tag:blogger.com,1999:blog-47847168147972780122018-07-16T03:20:21.445-04:00MATHEMATICAL MUSINGS by mathteacher24These are some of my thoughts about teaching high school mathematics. Trying a #teach180 blog this year and reflecting to become my best teaching self. #MTBoSLeigh Nataronoreply@blogger.comBlogger213125tag:blogger.com,1999:blog-4784716814797278012.post-88734171362685890092018-07-10T20:50:00.003-04:002018-07-10T20:50:39.538-04:00Ten Weeks of Summer: To Teach It or Not To Teach It (Week 4)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-jsMOFSUQTXA/W0VTeSyyknI/AAAAAAAAHX0/cQ8trq43SRswU6469MMSVD-GVRoUMekzQCLcBGAs/s1600/Screen%2BShot%2B2018-07-10%2Bat%2B8.46.24%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="579" data-original-width="482" height="320" src="https://4.bp.blogspot.com/-jsMOFSUQTXA/W0VTeSyyknI/AAAAAAAAHX0/cQ8trq43SRswU6469MMSVD-GVRoUMekzQCLcBGAs/s320/Screen%2BShot%2B2018-07-10%2Bat%2B8.46.24%2BPM.png" width="266" /></a></div><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif;">"To teach it or not to teach it. That is the quesiton. Whether 'tis nobler in the mind..." Whoops, slipped into Shakespeare's Hamlet for a second.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif;">I am part of an email group of math teachers and last week there was a question going around about should we teach and/or why don't we teach concepts about alternate exterior angles formed by two parallel lines and a transversal. Some argued via email that it was important to teach that concept for future engineers. Others explained that they did teach the concept and that they taught it through discovery. I said that it is probably not seen in many textbooks, because interior angles have more importance, especially when considering quadrilaterals. </span><span style="background-color: white; color: #222222; font-family: arial, sans-serif;">So, to teach it or not to teach it. Here are two stories I shared with my email group.</span><br /><div style="color: #222222; font-family: arial, sans-serif;"><br /></div><div style="color: #222222; font-family: arial, sans-serif;">Story #1: I got my master's degree at Iowa State and for one of my projects, I used a theorem about secants and tangents to circles. It is a theorem based on similar triangles. The professor (with a doctorate degree) questioned me on the use of the theorem. Either he had never heard of it or he couldn't recall it. However, he had enough understanding of geometry that when I explained why it was true, he understood. </div><div style="color: #222222; font-family: arial, sans-serif;"><br /></div><div style="color: #222222; font-family: arial, sans-serif;">Story #2: Over the past 10 years as department chair, I have to speak to groups of prospective parents/students and describe what is offered in the mathematics department at my school. Do I speak about specific theorems or concepts? No - they assume that I am teaching the math content that is typical of a high school mathematics curriculum, and let's be honest, a list of topics would be yawn-inducing. Instead, I say that in math class students make and test conjectures, combine concepts in new ways, form logical arguments and critique the reasoning of others (verbally and in writing), notice and describe patterns, and make connections between different ways to represent a concept. What I have just described is something that ALL students need no matter what major or career they go into.</div><div style="color: #222222; font-family: arial, sans-serif;"><br /></div><div style="color: #222222; font-family: arial, sans-serif;">Of course there are big ideas that matter in the teaching of mathematics and we can't gloss over those - function, inverse, transformations, proportionality, and rate of change are just a few. However, if we leave off some concepts, will our students be somehow lacking? Think back to Story #1. Was the Ph.D. mathematician lacking? Some might say yes and that he should have learned and remembered that concept. I would argue no, because he had a firm understanding of the big ideas and the ability to reason and apply his understanding. His ability to do this, and not recalling a specific theorem, is what makes him, and I hope my students, mathematicians.