tag:blogger.com,1999:blog-16974716106860077302017-02-21T10:53:42.965-05:00f(t)Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.comBlogger320125tag:blogger.com,1999:blog-1697471610686007730.post-45891314097304716102016-01-20T18:37:00.000-05:002016-01-20T22:23:56.730-05:00What I Think a Rate Is Right Now<h2>Stating a few assumptions before I get into this:</h2><div>I'm going to explain how I use the word "rate" and the phrase "unit rate" (and also throw around the word "ratio" somewhat recklessly) and it might not match what's in your textbook or how you use the words in your classroom. Some textbooks proclaim that ratios may only involve like units whereas rates use unlike units. In the physical sciences they typically use "rate" to refer to a measurement with respect to time, specifically. All names for things are conventions. I'm not trying to say that you or your textbook or the physics teacher are wrong. Here is a complete list of the arbiters of correctness when it comes to conventions: </div><div><ol><li>mathematical consistency</li><li>people in the act of communicating about the same situation understand each other</li><li>you're not setting up a person for massive confusion later on </li></ol></div><div>The definition of a trapezoid is a good example. Is a trapezoid a quadrilateral with one and only one pair of parallel sides, or is it a quadrilateral with at least one pair of parallel sides? Said another way, is a parallelogram a special type of trapezoid, or is a parallelogram by definition never also a trapezoid? Answer: <span style="background-color: white; font-family: "lyon display" , "georgia" , "times" , serif; line-height: 1.04;">¯\_(ツ)_/¯</span> It depends on a choice made by a person. Textbooks often present definitions like, "This is what the word means!" when they really mean something more like, "This is a choice we made in order to move forward." </div><div><br /></div><div>I'm working on a common core aligned math curriculum for sixth grade. So something to understand as a consequence of that: I'm thinking about how to make these ideas make sense to kids in middle school. So I don't want to write a post about mathematically ironclad definitions that would pass muster with research mathematicians; I want to write a post about stuff that it would be wonderful for kids age 11-13 to understand and is also flexible and useful to build on in later studies.</div><div><br /></div><div>And one last preliminary: sometimes it's important for teachers to understand some nuances and it's not as important for students to understand them at the same level of detail. So, I'm not suggesting that any of this post is appropriate for instructional or assessment purposes with students. For example, an appropriate question for a student might be "In a fruit punch, the ratio of cups of grape juice to cups of soda water is 2:5. How many cups of grape juice for every cup of soda water?" But this question would not be appropriate for sixth graders: "In the ratio 2:5, what is the unit rate?" Because, ew.</div><div><h2>Okay so here we go</h2></div><div>A tortoise travels 10 inches in 3 minutes. A snail travels 8 inches in 3 minutes. Are they traveling at the same rate? (Assuming they're both traveling at a constant rate.)</div><div><br /></div><div>No they are not traveling at the same rate, but I hope you didn't need to compute anything to know that. You can tell because they traveled different distances in the same amount of time. In this context, you have their distances traveled, you have the time it took, but then you have this <i>third thing</i> that means something concrete in the context -- how fast they are going. Their rates. We can express the tortoise's rate as 10 inches in 3 minutes or around 3.33 inches per minute or a foot-and-a-quarter every 270 seconds but the real live concrete in-context rate is the concept of how fast (or in this case, slow) he is moving.</div><div><br /></div><div>(Note for curriculum nerds: at some point you have to make it explicit to students that "are these happening at the same rate?" is structurally the same question as "are these equivalent ratios?" Not super relevant to this discussion but it seems worth mentioning.)</div><div><br /></div><div>At one store, 2 pounds of M&M's cost $14. At a different store, 2 pounds of M&M's cost $16.95. Which is a better deal? Did you have to compute anything to know that? No, you can compare the good-deal-ness without computing the cost of 1 pound or how many pounds you can get for $1. The rate is a <i>third thing</i> going on here capturing how-good-is-this-deal that could be expressed in different ways, one of which is a unit price.</div><div><br /></div><div>Those examples were different types of quantities (distance and time, weight and cost) but we can talk about rates with same quantities like volume and volume. </div><div><br /></div><div>Kate mixes 2 oz of gin with 5 oz of tonic water. Ashli mixes 3 oz of gin with 7 oz of tonic water. Are they sipping the same beverage? This is not so easy because none of the quantities match up. So now we need to find how many ounces of gin for one ounce of tonic water for each beverage right? We could, for sure, but we don't have to. We just have to compare equivalent ratios for the same amounts in the different concoctions:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-fvMjrpbBth4/VqAHDrcxzfI/AAAAAAAAFGU/ruGNDkd2MH0/s1600/gt.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="123" src="http://1.bp.blogspot.com/-fvMjrpbBth4/VqAHDrcxzfI/AAAAAAAAFGU/ruGNDkd2MH0/s320/gt.PNG" width="320" /></a></div><div class="" style="clear: both; text-align: left;"><br /></div><div class="" style="clear: both; text-align: left;">If I used 3 oz of gin to mix a beverage that tastes the same as my original drink, I would need 7.5 oz of tonic water. Ashli only mixed her 3 oz of gin with 7 oz of tonic water, so Ashli's was a bit stronger than mine. Here, that third hidden thing going on is the potency of the beverage, and I'm still asserting that it's a rate, and I still haven't figured out how many of anything per one of anything.</div><div class="" style="clear: both; text-align: left;">So, let's sum up what we have so far: in any set of equivalent ratios that represents a context, there is a third thing that characterizes something meaningful about those two things happening at the same time. It could be land speed, how much of a good deal you are getting, beverage strength, the tempo of a song (number of beats to number of minutes), how crowded my neighborhood feels (number of people to square miles)... This third thing hidden within a set of equivalent ratios is a concept I'm calling a rate.</div><div><br /></div><div>But then, it's often convenient to refer to the special equivalent ratio that is something-paired-with-a-one: "how many of these for every <i>one</i> of those?" It is convenient for at least two reasons (and probably more). First, it helps you solve equivalent ratio problems pretty quickly. For example, I know that I get a certain lovely shade of orange acrylic paint if I mix 3 teaspoons of yellow paint with 2 teaspoons of red paint, but I want to make the same shade and I need alot of it so I want to use up the 9 teaspoons of red paint I have on hand. How much yellow paint should I mix it with? I might approach that problem in any number of ways, but a good way is to reason that 2:3 is equivalent to 1:1.5, so to solve 9:? I just need to multiply 1.5 by 9. (This explanation would be clearer if I drew you a ratio table or another double number line but I am getting tired and it's almost cocktail hour.) It's convenient to use a word to name the 1.5, and "unit rate" is as good a name as any. I like how the "unit" part reinforces that it has something to do with 1. A question kids should be able to answer as part of their process is, "What does the 1.5 mean in this context?" and they should be able to say "there are 1.5 teaspoons of yellow paint for every 1 teaspoon of red paint."<br /><br />Second, it's a way to express that third thing in a set of equivalent ratios with just a single value which can be algorithmetized (like if you want to tell a computer how to do it.) In the gin-and-tonic example above, we could have computed that Kate's drink had 2/5 oz gin for every ounce of tonic water, and Ashli's drink had 3/7 ounce of gin per ounce of tonic water, and since 3/7 is greater than 2/5, Ashli's was stronger.<br /><br />Then actually later in seventh grade I could write an equation for the relationship that is my recipe, <i>g</i> = 2/5 <i>t</i>, where <i>t</i> is volume of tonic water and <i>g</i> is volume of gin, and 2/5 is re-named the constant of proportionality for the set of all gins and tonics of that particular strength, and I could graph this equation and an equation representing Ashli's recipe and note that the line representing her recipe is steeper, but we're really getting ahead of ourselves here.<br /><br />Okay, this was a long post, but we're almost done. I believe that my interpretation is supported by the CCSS standards and the RP progression document, although I also believe that those documents also allow you to conclude that rate only means "how many of these for every one of those" (because the only examples they give for "rate" are quantities per 1). But if you're going to use rate to mean how much of this for every one of that, I think you need to come up with another word for that third-thing physical quantity that I am calling rate.<br /><br />Alright. Comments are on. Come at me, nerds.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-48492846777666252092016-01-04T14:00:00.001-05:002016-03-21T17:38:02.809-04:00In Defense of UnsexyAt IM we're writing a sixth grade curriculum, and much of my time is spent writing, reviewing, and begging other people to write and review new <a href="https://www.illustrativemathematics.org/search?query=6." target="_blank">grade 6 tasks</a> that really just hit the fundamental stuff. After five-ish years of accepting task submissions, we have some holes. <i>Because nobody wants to write easy questions.</i><br /><br />Do you know what teachers have the hardest time finding?<br /><br />Quality, basic stuff.<br /><br />It is getting relatively easy to find the rather-complicated application problems, the projects, the whole new grading systems, the elegant warmups, the pinterest-worthy graphic organizers. Many people have invented and shared some very sexy, awesome stuff, and they are changing many teachers' and kids' experience of math. Hooray for that!<br /><br />But so many teachers aren't helped by sexy stuff. I think one reason is that they don't think the payoff is worth the time investment. Or maybe that changing the whole way they run class is too intimidating. I'd be happy to entertain alternative theories.<br /><br />Here's a cooking show analogy: in the early-mid 2000's people enjoyed Mario Batali's homemade gnocchi and Bobby Flay's 90-ingredient curries and Alton Brown's coconut cake that takes THREE DAYS (I'm not kidding. Three days.) But you know who I watched every day at 4:30? Rachel Ray. I suffered through her saying "yummo" and "EVOO-that-means-extra-virgin-olive-oil" approximately 19 times per episode, and she taught me how to get a reasonable meal on a plate in 30 minutes and how to chop a damn onion.<br /><br />This blog grew in popularity (and stays relatively popular even though I neglect it so) not because I invented something big and sexy but because it offered relatively easy swaps for practice worksheets and ugly, fresh-off-the-smartboard rewrites of high school lessons that made the kids do a tiny bit more thinking than usual.<br /><br />So I'm starting to hear my low-level angst echoed <a href="https://ihati.wordpress.com/2016/01/04/solving-teacher-problems-making-it-easier-to-be-better/" target="_blank">elsewhere</a> and it's bubbling over. I have a request for you if you are a math teacher and you have a blog.<br /><br />Share your kinda-borderline boring stuff. Your small tweaks that unloaded the right amount of cognitive lifting onto the kiddos. Your rather-basic task or set of tasks that don't seem that exciting, but your kids always seem to readily grasp that topic. Your snippets of classroom dialog where everybody ended up going OHHHH. Your artful arrangement of pieces of instructional units you found lying around. How you took that cool instructional idea you read in that book and figured out how to do it in a congruent triangles lesson.<br /><br />We need you. Your kinda-lame-but-seems-to-do-the-trick exponent rule investigation is going to make you somebody's superhero. If you share them with me (add a comment on this post, tweet them at me, whatever), I'll re-share them and compile them in new posts. (And probably they will get added to some of those wonderful virtual filing cabinets and wikis.)<br /><br /><h3>Update: Some gems in the comments. And some shared on Twitter. </h3><br /><ul><li><a href="http://limitsdne.blogspot.com/2015/11/constant-difference-building.html" target="_blank">Constant Difference: Building Understanding</a>, Bryan Dickinson's students engage in some fierce MP2 and MP8 extending a visual pattern.