ZRV6tcmM/UEpBOnraHTI/AAAAAAAAANQ/fWpGvPBSUCQ/s1600/Yoda%2BThe%2BEmpire%2BStrikes%2BBack.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="188" width="400" src="http://1.bp.blogspot.com/-336ZRV6tcmM/UEpBOnraHTI/AAAAAAAAANQ/fWpGvPBSUCQ/s400/Yoda%2BThe%2BEmpire%2BStrikes%2BBack.jpg" /></a></div><br />Some points to make about this lesson:<br /><br /><b>1)</b> They were all eager to see how tall they were.<br /><b>2) </b>Most walked away learning for the first time, how tall they were in centimetres.<br /><b>3)</b> The question 'How tall am I in feet?' (& inches) only came up after they were done.<br /><b>4)</b> In all 6 classes this same question came up instinctivly, without prompting.<br /><b>5)</b> I would like to thank the Empire and also my neighbours to the south for the long lasting influence and cultural dominance that provides me with this mathematical opportunity. I'm serious. No snark intended.<br /><br /><a href="http://perplexity.mrmeyer.com/">Perplexity</a> (as Dan likes to call it) was high and so too was engagement.<br /><br />I put 163cm on the board and asked students to convert it to metres.<br />1.63m appeared on most pages within the 30 second mark.<br /><br />But how about feet (and inches)?<br /><br />Any guesses?<br /><br />We started with the TOO HIGH guesses. Someone called out 11 feet, 100 feet (accompanied by laughs), and 6 feet.<br /><br />How about TOO LOW? 3 feet, 1 foot, and 5 feet were called out.<br /><br />So we had our range (5 feet to 6 feet).<br /><br />"What information can I give you that would help?" I asked the students.<br /><br />We agreed on two:<br />1 inch = 2.54 cm<br />and<br />1 foot = 12 inches.<br /><br />Small groups went to work. And the work that came out led to some great discussion. And this resulted in some lessons for the students as well as the teacher.<br /><br /><i>...here are a few</i><br /><br />Students<br />1) There is more than one way to get the answer.<br />2) We can learn from the mistakes we make along the way.<br />3) Feet is one measurement and inches is another (5.3 feet is not 5'3")<br /><br />Teacher<br />1) There is <i>way</i> more than one way to get the answer<br />2) I was surprised to see that no students used the "f-word", formula. (No one came up to me and asked for a formula to solve. Instead they went to work to solve it.)<br />3) 5 feet 3 inches was a common response and students had difficulty getting over that 1 tenth of a foot is not 1 inch. This came to light with a 0.5 feet discussion.<br /><br />Here are <strike>two</strike> some student samples to consider.<br />Which can we consider a candidate for a <a href="http://mathmistakes.org/">Michael Pershan</a> entry?<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Oxrk14KoXCI/UEy1Yi6dWXI/AAAAAAAAAQM/4vrB5HieWQk/s1600/20120909_093422.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://2.bp.blogspot.com/-Oxrk14KoXCI/UEy1Yi6dWXI/AAAAAAAAAQM/4vrB5HieWQk/s400/20120909_093422.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-yjtvDLbM3_M/UEy1oRcwkBI/AAAAAAAAAQY/PivHAyKhcck/s1600/20120909_103151.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-yjtvDLbM3_M/UEy1oRcwkBI/AAAAAAAAAQY/PivHAyKhcck/s400/20120909_103151.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-VAKIAEdWZ-0/UE0EXzI1McI/AAAAAAAAAQ4/nCWNGalTq78/s1600/20120909_165811.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="300" width="400" src="http://4.bp.blogspot.com/-VAKIAEdWZ-0/UE0EXzI1McI/AAAAAAAAAQ4/nCWNGalTq78/s400/20120909_165811.jpg" /></a></div><br /><br /><br />And now back to:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-dUroqTylkEY/UEy13-IXplI/AAAAAAAAAQk/6MmgBU1X5Sw/s1600/splash.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="216" width="320" src="http://1.bp.blogspot.com/-dUroqTylkEY/UEy13-IXplI/AAAAAAAAAQk/6MmgBU1X5Sw/s320/splash.jpg" /></a></div><br /><br />In answer to my first <a href="http://ynaughtmath.blogspot.ca/">my first blog post</a> where I posed this question, '<b>When I asked the 130 students "What is Math", what were the 6 most popular words used in their response/definition?'</b><br /><br />The top 6 were:<br />#6 - Life<br />#5 - Problems<br />#4 - Shapes<br />#3 - Equations<br />#2 - Operations (although I included the quartet of multiplying, dividing, adding and subtracting all in one. There is a 5th operation that was popular "and stuff" as in when you are multiplying and dividing and stuff.)