tag:blogger.com,1999:blog-6651514617266100245Sun, 23 Jul 2017 10:33:49 +0000math stuffgrade9schoolgrade7grade8ibtravelingel salvadorgrade11friendsGeoffmoving logisticsgrade10dancingfamilyfoodspanishES tourist trapsNYCberlincalculusgermanybeachdrivingbooksgrade12moviesmusicprecalculusDE tourist trapsdept chairweddingsPCMIgermanvolcanoweatherAsiaKlingSI11agisblogsbusseschinacrimehikingpovertyunited stateswcydwtyogaalgebra2bankingbarsbasic utilitiesguatemalahistorylakeluxuriessocial securityvisaatamibelizecosta ricafloodgoalsgoodbyesmom's storiespoliticsseattlesurfingtaxwomenargentinaaustinblog carnivalschristmasglacierherbal medicinehospitalkrakowmailmapsmtapnicaraguanorthwestpanamaperupolandpragueroadssoccerstressswine flutechnologyturkeyI Hope This Old Train Breaks Down...http://untilnextstop.blogspot.com/noreply@blogger.com (untilnextstop)Blogger552125tag:blogger.com,1999:blog-6651514617266100245.post-4764074768488246658Wed, 16 Nov 2016 16:31:00 +00002016-11-16T11:40:29.023-05:00Quizster: The Formative Assessment App<div dir="ltr" id="docs-internal-guid-5e9d86c8-6dd6-7650-7eb6-598cf0e50468" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;">(cross-listed on medium.com)</span></div><div dir="ltr" id="docs-internal-guid-5e9d86c8-6dd6-7650-7eb6-598cf0e50468" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: "arial";"><span style="font-size: 43.3038px;"><br /></span></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://quizster.co/wp-content/uploads/2016/11/Quizster-Video-Overlay-Cropped-Mascot-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="https://quizster.co/wp-content/uploads/2016/11/Quizster-Video-Overlay-Cropped-Mascot-2.png" width="320" /></a></div><div dir="ltr" id="docs-internal-guid-5e9d86c8-6dd6-7650-7eb6-598cf0e50468" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: "arial";"><span style="font-size: 43.3038px;"><br /></span></span></div><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: "arial"; vertical-align: baseline;">I teach high-school math, grades 9 through 12, with classes ranging from Geometry to Calculus. In the past few years, I have found myself increasingly interested in the structures of formative assessment, because although formative assessment is something we all aim to do in our classes, its specific method of implementation can have varying degrees of effectiveness on improving student learning. During the last two years, I have implemented weekly “No Big Deal quizzes” that contain a couple of questions targeting a recent topic, but without including any scaffolding in the questions or cues hinting at specific strategies. The students tended to fall on a spectrum on these formative assessments. They would take about 15 minutes at the start of class trying the questions; I would then collect their responses and go over the answers as a class as the students took careful notes and asked questions. The next day, I would return the graded “quizzes” (usually worth only 2 to 4 points for one or two questions) and assign similar problems on the board for the students to copy down and complete, in order to make up any points missed via showing an improved understanding on those topics. For some students, just turning this new assignment in is enough to show a thorough understanding and to address previous misconceptions, but for other students in the class, remediation frequently requires a continued back-and-forth dialogue on the new questions until they finally get it and are able to demonstrate that understanding on paper. If they have trouble revising the responses, they would first go to their notes on the initial quiz and ask me clarifying questions during class. If that wasn't enough, I would offer to sit down with them to go over the process again one-on-one. What this allowed me to do was to address real-time gaps in their knowledge, so that by the time I gave a summative assessment at the end of the unit (or halfway through a long unit), I could hold all the students accountable for higher-level complexity in problem-solving. This paper-based formative assessment process worked well and was critical in advancing my students’ learning, but I found myself having trouble with effectively following up with my students regarding their assessment results. They sometimes misplaced the graded papers with my written feedback on it, or they couldn't remember whether they still needed to revise a particular quiz. When they did sit down with me, I often wished that I had a more detailed record of their processes and misconceptions, instead of just a score in my gradebook and an unreliable memory of their errors. </span></div><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: "arial"; vertical-align: baseline;">My husband is a software developer, always on the lookout for ways to improve the existing ways of doing things. Sensing my frustration and seeing the hours I put into sorting through paperwork, he and I sat down to hash out the design for an app that would allow me to grade formative assessments in pieces, on the go, while I am on the bus or waiting for a meeting to start. The app we built takes photos of student work and allows the teacher to dialogue with the student, updating their assignment grades incrementally and providing additional feedback until the student completely masters the desired skill. It would also provide a photo record of the students’ progress, to enable effective one-on-one conferencing. I piloted the use of this program last year with 60 of my students during a full school term, and it easily complemented the formative assessment I was already doing. To maintain their focus on the math, the students would still complete their work on paper, but instead of handing me loose sheets during class, they would simply use their phones or tablets to take a photo of their work through the app for submission, which took a matter of seconds. After class, I would sit down and open the app to find all of their submissions, already sorted by question. I would grade a single question at a time using just my fingers and a touchscreen interface on a mobile device, across all students, and upon exiting, the scores and feedback would be published to the students instantaneously. Afterwards, the students would be able to submit further revisions and to receive further feedback until both they and I felt satisfied with the results. This app quickly gained popularity with my students. They loved the camera interface, the instantaneous feedback, and the ability to pull up an entire list of assignments to see which items still needed their attention. The more organized students liked not having loose sheets of paper floating around and to be able to keep track of all of their written work sequentially in a paper notebook, which stayed in their possession. I liked the flexibility to selectively collect a single question from a homework assignment, the freedom to grade anywhere without lugging around tons of papers and notebooks, and the ability to focus on providing quality feedback in real-time without fussing with the overhead surrounding the receipt, recording, and returning of assignments, because <a href="http://quizster.co/" target="_blank">Quizster</a> took care of all of those logistics for me.</span></div><br /><span style="font-family: "arial"; vertical-align: baseline;"><a href="http://quizster.co/" target="_blank">Quizster</a> has helped to streamline an important part of my students’ learning. In doing so, it has made me a more effective teacher, being able to focus my energy on what matters -- identifying student needs and personalizing my responses. I look forward to reading about how other teachers will implement this app in their classroom. Following 6 months of initial prototyping and testing with a small group of teachers, the app is now ready for release to a wider audience. If you wish to be part of the testing and evolution of this product, you can find more information at <a href="http://quizster.co/">http://quizster.co</a> .</span><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/E6tTEHAxOvE" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/E6tTEHAxOvE/quizster-formative-assessment-app.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2016/11/quizster-formative-assessment-app.htmltag:blogger.com,1999:blog-6651514617266100245.post-2392422819308767Fri, 08 Jan 2016 03:10:00 +00002016-01-07T22:10:55.984-05:00GoalMy goal for 2016 is to always respond to a potential conflict by killing it with kindness. I had a situation in class today which I had handled calmly, but in hindsight I don't know how kindly I had come across because I was actually feeling fairly upset in the moment. This goal extends beyond professional settings, but I do want to make sure it is something that I keep striving towards in the classroom. How do I consistently show love to a kid who is misbehaving in the moment? How do I do that with the people who are closest to me? As I prepare for parenthood*, that seems like an ever-important question to explore for myself.<br /><br />(*Yes, parenthood! Baby on the way, if all goes well between now and June!)<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/ox2Sm5k9Oxk" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/ox2Sm5k9Oxk/goal.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2016/01/goal.htmltag:blogger.com,1999:blog-6651514617266100245.post-4765158370013851575Tue, 10 Nov 2015 05:03:00 +00002015-11-10T00:18:51.896-05:00What it Means to Slow Down a ProblemWe did a really ambitious activity early this year in Geometry. We slowed down an optimization problem for start-of-year Grade 9 students, in order to get <i>every single kid</i> to understand the process. It worked brilliantly, and we are trying now to help them extend the idea to other optimization problems.<br /><br />I want to document and share this journey, because I think the experiment that we have started is SO challenging and SO worthwhile. We are trying to get the kids to think like mathematicians. By slowing them down.<br /><br /><a href="https://drive.google.com/file/d/0B9GuwbUfAT6MdWp3ckdIbXJ5Tnc/view?usp=sharing" target="_blank">Here</a> was the first time we formally introduced optimization (after having the kids play around with building their own popcorn containers). If you read it carefully, you might notice that we tried to emphasize a few things: 1. Tactile learning. 2. Justifying their thoughts. 3. Understanding what x and y represent. 4. Understanding how to analyze the domain. 5. Resourceful use of technology. 6. Interpretation of results back in context.<br /><br />A little while after going through this, we gave the kids an open-notes test, using a different sheet of paper to start. They did well, which showed us that they really understood the different pieces. Then, on the actual closed-notes exam, we worked backwards by giving them a factored cubic equation, and asking them some relevant questions: 1. What is the dimension of the piece of paper that we had started with? 2. What is the domain of this problem, and why? 3. What is the largest box that can be built here, and what are its dimensions? 4. How many different boxes can we build that would have a volume of _____, and what are their dimensions? Again, the kids did brilliantly!<br /><br />It has been amazing and humbling to see how far along these 9th-graders have come.<br /><br />The next question that we gave them, which they worked through in small groups, was an optimization problem involving a known perimeter of a rectangle and in trying to maximize its area. They needed to take the problem from start to finish, in writing a system, combining it into a single function, and doing domain and graphical analysis to find the maximum. Then, as usual, interpreting back in context.<br /><br />Now, the next problem they are tackling has to do with maximizing the area of an isosceles triangle whose perimeter is 30 units. Not easy. Some kids figured out right away that this follows the patterns of other similar problems, where they will try to write an area equation. Some kids started to make tables -- another really great habit of a mathematician! The class, as a whole, needed a nudge to help them figure out how to get the height of the triangles. We stopped discussions today at writing a general height equation as a class, in terms of x, the length of the two congruent sides, and I asked the kids to keep thinking about the rest of the problem.<br /><br />A worthwhile experiment, indeed! If by the end of the term, they can do even half of these problems completely independently of us, we will be so thrilled. I think the key is to slow them down. By feeding them the understanding in pieces and then giving them another similar problem, we are building the foundation that it takes to transfer the knowledge.<br /><br />Stay tuned! <img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/ZAzYI6LHWTU" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/ZAzYI6LHWTU/what-it-means-to-slow-down-problem.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/11/what-it-means-to-slow-down-problem.htmltag:blogger.com,1999:blog-6651514617266100245.post-5225613779375139569Fri, 30 Oct 2015 02:50:00 +00002015-10-29T22:58:38.445-04:00First Term ReflectionThis is a rough brain dump after the recent end of school term, but I am going to try to stay coherent and helpful.<br /><br /><br />1. This past school term, I decided to be tougher-than-usual and to cut off accepting make-up work a day before the end of the term. It seems like a small change, but it really saved my sanity by so much! I got to get all of that make-up work graded and returned by the last day of the term, and to just focus on the bigger assignments (like tests) once the kids went on term break and I was working on finalizing their grades. It probably didn't save me much time, but it saved me lots in terms of sanity and focus. I wasn't trying to grade a thousand different assignments all at once while trying to re-calculate their grades. <br /><br />2. Another thing that really saved me is that I created <a href="http://bit.ly/selfRubric" target="_blank">a learning rubric</a>, asking kids to rate themselves on their growth mindset, reflectiveness, responsibility, resourcefulness, and organization on the last day of the term. It was tremendously helpful to me in writing comments for them, looking at how they rate themselves in each category of the rubric! At this point of the year, it allowed me to really incorporate their own self-assessment into their evaluation, while keeping it somewhat objective (action-based, as my rubric was formatted to be, rather than opinion-based, like it would probably be if I gave them an open-form self-reflection). Again, in the end I don't think that I necessarily saved time in writing their evaluations, but I think that I wrote really detailed and comprehensive ones, considering that it is only the first term and we have only had 6 or so weeks of school. <br /><br />3. At the end of the term, I really liked ending Algebra 2 with choice assignments and ending Calculus with no-requiz-option exams. For Algebra 2, the choice assignments were all regression activities, and all groups ended up learning/practicing the same skills, but offering them the choice meant that we would potentially have richer discussions in coming weeks, and their interest level was also very high during the task. For Calculus, giving them a quiz that does not have any re-quizzing options was a great way for me to ask the students to step up to a "college-level" challenge after a term of slowly ramping them up to my expectations, and to show them that they could still do very well if they would commit to preparing and asking questions in advance. It worked! Although in general, I am a believer of re-quizzes, I think giving one no-requiz assessment every term will actually <i>reinforce</i> their confidence over time, even for the students who initially don't do well on them. It will also make for a more realistic preparation for college next year.