</li><li><a href="https://fractionfanatic.wordpress.com/" target="_blank">Intro to Integration</a>, Julie Morgan nails the productive struggle and the connections to prior learning by asking a backwards question.</li><li><a href="http://mathteachernerds.blogspot.com/2016/01/intro-to-angles-in-degrees-and-radians.html" target="_blank">Intro to Angles in Degrees and Radians</a>, Danielle Reycer increases mathematical coherence by reorganizing a series of lessons.</li><li><a href="https://twitter.com/MathMinds" target="_blank">Kristin Gray</a> likes to combine <a href="https://teacher.desmos.com/polygraph/custom/560aa8df9e65da561507a5ce" target="_blank">Polygraph: Points</a> by Robert Kaplinsky with <a href="https://www.illustrativemathematics.org/content-standards/6/NS/C/6/tasks/2227" target="_blank">this task</a> to orient students to the coordinate plane.</li></ul>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-52324112636448810702015-11-21T13:45:00.004-05:002015-11-21T13:45:48.106-05:00NCTM Nashville PresentationI had the pleasure of attending the NCTM regional meeting in Nashville this week. I learned some cool stuff that I'm still processing, and I got to do a presentation. In the presentation I tried to explore whether the way I would rewrite and rework lessons when I was a high school teacher can be generalized and communicated to other people. I was, I think, marginally successful.<br /><br />NCTM is trying this cool pilot where participants can engage with presenters after the conference. So instead of sharing stuff about my presentation here, I'm going to <a href="http://regionals.nctm.org/nashville/plan-a-killer-lesson-today/" target="_blank">send you over to the presentation page on their site</a>.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-9323694218862632352015-10-02T11:11:00.000-04:002015-10-03T11:19:55.380-04:00Friday Favorites 7Happy Friday! (It's really Saturday but I'm going to backdate this post and pretend it's Friday. Ha! Technology!) My reading and favoriting has slowed down because I have made the decision to limit my Twitter time, which is exceptionally mature of me, I think. (Using <a href="http://www.stayfocusd.com/" target="_blank">Stay Focusd</a>, which is a chrome plugin that yells at you for not working. It's brilliant.) What I'm mostly doing these days is a zillion math problems, which is pretty fun, actually... You know how when professional chefs see a bag of onions, they get excited because they get to chop a bag of onions? That's how I feel about doing a bunch of math problems. It's a little bit drudgery, but satisfying. Still and all, when something gets a little mentally difficult it can't be too easy to distract myself. Twitter needs to not be an option in those moments.<br /><br />This is not a favorite because I made it myself, but it's public, so I might as well share it. It's a place to <a href="https://www.tumblr.com/blog/mathspo" target="_blank">stash mathematically interesting artifacts</a> that I might turn into tasks or assessment questions or lessons. There's nothing worse than needing to write a question in a context and googling for hours. You're welcome, future Kate.<br /><br />Now here are real favorites:<br /><br /><h4><a href="http://cheesemonkeysf.blogspot.com/2015/09/proportional-reasoning-capture.html" target="_blank">Capture Recapture with Goldfish</a></h4>I did this lab in an Algebra 1 class ages ago. It reminds me of that illustration of statistics vs probability: If you know what's in the bag, reach in and grab a handful, and want to predict what's in your hand, that's probability. If you <i>don't </i>know what's in the bag, reach in and grab a handful, and use the handful to predict what's in the bag, that's statistics. It's a good activity, but my first or second year teacher self probably didn't do such a great job with it. Because, obviously, I didn't have Elizabeth and Julie's helpful writeups. I like the way Elizabeth frames how it fits into a bigger Algebra 1 picture. I could also see using it in a stats lab in a way that emphasizes sampling and sample proportions just as easily as a 7th grade-ish solving proportions lab.<br /><br /><h4><a href="https://crazymathteacherlady.wordpress.com/2015/09/16/constructions-was-much-better-this-year/" target="_blank">Problematizing Geometry Constructions</a></h4>I love everything about this. Using a popsicle stick as a straight edge: pro move.<br /><br /><h4><a href="http://www.brilliant-insane.com/2015/09/schooling-terrible-teacher-10-things-parents-never.html?utm_content=bufferc8e00&utm_medium=social&utm_source=twitter.com&utm_campaign=buffer" target="_blank">How Parents and Students and Teachers Can Work Better Together</a></h4>...is a better headline than the clickbaitey one they gave this article. Which is empathetic and treats everyone involved as a professional and a human. Forward anonymously to those parents whose first move is calling the Principal.<br /><br /><h4><a href="https://tle.soe.umich.edu/" target="_blank">Michigan's Teaching and Learning Exploratory</a></h4>Don't let the boring name fool you - Michigan has done an awesome thing here by posting hours and hours of unedited classroom footage. I learned in the last chapter of <a href="http://amzn.to/1M0Q96g" target="_blank">Why Don't Students Like School?</a> that looking at video of yourself or someone you know is too scary a place to start, and it's easier to watch and practice constructively critiquing someone you don't know. This resource makes that a whole lot easier.<br /><br />Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-56405876551426594382015-10-01T14:04:00.000-04:002015-10-01T14:04:57.327-04:00Every Bit of This<a href="http://www.stolaf.edu/people/steen/Papers/07carnegie.pdf" target="_blank">Link</a><br /><blockquote class="tr_bq">High schools focus on elementary applications of advanced mathematics whereas most people really make more use of sophisticated applications of elementary mathematics. … Many who master high school mathematics cannot think clearly about percentages or ratios.</blockquote>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-15887489652417997082015-09-30T09:22:00.004-04:002015-09-30T11:49:17.115-04:00Exponential Functions and also Area of a TriangleThat title is confusing, right? I know! I just wanted to alert y'all to some tasks that recently went up on Illustrative Mathematics that might address some of your needs, if you are teaching these things.<br /><div><br /></div><div><b>Exponential Functions</b>: These tasks involve negative exponents in a functional relationship in a context and are aligned with F-LE.</div><div><ul><li><a href="https://www.illustrativemathematics.org/content-standards/HSF/LE/A/2/tasks/2130" target="_blank">Decaying Dice</a> (It's like the penny lab for modeling half-life that kids often do in Earth Science... except with dice.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/HSF/LE/A/2/tasks/2127" target="_blank">Predicting the Past</a> (Making sense of negative integers in the domain of a simple exponential growth function.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/tasks/2129" target="_blank">All Your Base are Belong to Us</a> (Exponential decay and negative exponents, together at last. Bonus points if you get the reference.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/2128" target="_blank">DDT-Cay</a> (Interpreting the exponent in a half-life equation.)</li></ul></div><div><br /></div><div><b>Area</b>: These are meant to be used to build understanding as you're working toward a formula for area of a triangle in sixth grade (6-G.1). But they could be useful to reactivate knowledge at the beginning of a study of area in a later Geometry course.</div><div><ul><li><a href="https://www.illustrativemathematics.org/content-standards/6/G/A/1/tasks/2131" target="_blank">24 Unit Squares</a> (To remind kids what area means and stymie their attempts to use formulas they don't understand.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/6/G/A/1/tasks/2132" target="_blank">Areas of Right Triangles</a> (Depending on how you approach area of any triangle, this might be a necessary precursor.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/6/G/A/1/tasks/2133" target="_blank">Areas of Special Quadrilaterals</a> (Emphasizes decomposition into familiar figures.)</li></ul></div><div>And, hey, it is non-trivial for me to test stuff out with kids these days, so if YOU try them out and you notice stuff or have suggestions, you can comment here or better yet, right on the task on the IM site. (Please let me know if you do that - I don't think I get a notification. And thanks!)</div><div><br /></div><div>I did draft the initial versions but I can't take credit for these. Tasks published on IM are very much a team effort. Many thanks to <a href="https://twitter.com/mythagon" target="_blank">Ashli Black</a> who is an ace reviewer and helped me make these a ton better.</div><div><br /></div><div><br /></div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-57738796912872919652015-08-28T09:52:00.003-04:002015-08-28T09:52:41.881-04:00Friday Favorites 6<div>Happy Friday! I am elbows deep in Trello, of all things, but the cat is good company. Here we go...<br /><br /></div><div><h4><a href="https://twitter.com/Desmos/status/635081996704813056" target="_blank">Team Desmos</a></h4></div><div>I took a stab at Activity Builder with <a href="https://teacher.desmos.com/activitybuilder/custom/55d76883772721050e71663e" target="_blank">an activity</a> that deals with discovering pi and thinking of circumference vs diameter as a proportional relationship. And wow, it's so much better because of their Twitter interaction. I don't know if my favorite part is setting a table to make points draggable only vertically, or their suggestion to share Teacher Notes in a <a href="https://docs.google.com/document/d/1esXIrESWgJpIz_eVmiM-awZelNqb3ZeJHCTLniysH0Y" target="_blank">linked google document</a>.<br /><br />In case you haven't heard, there's also a repository <a href="https://sites.google.com/site/desmosbank/" target="_blank">of user-created Desmos activities here</a>. Mileage may vary.<br /><br /></div><h4><a href="https://drive.google.com/open?id=0B8XS5HkHe5eNfmNVSjYzXzRtTWVfUm1xWE9uRHdJbWZ6U05OdW9XLTc3ejV2OHdXYlQtSnM" target="_blank">All the Math Talking Points</a></h4><div>Are in this shared google folder. If you haven't grokked the magic of Talking Points yet, go read you <a href="http://cheesemonkeysf.blogspot.com/search/label/TalkingPoints" target="_blank">some cheesemonkey wonders</a>.<br /><br /></div><h4>OER Curricula and Curricular Outlines</h4><div>In case I haven't talked your ear off about it yet, I'm of the strong opinion that a school's math department should Decide on a Coherent Curriculum and riff off of that, rather than expecting their teachers to create a curriculum on the fly using random resources they find on the Internet. Some textbook series are good, and there are also decent OER (Open Educational Resource) ones are already out there, and too many people don't know about them.<br /><br /><ul><li><a href="http://math.newvisions.org/" target="_blank">New Visions for Public Schools (High School)</a></li><li><a href="https://www.carnegielearning.com/learning-solutions/curricula/middle-school/" target="_blank">Carnegie Learning (Middle School)</a></li><li><a href="http://collegeready.gatesfoundation.org/student-success/high-standards/literacy-tools/mathematics-design-collaborative/" target="_blank">BMGF Mathematics Design Collaborative (Middle and High School)</a></li></ul><div><br /></div><br /><h4><a href="http://www.doingmathematics.com/blog/the-teacher-partnership-origins-and-goals" target="_blank">This Coaching Model</a></h4></div><div>Where your team gets a Teacher Partner - someone who teaches a few classes but also coordinates your collaborative teacher learning. I love this.<br /><br /><div><h4><a href="https://algebrainiac.wordpress.com/2015/08/12/2015-2016-new-year-goalsexpectations/" target="_blank">Jessica's Practice is Wide Open</a></h4></div>I might be a little obsessed with other people's planning documents.<br /><br /><h4><a href="http://www.cultofpedagogy.com/classroom-icebreakers/" target="_blank">Icebreakers That Won't Make you Cringe</a></h4><div>You know what I'm talking about. <a href="https://twitter.com/tchmathculture/status/635242825261559808" target="_blank">h/t Lani</a>.<br /><br /></div></div><h4><a href="http://mathhombre.blogspot.com/2015/08/where-do-i-start.html" target="_blank">John's Exhaustive Tour of the Good Stuff</a> </h4><div>Where do I start? Here.