</div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-47615155030260106002018-07-03T14:02:00.000-04:002018-07-03T14:02:01.121-04:00Ten Weeks of Summer: Analyzing THE Exam (Week 3)Right before school ended, AP teachers at my school got an email from the person in charge of our AP program saying that the AP exam books from 2017 were available. They were put in my mailbox and I didn't have time to look at them at that point. I remember thinking that it probably would have been more helpful to get them before the students took the AP exam on May 17th. <div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-dfc0MQPXFZI/WzuyuVGfOhI/AAAAAAAAHXQ/M1_UY1en0mkvlI7v7aXoVBroH4qDJ_rbgCLcBGAs/s1600/Screen%2BShot%2B2018-07-03%2Bat%2B1.30.05%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="328" data-original-width="486" height="134" src="https://3.bp.blogspot.com/-dfc0MQPXFZI/WzuyuVGfOhI/AAAAAAAAHXQ/M1_UY1en0mkvlI7v7aXoVBroH4qDJ_rbgCLcBGAs/s200/Screen%2BShot%2B2018-07-03%2Bat%2B1.30.05%2BPM.png" width="200" /></a></div><div>At my most recent workshop, participants wondered how it worked to get AP exam books back. <a href="https://apcentral.collegeboard.org/ap-coordinators/after-exams/ordering-labels-booklets">Here is what I found</a>. It cost us about $300, because we got all the exam books back for all of our students for all our AP courses, and the exam books arrived at our school somewhere between October and December. This makes me wonder where they were from December through May. It is a bit frustrating, because I could have reviewed the material in the books and used what I learned with my current AP Stat students.</div><div><br /></div><div>Today I finally had a chance to look at my exam books and I used the rubrics to determine each student's score. I was happy to see that no student left a question blank and it appears that some students got the highest possible score for a few of the questions! But what were some of the big things I noticed? Do I need to modify my instruction or emphasize certain things more? Here is what I discovered.</div><div><br /></div><div>1) From question #1, I noticed that many students interpreted slope correctly, but did not mention a predicted change in the response variable. Using words the words on average, approximately or predicted, would have indicated the non-deterministic nature of a LSRL.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-PvWtU3w36Ic/WzuzKkJlJkI/AAAAAAAAHXY/Yif3ndDM59MBGY76T2kP3HTp5doLnuMywCEwYBhgL/s1600/Screen%2BShot%2B2018-07-03%2Bat%2B1.27.32%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="299" data-original-width="573" height="166" src="https://3.bp.blogspot.com/-PvWtU3w36Ic/WzuzKkJlJkI/AAAAAAAAHXY/Yif3ndDM59MBGY76T2kP3HTp5doLnuMywCEwYBhgL/s320/Screen%2BShot%2B2018-07-03%2Bat%2B1.27.32%2BPM.png" width="320" /></a></div><div>2) From question #2, almost all my students used their graphing calculator to do a one sample z-interval for a proportion. However, not all students named the procedure or checked to see if the conditions for doing the procedure were met. A few students did not have the correct confidence interval procedure at all, but all students gave a correct interpretation of the interval they found.</div><div><br /></div><div>3) From question #3, almost all of the students could do the normal distribution calculation correctly for part a. However, to get full credit for this, they needed to show boundary and direction. If the student had simply drawn a normal distribution and marked it appropriately, they would have received full credit for part a.</div><div><br /></div><div>4) In general, I noticed that students weren't showing their work for probability calculations. Students may get partial credit on probability calculations if you have a wrong answer, but the work must be shown. The directions on the exam even state, "Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations." I need to emphasize that more. </div><div><br /></div><div>5) From question #5, I had many students leaving off the expected counts condition entirely or comparing the sample size of 207 to 30. But on the plus side, almost all students were able to draw the correct conclusion in context and included linkage between their p-value and a chose alpha level.</div><div><br /></div><div><b><u>The Positives</u></b> - Students are using context, making correct inference conclusions and correctly interpreting confidence intervals!</div><div><br /></div><div><b><u>Areas to Work On</u></b> - I need to model drawing a normal distribution and require students to draw a well labeled/shaded normal distribution for any normal distribution calculation. We need to spend more time on mixed review of identifying inference procedures and their associated conditions.</div><div><br /></div><div><b><u>Consider</u></b> - I currently split up LSRL between the two semesters. LSRL, residuals, standard error, correlation and coefficient of determination are in the first semester and inference for LSRL is in the second semester. Interpreting slope (and y-intercept) with non-deterministic language can be challenging, because students don't really understand that one sample produced that one very specific LSRL and that it is not likely to be the true regression line for the population. In addition, they don't understand the concept of sampling variability early in the first semester. Perhaps moving all the LSRL topics to the second semester would make this better, I think. This is the approach used by some stats teachers I know and one that I may want to use myself.