<br /><br />4. In Grade 9 Geometry, we had given an open-notes exam half-way through the first term, and then a closed-notes final exam at the end of the first term. These tests were super helpful, in combination. The open-notes exam was a great informal check-in on how responsible they are as learners, in asking clarifying questions in class and making sure that they had understood a quite complex task. (Their open-notes task was to take a sheet of paper and to write a cubic equation modeling the volume that could be built by folding up the corners into a box. The 9th-graders needed to do domain analysis and to use Desmos to optimize the volume, and then to construct the box accordingly, individually.) The closed-notes exam at the end of the term, then, assessed how well they are practicing/preparing for exams. I loved this combination, particularly in Grade 9. I thought it was a very developmentally appropriate way to introduce them to high-school expectations.<br /><br />5. Math journals. Love them, but they're so much work! I am still looking for ways to cut down the work load that is to grade these concept journals all the time (often twice, if the kids are doing revisions on their entries). Any tips?<br /><br />6. Overall, a very exciting, albeit hectic, first term!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/GzFEZwHdtmo" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/GzFEZwHdtmo/first-term-reflection.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/10/first-term-reflection.htmltag:blogger.com,1999:blog-6651514617266100245.post-192435209889501445Sun, 11 Oct 2015 17:55:00 +00002015-10-11T14:09:47.308-04:00Wordle as a Tool to Respond to FeedbackThis year, particularly after reading <i>Thanks for the Feedback</i>, I am making a conscious effort to gather on-going feedback from my students, in order to address them in real time and to engage my students in a two-way communication. But, let's be honest, the thing that takes the time in class is not to <i>gather</i> feedback, but to go over it. It always feels tedious to go over kids' concerns and appreciation bullet point by bullet point, so this year I am going to try using Wordle (or one of the alternatives) to make the discussion take up less time.<br /><br />Here is an example of my Algebra 2 wordle, based on feedback for what <i>is</i> working well in the class thus far. I couldn't get the Java interface for Wordle creation to work either on my laptop or on the school's laptop, so I used TagXedo to make this in the end. I like TagXedo, because you have choice over both font and orientation of text. If I wanted to, I could have orthogonally-oriented text. As you can see, kids mostly thought that group work and their classmates were very helpful, as well as handouts that had some basic examples on them. Lots of kids mentioned fixing their errors as being very helpful as well, and the idea of growth mindset being weaved into that. With this picture, I can hopefully whittle down this part of the discussion into a couple of minutes, and then focus on discussing what isn't working well yet. (It didn't make sense for me to create a wordle for what isn't working well in this class, because they are only individual concerns and no real repeats.)<br /><span id="docs-internal-guid-f67bb8b8-5802-6308-1d10-1f33b1499d75" style="background-color: transparent; color: black; font-family: Arial; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"><img height="602px;" src="https://lh4.googleusercontent.com/S9swj1gn5BKejDT1ga-ibGf8ZeX2tD4xdWJcH_Jl9yExRMivaL1aNZiOVnX36OxbSbUA_WNWg2IjP5oIRws5xU_4v7ye3XyEr3jYFEHZ762DB2BRaWGUGgwp40roQFoEOxXMUxam" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="626px;" /></span> <br /><br /><br /><br />Incidentally, we had a great open house this week, which also weighs in as a source of feedback for me. I tried out an exercise where I asked my students' parents to write down on an index card <i>one</i> hope or excitement they have towards their child's math class this year, and <i>one</i> anxiety that they have towards the class this year. My student parents blew me away on this task! Their responses continue to reinforce my belief that our approach to teaching mathematics needs to consider, in every step, how we are impacting our students' mindset and attitude towards math.<br /><br />Here are their hopes or excitement. Believe me, their concerns are equally insightful!!<br /><br /><i>I would love it if [my child] developed a sense of the Beauty of Mathematics.</i><br /><br /><i>I am hopeful that [my child] can regain [their] confidence and enthusiasm for math.</i><br /><div class="text_exposed_show"><br /><i>Feel like a mathematician and enjoy math.</i><br /><br /><i>Hoping that [my child] sees the beauty of method and that it becomes a great mix of method, understanding, and simplicity as a meditative experience/tool.</i><br /><br /><i>[My child] continues to love learning.</i><br /><br /><i>I hope [my child] grows [their] confidence in Math and is able to solve problems with numbers.</i><br /><br /><i>Confident enough to not fear mistakes and attempts and iteration.</i><br /><br /><i>Learn algebra</i><br /><br /><i>I hope for [my child] not to hate Math. </i><br /><br /><i>To gain confidence by asking questions and talking more in class.<br /> </i><br /><i>Excited that [my child] is taking Calculus in high school.</i><br /><br /><i>Hope [my child] continues to enjoy math.</i><br /><br /><i>Deeply interested in math/Calculus both as theory and application.</i><br /><br /><i>I want [my child] to enjoy Calculus and math and to want to do more of it in college, to embrace quantitative theory and analysis.</i><br /><br /><i>Excited for [my child] to learn Calculus!!</i><br /><br /><i>[My child] is totally turned on by math. [They have] not expressed any anxiety.</i><br /><br /><i>Understanding the basics of differential equations and still keeping [their] interest/love/self-confidence in math.</i><br /><br /><i>Understanding the basics of Calculus.</i><br /><br /><i>I never took Calculus so I am excited for [my child] to learn something beyond what I took.</i><br /><br /><i>Glad [my child] has got classes and teachers that [they like].</i><br /><i><br /></i><i> I am excited that my student is in Calculus and I'm looking forward to [their] development of understanding derivatives.</i><br /><i><br /></i><i> Excited about [my child] building confidence with Calculus to ease the college course experience.</i><br /><i><br /></i><i> [My child] seems to love it.</i><br /><i><br /></i><i> My hope is for [my child] to appreciate that Calculus will be applied to Sciences, and necessary for [them] to succeed in.</i><br /><i><br /></i><i> Excited for [my child] about [their] continued exploration of new math concepts.</i><br /><i><br /></i><i> [My child] LOVES math and I'm excited that [they are] able to do 2 classes this year.</i><br /><i><br /></i><i> I am excited that no matter what [my child] learns, [they] will know more than me.</i><br /><i><br /></i><i> [My child] is taking charge of [their] work and meeting with you regularly.</i><br /><i><br /></i><i> Excited for [my child] to continue advancing in math as it is one of [their] favorite subjects.</i><br /><i><br /></i><i> That [they] can apply [their] learning to real life situations.</i></div><br />Thinking about incorporating some of this parental input into my discussion with the students about how class is going, where we are headed, and why.<br /><br />How do you manage gathering and addressing feedback in your classes, on an on-going basis? <br /><span id="docs-internal-guid-f67bb8b8-5802-6308-1d10-1f33b1499d75" style="background-color: transparent; color: black; font-family: Arial; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"></span><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/98lWecrZ6dg" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/98lWecrZ6dg/wordle-as-tool-to-respond-to-feedback.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/10/wordle-as-tool-to-respond-to-feedback.htmltag:blogger.com,1999:blog-6651514617266100245.post-5694683648532746072Mon, 05 Oct 2015 04:21:00 +00002015-10-05T00:27:35.665-04:00Changes to Class StructureThis year, I have made some exciting changes to my classes.<br /><br />1. I thought long and hard about how I do math journals. Last year, I got raving reviews of the math journal setup from my thoughtful and reflective students, but I didn't think that the way the math journal was run actually benefited the other students. It was a hard sell. They always procrastinated at the end of the term with revising all entries, and thereby it lost its value as a reflective tool throughout the term. Also, many students complained that I was "looking for something very specific" in their revisions, so it was very stressful. This year, I decided that more than accuracy in their math journals in reflecting upon the big ideas, I just want them to <i>engage</i> <i>in the process</i> of reflection. I also decided that the stress of trying to get the most accurate answer was turning some of them off from the process of reflection, which was having the opposite effect of what I was trying to achieve. So, this year I am de-coupling feedback from their grade. The first time they turn in a journal entry, I grade them based on a simple 6-point rubric that looks like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-9o7i_8GloGM/VhHXjFdvC8I/AAAAAAAAEGk/W0ej8p77oJY/s1600/journal%2Brubric.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="156" src="http://2.bp.blogspot.com/-9o7i_8GloGM/VhHXjFdvC8I/AAAAAAAAEGk/W0ej8p77oJY/s320/journal%2Brubric.PNG" width="320" /></a></div>So, at that point, they have some score out of 6 that I write down. I hand back their journal with both comments scribbled in the margins and a printed rubric with their score. They are then asked to revise it before I collect their journal again in a few days. (I go over in class the common errors, and I also post a couple of example entries on my wall, to help them with this revision.) The second time that they turn it in, although I still return a 6-point rubric to give them feedback, I don't penalize them for their errors. I give them full credit in my grade book for just <i>engaging</i> <i>in the process</i> of trying to revise their understanding.<br /><br />I have noticed three things from this: One, that most students get the revisions correct anyhow. Two, they are proactive about revising in order to get the points back. Three, they are turning the journal entries in more or less on time each time, which allows me to dialogue with them about their misconceptions in real time. The journal is serving its purpose!!<br /><br />2. Last year, I gave short one- or two-problem practice quizzes as we were learning the material, and I would grade them on a green, yellow, red sticker system. My intention was that it would be a low-pressure way to give students early feedback, without affecting their grade. But, I wasn't happy about how the kids with the red stickers would put off meeting with me, because they weren't feeling the urgency until the real, big quizzes rolled around. So, this year I still give practice quizzes, but with a few points attached. On the returned practice quizzes, I would write down a similar problem and ask all kids to show me via doing the similar problems that they now understand their error and the concept. I find that this is more productive towards encouraging improvement and a growth mindset, because in order for kids to practice a growth mindset, they have to actually do some work towards making progress!<br /><br />Thus far, I like the system a lot better. It's still reasonably low stress, but the expectation is clear that kids need to work on their areas of weakness, in order to make back those lost points.<br /><br />3. This year, I am working actively on building community in my classes. One thing that I did for the first few weeks was to always assign seats. Everyday, the kids would sit with a new partner. They would take the first couple of minutes of class to ask their partner for a fact that they didn't already know about that person, and they would write it down on their own name tag. This helps me to keep track of whom they have already sat with, and they started collecting lots of facts about lots of peers! I loved it! I have also started randomizing partners now that they have sat with a majority of the class. Each day, they come in and they pick a name randomly from the pile. If it's their own name, then they put it back and pick another. Otherwise, they sit with that person. (If they pick out names of people who already have picked a partner, then they give me those name tags back and pick another.) This is a low stress way of making sure that they rotate partners all the time. <br /><br />4. I am using Microsoft OneNote for two of my classes! Algebra 2 and Calculus. I kind of love it. I can lesson plan very far in advance, and make modifications in real time as I see fit. The OneNote websites serve as my course webpage, where kids can look up both notes and homework assignments. (For a long time I didn't believe in posting homework assignments, but this year I am doing the opposite after reading <i>Switch</i>, because I want to shift the entire classroom culture towards completing homework as a norm, rather than just something that is done by already-responsible students. Thus far, it seems to be working great!) I also upload video links for the kids who just want to use additional resources. Thus far, I absolutely love it.<br /><br />You can check them out, too! Please don't modify any of the content though, just look.<br /><a href="http://bit.ly/calcNoteboook" target="_blank">http://bit.ly/calcNoteboook</a> and <a href="http://bit.ly/alg2Notebook" target="_blank">http://bit.ly/alg2Notebook </a>are the two links I am using.<br /><br />That's it for now. Ciao!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/Z11T1aRi9-E" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/Z11T1aRi9-E/changes-to-class-structure.htmlnoreply@blogger.com (untilnextstop)2http://untilnextstop.blogspot.com/2015/10/changes-to-class-structure.htmltag:blogger.com,1999:blog-6651514617266100245.post-7290353317485369214Thu, 01 Oct 2015 04:53:00 +00002015-10-01T00:53:09.495-04:00The Growth Mindset Experiment Hi, blogosphere! I thought I'd check in quickly. This school year, I really wanted to make the focus to be on helping kids to transition to my class. One of the things I wanted to teach them about is growth mindset, and at the same time to be crystal-clear about why we do the things that we do. For example, why do a pre-test? Why do we have practice tests when not everyone is ready? Why play around and intuit things before we discuss them formally?