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-31441876342744457892015-08-19T22:13:00.000-04:002015-08-19T22:13:13.286-04:00And Then There Was Not Teaching Some MoreWaddup, nerds. Just a quick note about what is going on around here, which is that I've joined forces with <a href="https://www.illustrativemathematics.org/" target="_blank">Illustrative Mathematics</a> to do some very exciting curriculum work. I'll keep y'all posted here as I am able. <div><br /></div><div>Practically that means it's not a new school year for me, which sucks, because I love the first day of school. There's something so inspiring about a fresh start. And also because there won't be as much to report here. </div><div><br /></div><div>But it also means I work at home, which, I'm not going to lie, is pretty boss. I can get all the work done with a cat in my lap and also throw in a load of laundry and also prepare real food for dinner. </div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-69L2pVywGzQ/VdU3QZyxDjI/AAAAAAAAEx0/c8_-4eRjY_E/s1600/2015-08-14%2B11.06.35.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="180" src="http://1.bp.blogspot.com/-69L2pVywGzQ/VdU3QZyxDjI/AAAAAAAAEx0/c8_-4eRjY_E/s320/2015-08-14%2B11.06.35.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">My rig. I realize the television is dominant in this photo, but I haven't actually turned it on yet. It's just extra. That fridge is full of fizzy water.</td></tr></tbody></table><div><br /></div><div>I'm around, on the Internet, of course, and I want to keep doing the Friday Favorite thing. You are welcome to yell at me when I slack off.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-58417483458052509132015-07-24T12:01:00.000-04:002015-07-24T12:01:31.369-04:00Friday Favorites 5<div>Happy TMC, everybody! I know all the TMC-ers are busy TMC-ing right now, but it's Friday! Here we go...</div><div><br /></div><h4><a href="http://www.fishing4tech.com/fishin-solo-blog/the-mtbos-search-engine">John's </a><a href="http://www.fishing4tech.com/fishin-solo-blog/the-mtbos-search-engine">MTBOS search engine</a></h4>What a good idea. I don't know how I missed this.<br /><br /><h4><a href="https://docs.google.com/document/d/1zfRgtBc4n3Pf2xDhiEIwWwymhBEPi0OiINmhDALf7Pc/edit" target="_blank">Tracy's Proof Games</a></h4><div>Here at camp there's a "Games and Strategies" class running this week, and kids keep running up to staff proposing games like "We start at zero, take turns adding 1, 2, or 3, first one to 19 loses." These are kind of addicting, is what I'm saying, and motivate and "Support Generalizing, Conjecturing, Strategy, and Proof-Like Reasoning," as the title suggests. And here are a zillion of them in one document!</div><div><br /></div><h4><a href="http://edushyster.com/i-am-not-tom-brady/" target="_blank">I Am Not Tom Brady</a></h4><div>Just putting this out as a public service announcement that schools that pull shit like this exist, so you can walk away quickly if you get a whiff of it in an interview. h/t Lani for the share.</div><div><br /></div><h4><a href="http://mathbabe.org/2015/07/22/the-17-armed-spiral-within-a-spiral/" target="_blank">Cathy's Write-up of a 17-Armed Spiral</a></h4><div>Here's some recreational math for you, in the spirit of math camp.</div><div><br /></div><h4><a href="http://statteacher.blogspot.com/2015/07/teacher-binder-2015.html" target="_blank">Shelli's Teacher Binder</a></h4><div>Back in the days of student-ing, my life was all about my paper organizer. I had very specific requirements and shopped and shopped until I found it. These days I'm a more scattered leaving-digital-detritus-in-my-wake kind of organizer, but this makes me think maybe it's not too late. </div><div><br /></div><h4><a href="https://www.flickr.com/photos/133462526@N05/" target="_blank">Look How Pretty</a></h4><div>The #mathphoto15 Flickr stream. </div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-91073895123835320002015-07-17T17:46:00.000-04:002015-08-14T08:44:20.986-04:00Summer Problem-Solving CourseThis summer I have the privilege of teaching a problem solving class to mathematically-inclined rising eighth graders. The course is called Math Team Strategies because a big goal is to get kids more ready for contests like MATHCOUNTS and the AMC contests. But we are also looking to highlight problem solving strategies that are broadly useful, whether kids decide to participate in contests or not.<br /><br />I'm going to make this post pretty nuts and bolts just the facts ma'am - it's the nitty gritty details for people who want the ideas.<br /><br />I lovingly plucked from the work of, and want to give tons of credit to:<br /><ul><li>Matt Weber, who is teaching this same course at this program's other site</li><li><i>Crossing the River with Dogs</i> by Johnson, Herr and Kysh [<a href="http://amzn.to/1VaQ17w" target="_blank">Amazon</a> <a href="https://books.google.com/books/about/Crossing_the_River_With_Dogs.html?id=_JEcNfARtfUC" target="_blank">Google Books</a>]</li><li>MATHCOUNTS <a href="http://www.mathcounts.org/past-competitions" target="_blank">Past Competitions</a> and <a href="http://www.mathcounts.org/resources/school-handbook" target="_blank">School Handbook</a></li></ul><h2>Pacing</h2>Eight days, two hours a day, one focus strategy per day. On the final day, instead of a new strategy, students experience a somewhat-complete MATHCOUNTS contest.<br /><h2>The Strategies</h2>(Most of these are chapter titles in <i>Crossing the River with Dogs</i> - but that book has many, many more chapters. It's awesome. You should check it out.)<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ajftYFUV0Oo/ValosnVgABI/AAAAAAAAEuo/k7DHUdfVeSQ/s1600/2015-07-17%2B12.54.28.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-ajftYFUV0Oo/ValosnVgABI/AAAAAAAAEuo/k7DHUdfVeSQ/s320/2015-07-17%2B12.54.28.jpg" width="180" /></a></div><h2>The Lesson Flow</h2>For each day, I selected problems that lent themselves to that day's strategy. Some problems are from <i>Crossing the River</i>, some are from old MATHCOUNTS contests, and some I made up. Additionally, we developed a few mathematical shortcuts over the course of a few days, like counting permutations with repetition and the length of a diagonal of a square. I cut the problems up onto slips, so students would only have one problem at a time. (For a longer course, or perhaps for older students, I'd probably elect to use <i>Crossing the River</i> as a text.)<br /><br />All the students worked on the same problem at the same time, standing at chalkboards. I had anywhere from 6 to 12 students in a class, so this was manageable. I also had a TA who was a math-major undergrad. Nirvana. Before I left home I grabbed a handful of fridge magnets, thinking they might be useful for something, and we used them so students could stick the current problem to the chalkboard.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-oxeuEM8OLag/Valpd45XlCI/AAAAAAAAEu8/pf8Bi-5L9pw/s1600/2015-07-14%2B14.51.26.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://2.bp.blogspot.com/-oxeuEM8OLag/Valpd45XlCI/AAAAAAAAEu8/pf8Bi-5L9pw/s320/2015-07-14%2B14.51.26.jpg" width="320" /></a></div><br /><h2>The Posters</h2>The intention was for the whole class to go over each problem before everyone started the next one. (See <a href="http://function-of-time.blogspot.com/2015/07/a-magical-incantation.html" target="_blank">this post</a> about group discussions.) Of course, some students took longer and needed support. When I am helping, I tend to make the same suggestions and ask the same questions over and over. This poster was for students to refer to if both the TA and I were busy when they got stuck.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-CbjIZINOQnM/VakwfXlMmMI/AAAAAAAAEuQ/ya22L6DErfU/s1600/2015-07-07%2B11.57.20.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-CbjIZINOQnM/VakwfXlMmMI/AAAAAAAAEuQ/ya22L6DErfU/s320/2015-07-07%2B11.57.20.jpg" width="180" /></a></div><br />Also, of course, some students finished more quickly than the group. I also tend to always make the same suggestions when students say they are "done," so I made this poster for them, too.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-oQGAKfjhj7I/VakwfgZwY0I/AAAAAAAAEuM/uYapzVwc0M8/s1600/2015-07-07%2B11.57.27.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-oQGAKfjhj7I/VakwfgZwY0I/AAAAAAAAEuM/uYapzVwc0M8/s320/2015-07-07%2B11.57.27.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><h2 style="clear: both; text-align: left;">The Self-Assessment</h2><div class="separator" style="clear: both; text-align: left;">Before we went over each problem, I asked the students to turn in their problem slip with their name and a rating of the problem from 1 through 4. I did compile this data in a spreadsheet, but I'm not sure what to do with it. But I thought the self-assessment couldn't hurt.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-rkP-ROswAK4/VakwfcfLjzI/AAAAAAAAEuI/bpOzPCKE4pw/s1600/2015-07-07%2B11.57.06.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-rkP-ROswAK4/VakwfcfLjzI/AAAAAAAAEuI/bpOzPCKE4pw/s320/2015-07-07%2B11.57.06.jpg" width="180" /></a></div><br /><h2>The Resources</h2><a href="https://drive.google.com/folderview?id=0Bz4S-NJpJht7fmd5dC0tdDJPX3VjQXMybEpxLWUyWHNtdmk5M1ZUckVma0YzaUU0ejBvN2s" target="_blank">Will be here</a> until someone holding a copyright yells at me to take them down. Or maybe this is fair use. I dunno. I hope it's good advertising for the publications cited above. Some of the problems turned out to be too easy, and I'll be changing them if I'm back next year. Some were too hard, but I thought it was okay to give kids at most one problem a day that was a big stretch for them. When that happened, I invited the TA to share their solution.<br /><h2>And That's about That</h2>This was a really rewarding course. The kids loved it, I loved it, we all just had a grand old time talking about math for two hours a day! It was refreshing to not feel pressure to cover content at a breakneck speed, or sell kids on math (these kids already like math), or have to assign grades. (This morning when we did a sample MATHCOUNTS Sprint, a girl asked "Does this count? Oh, wait. We don't have grades." And she worked hard on it anyway.)<br /><br />Questions, feel free to throw them in the comments.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-961369125664493552015-07-17T06:00:00.000-04:002015-07-18T14:07:06.759-04:00Friday Favorites 4It's the second week of Math Camp... that means I have a little time to post. Yay! Things are still a tad chaotic here - long days, tween drama, field trips, little sleep, etc etc, but I finally have the class I'm teaching all planned out through the end of this week. Phew! Time to write and reflect and observe some great teachers in action.<br /><br />Also I taught some 13 year old boys how to juggle yesterday. Before I signed up for that duty, I did not consider how many times I would have to say the word "balls." The first time was awkward, but then we naturally took it to a ridiculous extreme. "MALACHI! CONTROL YOUR BALLS!" (Normally I wouldn't post photos of student faces, but this one is on <a href="http://spmps.weebly.com/july-141.html" target="_blank">the program's website</a>.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-YuT_U9UDAXQ/VaZ2tvRSEBI/AAAAAAAAEtQ/9YzT38fCoLE/s1600/2015-07-14%2B15.20.47.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://1.bp.blogspot.com/-YuT_U9UDAXQ/VaZ2tvRSEBI/AAAAAAAAEtQ/9YzT38fCoLE/s320/2015-07-14%2B15.20.47.jpg" width="320" /></a></div><br />And now for some fresh faves...<br /><h4><a href="http://kalamitykat.com/2015/07/05/ifttt-improves-my-daily-blogging-habit/" target="_blank">Megan's Easy Way to Start Blogging</a></h4>Although it's really valuable professional learning for lots of people, keeping up a blog during the school year can be a daunting proposition. An on-ramp can be a 180 blog - just take and publish one photo a day from your classroom. This practice has less overhead in terms of time, but gets you in the habit of noticing things to share. <a href="http://kalamitykat.com/2015/07/05/ifttt-improves-my-daily-blogging-habit/" target="_blank">This recent post by Megan Hayes-Golding</a> suggests one way to set this up using Instagram, IFTTT, and Wordpress to make it low friction so that you are more likely to stick with it. If you're unfamiliar with those platforms, don't worry - they are all pretty easy to get started. You could have this up and running in a few hours if you're new to it (a few minutes if you're not). Also, IFTTT works with lots of different services.<br /><br /><h4><a href="https://fivetwelvethirteen.wordpress.com/2015/06/30/teaching-and-intuition/" target="_blank">Dylan Builds His Intuition</a></h4>Dylan Kane has been chronicling his growth as an early-career teacher. If you haven't been following along, you should plug into that. I really enjoyed <a href="https://fivetwelvethirteen.wordpress.com/2015/06/30/teaching-and-intuition/" target="_blank">his post</a> about the ways he has to be attentive to avoiding pitfalls of bias and developing intuition that will be productive in his practice, because they paralleled some of the things I realized along the way (although he has articulated them much better).