</div><div><br /></div><div>The remainder of this week I will be working on reading the AB Calculus Course Description from the College Board and identifying my areas of weakness relative to AB topics. For example, I remember learning about slope fields in differential equations in college, but that was in 1990 - twenty-eight years ago!! Since I haven't seen that topic since then, it is safe to assume that I am rusty in that area. Perhaps next week I'll blog about AB Calculus or I'll blog about meta-analysis of my teach 180 blog from last year. (Spoiler: I am about 1/3 of the way through and definite themes are emerging!!)</div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-69810387821945621062018-06-24T23:21:00.002-04:002018-06-24T23:21:35.706-04:00Ten Weeks of Summer: Math Love! (Week 2)Last week I took my daughter to freshmen college orientation and she signed up for classes. I knew she was considering elementary education, but then she changed her mind. Her newly declared major - middle school mathematics education! (Of course I can't jump up and down physically beside her, that would be embarrassing. But inside, I was doing a happy dance.) Of course this study by the <a href="https://www.insidehighered.com/news/2017/12/08/nearly-third-students-change-major-within-three-years-math-majors-most">Education Department's National Center for Education Statistics</a>, says that about 1/3 of students change their major at least once. If changing major is genetically linked, we may be ok. Cassie's father was always a chemistry major from the first day of college and I was always a math major. Neither of us changed our majors EVER.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-S-KUI7Kcsy0/WzBfDP-va3I/AAAAAAAAHWs/SX_78GFulbsHhfmJrbBm4ZPsEgwhnFDxgCLcBGAs/s1600/Screen%2BShot%2B2018-06-24%2Bat%2B11.18.08%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="466" data-original-width="617" height="241" src="https://3.bp.blogspot.com/-S-KUI7Kcsy0/WzBfDP-va3I/AAAAAAAAHWs/SX_78GFulbsHhfmJrbBm4ZPsEgwhnFDxgCLcBGAs/s320/Screen%2BShot%2B2018-06-24%2Bat%2B11.18.08%2BPM.png" width="320" /></a></div>Then, I got a tweet from a fellow AP Stat teacher saying that his first ever AP Stat class created a class shirt and my name was on it because they found my TI-84 videos so helpful. I am glad to know that my work is being used by others. I thanked the teacher and wished his students well on the AP scores that are set to come out next week. Of course, I am keeping my fingers crossed that my students did well, too.<br /><br />Starting tomorrow, I will be leading an four-day AP Summer Institute for seven AP Statistics teachers. I spent two full days last week finalizing my plans for the workshop. I am definitely well prepared and hope it goes well. Sharing teaching ideas and learning from other math teachers is what keeps me teaching. It may seem strange that I get renewed energy in the summer by doing more teaching, but I do! Sharing the math love in workshops, over twitter, on my <a href="https://www.youtube.com/user/mathteacher24">YouTube</a> channel and in this blog motivates me. Plus I am always learning new ideas from my virtual math colleagues - shout out to #MTBOS, #iteachmath and <a href="https://www.bigmarker.com/communities/GlobalMathDept/conferences">Global Math Department</a> folks!Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-52280382105626343412018-06-17T17:11:00.001-04:002018-06-17T17:11:25.841-04:00Ten Weeks of Summer: The AP Stat Reading (Week 1)During the school year, I did a teach 180 blog and I will be spending some time next week reviewing my successes and failures of the previous school year. If you are a new teacher and you are reading this, you may be thinking, "Wait. You have taught since 1992. You still fail in your teaching?" Fail, fall short, learn and succeed. If I want to continue to be the best teacher I can be for my students, I will need to try new things to keep them engaged and help them learn. I think I may have even said in a previous blog post that once I stop learning and growing as a teacher it will be time for me to leave teaching. But I am nowhere near ready for that yet.<br /><br />This summer I will be blogging once each week to reflect on things I am doing over the summer relative to teaching. I might blog about something I learned from a participant at a workshop. Or I might blog about one of my summer goals, <a href="https://teacher-staging.desmos.com/computation-layer/documentation">learning the computational layer in Desmos</a>. Or I might blog about a book I am reading related to teaching.