<br /><br />and finally<br /><br />#1 - Numbers<br /><br /><i>honourable mention goes to the following student responses:</i><br />"Math is Math" ~ for humour<br />"Math is Life" ~ for philosophical<br />"Math is for problems that need to be solved." ~ for truth<br /><br />I suppose it was ambitious of me to try to get a tonne (metric) of comments to my first ever blog post, although I expected at least one. <br /><br />colon apostrophe open bracket <br /><br />In an effort to cheer myself up, here's a <a href="http://www.wordle.net/show/wrdl/5674295/What_is_Math%3F">wordle</a> cloud for most popular words my students used to answer <b>'What is Math?'</b>http://ynaughtmath.blogspot.com/2012/09/the-empire-strikes-back.htmlnoreply@blogger.com (Nico Rowinsky)0tag:blogger.com,1999:blog-6272447367682852770.post-5968323311525380585Tue, 04 Sep 2012 22:21:00 +00002012-09-30T10:53:13.627-07:00Defining MathDay OneIn the spirit of sharing and inspired by the likes of <a href="http://blog.mrmeyer.com">Dan</a>, <a href="http://function-of-time.blogspot.ca">Kate</a>, <a href="http://fawnnguyen.com">Fawn</a> and of course <a href="http://samjshah.com/">Sam</a>, this is my first blog entry. <br /><br />Day 1 is complete, and I wanted to share the first thing my students did as they entered the classroom. It was a simple enough activity. It included a half sheet of paper with the question, <b>What is Math?</b> written on the top. Students were encouraged to answer the question in the best way they saw fit. Sentences, point form, a paragraph or two. My reason for doing this was to get a sense of how the grade 8 mind sees math after 7 years of elementary school. So now I'll open it up to anyone who happens to read this, in the style of that old favourite TV show called, FAMILY FEUD.<br /><br />130 students were surveyed, top 6 answers are on the board, here is the question:<br /><br /><b>When I asked the 130 students "What is Math", what were the 6 most popular words used in their response/definition?</b><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-xEGBOYlobZY/UEZ9e5SP3PI/AAAAAAAAAM8/_Z6q_BJ41IY/s1600/Family-Feud.gif" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="273" width="400" src="http://3.bp.blogspot.com/-xEGBOYlobZY/UEZ9e5SP3PI/AAAAAAAAAM8/_Z6q_BJ41IY/s400/Family-Feud.gif" /></a></div><br /><br />I'll post the answers soon. Please take a moment to guess in the comments.<br /><br />Secondly, I asked each student to rank from 0 to 10 their ENJOYMENT level and their CONFIDENCE level, when it comes to math class. Results were quickly put together <a href="https://docs.google.com/spreadsheet/ccc?key=0AmBNdUXGO6rFdEp4Y19iTGJ2eVlyMmZqV0dmWlFSSkE">here</a>.<br /><br />I have lots of questions here in terms of the best way to show the results, what to talk about/emphasize, and how much importance I should put into the results. But primarily I wanted to show that I can take a question and represent the data in different ways (scatterplot is the one that came to mind here). I hope to do similar things in the future and even re-do this question at the end and see if there was a shift (hopefully, up and to the right).<br /><br />So there you have a couple of things to think about and perhaps comment on. Hopefully the comments will make me a better teacher. That is the goal after all, right? <br /><br />There are other things floating around that need mentioning. For example, I'm sure I will be getting in on the <a href="http://blogs.edweek.org/edweek/edtechresearcher/2012/08/khan_critiques_we_were_promised_jetpacks_got_lectures.html?cmp=SOC-SHR-TW">Khanversation</a> sooner or later. I'll also likely share things that I find way too cool to keep to myself like, <a href="http://flowingdata.com/2012/08/03/what-planets-would-look-like-if-they-replaced-our-moon/">what would the planets look like if they replaced our moon</a>.<br /><br />And finally I hope to tidy up the blog so that it includes all the buttons and links that make your blog look so nice. Yes, your blog. You know who you are.<br /><br />Thanks for sharing, and thanks for reading.<br />Oh, am I supposed to sign off with my name? #firstblogproblems<br />I will this one time.<br /><br />nico<br /><br />http://ynaughtmath.blogspot.com/2012/09/day-one.htmlnoreply@blogger.com (Nico Rowinsky)1