<br /><br /><h4><a href="http://www.megcraig.org/?p=703" target="_blank">Meg Encourages MTBoS Users to Make It Work for Them</a></h4>Much as I love our spirited army of awesome, folks can get a tad dogmatic and judgey from time to time. It can be a turn-off, when you come across some strident prose that makes you feel like you're doing everything wrong. <a href="http://www.megcraig.org/?p=703" target="_blank">Meg Craig's</a> post speaks to two audiences: seekers of resources and conversations, who are reminded to stick with it and make it work for them. Also sharers of resources and initiators of conversations, who she gently offers ways to phrase your sharing so that it's a bit more inviting and inclusive. <br /><br /><h4><a href="http://blog.mrmeyer.com/2015/your-conference-session-is-the-appetizer-the-internet-is-the-main-dish/" target="_blank">Dan Meyer is going to fix NCTM for Us</a></h4><a href="http://blog.mrmeyer.com/2015/your-conference-session-is-the-appetizer-the-internet-is-the-main-dish/" target="_blank">Here's how.</a> Thanks, Dan. (Adding some clarification here because I'm afraid this sounded snarky - I'm totally sincere. I'm really excited about the prospect of NCTM taking up the recommendations of the ShadowCon organizers. I think we all of us NCTM members realize that NCTM is not working well for many members and prospective members, and I wholly support these concrete proposals.)<br /><br /><h4><a href="http://mathmamawrites.blogspot.com/2015/07/playing-with-math-can-you-write-review.html" target="_blank">Please Review Our Book</a></h4>Have you read <a href="http://amzn.to/1M6BkhW" target="_blank">Playing with Math</a>? Are you going to? (You should! It's so awesome.) It's <a href="http://amzn.to/1M6BkhW" target="_blank">on Amazon now</a>, and it would be great to get some more reviews. (Since I wrote one of the essays I'm ambivalent about writing one myself.)Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-17950000220321736662015-07-15T19:18:00.000-04:002015-07-17T12:08:35.594-04:00A Magical IncantationSo this week I'm basically the luckiest girl in the world, because <a href="https://researchinpractice.wordpress.com/" target="_blank">Ben Blum-Smith</a> is on staff at <a href="http://www.artofproblemsolving.org/spmps/about.html" target="_blank">SPMPS</a>, and he observed me teach and then we had a conversation about it. (I know. Be jealous.)<br /><br />He offered a concrete suggestion enabling student dialog which I want to share. I am pretty good at getting kids to talk to each other about math in pairs or triples...<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-t1H7J0TuEYw/Vabo9dOQP0I/AAAAAAAAEts/WU-2KQihCtY/s1600/chalktalk.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://4.bp.blogspot.com/-t1H7J0TuEYw/Vabo9dOQP0I/AAAAAAAAEts/WU-2KQihCtY/s320/chalktalk.png" width="320" /></a></div><br />but I've always struggled with conducting good conversations with the whole group -- getting kids to talk to each other about math <i>in front of everybody</i>. (Aside from the two kids in every class who always raise their hand for everything.)<br /><br />What we have been doing in this class is having everyone work out solutions to a task on the board. (Classes are small enough (7-11 for my classes), I've partaken of the vertical-non-permanent-surfaces kool aid, and kids at camp are exhausted because it's a three week slumber party, so keeping them on their feet helps with the awakeness.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-dVZR2uNC6Z4/VabpZiSL5KI/AAAAAAAAEt0/aGTgi4xONfA/s1600/2015-07-10%2B17.24.43.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://2.bp.blogspot.com/-dVZR2uNC6Z4/VabpZiSL5KI/AAAAAAAAEt0/aGTgi4xONfA/s320/2015-07-10%2B17.24.43.jpg" width="320" /></a></div><br /><br />When everyone is done-ish, we gather around someone's solution and they walk us through it.<br /><br />So let's say that a student presents her solution or approach to a task to the whole group. Generally they speak too fast, and gloss over important bits. When they are finished, I have uncovered many, many unproductive questions to ask the rest of the class:<br /><br /><ul><li>Does anyone have any questions for Bianca? (crickets)</li><li>Miguel, what do you think of Bianca's solution? ("I don't know. It's fine.")</li><li>Does anyone have anything to add? (more crickets)</li><li>Did anyone approach it a different way? (Actually I never ask this anymore, because I pick usually two students with different approaches to present.)</li></ul><br />But here is the magical incantation that can pick this lock:<br /><br /><div style="text-align: center;"><i>Hey, so-and-so, would you explain your understanding of Bianca's solution?</i></div><br />This is a lovely question. I tried it out at every available opportunity today. Interpreting another student's written and verbal solution requires all kinds of nice cognitive work. I imagine that as kids come to expect that they might be asked this question, they're more likely to be more attentive to others' explanations. And, it offers a nonthreatening invitation into the conversation where a student is immediately clear on what she's expected to say.<br /><br />Ben mentioned that he didn't really grok the power of this move until well into his classroom experience, and I think I'm kind of in the same boat. I'm sure I've heard of it before, but now I'm in a place where I can really deploy it surgically. Well I mean today I deployed it in kind of carpet bomb fashion. It's like a new toy I can't put down. But I'm going to enjoy the process of integrating it.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-10934611250252529162015-07-03T06:00:00.000-04:002015-07-04T15:36:37.608-04:00Friday Favorites 3Hey! You thought I forgot, didn't you? DIDN'T YOU?! (Excusable. That would be totally in character.) I just arrived a few days ago at the best mathematical summer thing in the world, the <a href="http://www.artofproblemsolving.org/spmps/" target="_blank">Summer Program in Mathematical Problem Solving</a>, where I'm teaching a course called Math Team Strategies. It's so great, you guys. The staff is the bomb. The campers get here tomorrow. Now for some faves:<br /><br /><h4><a href="https://docs.google.com/document/d/1G0OMlzxSLCVtbf2XpjCquRAkefwb6JMY9Eq7ouXnVUs/edit#" target="_blank">Get Your Mathematical Modeling On</a></h4>...starting here. These are ideas for data students can easily collect, organized by function type. Compiled by <a href="https://learningandphysics.wordpress.com/" target="_blank">Casey Rutherford</a>.<br /><br /><h4><a href="https://www.youtube.com/watch?v=DhSjD5nLkjY" target="_blank">Sean Sweeney's New Video</a></h4><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/DhSjD5nLkjY" width="560"></iframe><br /><br />Who doesn't love a sweet math song? Okay there are people but you have to admit that this is delightful, even if you're not a singing-in-math-class type. Also if you need to catch up on Sean's (and other members of his school's) previous works of art: here's <a href="https://www.youtube.com/watch?v=TTYKcHJyLN4" target="_blank">Graph Shop</a>, <a href="https://www.youtube.com/watch?v=-chXvU4pza4" target="_blank">f(u)</a>, and the classic <a href="https://www.youtube.com/watch?v=IyfpM-ruafo" target="_blank">Slope Rider</a>.<br /><br /><h4><a href="http://nicoraplaca.com/pd-a-math-task-for-teachers/" target="_blank">Nicora Placa's Math Tasks for Teachers</a></h4>Nicora breaks down how she chooses or adapts mathematical tasks to use for teacher learning. This one is maybe a bit specialized for folks who work with groups of teachers, but if you are looking for good PD to sign up for as a math teacher, this kind of learning has had a huge impact on my practice.<br /><br /><h4><a href="https://medium.com/@PearDeck/pear-deck-and-google-classroom-715c9b109428" target="_blank">Google Classroom + Peardeck</a></h4>If you're at a GAFE school, and you haven't checked out what Google Classroom can do in the past three months or so, you really should get on that. (Especially if you're still using Doctopus. GC is way easier.) And Pear Deck is, of course, the money. And now they're more integrated. Go make some cool shit happen.<br /><br />(My favorite use of Pear Deck is asking kids to find numbers for (<i>x</i>, <i>y</i>) that make an equation true, and then each student plots that point on the Pear Deck slide. The collective points lie along a line, or a circle, or whatever. Connection between multiple representations: made. I used that move on them like a dozen times this year and it never got old. <a href="http://mrorr-isageek.com/logarithmic-warm-up/" target="_blank">Here's a straightforward post by Jon Orr</a> showing what this can look like.)<br /><br /><br />Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-34654083800066762532015-06-19T09:21:00.003-04:002015-06-19T09:26:11.224-04:00Friday Favorites 2Hey there! Two Fridays in a row! Whaddup! Here are some things that got my attention in a good way this week:<br /><br /><h4><a href="http://emergentmath.com/2015/06/06/necessary-and-sufficient-conditions-for-school-improvement/" target="_blank">Geoff Krall's Minimal Conditions</a></h4>Geoff Krall (of <a href="http://emergentmath.com/my-problem-based-curriculum-maps/" target="_blank">PBL Curriculum Map</a> fame) gives an excellent wide-angle view of <a href="http://emergentmath.com/2015/06/06/necessary-and-sufficient-conditions-for-school-improvement/" target="_blank">practices school staff should engage in</a> when they get serious about improving instruction. My favorite thing about this is it seems so do-able. There are things small groups of teachers can start doing with the PD time that's in their control, or if that time isn't yet in their control, suggests some concrete practices to start advocating for.<br /><br /><h4><a href="https://picrust.wordpress.com/2015/06/10/tinkering-with-virtual-patty-paper/" target="_blank">Allison Krasnow's Virtual Patty Paper</a></h4><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-yGcrQRqW_dA/VYQYOL06wgI/AAAAAAAAErA/SevQZ0gABk4/s1600/ss2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="http://4.bp.blogspot.com/-yGcrQRqW_dA/VYQYOL06wgI/AAAAAAAAErA/SevQZ0gABk4/s200/ss2.png" width="200" /></a></div><br />Allison rediscovered a great <a href="http://amzn.to/1I1EnXq" target="_blank">patty paper book by Michael Serra</a>, and noticed that all of the activities could be recreated on Geogebra. I love this! It demonstrates that ways for students to tinker with ideas -- the important part -- is somewhat independent of choice of technology. Use the patty paper, create a Geogebra version, use both, or give students a choice.<br /><br /><h4><a href="https://www.teachingchannel.org/videos/illustrative-mathematics-sbac" target="_blank">What Collaborating Looks Like</a></h4>Many of us know that we should be collaborating with building colleagues on the nuts and bolts of planning and instruction, but if you've never done this before, it can be hard to imagine what it looks like. <a href="https://www.teachingchannel.org/videos/illustrative-mathematics-sbac" target="_blank">This video series</a> (a collaboration between Teaching Channel, Illustrative Mathematics, and Smarter Balanced) is a really excellent resource including teachers working in elementary, middle, and high school math before, during, and after instruction.<br /><br /><h4><a href="http://jhhs.d214.org/staff_resources/default.aspx" target="_blank">Jackie Ballarini's School's Starting Page</a></h4>Hey, if you haven't put all the stuff your new teachers need to know in one place, like this, you should! <a href="http://jhhs.d214.org/staff_resources/default.aspx" target="_blank">This page</a> was shared during a conversation initiated by <a href="https://sonatamathematique.wordpress.com/" target="_blank">Rachel </a>about supporting new teachers, and everybody drooled over it.<br /><br /><h4><a href="http://infinitesums.com/commentary/2015/youll-have-to-drag-me-out" target="_blank">Jonathan Claydon is Not Leaving</a></h4>I really enjoyed reading <a href="http://infinitesums.com/commentary/2015/youll-have-to-drag-me-out" target="_blank">Jonathan's piece</a> about why he intends to remain a classroom teacher. In this environment it's contrary to so many other articles coming out about folks throwing in the towel, and I think Jonathan shares important sentiments that usually go unarticulated, or at least don't go viral. But should.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-14447134549824247532015-06-18T09:18:00.002-04:002015-06-18T21:59:59.285-04:00Surprises in Scatter PlotsOn Derby Day, in my living room:<br /><br />"Do you think horse races have gotten faster over time, like people races?"<br /><br />"We can find out!"<br /><br /><a href="https://en.wikipedia.org/wiki/Kentucky_Derby#Winners" target="_blank">Heads to wikipedia</a>. Does <a href="https://docs.google.com/spreadsheets/d/1nY7qPT0O1XzUV7jC31RtNxv_3wUvOOXDOlGXve5eXrg/edit?usp=sharing" target="_blank">some fancy footwork in drive</a> to convert units of time from M:SS.SS to seconds. Heads to <a href="https://plot.ly/~k8nowak/3" target="_blank">plot.ly</a>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-tLahV8_5a6g/VYLB5cJOYFI/AAAAAAAAEqY/uWWph4bNnEM/s1600/Winning%2BKentucky%2BDerby%2BTimes.