<br /><br />I did a brain talk on Day 1, which had elements of the growth and fixed mindset as part of the talk, including asking the kids to practice re-framing certain negative sentiments as growth mindset statements. For homework, I asked the kids to write about an example of growth mindset in their lives, outside of math. I learned so much about their lives through this first assignment! I learned about their interests and also how they respond to setbacks. Subsequently, I have shared my own story of a growth mindset with them, and continue to think daily in terms of what I need to do or say in my class in order to generate and sustain more growth mindset. Now every time a kid asks to meet with me, I thank them for having a great growth mindset. When a kid fails an assignment, I ask them to keep working on it to revise their understanding, and I write, "Growth mindset!!" on their paper. In our Geometry class, on the day when we had individual algebra tasks, I asked the kids to write on a post-it one thing that they had learned during that class. (Some kids said, "I didn't learn anything new! I had to review..." and it was a great opportunity for us to chat about how learning new things sometimes requires re-learning old stuff first.) I am thinking about doing a growth mindset check-in, where each kid is asked to write down one thing that they recently learned or improved on and one thing that they are still working on or challenged by. This would hopefully remind them that learning is a spectrum, in order to reinforce their confidence in the process.<br /><br />Anyway, all this thinking about growth mindser is directly feeding into the classroom culture, I think!! Thus far, I am loving my classes and find the kids to be actively engaged everyday.<br /><br />Next time, I will talk about actual changes I have made to my classes...Some good ones, I think!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/e-nx2tWnHII" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/e-nx2tWnHII/the-growth-mindset-experiment.htmlnoreply@blogger.com (untilnextstop)2http://untilnextstop.blogspot.com/2015/10/the-growth-mindset-experiment.htmltag:blogger.com,1999:blog-6651514617266100245.post-847037049701636806Fri, 28 Aug 2015 20:10:00 +00002015-08-28T16:11:49.950-04:00Summer Reading #2: SwitchWow, this week has passed rather quickly. It's the last week of holidays before we head back to school to prep for students to return. (But, don't be jealous if you have already returned to work; in order to make up for the late start of school this year, we are shaving off holidays <i>during</i> the school year, ie. the buffer days that used to exist in between terms, which I would have probably rather kept.) Since I am nursing a foot injury (plantar fasciitis on both feet, sigh...), I did some planning, read some books, and hung out with the hubby*. It is so luxurious to be able to have lunch and coffee dates with my husband during the week! I love it.<br /><br />*The hubby works from home. <br /><br />Anyhow, I thought I'd jot down some notes about <i>Switch</i>, which is written by Chip and Dan Heath of <i>Made to Stick</i> fame. <i>Switch</i> is a book about how to initiate change in other people or in a system, when people's natural inertia is to resist change. I wanted to read this book because I think that as teachers, we are constantly trying to change our students' approaches and attitudes towards their learning. In our minds, there is a vision of what an ideal student does, and we are striving to move all of our students a bit closer to that model. For example, for me, an ideal student is always actively engaged and reflective of their learning. They are always asking questions and trying to draw connections between topics. They are risk-takers, communicators, and they have a deeply rooted growth mindset. The ideal student does not necessarily always enter the class with all of the skills from previous classes (...in fact, sometimes they can be missing significant skills...), but they have a big, open heart, ready to take on feedback and to re-tool their learning processes as needed.<br /><br />The book outlines a deceptively simple framework for initiating change, and then it illustrates the use of this framework through a variety of almost unbelievably successful stories. The framework is basically this:<br /><br />* <i>Motivate people emotionally</i>. In order for someone to change, they have to want to change. You don't motivate people using numbers or research, because our intellect is not what causes us to change; you have to find a way to give them a vision and trigger their emotions. Sometimes, this can mean helping them to envision or build a new <i>identity</i>, because we tend to behave in a way that we wish to see ourselves. (This is a silly example, but have you ever wanted to buy something from a store that looks really stylish, but whose clothes are not very flattering on you? You're subconsciously trying to assume the identity of the type of person who would shop at that store, and you're shifting your behavior to match that identity. Another example is how Geoff and I started composting. In Seattle, it's part of our collective identity because the city provides infrastructure and even a small financial incentive to compost, as compost-collection costs less than regular garbage. Now that we have grown to see ourselves as composters, we can't seem to break the habit of looking to compost everywhere, even when we're away traveling. It has changed our behavior permanently!)<br /><br />* <i>Give clear directives and "shrink the change"</i>. Once you get people on board emotionally to change (which, I know, isn't easy), you have to give them clear, achievable, and black-and-white goals in order for them to get started and feeling successful. The example given in the book is that instead of telling someone to "eat healthy", to say instead to "drink 1% milk." This clear health directive has completely shifted the way America now consumes milk. When you script these directives, think in terms of something immediately achievable, although you can link it to a longer-term vision that it will hopefully pave way to. Give the brain a reason to follow the change and give it also a clear path to follow.<br /><br />* <i>Follow the bright spots</i>. Change is hard and change takes time. Instead of focusing on what isn't working, look for what is working and highlight those consistently. Find ways to duplicate the success until it creates a positive momentum. For example, are 70% of the kids turning in their projects on time? Praise those kids and ask them to share what strategies are working for them to help them stay ahead of schedule.<br /><br />* <i>Shape the path</i>. Are you making it as easy as possible for people to create and maintain the change? For example, if your students are not in the habit of doing homework, are you posting homework online and giving them organizers in order to help them ease into the habit? Are you finding ways to create a positive group culture wherein the norm is to do homework? If the change you wish to see is kids volunteering to speak in class, are you providing a structure wherein it's the norm to speak up?<br /><br />I liked the book, but what I would like to hear is what other teachers think are challenging changes to institute in their classroom, and how we can use this framework as a lens to help us think about transitioning students into more successful learners over time. For example, one incentive that I want to try this year is the ability for the class to earn "homework passes" by showing consistency in completing homework. It's not so much the homework that I care about, as much as I care about them doing and thinking about math <i>consistently</i> outside of the classroom and really attempting problems when I am not there by their side. I figure that if I start with an incentive system, over time the learning will hopefully become its own reward, and I can wean them off of "homework passes." But, in the mean time, it can help me to shrink the change from "be a more proactive learner" to "try your homework", to help the kids who don't already have good study habits to start to build them.<br /><br />Thoughts??<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/LA7oaf67kaQ" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/LA7oaf67kaQ/summer-reading-2-switch.htmlnoreply@blogger.com (untilnextstop)1http://untilnextstop.blogspot.com/2015/08/summer-reading-2-switch.htmltag:blogger.com,1999:blog-6651514617266100245.post-6022482455470854505Tue, 25 Aug 2015 00:43:00 +00002015-08-28T16:20:55.742-04:00Summer Reading #1: Thanks for the FeedbackOK, web, I am back! I had a fabulous summer. My husband and I did a big trip! We started in New Zealand and ended in Iceland, covering a total of 12 countries in 60 days, so it was quite a sprint. <br /><br />Anyhow, I have been doing some summer reading after returning from our trip. I had picked up two professional books prior to the start of the summer, but didn't get around to reading them until I got back. (Traveling with a smallish backpack made carrying paperback books unrealistic, and as it turned out, it was hard to find places to reliably charge our phones while staying in hostels, so I did relatively minimal digital reading also while away!) I thought I'd jot down some notes about the books I read, as the books were very useful to me!<br /><br />One book I read this summer is called <i>Thanks for the Feedback,</i> and I had picked it up because I wanted to get better at hearing and parsing through feedback myself, as well as giving feedback effectively. I highly recommend the book! It was a great lesson in thinking about nurturing my own growth mindset, because how we receive feedback has everything to do with our own growth mindset. The book, for me, contained a lot of valuable information regarding why hearing negative feedback is challenging and how we can frame our minds around this more effectively.<br /><br />To give you a small taste of why I really liked the book, it talks about how sometimes someone we really don't get along with would try to tell us what they think about us. They would probably present it at the wrong time and in the wrong manner, and you can easily, based on many legitimate reasons, write them off and think to yourself that what they have to say does not apply to you, and that they're in the wrong for X, Y, Z reasons or disqualified to give you feedback for M, N, R reasons. Well, the book encourages you to put aside all those factors and to ask them questions about why they feel this way. Dig further into the data that they're looking at. Do they have information that you don't have? Is their role giving them a reason to consider other factors that are not on your radar? Instead of deflecting negative feedback as is tempting for most people to do, embrace it actively and ask probing questions in order to start a conversation. In the end, you don't have to accept all parts of their feedback, but the first step to growing is to understand where they are coming from, particularly because the people you get along with the least are most likely to offer you an honest look at yourself.<br /><br />The book also talks about examples of when someone we are in a close relationship with tries to give us feedback. While reading this, I related this in my mind to my husband, who from time to time tries to give me critical feedback about something in my personality that he thinks needs some work. The book talks about how people often react to this by essentially redirecting the conversation to include a new thread about how you are also dissatisfied about something that the other person does. I know I'm certainly guilty of this, and the book gives specific strategies about how to tackle this type of impulse / conversation trainwreck to guide it towards a productive conversation, wherein you focus on one conversation at a time and really try to hear the other person's point of view.<br /><br />One of the things that I really liked also is a diagrammed model in the book about how our own behaviors are invisible to ourselves. We, as individuals, are only aware of our intentions and the impact that we intend to make, but we have no visibility into our outward behaviors (particularly our micro-movements and our body language) and their impact on others. On the flip side, the people who are experiencing our actions have no visibility into our intentions, even though both parties think that we are seeing the full picture. For example, I am only aware of what I say to my students and why I am saying it, but I'm not aware of how it is coming across in the moment and whether it has the intended effect. This is why getting feedback from our students is very, very important. It helps to close that feedback loop and to make sure that the messages that I hope to send are actually aligned with the messages that the students are receiving, particularly when it comes to my instructional choices and what I think are important aspects of their learning!<br /><br />The book also helped me reflect upon how I give feedback to my students! The book distinguishes between feedback forms that are evaluation, coaching, and appreciation, and gives examples for why it can cause a lot of frustration for the receiver if they are constantly missing a certain type of feedback. It also talks about the importance of giving feedback in the manner that a person prefers to hear it. So, one of the things I will do this year is to find out, from each student, how they prefer to receive feedback. I'll have to think about how best to phrase this, but the book has some good suggestions for probing questions.<br /><br />Wow! It is turning out to be a fairly hefty entry here, particularly because I have gone for so long without posting much at all. I'll come back and talk about the next book tomorrow, but if this quick summary sounds intriguing to you, I highly recommend checking out <i>Thanks for the Feedback</i> as written by Douglas Stone and Sheila Heen. Worth a read and worth taking notes!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/L7n6vgb61-k" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/L7n6vgb61-k/summer-reading-1-thanks-for-feedback.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/08/summer-reading-1-thanks-for-feedback.htmltag:blogger.com,1999:blog-6651514617266100245.post-6499107925716282777Wed, 25 Mar 2015 05:43:00 +00002015-03-26T23:25:51.451-04:00Circumference of the MoonSo, as a follow-up to Erastothenes using geometry to calculate the circumference of the Earth, this week I plan to go over how we can use the ratio to Earth to calculate the circumference of the moon!