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dyOkE1lSfz28nwx3Xa2BVQzF1R1pQme9-68tnLZf-e4jYy2ucxRvHkDbFMsH_CZD16QkEYYjY75TQLNtzA0Kw' class='b-hbp-video b-uploaded' frameborder='0' /></div>Let's begin. Week 1: The AP Statistics Reading. Today is Day 7 - a.k.a. the last day of the AP Statistics Reading. It is one of the best professional development experiences of the year and this is my 9th year as an AP Reader. Each year I learn more about how to help my students become better statistical communicators and better statistical thinkers. I pick up teaching ideas, book recommendations and share stories with old friends and new friends. This year I was able to escape from my third Escape Room in Kansas City and a bunch of us tried ax throwing. (It's ok to throw axes in statistics, but you don't forget to label them.)<br /><br />This year at Best Practices night I presented about a teaching technique called "Stand and Talk". It was based off of <a href="https://saravanderwerf.com/2017/08/09/stand-talks-the-best-thing-i-ever-did-to-get-students-talking-to-one-another/">Sara VanDerWerf's blog</a> and talk that I saw at NCTM. (My blog posts about this technique can be found at <a href="http://mathteacher24.blogspot.com/2018/04/teach-180-stand-talk-some-success-day.html">Day 150</a> and <a href="http://mathteacher24.blogspot.com/2018/05/teach-180-stand-and-talk-again-day-151.html">Day 151</a> in my blog from the 2017-2018 school year.) I also picked up some ideas from other teachers at that session. Specifically, I want to try Kelly M. Spoon's (@kellymspoon) Desmos card sort activities. (As I was writing this blog, I tried to search for it and could not find it online.) So, I sent her a tweet and will modify my blog with the link once she sends it to me.<br /><br />Stay tuned for week 2 - prepping for some workshops and looking ahead to teaching at my new school.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-3125469194141105012018-06-08T17:14:00.002-04:002018-06-08T17:14:16.274-04:00Teach 180: The Last Day (Day 178)Today was the last day for all faculty. It was sad to think that this was my last time to be with this particular group of teachers and friends. I took pictures of my empty classroom to remind me of where I have taught for the past 12 years. I also took a picture of this one special frame I have hanging on my wall.<br /><br />My classroom has no real windows and has served as "the detention room" for many years. About a year or two after being at Moravian Academy, the photography teacher, Nancy, asked me "If you could look out a window, what would you see?" <br /><br />"A beach scene," I told her. "It doesn't need to be tropical, but it needs to be a beach scene." <br /><br />It was right before spring break and many of her photo students would soon be spending their days relaxing at a beach. So, Nancy held a contest for her students. The winning photo would be enlarged and framed behind this window that had been salvaged from Illick (a building on campus). Thank you Lauren Comes (class of 2009?) for giving me something beautiful and relaxing to look at each day. Thank you colleagues and friends; I will miss you greatly.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-zhfeaHkfy9o/WxrwJ1n4DpI/AAAAAAAAHVs/3abb11CbwdckT73uQQrqz4pnBiDooxXMwCLcBGAs/s1600/Screen%2BShot%2B2018-06-08%2Bat%2B5.07.21%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="517" data-original-width="543" height="304" src="https://3.bp.blogspot.com/-zhfeaHkfy9o/WxrwJ1n4DpI/AAAAAAAAHVs/3abb11CbwdckT73uQQrqz4pnBiDooxXMwCLcBGAs/s320/Screen%2BShot%2B2018-06-08%2Bat%2B5.07.21%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-10050974658496216872018-06-07T13:08:00.000-04:002018-06-07T13:08:22.177-04:00Teach 180: Write About a Time... (Day 176 & 177)Yesterday I proctored a chemistry exam, prepared sixteen different midterm and final exams for summer acceleration, wrote three letters to parents about summer acceleration, attended a faculty luncheon honoring about 15 people (of whom 8 were retiring) and took some time for a singing "jam" session with two colleagues. At the end of the day, I left feeling like three or four days had elapsed. In fact, I even forgot where I had parked my car. It was parked safely down by the science labs, where my day had started!<br /><br />Today we have a final meeting as an Upper School faculty and we have been asked to take ten minutes to journal about one of the following topics. (We subsequently had to write for two additional seven minute bursts.) We are to write about at time this year when:<br /><br /><ul><li>you were empowered to make a choice.</li><li>the power of choice was taken from you.</li><li>you felt heard.</li><li>you felt dismissed.