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="256" src="http://1.bp.blogspot.com/-tLahV8_5a6g/VYLB5cJOYFI/AAAAAAAAEqY/uWWph4bNnEM/s400/Winning%2BKentucky%2BDerby%2BTimes.png" width="400" /></a></div><br />"Whoa, something weird happened in 1896."<br /><br />"Is that when they figured out jockeys should be tiny?"<br /><br />"Oh, <a href="https://en.wikipedia.org/wiki/Kentucky_Derby#History" target="_blank">look</a>, they made the track shorter."<br /><br />"Oohhhhhh."<br /><br />"Let's only look at times since 1896."<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-FvhAanrSmA4/VYLDdM_ldAI/AAAAAAAAEqk/tE9fIemDejc/s1600/Winning%2BKentucky%2BDerby%2BTimes%2Bsince%2B1896.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="241" src="http://4.bp.blogspot.com/-FvhAanrSmA4/VYLDdM_ldAI/AAAAAAAAEqk/tE9fIemDejc/s400/Winning%2BKentucky%2BDerby%2BTimes%2Bsince%2B1896.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">"So, yes, but it's leveling off?"</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Looks that way."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Huh."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This data could be fun to build out for an activity to get kids using whatever scatterplot-creating tools you want them to use. It's also nice for interpreting plots -- it smacks you in the face that something changed in 1896, and there's a quick and satisfying explanation. Enjoy!</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-89130131241365811962015-06-12T07:49:00.002-04:002015-06-12T07:49:29.552-04:00Favorites Fridays 1Hi! Welcome to Favorites Fridays. Instead of just sharing or retweeting on Twitter, which is ephemeral and misses lots of people, I'm going to start collecting my favorite stuff from the mathematical educational Internet from the week here. I've never been one for regular publishing or weekly series-es, but we're going to give this a try. (This may be a dumb time to start this because I'm heading off on vacation next week and I promised my boi-freeeen I'd give Twitter a rest, so I'll skip a week soon but anyway.) I hope you find it useful, but this is also for my personal archival use too. Here goes!<br /><br /><h4><a href="http://recursiveprocess.com/mathprojects/" target="_blank">Dandersod's Calculus Projects</a></h4><div><blockquote class="twitter-tweet" data-cards="hidden" lang="en"><div dir="ltr" lang="en">Calculus Project Day 2! "The Math of Twitter" <a href="http://t.co/ncqFwJpE7u">http://t.co/ncqFwJpE7u</a> Benford's law and the Friendship Paradox</div>— Dan Anderson (@dandersod) <a href="https://twitter.com/dandersod/status/608264630956376064">June 9, 2015</a></blockquote>Dan Anderson (<a href="https://twitter.com/dandersod" target="_blank">@dandersod</a>) (does anyone else just think of him in their head as "dandersod?") set a project for his calculus kids, live-tweeted it, and <a href="http://recursiveprocess.com/mathprojects/" target="_blank">published their reports</a>. You might have mixed emotions about the phrase "calculus projects," but I found these to be super fun, interesting, entertaining reading.<br /><br /><h4><a href="https://teachingmathculture.wordpress.com/2015/06/09/facilitating-conversations-about-student-data/" target="_blank">Lani's Memo</a></h4><blockquote class="tr_bq">This memo focuses on research-based ideas on how to support common planning time so that it has the greatest potential for teacher learning about ambitious mathematics teaching. To that end, we provide a framework for effective conversations about mathematics teaching and learning. We develop the framework by using vignettes that show examples of stronger and weaker teacher collaboration.</blockquote>"Sometimes, you ask and the internet answers." Lani Horn came through with what <a href="https://twitter.com/JuliaTsygan/status/608260045252534272" target="_blank">Julie, and many teachers are looking for</a>: nuts and bolts direction for teachers hungry for useful professional conversations. We're tired of wasting collaboration time and "PLC time" (a now-meaningless name if there ever was one) on aimless, unhelpful activities that don't have an impact on our practice, and we know there's a better way. <a href="https://teachingmathculture.wordpress.com/2015/06/09/facilitating-conversations-about-student-data/" target="_blank">This post is going to be a huge help</a>. Bonus: <a href="https://teachingmathculture.wordpress.com/2015/04/28/making-sense-of-student-performance-data/" target="_blank">a summary on research about using student performance data.</a><br /><br /></div><h4><a href="https://vimeo.com/129522026" target="_blank">Tracy Zager's ShadowCon Talk</a></h4><div><blockquote class="twitter-tweet" data-conversation="none" lang="en"><div dir="ltr" lang="en"><a href="https://twitter.com/k8nowak">@k8nowak</a> <a href="https://twitter.com/TracyZager">@TracyZager</a> "In a class where math is taught in an authentic way, confusion is a good thing." Boom.</div>— Matt Enlow (@CmonMattTHINK) <a href="https://twitter.com/CmonMattTHINK/status/606933431608557568">June 5, 2015</a></blockquote><script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script><br /></div><div>It will blow your doors off. Tracy is dazzling. <a href="https://vimeo.com/129522026" target="_blank">Just go watch it</a>. Best use of word clouds in history.<br /><br /><h4><a href="http://www.michaelkaechele.com/how-to-build-a-pbl-culture/" target="_blank">Mike's How to Build a PBL Culture</a></h4><blockquote class="twitter-tweet" data-cards="hidden" lang="en"><div dir="ltr" lang="en">How to build a PBL Culture -my compilation of activities to start the year with students new to <a href="https://twitter.com/hashtag/PBL?src=hash">#PBL</a> <a href="http://t.co/7YRDKkyQc5">http://t.co/7YRDKkyQc5</a> <a href="https://twitter.com/hashtag/edchat?src=hash">#edchat</a></div>— Mike (@mikekaechele) <a href="https://twitter.com/mikekaechele/status/609316678510493696">June 12, 2015</a></blockquote><script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>Mike's PBL is Project Based, but I think this fab collection of activities and recommendations for kicking off a school year would work just as nicely if your PBL is Problem Based.<br /><br />And that's a wrap! Somebody hold me accountable for doing this next Friday! </div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-87093756092368544882015-05-07T15:24:00.000-04:002015-06-11T18:12:10.312-04:00Pretty Painless GamificationToday I was at a loss for something fun-ish to review circles in Geometry. I hastily searched my Evernote for "review games" and came across <a href="http://mathtalesfromthespring.blogspot.com/2009/10/ghosts-in-graveyard.html" target="_blank">this gem from 2009 from Kim</a>. She called it Ghosts in the Graveyard for Halloween, but since it's springtime I went with a garden theme. I modified the activity slightly.<br /><br />1. Set up a Smartboard file like so, for six groups to play. The ten objects to populate their gardens are infinitely cloned, and the fences are locked in place so they can't be accidentally moved. (This screenshot shows the "gardens" in the middle of a class.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-zI47e_jzECg/VUu4nHyaGJI/AAAAAAAAEmc/PV3QZhOzgyc/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="310" src="http://1.bp.blogspot.com/-zI47e_jzECg/VUu4nHyaGJI/AAAAAAAAEmc/PV3QZhOzgyc/s400/Capture.PNG" width="400" /></a></div><br /><br />2. Students in groups of 3-4. I wrote students' initials (in red) next to their garden.<br />3. Every student gets a copy <a href="https://www.dropbox.com/s/x2c3erjgijtgkj4/Quiz%20Review%20Questions.pdf?dl=0" target="_blank">of ten problems</a>.<br />4. When all group members understand a problem, they call me over. I randomly choose one student to explain how they did it.<br />5. If she can explain their process sufficiently, she can go up to the Smartboard and add the corresponding item to their garden. (If not, I just say okay, I'll be back in a couple minutes.)<br />6. They were instructed to use the review problems to help them study for the quiz tomorrow, so if they didn't get to all the problems in class, it was okay.<br /><br /><b>Why I liked it:</b><br /><br /><ul><li>It did not take forever to set up. I used the review questions I was planning on giving them anyway, and just had to whip up a smartboard page which took all of 5 minutes.</li><li>You wouldn't think that the state of an illustrated garden on a smartboard file would be very motivating, but they all worked diligently for the entire 30-ish minutes we did this. Thanks, Zynga.</li><li>I heard lots of good discussion as they made sure all of their group members understood a problem before calling me over.</li><li>Nobody could slack off and nobody got bored.</li></ul>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-34919352447379020262015-03-02T19:49:00.000-05:002015-06-11T18:12:46.789-04:00Kicking Some Serious Triangle BootyThe children understand that sin, cos, and tan are side ratios. The children! They understand! They are not making ridiculous mistakes, and they can answer deeper understanding questions like, "Explain why sin(11) = cos(79)." I think right triangle trig is a frequent victim of the "First ya do this, then ya do this" treatment -- where kids <a href="http://mathwithbaddrawings.com/2015/02/11/the-church-of-the-right-answer/" target="_blank">can solve problems but have no idea what is going on</a>. There's often not a ton of time for it, and it responds well to memorized procedures (in the short term). So, if your Day One of right triangle trig involves defining sine, cosine, and tangent, read on! I have a better way, and it doesn't take any longer.<br /><br />First, build on what students have already learned about similar triangles. Ideally, this unit immediately follows that one. On Day One, I assign each pair of students an angle. (You guys have 20 degrees. You all have 25. etc etc, all around the room, so each pair of students is responsible for a different angle.) They work through this document (<a href="https://www.dropbox.com/s/jn1qw86o71flq2d/Measuring%20Angles%202015.docx?dl=0" target="_blank">docx</a> <a href="https://www.dropbox.com/s/nc5p1e6nb47xgd7/Measuring%20Angles%202015.pdf?dl=0" target="_blank">pdf</a>), using <a href="http://tube.geogebra.org/student/m686897" target="_blank">Geogebra to do the measuring</a>. They write down the length of the side opposite and adjacent their angle, for triangles of five different sizes. It's important that they write down the lengths, divide them with a calculator, and experience surprise and wonder why they are all exactly the same. (Geogebra made this soooo much better and easier than when I did this with rulers and protractors. So much better. In fact, one of my Matt colleagues basically deserves a medal for all the times he's said "Why don't we just do this with Geogebra?" this year.)<br /><br />On this day, they just do opposite/adjacent ratio, share the ratio for their angle in a shared spreadsheet, and then everyone has access to the shared spreadsheet (an opp/adj-only trig table) to solve some problems (in that same document). The thing is, they are figuring out how to use what they have learned to solve the problems; they're not just repeating a procedure that was demonstrated. This took one 45-minute period, including checking Chromebooks out and in. I collect the sheets and look for students who had a strategy for #11 (how to solve when the variable is in the bottom of the ratio) so they can share their strategy next class.<br /><br />Next two classes, I provide them with <a href="https://www.dropbox.com/s/uahqj9aeq0et8mo/Table%20of%20Side%20Ratios%20-%20Google%20Sheets.pdf?dl=0" target="_blank">a table of all three ratios</a> (to tape in their notebook) for angles of 5-degree increments, and they work their way through <a href="https://www.dropbox.com/s/6qz93lbxu41t9tr/7.7%20Exploring%20ratios%20in%20right%20triangles.pdf?dl=0" target="_blank">this page</a> with appropriate help. For example, in the first set of problems, I just had them label the sides first. Then choose the ratio for all the problems, then solve for an answer. This particular document is not terribly pretty, because I had limited time to put it together. In every class, someone wanted to know why they couldn't use like hyp/adj if that equation was easier to solve. For those that asked, I pointed out that we couldn't look up hyp/adj in the table, BUT, they could use the other angle in the triangle. (And yes I'm aware they could use 1-over the ratio in the table, but that seemed like an overly complicated strategy to suggest.) I gave them a few find-sides and find-angles problems (limited to the angles in their table) to practice for homework. They did not all get to the back, but the kids to catch on/work quicker had something to do after the basic problems.