<br /><br />The lesson idea came from my colleague John. I fleshed it out to scaffold it for my kids. It looks like <a href="https://drive.google.com/file/d/0B9GuwbUfAT6Mbm11aVl6Mi1EeDg/view?usp=sharing" target="_blank">this</a>, and it ties in nicely both with our school's Grade 9 science curriculum (which teaches astronomy for all of next term), and our current circles unit. I plan to use this lesson the day after our basic circles quiz, when a few of the students will have family members visiting our class. (At our school, this is called Grandparents' Day, even though it is quite possibly not the grandparents that are coming.)<br /><br />I'm excited!! I've never done this lesson before, but I like how it revisits perpendicular bisectors and makes them seem useful in application.<br /><br />Addendum 3/26/15: I prepped for this lesson today and it REALLY BOTHERED ME that I got the estimate that the Earth's circumference was about 2.5 times bigger than that of the moon, when in reality it should be about 3.6 times bigger. I did some more digging and worked out a ratio to find out how big the Earth's shadow would be by the time it reaches the moon, and I think it's 9200 km in diameter at that point! That makes my ratio make a lot more sense, because this is only 2.6 times bigger than the moon!!! Go Geometry!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/KseRqNyy3IE" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/KseRqNyy3IE/circumference-of-moon.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/03/circumference-of-moon.htmltag:blogger.com,1999:blog-6651514617266100245.post-296473433682769152Sat, 21 Mar 2015 20:27:00 +00002015-03-21T16:29:10.793-04:00Geometry to Algebra TransitionOur school is trying out a new thing this year (one that I think is fabulous). We're re-shuffling 9th-grade kids' Math classes for the last 8 weeks of school this year, depending on whether they intend on taking Algebra 2 or Precalculus after finishing Geometry this year. The classes in our last term of the school year will prep the kids for transition into their choice of algebra classes for next year, and we'll assess them at the start of the term, end of the term, and again at the end of summer to determine whether their achievement and commitment-to-hard-work together seem to predict success in their choice of classes (particularly those intending on skipping Algebra 2), in order for us to advise them and their parents about whether they should be working over the summer, and what seems to make sense for their course placement. In prepping for this transition, we are including lots of algebra into our current circle unit to help kids "warm up" in thinking about algebra skills.<br /><br />Below is what I have so far. The kids are definitely hitting their edge, but I am able to motivate them by explaining that quadratics is the next logical thing for us to practice, since we have done already a lot of work with lines and systems this year. Even those who have taken Algebra 2 in Grade 8 and who are intending to take Precalc next year did not have an easy time solving for points on a circle, so this is great stuff for all of them!<br /><br /><a href="https://drive.google.com/file/d/0B9GuwbUfAT6MelBLZWRJbGNPejA/view?usp=sharing" target="_blank">Here</a> is my intro to circular equations, which most of the class is about finished with. Following it, I plan to spend a few days doing <a href="https://drive.google.com/file/d/0B9GuwbUfAT6McDlaUjBjbDZ4RVk/view?usp=sharing" target="_blank">this</a>, which is a modified version of a worksheet that two of my colleagues had created. We want the kids to get familiar with circular vocabulary (as preparation for Calculus) and to do some algebra practice involving circles, but besides it, we're not too attached to teaching all of the circle theorems, since we only have 7 more school days left of this term. I am excited to see the kids' transition to algebra after all the work we've done with them this year in terms of problem-solving. I hope it'll pay off when they get to Algebra 2 or Precalc next year!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/X0BoMF8tCTI" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/X0BoMF8tCTI/geometry-to-algebra-transition.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/03/geometry-to-algebra-transition.htmltag:blogger.com,1999:blog-6651514617266100245.post-742918028603802442Mon, 16 Mar 2015 00:59:00 +00002015-03-15T21:04:13.471-04:00Some Calculus WorksheetsI took some time during the previous school term to observe some of my colleagues, in order to educate myself about the ways in which they encourage inquiry in their classrooms. Among other observations, I enjoyed seeing how the other Calculus teacher (who is about 20 years more experienced than me) structures his worksheets to always circle back to applications and interpretation of answers. Since my visit to his classroom, I have been working on modifying my handouts from the previous year in order to put in more application into every concept.<br /><br />Here are two of them: I used <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MV0hFcXdlQXlmbmM/view?usp=sharing" target="_blank">this</a> to help kids wrap their minds around basic integral Calculus applications, and <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MZm1IZ21COURfTFU/view?usp=sharing" target="_blank">this one</a> reviews some algebra skills from earlier this year, plus introduces the necessity of going in between algebra and the graphing calculator sometimes. The problems are not ground-breaking, but I think they've definitely helped to break up the skills practice, so I'm happy to share them if they might be useful to someone else.<br /><br />From earlier months of this year, one thing that I did that totally helped with teaching Related Rates is that I first taught implicit differentiation with respect to time, and formally assessed students on this skill, prior to starting Related Rates word problems. (Sorry if this sounds obvious; it wasn't that obvious to me last year, teaching Related Rates for the first time!) <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MdWI3NUNhajhGZDg/view?usp=sharing" target="_blank">Here</a> is how I introduced implicit differentiation, using the analysis of non-functional relationships as a premise. After this, I had the students do some pure skills practice in <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MS0kxM2xQdTJZYkE/view?usp=sharing" target="_blank">converting geometric formulas to differential equations with time as the domain</a>, before introducing my <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MWFlnQzY5RWE3SG8/view?usp=sharing" target="_blank">scaffolding for related rates problems</a> and the many related problems I took from <a href="http://bowmandickson.com/category/teaching/calculus/related-rates/" target="_blank">Bowman</a> last year. I felt really good about this sequence of skills this year, because I noticed that it really made the problems more accessible to ALL students (as in, by the time they got to the word problems, they were really only focused on parsing the word problems process, rather than simultaneously struggling with the algebraic skills of differentiating implicitly). I recommend trying this, if your students get baffled by Related Rates problems.<br /><br />I am also trying to place a general focus on vocabulary and communication this year. I've been doing this in all classes by giving the kids a list of essential questions at the start of a unit, and then having them journal their responses to those essential questions throughout the term. For example, for our current term in Calculus (which is short, only about 5+ weeks), I gave the kids the following questions. The questions are a mix between related rates (which we did at the start of the term) and intro to integral Calculus.<br /><br /><ul><li><i>How are "implicit differentiation", "chain rule", and "related rates" all related? Illustrating this with a simple algebra example may help to clarify your thinking.</i></li></ul><ul><li><i><span style="color: black;">Take one of our Level 2 or Level 3 Related Rate problems from class and explain/describe, step-by-step, how you are able to find the missing information.</span></i></li></ul><ul><li><i><span style="color: black;">What is integral Calculus? Describe a couple of situations where this concept is useful.</span></i></li></ul><ul><li><i><span style="color: black;">Choose an exponential function of the form f(x) = a*e^(x - k), by assigning values for a and k. Estimate the area underneath the curve of f, from x = 0 to x = 5, using a total of 10 rectangles. Show both left-hand sum and right-hand sum, and draw labeled diagrams to show what your numbers mean.</span> </i></li><li><i>Show, step-by-step, how you would calculate the enclosed area that lies between two functions f and g, where f is a quadratic function of the form f(x) = ax^2 + bx + c and g is a trigonometric function of the form g(x) = m*sin(n(x - k)) + p, where the value of n is not 1. You get to choose the parameters a, b, c, m, n, k, p to start, but make sure n is not 1.</i></li></ul><span style="color: black;"></span><br />Students have shown a varying degree of enthusiasm about the journal assignment, even though I have been doing it since the start of the school year and explaining periodically its purpose. Part of the purpose of this journal is to get them to record, in their own words, examples and explanations to important concepts, so that they can have a succinct set of notes for future years. Another purpose is for me to see what they write periodically, so that I can informally gather information about common misconceptions for the topics that we have finished learning, and clarify them with the class. As it turns out, however, the naturally reflective students are thoroughly utilizing the journal to dialogue with me about their understanding, and the rest of the kids see it as a drag to have to keep revising their explanations until the end of the term, so the work that I receive is kind of a mixed bag in terms of quality. It has been a somewhat tough sell, but one that I think is important, because from time to time, students would comment on how they notice that by answering questions in their journal while learning the concepts (instead of putting it off until the end of the term), their understanding improves in real time. Do you do something like this in your classes? How do you drum up enthusiasm for such a revision-based assignment?<br /><br />That's it for now! My Geometry students are wrapping up their 3-D project, which is very interesting as per usual. They have some really neat designs this year, which I might share at some point. Algebra 2 kids are knee-deep in thinking about the domain and range of different function types, and thinking about transformations on the various functions. It is nice to hear them go, "Ooh, ahh..." as they realize that they can connect information from different types of functions. Not much to write home about, but a productive time of the year nonetheless!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/PUVXWelsgnU" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/PUVXWelsgnU/some-calculus-worksheets.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/03/some-calculus-worksheets.htmltag:blogger.com,1999:blog-6651514617266100245.post-8540502590158688334Sun, 15 Mar 2015 00:12:00 +00002015-03-14T20:12:19.524-04:00Hello, World!Sorry, web, I have been away! It has been a busy few months.<br /><br />Geoff and I went to Hawaii in December, followed by some busy weeks at work for both of us while shopping in our off-time for a house. In February, I went home to visit my parents, and almost immediately afterwards, my in-laws came to stay with us for 10 days in our 700-square-foot apartment. They have just left, and Geoff and I have finally begun planning for our big summer trip. This summer, Geoff plans to take off 2 months from work and travel with me. Tentatively, we will start in New Zealand, then go through Philippines, Indonesia, South Korea, Japan, Estonia, Russia, Greece, Italy, Germany, England, Ireland, Iceland before coming home to Seattle. Our house's closing date is still set for the end of March, so in the mean time, there are just a lot of things keeping us busy. Hence, the radio silence...<br /><br />But, I have been reminded recently that I have a blog! Through the grapevine, two friends of friends have mentioned this blog to me. Funny, small world. So, let me think about what I can put on the blog that is worth sharing. Stay tuned.<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/aPbhBkNGQuI" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/aPbhBkNGQuI/hello-world.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2015/03/hello-world.htmltag:blogger.com,1999:blog-6651514617266100245.post-4935831762157189183Fri, 12 Dec 2014 05:22:00 +00002014-12-12T00:31:25.147-05:00A Fun Lesson on SimilarityThis term, I have been mixing in some old Geometry lesson material into my current Geometry class, now that we're no longer doing purely Exeter problems. It has been so much fun!!! My kids are delighted every time I throw in something that I had used and liked in El Salvador. It makes me very happy to observe.<br /><br />The last couple of days, we have been doing a fun little similarity activity on the Geoboard, that I had made back then and then revised for this year. See it <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MeF9uWGJ6QUdaVnM/view?usp=sharing" target="_blank">here</a>. I am using it to introduce similarity, as a lead in to special right triangles and right-triangle trigonometry. The kids are having so much fun with rubber bands that they don't realize I am sneaking in significant learning.<br /><br />I still like the Exeter problems for their incredible richness, but balancing them out with other modes of learning is the way to go, I think! <br /><br />PS. This has been a good teaching week. Two of my low-confidence students who have been working their BUTTS OFF for weeks each got an 100% on their requiz. HOT DAMN! I'm so, so proud! <img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/2In1yzaIt7c" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/2In1yzaIt7c/a-fun-lesson-on-similarity.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2014/12/a-fun-lesson-on-similarity.htmltag:blogger.com,1999:blog-6651514617266100245.post-2732520975616998398Fri, 31 Oct 2014 06:13:00 +00002014-10-31T02:24:08.135-04:00How it Must FeelImagine yourself as a student who has always had a hard time with math. Honestly, you are not even sure why there are sometimes multiple variables in the same formula. When the teacher starts a new topic, you may need to see it 4 different ways and have it be repeated a lot of times in order to become somewhat proficient at the skills involved, even though the general idea is sometimes (not always) accessible to you. When other people discuss mathematics in groups, the speed at which they are discussing the ideas often flies right over your head, but you feel embarrassed to ask them to explain every part to you, because other people all seem to understand it fine. So, you do ask some questions during class, enough to make some progress on the work, and then you wait until you can find the teacher to ask for more help. You can only find the teacher to meet outside of class about once a week, and by then you already have more questions accumulated than can be answered in one session, so that although you think the concept is probably important, you just want to know the bottom line of how to get through the various problems you are going to see on the quizzes. So, the quiz comes and goes and you feel defeated by your score, even though you are working hard everyday in class and your teacher can also see that, too. You go home with the quiz and you don't want to even look at it for now, even though you know that you probably should start thinking about the re-quiz and that the teacher will probably start something new either tomorrow or the next day, and that the cycle will likely repeat itself.