</li></ul><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-_Fyx80Cjvyo/WxlHxnYRAcI/AAAAAAAAHU8/TLPPWrd0jSQhwlLr4SlCbDAs1jP6BkPzQCLcBGAs/s1600/Screen%2BShot%2B2018-06-07%2Bat%2B10.56.17%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="456" data-original-width="640" height="228" src="https://4.bp.blogspot.com/-_Fyx80Cjvyo/WxlHxnYRAcI/AAAAAAAAHU8/TLPPWrd0jSQhwlLr4SlCbDAs1jP6BkPzQCLcBGAs/s320/Screen%2BShot%2B2018-06-07%2Bat%2B10.56.17%2BAM.png" width="320" /></a></div><div>After this school year, I could write on each of these topics, but the two that I could write the most about are the power of choice being taken from me and feeling dismissed. Based on many of my friends being in states where they have gone on strike, I feel like being dismissed and having choice taken away are happening way to often, especially in <a href="https://www.youtube.com/watch?v=nva8UofZ3fY">public education</a>. </div><div><br /></div><div>This blog entry will focus on the feeling of being dismissed. (It is partly because of this feeling that I chose to leave Moravian and took a job at Kent Place School.) At the beginning of the year, I suggested two major curriculum changes, one was a re-ordering of courses. Another was lengthening courses to accommodate the decreased instructional time that was happening relative to the schedule change this year. Both of those ideas where shot down. The middle school math department did not want to consider the change in the ordering of courses and the Headmaster (by way of the Upper School Director) told me that we could not lengthen courses. In essence, we needed to work within the structure we were given. The Headmaster suggested that new ways of teaching should be used, but in reality, the curriculum itself needed to be restructured first. (In hindsight, we also should have been given two full days at the start of the school year to meet as a department to consider the impending changes.) After my meeting with him to discuss the impact of the schedule on the math curriculum, I walked away feeling like I was dismissed and not heard. </div><div><br /></div><div>So, the math department worked with what we were given. We had less review before tests and covered topics in less depth and we created a three page document to show what has been removed or reduced in the curriculum. The "surfacy" treatment of topics led to decreased student understanding in many areas. We didn't have time to look at more nuanced problems than we had in the past and students said that they wished they had more practice with math on a daily basis. I am sad that there is a mess that is being left behind in mathematics and that the new department chair will have to figure out how to clean up the mess. I am also sad to be leaving many wonderful colleagues. Keep fighting the good fight, LC and LG! (If you are reading this MR, don't worry. I am a phone call away and will gladly help you in any way I can.) </div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-8480860259848926272018-06-05T23:03:00.000-04:002018-06-05T23:03:12.331-04:00Teach 180: 26 Boxes and a Gem (Days 174 & 175)<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-M5hORexsMSI/WxdFtTulP8I/AAAAAAAAHTQ/gOT01AbRGoIkntCUS7JrpLupF3ldaRzGwCLcBGAs/s1600/Screen%2BShot%2B2018-06-05%2Bat%2B10.21.59%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="678" data-original-width="552" height="400" src="https://2.bp.blogspot.com/-M5hORexsMSI/WxdFtTulP8I/AAAAAAAAHTQ/gOT01AbRGoIkntCUS7JrpLupF3ldaRzGwCLcBGAs/s400/Screen%2BShot%2B2018-06-05%2Bat%2B10.21.59%2BPM.png" width="325" /></a></div>Yesterday I finished organizing some files and I am down to my final "To Do" list as I prepare to move from my job at Moravian Academy to a job at Kent Place School. This afternoon I counted the number of boxes and bins in my basement and there was a whopping 26! As I sorted through one box today, I found this gem from an NCTM Math session from 2006.<br /><br />I would say that if you asked my students that it means to do mathematics they would say things like find examples, look for patterns and solve problems. I am not so sure they would say the ones I have highlighted - "take chances, make mistakes", "discover more elegant solutions" and "explain, validate, convince". Why would they not say these? Here are my thoughts.<br /><br /><span style="color: red;"><u style="font-weight: bold;">Take Chances, Make Mistakes</u> - Although I try to have a culture where mistakes are seen as part of the learning process, students often don't want to make mistakes for fear of "losing face" in front of their peers. However, taking chances and making mistakes are a huge part of learning. Desmos has helped my students to take chances, try things out, make mistakes, revise their answers. Desmos gives feedback in a non-judgemental way. It says your wrong, but without using the word "wrong". I don't use the word "wrong", but what do I say specifically? I try to say things like "Why do you think that?" and "Can you explain your thinking?" Do I really value mistakes? I'll need to be more cognizant of this next year.