<br /><br />Today I spilled the beans that these ratios have special names, and we could look them up in our calculator. We mostly <a href="https://www.dropbox.com/s/g60iwfeclqh7fsq/7.8%20Notes.pdf?dl=0" target="_blank">spent the period getting used to looking stuff up in the calculator</a> including some <a href="https://plickers.com/" target="_blank">hot Plickr action</a>, and <a href="https://www.dropbox.com/s/re1kc1d1cd32z7p/Find%20someone%20who%20can%20Bingo%20-%20Google%20Docs.pdf?dl=0" target="_blank">working on these problems</a> which they are finishing for homework. I told them they only had to do one "Explain why," but they had to complete all the rest.<br /><br />Tomorrow on our block day, we are going to go outside and figure out the heights of some really tall things (<a href="https://www.dropbox.com/s/9c3xeaf2aps4a52/Measuring%20a%20Really%20Tall%20Thing.docx?dl=0" target="_blank">docx</a> <a href="https://www.dropbox.com/s/dfycvyve71uc7yr/Measuring%20a%20Really%20Tall%20Thing.pdf?dl=0" target="_blank">pdf</a>). There are lots of "measuring tall things" activities out there, but I heavily adapted <a href="http://www.wmich.edu/engineer/ceee/edcsl/pdf/Indirect%20height.Tangents.pdf" target="_blank">this document</a>, so thanks to Christopher Conrad for posting it.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-86174664632238230902015-01-23T14:36:00.000-05:002015-06-11T18:13:04.261-04:00On Making Them Figure Something OutOften when I don't really know a great way to teach something, I end up defaulting to Making Them Do Something and Making Them Notice Something, and then finally Making Them Practice Something. I think lots of people do, and it's not bad. It's loads better then Telling Them Something, and it's certainly better than Children with Nothing To Do. But lots of times, the learning that comes out of MTDS and MTNS doesn't really stick that great. They can maybe do an exit ticket, but ask them a question that relies on The Thing in a week, and you just get a bunch of blank stares. So in my planning, I'm taking as a signpost Making Them Figure Something Out or MTFSO. I think the best existing curricula <i>depend</i> on MTFSO, and it must be nice to be working with one of those.<br /><br />So here's an example: the discriminant in Algebra 2, or said another way "What kind of roots does this quadratic equation have?" We're not particularly concerned in Virginia with them being able to define the word "discriminant," but they should be able to recognize whether the solutions are rational or irrational, real or non-real. Given a quadratic equation, they should be able to figure out what kind of solutions it has, and know how to describe those kinds of numbers. Here is a sample released item (<a href="http://www.doe.virginia.gov/testing/sol/released_tests/2013/algebra_2_released_in_spring_2014.pdf" target="_blank">question 50 of 50 in this document</a>).<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-G-9xtsIZ488/VMKYv-go_BI/AAAAAAAAEJI/L3L83EjhsOI/s1600/Screenshot%2B2015-01-23%2Bat%2B1.53.12%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="135" src="http://1.bp.blogspot.com/-G-9xtsIZ488/VMKYv-go_BI/AAAAAAAAEJI/L3L83EjhsOI/s1600/Screenshot%2B2015-01-23%2Bat%2B1.53.12%2BPM.png" width="400" /></a></div><br />You can find all kinds of MTDS/MTNS lessons about the discriminant. <a href="http://hchsmath.weebly.com/uploads/2/4/9/3/24930428/day_1_-_10.7_notes_key_.pdf" target="_blank">Here</a> and <a href="http://roensmath.edublogs.org/files/2013/04/MHF4U-Unit-2-6-disc-inv-spring-2013-1t1ddlg.pdf" target="_blank">here</a>, for example. But here's what I came up with to turn it into MTFSO. We had already spent a day on simplifying radicals (including with imaginary results), a couple days on solving by undoing and solving with the quadratic formula. Then we made a big map of different kinds of numbers with special attention to recognizing rationals vs irrational and real vs non-real.<br /><br />For this particular lesson, they first solved four equations using the quadratic formula: x<sup>2</sup> - 10x + 9 =0, x<sup>2</sup> - 6x + 9 = 0, x<sup>2</sup> - 7x + 9 = 0, and x<sup>2</sup> - 4x + 9 = 0. When we debriefed their solutions, we spent time describing the types of solutions, but we did not belabor the point about why the roots came out each way.<br /><br />Then, they sat in groups of 2-3 at big whiteboards and got one of these sets of questions:<br /><blockquote class="tr_bq">A1) Come up with a new, original quadratic equation whose roots are <b>real and irrational</b>. Demonstrate that your equation works by using the quadratic formula to solve it.<br />A2) Come up with a new, original quadratic equation whose roots are <b>real, rational, and unequal</b>. Demonstrate that your equation works by using the quadratic formula to solve it. </blockquote><blockquote class="tr_bq">B1) Come up with a new, original quadratic equation whose roots are <b>imaginary</b>. Demonstrate that your equation works by using the quadratic formula to solve it.<br />B2) Come up with a new, original quadratic equation whose roots are <b>real, rational, and equal</b>. Demonstrate that your equation works by using the quadratic formula to solve it.</blockquote>(In the first class to do this, I gave out the tasks haphazardly. But from that experience, learned that "rational" and "equal" are harder to find than "irrational" and "imaginary." So I adjusted accordingly for the second class. Each set above consists of one of the easier ones, and then one of the harder ones.)<br /><br />I saw different approaches... start with a desired answer and try to work backwards. Write out the quadratic formula with blank spots and repeatedly fill in and try values. Start with an equation very close to one we had just worked with and see how it worked out. And of course the bane of all group work reared its head - one kid grabs a marker and takes over while everyone else is happy to let them.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-JY5p8w51GjU/VMKgkf2xy6I/AAAAAAAAEJo/AH85VKipLFI/s1600/20150122_135425.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://1.bp.blogspot.com/-JY5p8w51GjU/VMKgkf2xy6I/AAAAAAAAEJo/AH85VKipLFI/s1600/20150122_135425.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-7zI66pdZ2B0/VMKgfrwe2uI/AAAAAAAAEJg/OXuE6iqClHo/s1600/20150122_135405.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-7zI66pdZ2B0/VMKgfrwe2uI/AAAAAAAAEJg/OXuE6iqClHo/s1600/20150122_135405.jpg" width="180" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-EkIYiNLjQnU/VMKgbAVf8lI/AAAAAAAAEJY/AahC9rFHFLI/s1600/20150122_135355.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-EkIYiNLjQnU/VMKgbAVf8lI/AAAAAAAAEJY/AahC9rFHFLI/s1600/20150122_135355.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">After groups had a chance to figure stuff out and explain how they did it (it all came down to paying attention to what kind of number was under the radical, of course), we summarized with some notes:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Lwc4AkyXM2U/VMKhpBxNk2I/AAAAAAAAEJs/MiSdO53kk8w/s1600/Screenshot%2B2015-01-23%2Bat%2B2.30.14%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="258" src="http://2.bp.blogspot.com/-Lwc4AkyXM2U/VMKhpBxNk2I/AAAAAAAAEJs/MiSdO53kk8w/s1600/Screenshot%2B2015-01-23%2Bat%2B2.30.14%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">And that is that. A general understanding of what was going on persisted to the next day... We'll see how Monday goes.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-51840271315767684652015-01-11T18:31:00.001-05:002015-06-11T17:42:56.697-04:00SSA ASA and All the RestI finally, Finally, FINALLY have a plan I like for introducing triangle congruence theorems.<br /><br />For a few years there (2? 3?) I had been trying to make a go of Triangles a la Fettucine <a href="http://www.nctm.org/publications/article.aspx?id=18305" target="_blank">as described in this MT article</a>. And I just couldn't work it! The principle of the thing is sound, and it always started out okay, but quickly got tedious and the kids would both lose interest and not get the point. Many students just would stubbornly not get the memo that you had to use the entire length of the colored-in sides, but you could use any length of the uncolored sides. The mechanics of the thing got in the way of seeing the larger picture.<br /><br />In this new lesson, I deliberately separated the triangle-creating phase from the seeing the larger picture phase.<br /><br />Before the lesson, I gave students the first page of the pre-assessment from this <a href="http://map.mathshell.org/materials/lessons.php?taskid=452" target="_blank">Shell Center Formative Assessment Task</a>. (Thanks someone on Twitter who suggested that.) I did not use the lesson itself, because I felt my students weren't at the point of being able to understand what it was asking. The pre-assessment was compared to what they could do afterward, of course, but also to kind of get their juices flowing about angles and side lengths and what congruent means. They worked on the pre-assessment for about 15 minutes.<br /><br />Phase 1 was constructing some triangles out of construction paper based on various given information using straight edge, protractor, and occasionally a compass. <a href="https://www.dropbox.com/s/96n62re5uhfgsn2/CongruentTrianglesIntro.docx?dl=0" target="_blank">Here is the instruction page</a>. (Thanks again, Twitter, for some helpful feedback making it better before it went live to the children.) Students were in groups of 3 or 4, and <i>the group</i> was responsible for creating the nine triangles (There are ten on the sheet, but the last one is impossible.) For thoroughness, I'd love if every student had to create all the triangles, but I was afraid 1) that would take way way too long and 2) many students would tire of it midway through and check out. I think my instincts were right on both counts. The more difficult constructions were marked with a *, which I told the students, which allowed for some self-moderated differentiation (by less-confident students quickly claiming responsibility for non-* triangles).<br /><br />Before they started I ran a quick protractor clinic, and they were off. The groups worked on creating triangles for 30-40 minutes. Beforehand, I created a set of reference triangles out of cardstock, so I could quickly assess their products for accuracy when they were done. I just kept the correct ones and discarded the inaccurate ones (where an angle was not measured correctly, for example.) (Instead of making them re-do incorrect ones, but I think that would be a fine thing to do if you can swing the time.) The groups that finished first, I handed off the cardstock triangles and put them in charge of assessing other groups' work. By the time triangle construction was complete in all three of my Geometry sections, I had 10+ correctly-made copies of each triangle.<br /><br />I suppose you could try to recreate this experience with dynamic geometry software somehow, but I'm dubious that there's a replacement for creating physical triangles with your hands. Students directly encountered having choices for how to finish making the triangle (when they were only given the lengths of two sides, for example) vs being locked into only one possible triangle (when they were given ASA, for example). In the future, I'd like to be more deliberate in assigning each student one of each.<br /><br />After triangle construction, we ran through a quick lesson on notation and naming conventions for congruent polygons. Then I put up a kind of standard-looking congruent triangle proof, where enough information was given (or able to be inferred) to show that all three pairs of sides and all three pairs of angles were congruent. We wrote out a 9-step proof that proved the triangles congruent by SASASA. "Wasn't that a pain?" I said. "Yes," they said. "Wouldn't it be nice if we could know triangles were congruent to each other with a smaller set of information?" I said. "Yes," they said.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/DgY5jMYQLmsjYtkgwzaNLkusEGgeztSIOWeAFRxLD-4=s315-p-no" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/DgY5jMYQLmsjYtkgwzaNLkusEGgeztSIOWeAFRxLD-4=s315-p-no" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">after school gluestick par-taaaayyyy</td></tr></tbody></table><div class="separator" style="clear: both; text-align: left;">In the meantime (the lesson covered by this post spanned five days, FYI) I had glued all the triangle A's to a poster, all the triangle B's to a poster, etc. Before they looked at them, I had them predict (using the original instruction sheet) whether they thought all the copies of each triangle had to be congruent. Will all the triangle A's be congruent? Will all the triangle B's be congruent? etc. Reminders that for congruence, it's okay if you have to reflect or rotate one to make it look exactly like the other. </div><br />Justin Lanier and Pershan? possibly others? brought up the issue on Twitter that triangle-uniqueness (will this given information only allow you to make one triangle?) is a cognitively different thing from triangle congruence (can I be sure these two triangles are identical?). That was a slippery thing that always poked at the edges of my thoughts this time in the school year, but I'd never thought to explicitly address it. I think this lesson does a nice job of bridging those two related understandings. The triangle-constructing compels one to think whether there are choices to be made in what this triangle looks like... or is it unique? But putting all the triangles together on a poster highlights the question of do all these triangles have to be identical, using certain given information? Pretty seamlessly, I think, and without having to dwell on it.