<br /><div><br /></div><div>What do you do?</div><div><br /></div><div>In every class I teach, there are some kids whom I meet outside of class on a weekly basis. For many of them, that is enough. For some, that isn't. I love the idea of heterogeneous grouping, because I believe in all the things that other math educators believe, which is that it promotes safer learning environments, teaches diverse students to work together, promotes growth mindset, etc. But, at the end of the day, I don't know what to do in a practical sense to help these kids who experience those cycles of disappointment in every unit. I try to vary up what I do to help them build the conceptual understanding of underlying topics (ie. Math Talk, problem-based learning, visual representations), but realistically I have to balance too much going-back-to-basics against the majority of the class's need to develop other, more sophisticated, skills and concepts. Open-ended problems sound awesome in theory, but in reality we still need coherent conceptual and skills development the majority of the time. </div><div><br /></div><div>So, what do you do besides trying to be empathetic?</div><div><br /></div><div>I don't have an answer right now. In the past, I have had great success in homogeneous grouping at helping the slower-paced classes build confidence and feel successful with a smaller set of topics, but I find that goal very challenging/elusive when those lower-confidence students are situated in an environment that is perhaps just faster-paced than they can handle. I am fortunate that most of my students give me 100%. I have no doubt about that. But, how can I grade them all on an absolute scale based on what they know, if they are all starting off at very different places along the spectrum of prior algebra experience? And, more so than grading, how can I serve them all? </div><div><br /></div><div>Those are open questions in my mind. Would love if you could chime in to enlighten me in your thoughts about this. </div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/M8r-_H_DzRs" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/M8r-_H_DzRs/how-it-must-feel.htmlnoreply@blogger.com (untilnextstop)2http://untilnextstop.blogspot.com/2014/10/how-it-must-feel.htmltag:blogger.com,1999:blog-6651514617266100245.post-487306465627078231Thu, 02 Oct 2014 06:49:00 +00002014-10-02T02:55:29.982-04:00What to Do When You Are BusyToday, one of the things I read over the summer came back to me. The thing I read was written by a therapist, regarding his patients who are stressed and unhappy because they feel like they're spread too thin among work, family, friends, etc. The therapist's advice is that you cannot make more minutes in a day, but you can increase the <i>quality</i> of your minutes in the day. If you're thinking of work when you are at home, and thinking of home when you're at work, then you're not making anyone feel valued and therefore the quality of your minutes spent with them is low. This way, everyone around you will feel dissatisfied, and you'll feel unhappy as a result. The wisdom he shares is to focus on making the people you are presently with feel like you are 110% present. Be proactive about it, instead of trying to coast by with little engagement. (In terms of active engagement, he distinguishes between your son telling you about his upcoming soccer game, and you remembering it yourself and mentioning, without being prompted, how much you already look forward to it. In both cases, your time commitment is the same -- you're still going to watch his game -- but just by being a little bit more thoughtful, the reception of your time spent will be very different.)<br /><div><br /></div><div>I thought of this today because I noticed that I have been working a lot of extra hours. But, when I meet with my students outside of class, it's not helpful for me to think even fleetingly about the many things that I still need to do that day. Instead, being fully present and taking an extra 10 minutes to let the kid slowly organize their binder or to tell me about their week before we start to look at where they need help in math, is so valuable. It slows me down and it slows them down, and it allows me to make them feel important before, during, and after our meeting. Maybe I cannot increase the amount of time that I have for them on a regular basis (and maybe I cannot solve all of their math troubles in one meeting), but I can make each encounter outside of class more meaningful by just taking a little extra time.</div><div><br /></div><div>So, here is a reminder to myself to keep building the quality of my minutes in a day, especially as it gets really busy. </div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/1oLITqwH6Rs" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/1oLITqwH6Rs/what-to-do-when-you-are-busy.htmlnoreply@blogger.com (untilnextstop)1http://untilnextstop.blogspot.com/2014/10/what-to-do-when-you-are-busy.htmltag:blogger.com,1999:blog-6651514617266100245.post-882414641427581348Sat, 20 Sep 2014 23:42:00 +00002014-09-20T19:55:24.688-04:00The Start to a Great Year!It has been 2.5 weeks since we got back to school! During this time, because of various back-to-school grade-level bonding activities, we only had 2 good weeks' worth of classes in each grade. Still, I am feeling somewhat settled into my classes, having learned all the kids' names, and am starting to figure out who needs help and to meet regularly with some students outside of class. So, things are good!<br /><div><br /></div><div>In addition to revising curricula, I am trying a lot of new classroom procedures this year. At first I thought these changes wouldn't be manageable, but so far, although I am working a fair amount, my work time has been very productive! I can clearly see the impact of various small changes on my classes, and that leaves me to spend my energy where it counts the most (ie. trying to tackle the big challenge of having many more high-need students this year).</div><div><br /></div><div>Here are some things that I am trying this year:</div><div><br /></div><div>* <b>Assigning group roles for each group member:</b> The one benefit that has stood out the most to me is the role of the recorder. The group's facilitator and time-keeper help me ensure that every student in the group tries the problems independently and jots down their individual work, but at the end of the activity/discussion, I only collect 1 copy of the worksheet and notes (from the recorder) to make sure the answers they went over are in fact correct. I also ask the recorder to write down additional items as per their discussion, such as a description for the algebra process used and definitions for key terms. I then correct just that one copy (from a group of 3 or 4), and then make copies for the other group members to keep. This way, everyone has the conceptual explanations, to help them remember what was discussed in the group besides just the answers they have on their paper. It also cuts down on the amount of grading I have to do, while being able to give written feedback to the entire group. Playing the role of recorder has also been very motivating for the kids who are the weakest performers, stay actively engaged. They know that their role is highly relied upon, so there is some unsaid peer pressure for them to make sure that they understand the process enough to write down important pieces of it on paper. Besides the recorder, I also have a facilitator, a time-keeper, and a questioner in each group. (The basic construct comes from Complex Instruction, but I am not sure if I am following their protocol exactly.)</div><div><br /></div><div>* <b>Green, yellow, red stickers:</b> In lieu of grades, this year I have been, thus far, giving feedback in terms of colored stickers. Eventually we will have quizzes that will count for points, but in the mean time, "mock quizzes" are just about 3 questions I write on the board, and I collect them to see how the kids are doing (and to take notes on who needs reinforcement on what), and we discuss the answers as a class. The next day, I return the mock quizzes with a colored sticker to show whether each student needs to keep working on that concept. It's de-coupled from grades, but my secret hope is that kids will just <i>want</i> to get green stickers. And, so far, that is pretty true!! I have been using this colored-sticker system on the collected discussion notes (from recorders), mock quizzes, and conceptual "check-in" homework assigned individually. The kids who got red stickers from me on a quiz or an important assignment have approached me to automatically resubmit the assignment after revision, to ask for help, or have been very happy when I proactively approached them to set up a time to meet outside of class. They treat the red stickers very seriously, which is great. It's such a visual way to alert them that there is a gap. The fact that the assignments are de-coupled from grades makes it possible for me to focus on meeting with them for their learning, rather than the conversation being about grades. It also makes it possible for me to have multiple "mock quizzes" in the same week without it being stressful for the kids. So far, I love it! I got a pack of round stickers from a local drug store's stationary/school-supplies section, and it already has all the colors I needed. One of the science teachers started using the same system with his students after hearing me talk about it, and he thinks it's much more clear than giving kids check plus, check, and check minus. Eventually, my goal is to have the kids self-assess via a sticker when they turn in certain assignments. And then I'd give a second sticker upon returning it, to confirm or correct their self-assessment. </div><div><br /></div><div>* <b>Green, yellow, red cups: </b>I know this is not news for other teachers, but I wanted to try the green, yellow, red cups in groups as a passive indicator to me whether the students need help. Sometimes, I think as a teacher, it's hard for me to tell when students are positively frustrated or negatively frustrated by a task. The cups are a clear way to indicate to me whether I should intervene. My secret hope in introducing this was that if they looked around the room and the other students all have green cups, maybe the group that was already about to give up would push themselves to persevere just a little more. Also, because they could only indicate green, yellow, or red to me as a group, they would be forced to communicate amongst themselves before they reached out to me for help on a problem. I have thus far only used this in my lower-grade classes (Algebra 2 and Geometry), because I am not sure whether it's a bit too cheesy for my 11th- and 12th-graders. But, I may roll it out to them eventually.</div><div><br /></div><div>* <b>Self-reporting math efforts outside of class: </b>I feel strongly that as juniors and seniors in my Calculus class, the students need to be doing self-directed learning outside of class. Practically, that means that I don't assign problems everyday, because I want them to use their at-home time to make flashcards, concept outlines, re-do tricky problems, do new problems on their own, see me for help, etc. It does <i>not</i> mean that on those days when I don't assign specific problems, they shouldn't be working on math at home!!!! But, I am also a realist; I know that most teens will do the minimum unless there is some visibility into their action. So,<a href="https://docs.google.com/forms/d/1SlAjurBypmYH4xP8sGxF-jvNVocmJjlrc0U6jtANtFI/viewform" target="_blank"> I made a Google Form that collects data about what the students are doing</a>. (<i>You can click on it. This is a copy of the link I sent out to the kids, so even if you fill it out, it won't mess up their data.</i>) I created a bit.ly link to the form I made for my students, and I asked them to fill it out every time that they do math outside of class. I stated that, as "homework", I expect that they're doing/logging 20 minutes of math outside of class, 5 times a week. Of course, they can clump the times if that is not possible, but if they are doing 100 minutes of math once a week, that is probably something I should talk to them about. Anyhow, I think the information collected this way will be a great resource to allow me to have productive learning dialogue with my students, while encouraging them to be self-directed learners, asking me questions like, "What else can I be doing with my studying time?" I've only rolled this out on Thursday, and already I can see some data being logged by some students, with comments on what they might need from me as next steps. I'm very excited about this!!!!! If this is successful, I will extend it to my Algebra 2 class. Throughout the term, it'll be a very valuable resource for me in terms of giving them specific feedback on their learning strategies. It'll also make writing narrative report cards a breeze at the end of the term.</div><div><br /></div><div>So far, these are the "systematic" changes that I am trying to make. I have already noticed a tremendous difference in my classroom culture this year, in terms of how equitably and actively students participate, and how positive they are. I will observe a bit more and write a big post about that, maybe next week! </div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/F54PcXDivM0" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/F54PcXDivM0/the-start-to-great-year.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2014/09/the-start-to-great-year.htmltag:blogger.com,1999:blog-6651514617266100245.post-2537606083578120142Fri, 05 Sep 2014 02:05:00 +00002014-09-05T09:43:54.943-04:00Experimenting with Structured Group WorkI am trying to change my classes this year drastically in the way that I handle group work! I want to be intentional, intentional, intentional.<br /><br />So far, it has been one day and that one day has been excellent. I decided that for Day 1, I would talk to the kids about inquiry-based learning vs. direct instruction, and then pass out a task that is accessible for everyone to start. All they had time for on Day 1 was the individual thinking time, which I am going to build into every group activity this year.<br /><br />In Geometry and Algebra 2, I did <a href="http://youcubed.org/teachers/2014/back-to-school-tasks/" target="_blank">Mark Driscoll's folding task</a> on Day 1, but emphasized that the content of the task is not nearly as important as the students practicing the expectations for group work. In Calculus, I made a custom sequence from <a href="http://teacher.desmos.com/" target="_blank">Desmo's Function Carnival</a> that consisted only of the parachute height vs. time, followed by misconception analysis, and parachute vertical velocity vs. time, with subsequent misconception analysis. The students in all classes then were asked to go home and finish the rest of their individual thinking.