<br /><br />Using <a href="https://www.dropbox.com/s/0wz6qmjrfrhwdd4/5%20Congruent%20Triangle%20Summary%20Sheet.docx?dl=0" target="_blank">a recording sheet</a> (created by my colleague, Matt), they did a gallery walk, recording whether, in fact, all the triangles were congruent, which parts were the given information, and drawing a sketch. (He provided the triangle outlines in the rightmost column -- I took those off.) Then we had a quick discussion and they made their first foray into identifying which theorem applied based on given information (the back of that sheet).<br /><br />And that was that! On their quiz Friday (usually we have a half-period quiz on Fridays), I asked another question very similar to one of the pre-assessment questions, and every single student showed growth in their ability to explain why the given information was not enough to guarantee unique triangles.<br /><br />(I'm not sure if this lesson is useful in Common Core land -- over there, you're supposed to link congruence to rigid transformations. Which I do here in Virginia, informally, but it doesn't rise to the level of students performing transformational proofs.)Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-82351303126016105492014-11-19T21:08:00.001-05:002015-06-11T18:13:25.199-04:00Graphles to GraphlesNew game! My Algebra 2 students struggle with stating the domain and range for reasons including: trouble understanding and writing inequalities, and a lack of comfort with the coordinate plane. We spent a day on looking at graphs and identifying their domain and range. We learned to deploy our wonderful domain meters and range meters that <a href="http://samjshah.com/2009/12/10/phrases-actions-rituals/" target="_blank">I learned about from Sam</a>. But for maybe 25% of the students, the cluebird was stubbornly refusing to land.<br /><br />So, I thought asking the question backward might be a good way to attack it. Instead of here's the graph, what's the D and R? ask, here's a D and/or R, draw a graph. I mean, I know this is pretty standard fare. The thing is, I didn't want to do examples and a worksheet, or hold-up-your-whiteboard so I could somehow assess 22 graphs in a split second. It seemed like there should be a better way.<br /><br />So I did what I do, which is ask on Twitter. And I got lots of helpful ideas, but this was the one that I latched onto and ran with:<br /><blockquote class="twitter-tweet" lang="en"><a href="https://twitter.com/k8nowak">@k8nowak</a> make it a game? Charades-ish: Each student puts in 2 requirements. Team draws two and has a minute to make a function that fits.<br />— John Golden (@mathhombre) <a href="https://twitter.com/mathhombre/status/534486496884326400">November 17, 2014</a></blockquote>The end result is, I'd argue, more like Apples to Apples than Charades (hence the title). <br /><br />To prep: Make game cards. I printed each page (<a href="https://www.dropbox.com/s/hwvbzes9i4vc8i4/GraphlestoGraphlesGameCards.docx?dl=0" target="_blank">docx</a> <a href="https://www.dropbox.com/s/81d3eblg6k1mkw2/GraphlestoGraphlesGameCards.pdf?dl=0" target="_blank">pdf</a>) on a different color card stock. Student play in groups of 4-ish, so plan accordingly. I printed 6 sets. (John suggested having students submit constraints, but, for this crew, I decided to unload that part and create cards with the constraints.) You'll also need a mini-whiteboard, marker, and eraser for each student. Check your dry erase markers, because nothing kills a math game buzz like a weaksauce marker. (I'll admit to a minor teacher temper tantrum where I uttered (okay, yelled) the words "I'M NOT THE MARKER FAIRY! I DON'T POOP MARKERS!" Teacher of the year, right here, folks.) Also, you'll need some kind of token that players can collect when they win a turn. I use these plastic counting chips that I use for everything, but anything would work, candy, whatever.<br /><br />Doing a demo round with a few kids playing and everyone watching will pay off, in the more-kids-will-know-what-is-up sense.<br /><br />Here's how the game plays:<br /><ul><li>Someone is the referee.</li><li>To begin the turn, the referee turns over two (or one, or three) different-colored cards, <b>and reads them out loud</b>. (I feel the reading aloud is important practice for interpreting inequalities.) You could do, like, first round is one card, second round is two cards, third round is three cards. Whatever suits your needs.</li><li>The other players have one minute to sketch a graph meeting the constraints on the cards. The referee is responsible for timing one minute.</li><li>The players hold up their mini-whiteboards so the referee can see. </li><li>The referee disqualifies any graphs that don't match the cards, and explains why. Other players should police this, too.</li><li>Of the remaining graphs, the referee picks his favorite. This player wins a token.</li><li>The turn is over, and the player to the referee's right becomes the new referee.</li><li>Repeat.</li></ul><div>It was great! Here are things I liked about it:</div><div><ul><li>100% participation 100% of the time. At no point should anyone be kicking back.</li><li>Nowhere to hide. There were a couple kids who had to come to me and say, "Miss Nowak, I really don't know what's going on." which I don't think they'd be compelled to do if we were just doing some practice problems.</li><li>Good conversations. Especially reasons for why graphs were disqualified. "You need an arrow there! The domain goes to infinity!" That sort of thing.</li><li>Students were necessarily <i>creating</i> and <i>evaluating</i>. <a href="http://en.wikipedia.org/wiki/Bloom's_taxonomy" target="_blank">Take that, Bloom</a>! </li><li>Built-in review of what makes a graph a function vs not a function.</li><li>My chronic doodlers had a venue to express themselves. Especially if the graph didn't have to be a function.</li><li>Authentic game play. You could use your knowledge of what a referee liked to curry favor.</li></ul>Here are some action shots. Let me know if you try it, and how it goes!</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-BtSxfVGg7Ic/VG1Lo141oXI/AAAAAAAADvA/shpQ_C4ykwk/s1600/20141119_134734.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://4.bp.blogspot.com/-BtSxfVGg7Ic/VG1Lo141oXI/AAAAAAAADvA/shpQ_C4ykwk/s1600/20141119_134734.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-7BZ6I6Fca-A/VG1Lhv5aKkI/AAAAAAAADu4/L5PkXQq-AY8/s1600/20141119_135249.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-7BZ6I6Fca-A/VG1Lhv5aKkI/AAAAAAAADu4/L5PkXQq-AY8/s1600/20141119_135249.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-IPh0o_4t1SQ/VG1LvpITmoI/AAAAAAAADvI/fZgCR4ig298/s1600/20141119_134644.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-IPh0o_4t1SQ/VG1LvpITmoI/AAAAAAAADvI/fZgCR4ig298/s1600/20141119_134644.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-4KSmIjh1hvY/VG1L2kgMN2I/AAAAAAAADvQ/wASYozJmSsc/s1600/20141119_134708.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-4KSmIjh1hvY/VG1L2kgMN2I/AAAAAAAADvQ/wASYozJmSsc/s1600/20141119_134708.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-57939903865275278852014-11-13T20:14:00.000-05:002015-06-11T18:13:51.931-04:00We Got a ProblemWe spent practically the whole period (35-ish minutes) on one problem today. This one, that <a href="http://relearningtoteach.blogspot.com/2014/11/day-52-single-problem.html" target="_blank">Justin wrote about recently</a>, that he found on <a href="http://fivetriangles.blogspot.com/2014/11/202-folded-rectangular-strip.html" target="_blank">Five Triangles</a>:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-zZcrKOy-OpU/VXoDMhwsOJI/AAAAAAAAEpQ/6shFxrnY2x8/s1600/paperstrip.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="270" src="http://2.bp.blogspot.com/-zZcrKOy-OpU/VXoDMhwsOJI/AAAAAAAAEpQ/6shFxrnY2x8/s320/paperstrip.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Since we just spent a few days naming pairs of angles made by parallel lines and proving what's congruent and what's supplementary, I was provisionally hoping we'd get, at the end, 10 or so minutes for students to present various solutions to the class. That did not happen in any of my three class periods. Because I didn't have the heart to interrupt them. At the 10-minutes-left mark, too many were still making passionate arguments to their small groups about why they thought their solution worked.<br /><br />Here's what we did: The big whiteboards were on the tables. Groups of 3 or 4. I stated the problem while showing this diagram. Made a big deal of starting with a regular old piece of copier paper and making a single fold. 3-5 minutes silent, individual think time. No class made it to 5 minutes without a buzz starting. I wrote a time on the board and said, by this time, everyone in your group needs to be prepared to present a solution to the class. Also, the answer is not as important as the reasoning that got you there. If you are saying something like "this angle has this many degrees," you have to explain the reason why that must be true.<br /><br />Then I started listening and circulating. There was almost 100% engagement, and I have to think it's due, to a high degree, to the problem itself. This problem just felt do-able, but not obvious, to every learner in the room -- the sweet spot.<br /><br />When a group would start crowing, in their 9th grade way, that "MISS NOWAK. WE GOT IT," I refused to confirm or deny that their answer was right, and asked a randomly selected group member (everyone was supposed to be able to explain the solution) to walk me through the reasoning. I played the role of highly annoying and dense skeptic. "Wait, how did you know that angle was 90?" "Because it's a RIGHT ANGLE." "Wait, how do you know it's a right angle?" "... ... ...BECAUSE A SHEET OF PAPER IS A RECTANGLE." "Oohhh, right." And then, when they got to a part that was not justified (an assumed bisector, an assumed isosceles triangle, trying to use two sets of parallel lines to leap to a conclusion about congruent angles), I wasn't shy about saying I wasn't convinced. In their groups, they had already harvested the low hanging reasoning fruit. I figured my experienced eye was valuable for training a spotlight on flaws in their arguments. And they responded well, in a back-to-the-drawing-board kind of way.<br /><br /><b>Props</b><br />I'm in the habit of slagging myself on here, but I'm going to take a moment and describe a few times I witnessed and celebrated some great, inspired ideas with at least one learner today:<br /><ul><li>You extended that line to make it intersect another line!</li><li>You marked those lines as parallel with some arrows! And those other ones too!</li><li>You made an estimate of a reasonable answer!</li><li>You grabbed a piece of scrap copier paper and made a physical model to look at and play with!</li><li>You suggested your group start over and draw a clearer diagram, so more people would know what you were talking about!</li></ul><b>The Value</b><br />This seemed evident: the importance of being able to articulate how you know things are true. I think that was one of the purposes of proof that Pershan hit on last summer... knowing why a right answer is right. In a world (of school Geometry) where if I'm careless, I'm too often asking kids to "prove" things that they think are already obvious, I want to make as much room as possible for problems like this where something is not obvious and needs justification.<br /><br /><b>Questions</b><br />I'm not entirely comfortable with leaving 72 kids hanging about what the correct solution was, and why. What I <i>want</i> is for this to keep bugging them, and for them to make little doodles and sketches of it in their spare time, and for them to not be able to leave it alone. I did not want to ruin anyone's fun. At the same time, I think many kids could have benefited and learned from seeing a few different correct arguments for why the angle had to measure 140 degrees. This is still an open question for me -- what's the best way to handle kids/groups that don't arrive at the correct answer? Do you let the question hang, or do you interrupt everybody so groups who made it down a valid path can have time to show what they did? I'd love to pick it up tomorrow, but I'm going to be out (whaddup, #NCTMRichmond!) so we'll have to see if anyone has any memory of what happened today on Monday.<br /><br /><b>Regrets</b><br />They should have sent a poet. I should have done this on a block day.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-bmWqEPcyKfA/VGVW6PBGwcI/AAAAAAAADtA/1-nSy1hp3fg/s1600/20141113_125857.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://2.bp.blogspot.com/-bmWqEPcyKfA/VGVW6PBGwcI/AAAAAAAADtA/1-nSy1hp3fg/s320/20141113_125857.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-AfIWz2nwqGk/VGVWCVDcjbI/AAAAAAAADso/Focc7PqvHfQ/s1600/20141113_125925.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://3.bp.blogspot.com/-AfIWz2nwqGk/VGVWCVDcjbI/AAAAAAAADso/Focc7PqvHfQ/s320/20141113_125925.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-BFOnEZ0kGXM/VGVWCvHKc3I/AAAAAAAADsk/Y0uOsoFkTG4/s1600/20141113_125945.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-BFOnEZ0kGXM/VGVWCvHKc3I/AAAAAAAADsk/Y0uOsoFkTG4/s320/20141113_125945.jpg" width="180" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-8mYPW-8Zso0/VGVWC4RMoRI/AAAAAAAADss/RJUM7hsEFRk/s1600/20141113_130013.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://3.bp.blogspot.com/-8mYPW-8Zso0/VGVWC4RMoRI/AAAAAAAADss/RJUM7hsEFRk/s320/20141113_130013.jpg" width="320" /></a></div><br />Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-20262527777827355082014-11-10T18:26:00.002-05:002015-06-11T18:14:11.337-04:00DDT, Y'allToday in Geometry we tried Dance, Dance Transversal as popularized by <a href="http://algebrainiac.wordpress.com/2013/10/22/dance-dance-transversal/" target="_blank">Jessica</a> and <a href="http://ispeakmath.org/2014/10/24/transversals-parallel-lines-and-discovering-angle-properties/" target="_blank">Julie</a>. The kids dug it, and nailed an exit ticket identifying names of pairs of angles. I followed Julie's plan pretty closely. I loved that the kids were up and moving around for a good 20 minutes of class. (Was anyone else traumatized by that <a href="http://grantwiggins.wordpress.com/2014/10/10/a-veteran-teacher-turned-coach-shadows-2-students-for-2-days-a-sobering-lesson-learned/" target="_blank">Grant Wiggins article</a>? I'm very on the lookout for ways to make kids move.)<br /><br />I just want to add one more resource to the arsenal: a <a href="https://www.dropbox.com/s/sutl9oz846t2ma2/Dance%20Dance.pptx?dl=0" target="_blank">powerpoint</a> I made to show while playing. The slides auto-play the different moves. There are some initial slides that demonstrate where to put your feet for each cue. Then, the first two game slides are timed with a 1.5 second delay and worked well with <a href="https://www.youtube.com/watch?v=iS1g8G_njx8" target="_blank">Problem</a>, and the second two game slides are timed with a 1 second delay and worked well with <a href="https://www.youtube.com/watch?v=Vysgv7qVYTo" target="_blank">Dynamite</a>.<br /><br />Here's video. In case you're wondering, I did also play along, every period. Because their dancing did not have enough FLAVOR, and I had to demonstrate. Try to ignore the one kid doing some kind of demented hopscotch:<br /><br /><div class="separator" style="clear: both; text-align: center;"><object width="320" height="266" class="BLOG_video_class" id="BLOG_video-79702e3a419e783b" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,40,0"><param name="movie" value="https://www.youtube.com/get_player"><param name="bgcolor" value="#FFFFFF"><param name="allowfullscreen" value="true"><param name="flashvars" value="flvurl=https://redirector.googlevideo.com/videoplayback?id%3D79702e3a419e783b%26itag%3D5%26source%3Dblogger%26requiressl%3Dyes%26app%3Dblogger%26cmo%3Dsecure_transport%3Dyes%26cmo%3Dsensitive_content%3Dyes%26ip%3D0.0.0.0%26ipbits%3D0%26expire%3D1489880862%26sparams%3Dip,ipbits,expire,id,itag,source,requiressl%26signature%3D32472A8D5D6532DEDDB528EC085ACBA460F601FA.52BA5593B89C370BA2DEB1AA5FAB7D28B29C3ABE%26key%3Dck2&iurl=http://video.google.com/ThumbnailServer2?app%3Dblogger%26contentid%3D79702e3a419e783b%26offsetms%3D5000%26itag%3Dw160%26sigh%3DByfBK2e0DNbfw4J0FszVpRAqdmU&autoplay=0&ps=blogger"><embed src="https://www.youtube.com/get_player" type="application/x-shockwave-flash" width="320" height="266" bgcolor="#FFFFFF" flashvars="flvurl=https://redirector.googlevideo.com/videoplayback?id%3D79702e3a419e783b%26itag%3D5%26source%3Dblogger%26requiressl%3Dyes%26app%3Dblogger%26cmo%3Dsecure_transport%3Dyes%26cmo%3Dsensitive_content%3Dyes%26ip%3D0.0.0.0%26ipbits%3D0%26expire%3D1489880862%26sparams%3Dip,ipbits,expire,id,itag,source,requiressl%26signature%3D32472A8D5D6532DEDDB528EC085ACBA460F601FA.52BA5593B89C370BA2DEB1AA5FAB7D28B29C3ABE%26key%3Dck2&iurl=http://video.google.com/ThumbnailServer2?app%3Dblogger%26contentid%3D79702e3a419e783b%26offsetms%3D5000%26itag%3Dw160%26sigh%3DByfBK2e0DNbfw4J0FszVpRAqdmU&autoplay=0&ps=blogger" allowFullScreen="true" /></object></div><br /><br /><br />Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-50281571787495952032014-11-08T10:52:00.001-05:002015-06-11T18:14:25.371-04:00Gallery Walk for Noticing Features of Inverse FunctionsI put a call out on Twitter last week for good things for inverse functions. I got a few helpful responses but nothing that was really <i>the thing</i>. So here's what I made.<br /><div><br /></div><div>The day before, we had worked with inverse functions as doing and undoing equations. I started with ciphers. Students walked in and the board said, IQQF OQTPKPI, DGCWVKHWN DTCKPU! with no explanation from me. I just greeted them and took attendance and acted nonchalant. One kid sidles up and goes all sotto voce, "Miss Nowak, does the first part say <i>Good Morning</i>?" Since the good morning part was a pretty easy crack using context, after a minute or two someone notices that all the letters are shifted over by two, and can't keep from blurting it out, and we're off.<br /><br />I had one of them explain how the encoding was done with this example. Then, they wrote secret messages using their own shift <i>n</i> cipher, traded, decoded. I babbled a little bit about Caesar and Enigma (I really want to show them <a href="https://www.youtube.com/watch?v=G2_Q9FoD-oQ" target="_blank">this Numberphile video</a>, thanks for the tip <a href="https://twitter.com/mikeandallie" target="_blank">Mike Lawler</a>). The encoding and trading and decoding only took about ten minutes, we went through one from beginning to end: what was your message, how did you encode it, how did you decode it. The alphabet was written on the board along with a counting number under each letter, the idea being that if your encoding added 5, the decoding would be subtract 5.</div><div><br /></div><div>We spent the rest of that day couching inverses in terms of equation rules. <i>x</i> + 5 and <i>x</i> - 5 is fine and pretty obvious, but what about more complicated rules. Kids had mini whiteboards, I'd throw a function on the board and they'd try to write the inverse. Each time, they wrote down operations done in the original function, inverse operations in reverse order, then do that to an x. So for example if the given function is 3<i>x</i><sup>2</sup> - 5, they write down "square, multiply by 3, subtract 5" and then write down "add five, divide by 3, square root." Plop down an <i>x</i> and do those things to it. The biggest hurdles were order of operations (so they might write down "multiply by 3, square, subtract 5"). Also, always undoing the whole of what came before. So in this example, they'd be likely to write <i>x</i> + 5/3 instead of (<i>x </i>+ 5)/3. But we just kept going and honestly, they didn't want to stop until they were getting them right. (I just got an idea about how to make this part better. Compute like f(3) (or something) and run the result through their inverse to see if a 3 comes out. (Instead of just you're right or wrong because I say so.) Have to figure out how to make that manageable.)<br /><br />The next day, I wanted them to notice all the nice things that are true for functions and their inverses: the symmetry over <i>y</i> = <i>x</i>, that the inputs and outputs trade places, that <i>f</i><sup>-1</sup>(<i>f</i>(<i>x</i>)) = <i>x</i>. So, each student got <a href="https://www.dropbox.com/s/7091w2ewgfos6vv/3C%20inverse%20cards.docx?dl=0" target="_blank">one of these cards</a>. They figured out the inverse of that function using the technique from the day before. There was another student in the room with the inverse of their function, so they had to get up, talk to people, and then sit with their partner.<br /><br />Each pair of students got <a href="https://www.dropbox.com/s/1xt8emfw7jxq4vy/3C%20inverse%20cards%20template.docx?dl=0" target="_blank">one of these</a> (the first page). They tacked their cards to the paper, completed the tables, graphed each function in a different color, and computed <i>f</i><sup>-1</sup>(<i>f</i>(0)) and <i>f</i><sup>-1</sup>(<i>f</i>(1)). They needed various levels of support interpreting instructions, but it helped to have them working in pairs on the same piece of paper - there was a natural reason for them to talk to each other to figure it out. My colleague <a href="https://twitter.com/lbburke" target="_blank">Lois</a> is teaching the same course, and got a coach to come in for one of her sections, which was a great move.<br /><br />As the mini-posters were completed, they were hung up around the room. I said, hey, you all had different functions and now they're up there with their inverses. There are some neat things that are always true about a function and its inverse. Walk around and look at them all, and write down at least two things you notice. If you look at <a href="https://www.dropbox.com/s/1xt8emfw7jxq4vy/3C%20inverse%20cards%20template.docx?dl=0" target="_blank">page 2 of this same document</a>, the first question has space for them to write down observations.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JQY17QKjBok/VF47s2X67ZI/AAAAAAAADoI/3bfViga7rZc/s1600/New%2BSkitch.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="225" src="http://3.bp.blogspot.com/-JQY17QKjBok/VF47s2X67ZI/AAAAAAAADoI/3bfViga7rZc/s1600/New%2BSkitch.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>They sat back down, and they shared their noticings with the class. I had Desmos up on the projector with some pre-loaded functions, so we had a concrete thing to point to as they were sharing.<br /><br />Then they got to work on the rest of that page 2, which is lifted directly from lesson 6 of <a href="http://www.emathinstruction.com/id7.html" target="_blank">this eMathInstruction textbook</a> (thanks Sam for pointing me to this resource). Some of them were able to just do those problems, some needed help restating the given information and what the problems were asking.<br /><br />So there you have it. I especially liked this lesson for the social, discussion, get-up-and-move-around aspects. These Algebra 2 classes have not responded positively to problem-posing when they haven't been "shown how" to do a problem first, but, we have been successful with lessons like this where we break the questions into clear chunks while still requiring that they do some thinking and figuring out. It's a bit of a tightrope walk but that's how you get down a tightrope, right? One tiny step at a time?</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-48854809373420637442014-11-07T05:19:00.002-05:002014-11-07T05:22:00.162-05:00Fire Up Blogging MachineYo. This year is hard. New building, blah blah. I'll pause a minute for no one to be surprised.<br /><br />I feel, very often, like I'm not that good at this. I know that everyone does sometimes. I know, I know. I think it has very much to do with attending to formative assessment every day. (Every. Damn. Day.) Measurement: making it hard to lie to yourself since... measuring was invented. <br /><br />The children are charming and testy and pathetic and confident and devious and brave, all in the same day, all in the same 45 minutes. There are 70 ninth graders that move through my room, and the thing is that a ninth grader is like the weather in Buffalo in April -- if you don't like it (or if you do), just wait five minutes.<br /><br />I have lessons that I want to write up for this blog, the problem being my artifacts (documents, pictures, student work, etc) are all over the place. The file system on the school network is unreliable, so teachers all use either Dropbox or Drive or flash drives to store and/or keep a backup of everything (even though Dropbox isn't installed at school -- it's web interface only, 2005-style). Colleagues have been very generous sharing (bewildering Virginia-standards-based) materials with me. So all the stuff I've modified or created and used is on... the school file system and Dropbox and Drive and a flash drive. That puts just enough of an annoying-barrier in the way of assembling blog posts. I have got to get my computer file organizing act together.<br /><br />So, those are some lame excuses for the radio silence. More coming. I'll figure this out.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0