<br /><br />Tomorrow, when we come back, we will do verrrry structured group work, and I will ask the students to do first one round of just making observations with no comments from their peers (Round 1). Then, they will go around and ask clarifying questions or challenge each other to justify their thinking (Round 2). Then, the group will engage in an open discussion while the recorder continues to take careful notes, to turn in later and to distribute to the group. Afterwards, they will summarize the findings and record what questions they still have as a group. To help the Calculus students focus in on the misconceptions I saw today, I will give them <a href="https://drive.google.com/file/d/0B9GuwbUfAT6MNkpXQkVZZkg2Yjg/edit?usp=sharing" target="_blank">this handout</a> as generated from their parachute graphs, and ask them to brainstorm as many observations as possible for each graph before we discuss as a whole class. The recorder of each group will record all the accuracies and inaccuracies that they notice about each graph.<br /><br />Here is some language I will offer the groups tomorrow to help them with their structured discussions:<br /><br /><div class="MsoNormal">Beginning of discussion (Round 1):</div><div class="MsoNormal">"What have you tried so far?"<o:p></o:p></div><div class="MsoNormal">"I noticed that ______________"<o:p></o:p></div><div class="MsoNormal">"I tried __________ and found that to be (un)helpful, because _____________"<o:p></o:p></div><div class="MsoNormal"><br /></div>Middle of discussion (Round 2):<br />"I was confused about how to _______________"<br /><div class="MsoNormal">"Are you saying that _______________?"<o:p></o:p></div><div class="MsoNormal">"Can you explain why you think ___________ ?"<o:p></o:p></div><div class="MsoNormal">"I don’t get that. Can you explain it in another way?"<o:p></o:p></div><div class="MsoNormal">"If we changed __________, then what would the result look like?"</div><div class="MsoNormal"><br /></div><div class="MsoNormal">End of discussion:</div><div class="MsoNormal">"In conclusion, we agreed that ______________"</div><div class="MsoNormal">"We found it hard to agree on _____________"</div><br /><div class="MsoNormal">"As a group, we still have trouble understanding ____________"<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal">I am really excited about this! Having structured group work is allowing me to slow down the pace at the start of the year and to emphasize quality over quantity of work done, in order to set the right tone for the rest of the year. I plan to have the kids stay in these groups for about a week, and then we'll discuss what have been the most helpful parts of the structure, and then switch into new groups. Wish me luck!!! </div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/8vGKGtqX4b4" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/8vGKGtqX4b4/experimenting-with-structured-group-work.htmlnoreply@blogger.com (untilnextstop)5http://untilnextstop.blogspot.com/2014/09/experimenting-with-structured-group-work.htmltag:blogger.com,1999:blog-6651514617266100245.post-6468703267614267653Fri, 01 Aug 2014 21:31:00 +00002014-08-01T17:31:20.287-04:00One Resource a (Week)Day #19: Using Desmos in CalculusI am starting to focus in to think about what I want to do with my classes during the first week of school. For Calculus, I think the choice is obvious. I should start them off by playing with Function Carnival, over at Desmos.com!! (For those of you who, like I, did not attend Twitter Math Camp and have missed the demo / all the blogosphere buzz, you should sign up for an account at teacher.desmos.com to play with their interactive applets immediately. The interface is thoughtful and it allows you to collect data regarding what your students are creating as graphs and how they analyze other people's errors, and you can use it at the start of a unit to build understanding from recognizing and addressing misconceptions.) I will be starting them off easy, on Day 1, by playing with the regular algebra version to warm up. And then, on Day 2, plunge them right into predicting and testing velocity function graphs in the Function Carnival, in order to jump-start their interest in thinking about rates of change.<br /><br />To reiterate things that others have said, I really like the ability to both toggle through individual students' responses and to look at the whole class's responses at once. To start with this activity at the very beginning of the year can help me to set the tone that it is okay to experiment and to make mistakes, because that is how we learn math.<br /><br />Yay Desmos! This is really a fantastic tool that incorporates real mathematical instruction, <i>not </i>by replacing the whole-group experience but to reinforce it by giving each student a structured think-time before the whole-group discussion. I really hope that this is the future of digital instruction!<img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/0_j0ssfF1Qg" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/0_j0ssfF1Qg/one-resource-weekday-19-using-desmos-in.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2014/08/one-resource-weekday-19-using-desmos-in.htmltag:blogger.com,1999:blog-6651514617266100245.post-5471918616214732448Thu, 31 Jul 2014 21:53:00 +00002014-07-31T18:04:34.728-04:00Thinking About CalculusIf you teach Calculus in high school, you probably have a similar observation as mine that the student population is highly heterogeneous (a fact that is perhaps surprising to outsiders). Many students feel pressure from their parents and peers (ie. social pressure) to stay on the Calculus track, so within the same group they can span truly a wide range, from those who really could have benefited from another year of Precalculus-type skills-building, to those who are somewhat okay with procedures but who really need more experience/exposure with integrated problem-solving (something like a "Math 4" course that involves college-level math language), to those who are truly ready -- and eager -- to tackle new topics, both conceptually and algebraically.<br /><div><br /></div><div>I stumbled across <a href="http://www.ericdigests.org/pre-9217/calculus.htm" target="_blank">this research summary</a> from years ago (dated 1988) that noted a shift in college Calculus curricula away from conceptual understanding and toward procedural manipulation. Even though the article is dated, I find myself torn in the middle of a similar tug-of-war in my own thinking of Calculus. Fortunately, I teach a non-AP class, but my juniors (within a class mixed with seniors) do largely feed into another year of Multivariable Calculus after my class (combined with students from another class), so I have to take that into consideration in organizing my course. But, the statement from the article that we're trying to make the course "be all things to all people" really resonates with me. In the same class, I am supposed (and trying!) to differentiate so much as to help all the kids who need an extra year of algebra practice; to help all the kids who need an extra year of problem-solving; to cover all the major Calculus concepts; <i>and</i> to develop all of their skills as learners and to nurture their mathematical practices (all of which, mind you, take time). It was really half a miracle that many of the kids ended up enjoying and appreciating the experience of my first year of trial-by-fire to accomplish ALL of this at a new school; the task set for us can be daunting, frustrating, or exciting, depending on how you choose to look at it. </div><div><br /></div><div>One of the things that I did this year which actually helped a lot with teaching a heterogeneous group was to make sure that I made time for projects in this class. The projects gave the kids time to self-differentiate. Many of the kids kept working on their projects until the last minute, throwing in extra features or sometimes even working on it after it was due and presented. The next year, I will keep most of the projects but re-arrange the pacing so that they are better timed and hopefully a little less stressful for the kids. </div><div><br /></div><div>Projects I did:</div><div>* Economics mini-project (but this one I will switch out for something better, more individualized)</div><div>* Graphical organizer showing connections between all the learned skills</div><div>* Rollercoaster design using piecewise functions</div><div>* Function pictures including shaded definite integrals (calculated via GeoGebra but also by hand)</div><div>* 3D-modeling using vases that they created and volumes of revolution / scaling up to find and test against real volumes</div><div><br /></div><div>I also did a group quiz this year on the minimization of coordinate distances (to help review the distance formula) and on related rates, which was assigned to be completed mostly outside of class. It was such a great experience for me and most of the kids, that I'll definitely have to find some opportunity to repeat it. I decided that next year, I'll have to bump up the Function Pictures project to the first grading period, to help the kids review functional forms and inverses. That way, when we revisit their projects to fill in the integrals later on in the year, their focus will be more on the integrals and less on the outlines.</div><div><br /></div><div>Since I need to replace the economics mini-project, I am going to try doing a sustainability project this year, that involves some regression and rate-analysis at the start of the year, and then have an individual component where the kids look for something else (an interesting data set) that does not have a time domain, to extend their understanding of rates beyond the time domain.</div><div><br /></div><div>Another choice that really helped me with approaching the heterogeneity was teaching Calculus in reverse. First, we did a lot of graph sketching and graphical analysis by calculator. This evened out the playing field because the kids who were weak with prerequisite algebra skills could still access and feel successful immediately about the new concepts. Then, we learned differential Calculus skills via exploration, which helped the kids build some of those valuable mathematical practices and to build their confidence in playing around with math. I took our time on this part of the course, to go through old algebra skills and to practice things that were challenging as they came up. The really intuitive students started during this unit to peek ahead on their own at how to procedurally un-do differentiation. After the derivative skills, I took a significant amount of time to work on related rates with the kids, to give everyone some quality time with problem-solving. (It wasn't nearly enough time though. Obviously, you can never spend enough time on teaching and practicing problem-solving.) Later, through explorations, the students were able to gather most of the core concepts about integrals. We then used projects to reinforce their algebra skills. Limits came last in my course (because it is the most abstract, and I think it made the most sense to introduce it last as a way to <i>prove</i> the things we thought were true), and at that point, it was a really nice tie-up of the whole year, reinforcing the definitions we had learned throughout the year regarding derivatives and anti-derivatives by proving them via limits.</div><div><br /></div><div><span class="">Even though I have been thoughtful with my course, I wish I could feel more certain that I am making the right choices for my class. For everything that I decided to spend more time on, it was a choice to leave something out. I tried to keep the class fluid, so that if an interesting question came up during a project with a bunch of kids, I expanded that during the next class to go into the relevant material, even if it's not part of the classic Calculus 1 curriculum. (For example, my students really wanted to know how to integrate circular areas on their projects, so we did an example together and about a third of the class then followed suit to use trig-substitution to help them on their projects.) Never in my class did I feel like I had wasted time, but I couldn't shake the feeling that Calculus 1, the way that I was teaching the class, could easily have spanned 1.5 years.</span></div><div><br /></div><div>What do you think? What choices have you had to make for your own Calculus classes? Do you think they are worth it?</div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/LnDJ0E3sfuM" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/LnDJ0E3sfuM/thinking-about-calculus.htmlnoreply@blogger.com (untilnextstop)1http://untilnextstop.blogspot.com/2014/07/thinking-about-calculus.htmltag:blogger.com,1999:blog-6651514617266100245.post-3635872025489620756Wed, 30 Jul 2014 23:30:00 +00002014-07-31T00:50:34.505-04:00My Work-in-Progress Algebra 2 Sequence (for Next Year)Last year, I felt pretty good about my Algebra 2 sequence. Some of the students struggled with the formal assessments in the class, but I don't think the fact that they struggled was tied to the way the material was sequenced. (Re-quizzes definitely helped them, but I also saw them learning things like managing their responsibility as the year went on and the material ramped up in complexity.) This year, as usual, I want to do better.<br /><div><br /></div><div>I think that my ideal Algebra 2 sequence, assuming that the students come in only knowing how to solve basic one-variable equations, would look like this:</div><div><br /></div><div>Unit 1. <i>Review solving equations, clearing fractions, and manipulating formulas. </i>The reason why I would start with this topic is that it allows me to see immediately who is struggling with topics from the years prior, and who isn't. Introducing fractions at this point of the year also gives me an opportunity to keep spiraling back to it in every other topic. A nice problem to use at the very start of the year is the classic pool border problem or any <a href="http://www.visualpatterns.org/" target="_blank">visual pattern</a> that extends linearly (to review the idea of inductive thinking and what a variable means for generalizing patterns).</div><div><br /></div><div>Unit 2. <i>Linear Functions. </i>After reviewing the meaning of a solution in Unit 1, I feel that it's super important for kids to see that when there are multiple variables, you can now intuit an <i>infinite</i> number of solutions to an equation! Using exploration to plot some of those solutions allows us to see a linear pattern emerge. Unit 2 is all about understanding the connection between predictability of elements and algebraic forms. In Algebra 2, I cover both slope-intercept and point-slope form, the latter I start with <a href="https://drive.google.com/?ddrp=1#folders/0B9GuwbUfAT6MbTlXSi1VWU40bkk" target="_blank">letting the kids figure out that collinearity has everything to do with slope</a>, and then from there they can simplify the slope formula m = (y<sub>2</sub>-y<sub>1</sub>)/(x<sub>2</sub>-x<sub>1</sub>) into the point-slope form. This year, I think I am also going to throw some unit analysis in there to help explain why slope m <i>has to be</i> change of y over change of x, and not vice versa (in order for mx + b to work out to have the same units as y.) Along with linear functions last year, we did a bungee-jumping regression project including a significant lab write-up, and I plan to repeat that next year.</div><div><br /></div><div>Unit 3. <i>Systems of Equations Setup and Solving via Graphing Calculator</i>. Following basic review of things from Algebra 1, the most important baseline skill for a student's algebra success is the ability to go from language to symbols. I always take some time (even in Algebra 2) to go through how to write basic equations of the form PART + PART = TOTAL or PART*PART = TOTAL, and then I give them a chance to put that to use by writing various systems equations and solving by graphing. A trick I learned from a former rock-star colleague is that you have to always teach and thoroughly practice the graphing skills first, if you want to have a fighting chance of the kids using the graphing calculator later on. If you make the choice of teaching algebraic approaches first, most kids who are afraid of thinking flexibly will always resort to the algebra, even in the cases when the calculator is clearly more efficient and less error-prone. Similarly, kids will be reluctant to check their answers using technology, unless the mechanics of doing so is already second-nature. In this unit, I teach them how to graph, zoom, trace, find intersection, and look at the table to help them with figuring out the appropriate zoom. (I don't like the Zoom Fit feature of TIs, since they're a bit buggy.) For you GeoGebra-lovers, don't worry, the kids will use graphing software later on. </div><div><br /></div><div>Unit 4. <i>Systems of Equations Algebra</i>. Now that the kids already know how to set up word problems and to solve by graph, we are ready to delve into the various methods of manually solving a system. This year, I will start with the <a href="http://untilnextstop.blogspot.com/2010/07/my-take-on-using-puzzles-to-teach.html" target="_blank">puzzle explorations</a> for systems to help the kids really <i>get</i> what it means to substitute. After they learn both elimination and substitution methods, they will then practice setting up and solving systems involving fractions (spiraling back to fractions is always a good idea) and word problems, and to use their graphical solution from the calculator to check. I wrote about this before, but <a href="http://untilnextstop.blogspot.com/2013/11/graphically-analyzing-inequalities-and.html" target="_blank">I always require on tests that kids solve each complicated problem twice</a>, using two different methods, to reinforce their understanding of graphical and algebraic connections.<br /><br />Unit 5. <i>Inequalities in the Coordinate Plane</i>. If time allows, I want to spend a short amount of time on inequalities this year. (I did so last year as well, but it was sort of scattered.) Following systems is a good time, because I can then use linear programming problems to drive home the usefulness of the constraints and the graphing.</div><div><br /></div><div>Unit 6. <i>Quadratic Functions. </i>The way I teach quadratics is by <a href="http://untilnextstop.blogspot.de/2011/09/linking-linear-functions-and-quadratic.html" target="_blank">building it up from linear patterns</a>, and I drive home the connection between dimensionality and degrees via <a href="http://untilnextstop.blogspot.de/2011/09/fun-pattern.html" target="_blank">this type of side-by-side comparison</a>. The recurrent problem sometimes is that kids don't really understand dimensions from Geometry. (If you're a Geometry teacher, please give some TLC to this very important idea!) We do go into various forms of quadratics and I teach them both how to factor and <a href="http://untilnextstop.blogspot.de/2012/11/solo-performance-and-remembering-formula.html" target="_blank">how to sing and apply the quadratic formula</a>. We do some completing the square, but not enough to master it in Algebra 2, only to see that you <i>can</i> get things from standard form into vertex form. It's important for them to recognize that the quadratic formula can be broken down into various useful parts (discriminant <i>and</i> axis of symmetry) before we move on, so that they could sketch graphs based on any given function equation. Last year, I really drilled the kids to be able to sketch linear-quadratic systems, which, although they probably will not remember the specifics of the procedure, definitely helped to reinforce the idea of connections between graphs and algebraic forms. I didn't do a quadratic-specific project last year, but this year I plan to do a bridge modeling project using all three forms of the quadratic function, as I have done previously in other classes. Dan Meyer's pennies and circles task is also nice to use during this unit to review the idea of regression in the context of quadratics. Sometime early in the quadratics unit, I feel that it is very important to explore the idea of constant second differences between the sequence elements. This sets the stage for other types of patterns to come and helps to reinforce the difference between linear and quadratic patterns.<br /><br />Unit 7. <i>Transformations.</i> Following quadratics is a good time to talk about general function transformations. The same rock-star colleague had advised me that kids think this topic is too abstract. They will not retain it if you start by teaching g(x) = a*g(x - c) + d, but they <i>will</i> retain it if they can think of a concrete (ie. quadratic) pattern that they already are familiar with. I do these with explorations on the computer, and I have a <a href="https://drive.google.com/?ddrp=1#folders/0B9GuwbUfAT6ManBkSzhVZ1ZjMEk" target="_blank">pretty scaffolded plan</a> if you want to grab it to take a look.</div><div><br /></div><div>Unit 8. <i>Exponential Functions</i>. I think after quadratics as a big unit, the most natural next major topic is exponential patterns, if your students are following the trajectory of discussing various sequences. Kids can see geometric sequences everywhere, and it is so useful in their lives to understand compounded growth, that I think exponential sequences should be introduced as early as possible to contrast with linear patterns. Here, sustainability issues should really be discussed, both in terms of inflation of costs (of living and education and debts) and our unsustainable human growth / depletion of resources. I teach exponent rules inductively, and they go along with this unit but are assessed separately. I had the idea last year of asking the students to do a sustainability PSA (public service announcement) project, and I will really try that this year with more careful planning / pacing. I have not decided if I will teach logs this year yet, only because not all the teachers in our department can agree where in our curricula (Algebra 2 vs. Precalc) that should be taught.<br /><br />Unit 8. <i>Inverses of Functions</i>. I didn't do a full unit on this last year, but I think a full unit on inverses, domain, and range should logically follow the introduction of basic function types, because it helps to introduce all the interesting forms that the kids may wish to use in their function pictures project. The wrap-up of this unit should be a functions picture project (via GeoGebra or Desmos), in which they show at least two types of "sideways" functions as part of their included functions. For me, the functions pictures project in Algebra 2 <i>needs</i> to be accompanied by explanations of the transformations, to reinforce the connections between algebraic form and graph.<br /><br />Unit 9. <i>Polynomials. </i>I taught polynomials as the very last topic this past year, and absolutely loved it! I loved the particular placing of this unit at the end of the year because it allowed us to spiral back to factorization and quadratic formulas, while covering deeper ideas like u-substitution and complex roots. (I didn't do complex roots during the quadratics unit, since there were already so many skills there.) The kids also learned to apply the Rational Root Theorem, of course, and reinforced their understanding of the root. We did a quick maximization problem and <a href="http://untilnextstop.blogspot.de/2014/07/one-resource-weekday-9-open-middle.html" target="_blank">some backwards problems working from remainders and factors to finding missing coefficients</a>, and I was so happy to see how well the kids did! If we have time, we'll do a stocks project along with this unit. If so, I'll have to dust that one off from the archives...<br /><br />Ok, that is a lot! I've just pretty much laid out my entire Algebra 2 curriculum for next year. Whew! Let me know what you think and where you think the missing corners are! xoxo.</div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/5Owlr-Rc_VI" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/5Owlr-Rc_VI/my-work-in-progress-algebra-2-sequence.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2014/07/my-work-in-progress-algebra-2-sequence.htmltag:blogger.com,1999:blog-6651514617266100245.post-535387851052566107Tue, 29 Jul 2014 21:56:00 +00002014-07-29T17:56:15.457-04:00One Resource a (Week)Day #18: Geometry ActivitiesWhen I taught Geometry last, I found that it was very feasible to structure most of the geometry class like this:<div><br /></div><div>* First, some exploratory activity meant to introduce a new topic and important vocabulary terms</div><div>* Some project- or lab- based learning that lasts about 2 or 3 weeks, interleaved with skills taught as needed</div><div>* Concentrated skills practice / "review" after the project</div><div>* Quiz or test on the skills</div><div><br /></div><div>Just off the top of my head, the units where this learning structure was very applicable included: </div><div>* Tessellations (we made triangular, quadrilateral, and custom tessellations using rulers and protractors, which motivated some triangular congruent properties) </div><div>* Measurement and conversions (we learned to measure everything from lengths to volume to mass, and practiced some unconventional or indirect methods as well)</div><div>*Right-triangle trig (lots of outdoors measurements involving angles of elevation and depression and inclinometers)</div><div>* Quadrilateral trig (using KFouss's problems and some paper folding to see why quadrilaterals are built from non-right triangles, which are built from right-triangles)</div><div>* Scaling (we did logo projects and calculated how that impacted the perimeters and areas) </div><div>* Perimeter and area (using blueprint of houses on coordinate planes, with circular and concave portions)</div><div>* Surface area and volume (kids designed and built their own 3-D composite solids)</div><div>* Construction of reflections (mini-golf course designs)</div><div>* spatial projections (going from 3-D views to drawing 2-D views, and vice versa, using the computer to verify their hypotheses)</div><div><br /></div><div>Some of the other traditional topics (integration of algebra with geometry; some coordinate-plane concepts; proofs and counterexamples; and basic geometry visualization based on language) we didn't do through projects, but I tried to still make those parts of the class as interactive as possible. Most of the topics you can illustrate through patty paper, move-around demos, and just plain fun things. Geometry is definitely my favorite class to teach, but I am always looking for new ways to spice it up! I find that after doing a project, the kids are solid with the basics and are ready for me to push them a bit farther on the paper assessment.</div><div><br /></div><div><a href="http://hs.nburlington.com/ourpages/auto/2009/2/13/55136186/Honors%20Geometry%20project%20packet.pdf" target="_blank">Here are some more ideas of projects</a>, from a school in Columbus, NJ. (Sorry but I couldn't find the teacher's name!) I like this list. It has a variety of ideas, so that if our Geometry team decides to do different sets of core topics this year, I can still incorporate projects into my class. I think that it will make a nice complement to the visual / artistic activities from the beginning of <i>Discovering Geometry: An Inductive Approach. </i></div><div><br /></div><div>That's it for today. Till tomorrow! </div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/r2b2oNMJGl4" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/r2b2oNMJGl4/one-resource-weekday-18-geometry.htmlnoreply@blogger.com (untilnextstop)2http://untilnextstop.blogspot.com/2014/07/one-resource-weekday-18-geometry.htmltag:blogger.com,1999:blog-6651514617266100245.post-4239055440871599613Mon, 28 Jul 2014 22:23:00 +00002014-07-28T18:32:28.806-04:00Math and History: A Look at SlaveryGeoff and I decided to go on a plantation tour on Saturday. Before we had committed to this, I was feeling uneasy about how morally gray this experience might be, so I did some research and found out online that there is a plantation that runs its tours with historical accuracy and talks about slavery with candor. So, that was the one we decided to go on. The tour was combined with going to another plantation also in the area, and the experience was one that I will not forget.<br /><div><br /></div><div>This is what I gathered from both plantations tours:</div><div><br /></div><div>* During the Antebellum period, the sugar cane industry down in New Orleans had boomed. The farmers invested in slaves to help them expand their business. Both plantations we visited had roughly 100 slaves in the 1830s. Most of them lived in small slave quarters a short distance away from the main house, but the house slaves lived and worked closer to their masters.</div><div><br /></div><div>* Among the slaves on the plantation, there existed a hierarchy both depending on their skills and where they came from. The Creole slaves spoke French, and therefore were able to work inside the house and/or communicate with their masters, and therefore were valued more highly (and bought/sold for more money, especially if they were also highly skilled in things like metalsmith). The "American" slaves that were brought in after the Louisiana Purchase were generally valued less, because if they did not happen to speak the same African language as the other Creole slaves, the owners often had trouble communicating with them and they would struggle on the job.</div><div><br /></div><div>* The slaves worked in grueling conditions, sometimes for up to 16 hours a day. I cannot imagine working in the fields when it was 90 degrees and super humid. I was sweating up a storm just from the short walk in between the buildings during the tour. At one of the plantations, we saw a huge shackle that the slaves would wear around their necks to prevent them from running away.</div><div><br /></div><div>* The slaves who were considered of least value slept on the floor of their slave quarters, without a bed. Those who were valued more, had more complete furnishing.</div><div><br /></div><div>* The slaves grew their own foodstuff on the farms, in order to feed themselves and to live off the land. </div><div><br /></div><div>* During the Postbellum period, the slaves were essentially kept in slavery because each plantation only paid the "freedmen" in tokens that only worked on the plantation, at the plantation store. This way, the freedmen could never really leave because they could not save up money to do so. This continued on one of the plantations until 1900s and on the other all the way until 1940s. The latter plantation, for this reason, still has the original slave quarters that you can visit today. When the current owner bought the plantation in 1940s, the tenement farmers who still lived in those quarters were still largely the descendants of the former slaves. The big difference before- and after- the war was that the freedmen could send their kids to school.</div><div><br /></div><div>* Both plantations have a list commemorating the slaves who once lived there, first names only (because they didn't have last names as slaves). The slaves were listed with values, some as little as $25 (in the 1830s) and others listed as $1500 if they had specialized skills.</div><div><br /></div><div>* The tour guide at one of the plantations told us that after the war, the public schools in New Orleans were actually initially desegregated until the Jim Crow laws came into effect in the 1880s. Afterwards, the schools remained segregated until the Civil Rights era. Although I found <a href="http://www.nola.com/politics/index.ssf/2010/11/fifty_years_later_students_rec.html" target="_blank">this article</a> about the re-integration in schools in 1960, the tour guide had explained that the schools weren't integrated here until the 70s. (Anyway, now everyone goes to charter schools in NOLA, and people who could afford it send their kids here to Catholic schools.)</div><div><br /></div><div>Although I was ambivalent about these trips, and I was bothered by the way one of the plantations seemed to brush the slavery issue under the rug, I still think that going to see a plantation firsthand was a very educational experience. If my students end up learning about slavery this year (and they must, in one of the grades I teach), I could tell them about this experience and when I tell them that people were bought and sold for as little as $25, we could do the math to figure out how little that money is in today's terms.</div><div><br /></div><div>So, anyway, that was my rumination on math's role in history.</div><div><br /></div><div>PS. I found this interesting article about <a href="http://libres.uncg.edu/ir/uncg/f/L_Schweninger_Antebellum_1989.pdf" target="_blank">Creole slave-owners who were themselves black</a> in Louisiana. It definitely helps to explain some of the things people have said about Creole blacks feeling superior to African-Americans in NOLA. There are lots of numbers in this article to use for calculations of current-day value.</div><div><br /></div><div>PPS. I've been doing some <a href="http://www.amazon.com/Girls-Atomic-City-Untold-Helped-ebook/dp/B008J4GTU4/ref=sr_1_1?s=books&ie=UTF8&qid=1406586698&sr=1-1&keywords=the+girls+of+atomic+city" target="_blank">recreational reading about WWII</a> for my book club, and one of the factoids I learned was that the initial funding for the Manhattan Project was $6000. This was in 1940. This is also a real exponential growth application, to figure out how much that funding would be worth today.</div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/OdFClig5rDY" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/OdFClig5rDY/math-and-history-look-at-slavery.htmlnoreply@blogger.com (untilnextstop)0http://untilnextstop.blogspot.com/2014/07/math-and-history-look-at-slavery.htmltag:blogger.com,1999:blog-6651514617266100245.post-3579203745208511026Fri, 25 Jul 2014 21:46:00 +00002014-07-25T17:54:38.245-04:00When Linear Sequences CoincideI was playing around with this idea yesterday: How do we figure out when two linear sequences will eventually have the same value? How do we know when they will not? Randomly, I came across Amy Gruen's question on Twitter from a while back (but it also relates to the last NRich task I had posted yesterday, in trying to figure out how to find numbers that would light up multiple colors in the applet):<br /><div><br /></div><div><span style="background-color: white; color: #292f33; font-family: 'Helvetica Neue', Arial, sans-serif; font-size: 14px; line-height: 18px; white-space: pre-wrap;">I am a number less than 3000, Divide by 32, the remainder is 30. Divide by 58, the remainder is 44. Who am I?</span><br /><br />(If you haven't had a chance to play with this problem, I encourage you to do so and to get back to me if you have a different method than the one I have described below!)<br /><br />I started by listing some multiples of 32 (32, 64, 96, 128, 160, 192, 224, ...) in Excel and then adding 30 to find numbers that satisfy the first condition ("Divide by 32, the remainder is 30"). These are the numbers in the first linear sequence, we'll call it sequence A: 62, 94, 126, 158, 190, 222, 254, ... A 5th grader can do this as well (as Amy had stipulated), but probably not in Excel but by hand.<br /><br />And then, similarly, I listed out some of the sequence of numbers that would satisfy the second condition ("Divide by 58, the remainder is 44"): 102, 160, 218, 276, 334, 392, 450, ... We'll call this Sequence B.<br /><br />My first instinct was to write two expressions 32n + 30 and 58n + 44 and to set them equal, but of course that doesn't work because the sequence values are likely not going to coincide at the same position n. (This is probably a common misconception, so I thought I would point out that it's a natural one to make.) Also, algebra isn't part of the Grade 5 curriculum.<br /><br />Then, I thought if I started iterating through elements of Sequence B, I would probably reach the coinciding element faster, only because sequence B takes "bigger steps" and skips more of the in-between, irrelevant values. And, instead of listing every element from each sequence, I thought that maybe keeping track of how far "off" sequence B is from the closest element of sequence A might help me.<br /><br />I made a table that looked like this. I decided to use shorthands in column 3 to help me focus on seeing a pattern. Originally I didn't have the a and b, but the numbers by themselves didn't seem helpful. Once I added the a and b (for above and below), the pattern was much more recognizable, because I was essentially assigning positive and negative signs to the distances.<br /><br /></div><table border="1" cellpadding="2"><tbody><tr><td>Sequence B Element</td><td>Distance from Nearest Sequence A Elements</td><td>Shorthand representation of distance</td></tr><tr><td>102</td><td>8 above 94; 24 below 126</td><td>8a (24b)</td></tr><tr><td>160</td><td>2 above 158; 30 below 190</td><td>2a (30b)</td></tr><tr><td>218</td><td>4 below 222; 28 above 190</td><td>4b (28a)</td></tr></tbody></table><br /><div>I observed what I think was a linear pattern by this point, and decided that I could predict what the next few sequence B elements' distance would be from the nearest element of sequence A. I also noticed that the above and below nearest distances added up to 32, as you would expect. (Since sequence A elements are separated by steps of size 32, if you're 8 values above the nearest value in that sequence, you must be 24 values below the next one, like being suspended in between two rungs of a ladder of fixed space between the rungs.)</div><div><br /></div><div>So, based on this I hypothesized and tested that the next elements of sequence B will continue to follow this pattern and be located at a predictable distance from the nearest sequence A elements. </div><div><br /></div><table border="1" cellpadding="2"><tbody><tr><td>Sequence B Element</td><td>Shorthand representation of distance from nearest Sequence A elements</td></tr><tr><td>276</td><td>10b (22a)</td></tr><tr><td>334</td><td>16b (16a)</td></tr><tr><td>392</td><td>22b (10a)</td></tr><tr><td>450</td><td>28b (4a)</td></tr></tbody></table><br /><div>A pause here. I haven't reached any repeats yet. If I had reached any repeats in my table in terms of distance from the nearest elements (for example if I saw 8a and 24b appear twice in the table before reaching 0a or 0b), I would conclude that the two sequences will <i>never</i> meet. In the case of this problem, I should continue the table since we haven't reached any cycles yet.</div><div><br /></div><div><table border="1" cellpadding="2"><tbody><tr><td>Sequence B Element</td><td>Shorthand representation of distance from nearest Sequence A elements</td></tr><tr><td>508</td><td>30a (2b)</td></tr><tr><td>566</td><td>24a (8b)</td></tr><tr><td>...</td><td>... skipping some rows here, since I can see that 24a is a multiple of 6 and it will eventually decrease to 0a perfectly...</td></tr><tr><td>798</td><td>0a (32b)</td></tr></tbody></table></div><div>So, if this pattern holds, 798 should be the first time that sequences A and B converge, which means this number should leave me a remainder of 30 when divided by 32 and leave me a remainder of 44 when divided by 58. And it does! </div><div><br /></div><div>To find the subsequent elements is much easier (more of a standard math problem), because we know that the two sequences move by paces of 32 and 58, respectively. All we need to do is to find the least common multiple of their steps, which will be the distance that separates pairs of coinciding elements. The factors of 58 are 29 and 2, the (partial) factors of 32 are 2 and 16. So, 29x2x16 = 928 should be the least common multiple. That means that after 798, the next time the sequences converge to satisfy both conditions is at 798 + 928 = 1726, and the next time is 1726 + 928 = 2654. Since 2654 + 928 > 3000 and 798 - 928 < 0, our complete set of solutions is {798, 1726, 2654}.</div><div><br /></div><div>Just in case, I tested all three values against the two given conditions (since I don't trust myself with arithmetic and book-keeping). Also just in case I didn't mess up the LCM calculation, I tested the value halfway in between 798 and 1726 to make sure that it <i>doesn't</i> satisfy both given conditions. </div><div><br /></div><div>Now, here are the follow-up questions: Do you think this problem is doable by a student? What type of scaffolding would they need in order to accomplish this type of task? Are there other ways of doing this problem?</div><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/IAewCIRg1kU" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/IAewCIRg1kU/when-linear-sequences-coincide.htmlnoreply@blogger.com (untilnextstop)1http://untilnextstop.blogspot.com/2014/07/when-linear-sequences-coincide.htmltag:blogger.com,1999:blog-6651514617266100245.post-6526733737496673900Thu, 24 Jul 2014 22:40:00 +00002014-07-24T20:51:31.315-04:00One Resource a (Week)Day #17: Interactive Tasks from NRichI have been doing some more playing of the secondary-school tasks from NRich, and I noticed in that process that they actually have some really nice interactive applets. I think that making an effective teaching applet is tricky, because:<br /><br />1. If you make an applet that has too many features, even if you have the best of intentions, it can end up distracting from the actual mathematics.<br /><br />2. If you make an applet that has too few features, on the other hand, it does not necessarily support the student's need to generate more data points and to test their conjectures.<br /><br />Anyhow, here are a few tasks that have quite nice connections to high-school topics, each with a useful interactive applet.<br /><br /><a href="http://nrich.maths.org/2293" target="_blank">http://nrich.maths.org/2293</a> has to do with finding (and predicting) areas of tilted squares, with a specified tilt k. The problem is accessible with just basic geometry, but it is extendable to a function of two input variables. You can generalize the pattern A(t, h) to describe the area of a square with a tilt t and whose two leftmost vertices differ by h units in height. The applet at the bottom of that page is very user-friendly. It only has two togglable points for you to construct squares of a certain tilt and height, and it is only there to help students construct newer instances and to observe their resulting shape and area concretely.<br /><br /><a href="http://nrich.maths.org/2281" target="_blank">http://nrich.maths.org/2281</a> is a super easy-entry puzzle on building a pyramid of numbers. The guiding questions are gentle but they effectively get the kids to start thinking about how the position of a bottom number affects the final value at the top of the pyramid. They can make conjectures and test them repeatedly using the applet, thereby deepening their observations along the way. And then, the plot thickens when the pyramid gets to be bigger -- with 4 or more elements at the bottom level. Eventually, it could be generalized to show connections to Pascal's Triangle, a topic often touched upon in Algebra 2. Tres cool!<br /><br /><a href="http://nrich.maths.org/7016" target="_blank">http://nrich.maths.org/7016</a> is a quite high-level task suitable for thinking about sequences. The applet is there for the students to try and gather data about which numbers will light up each color, and the really nice thing is that each group can be working on different patterns, without extra work on your part to generate different data. The entry to this task is a fairly straight-forward practice of linear equations / sequences, but when you start asking questions about how to light up multiple colors, the question gets rich really fast. When we dig even deeper into how to generalize relationships between sequences, I at least found myself in a quick sand. Besides some trial-and-error, I couldn't find a systematic way of predicting the first sequence element where two lights (of known pattern) will both light up. (After the first coinciding lighting, the rest is easy to obtain.) Can you help?<br /><br />By the way, I am loving the various Twitter quotes from the Twitter Math Camp y'all are at. Keep them coming! You guys are so inspiring!<br /><br />Addendum 7/24/14: I did a bit of playing and figured out how to find the first coinciding element of two linear sequences! For example, this problem from Amy Gruen goes nicely with the problem #7016 from above (hits the same type of idea). I leave it for you as an exercise to find all the numbers that satisfy this within the range 1 to 3000, but I'll answer it in a few days if you haven't already figured out how to do it...<br /><br /><span style="background-color: white; color: #292f33; font-family: 'Helvetica Neue', Arial, sans-serif; font-size: 13.63636302947998px; line-height: 18px; white-space: pre-wrap;">I am a number less than 3000, Divide by 32, the remainder is 30. Divide by 58, the remainder is 44. Who am I?</span><br /><br /><br /><img src="http://feeds.feedburner.com/~r/IHopeThisOldTrainBreaksDown/~4/UopFIROmm-0" height="1" width="1" alt=""/>http://feedproxy.google.com/~r/IHopeThisOldTrainBreaksDown/~3/UopFIROmm-0/one-resource-weekday-17-interactive.htmlnoreply@blogger.com (untilnextstop)2http://untilnextstop.blogspot.com/2014/07/one-resource-weekday-17-interactive.html