</span><br /><span style="color: red;"><br /></span><span style="color: blue;"><u style="font-weight: bold;">Discover More Elegant Solutions</u> - Rarely do we take the time to have students discover more solutions, let alone more elegant solutions. This is especially true in the public school system. At times this has also been true in my classroom. To have more elegant solutions, you need to have problems that are interesting and can be solved by multiple perspectives. As I look to next year, can I make this part of my classroom culture? As I plan for my classes for the fall, my goal will be to find one or two problems per unit that lend themselves to discovering more elegant solutions.</span><br /><span style="color: blue;"><br /></span><u style="font-weight: bold;">Explain, Validate, Convince</u> - This is something that should be happening in every classroom (not just math classrooms) daily. "Convince us." and "How can you tell if your answer is right/reasonable?" needs to also happen more frequently in my classroom. In the name of "covering" content, we often rush past this part of a lesson to do more examples. The answer is "right" and for many teachers that are pressured by state testing, a "right" answer is all that matters. However, I believe that explain, validate and convince will have a stronger and more lasting impact on understanding than drill and kill in the name of "right" answers.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-8417489516880299832018-06-03T13:39:00.000-04:002018-06-03T13:38:59.984-04:00Teach 180: Celebrating the End (Day 173)Friday was the beginning of the end. Although we still have underclass exams for three more school days and two days of teacher meetings, we celebrated the graduation of this year's senior class on Friday evening with a baccalaureate service. It is especially bittersweet this year, because my own daughter is in the graduating class. After the service, we were milling on the lawn outside of the church with many pictures being taken. This one was taken of me and my first period AP Stat students. I will especially miss this group. Teaching them was a joy. They asked questions, made me laugh and hopefully, they will think more critically about any statistics headlines they see in the news. Congratulations to the class of 2018!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-HDza6w_zXj8/WxQg3WxDEpI/AAAAAAAAHS4/OkRONJL8_Qs9A6RZBzEb_6OtftkC2r-GwCLcBGAs/s1600/Screen%2BShot%2B2018-06-03%2Bat%2B1.09.00%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="514" data-original-width="824" height="199" src="https://2.bp.blogspot.com/-HDza6w_zXj8/WxQg3WxDEpI/AAAAAAAAHS4/OkRONJL8_Qs9A6RZBzEb_6OtftkC2r-GwCLcBGAs/s320/Screen%2BShot%2B2018-06-03%2Bat%2B1.09.00%2BPM.png" width="320" /></a></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-50672239874025680452018-05-31T19:04:00.000-04:002018-05-31T19:04:40.028-04:00Teach 180: Last Day of Classes (Day 171 & 172)<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Ah9FCFQBT7k/WxB-TTDqsUI/AAAAAAAAHSs/gkZrBroY78wUtFJlWMz2Eyz0iSSjtjmrQCEwYBhgL/s1600/Screen%2BShot%2B2018-05-31%2Bat%2B6.57.58%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="308" data-original-width="822" height="119" src="https://2.bp.blogspot.com/-Ah9FCFQBT7k/WxB-TTDqsUI/AAAAAAAAHSs/gkZrBroY78wUtFJlWMz2Eyz0iSSjtjmrQCEwYBhgL/s320/Screen%2BShot%2B2018-05-31%2Bat%2B6.57.58%2BPM.png" width="320" /></a></div>I don't normally combine two days of teaching into one blog post, but here are two posts for the price of one. Yesterday was not a good teaching day. About an hour into the school day, I got a call from my daughter, Cassie, saying that my dad (Pop-pop to her) was in ICU with chest pains. I wan't teaching at the time and had the next hour to make phone calls and figure out what was happening. He is home now and fine (tests came back negative), but the scare gave me time to think about how precious time is with family. After school, I spent time with Cassie using her camera to take a "photo shoot" of her at the local park in her white dress for graduation. We had fun and not once during that time did I think about teaching/school. Plus I have some new photos for the background on my computer and phone to remind me of her when she goes to college.<br /><br />Today was the last day of classes and final exams start tomorrow. My PreCalc students requested a Kahoot to review for their last 35 minutes with me. They chose to review logarithmic and exponential functions and based on the Kahoot, they did need more work in this area. The fact that the students realized that they need more work in a certain area is good. Hopefully, this will help them target their review. I encouraged them to send me emails with questions they might have over the next few days. As several of them left, they shook my hand and told me to "have a nice summer" and "good luck next year". All my students have been wonderful this year and I will miss them.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-10517221721869365052018-05-30T00:11:00.002-04:002018-05-30T00:11:53.562-04:00Teach 180: Grow or Stagnate (Day 170)Beginning in the fall, I will be teaching at Kent Place School in Summit, New Jersey and I spent a good portion of my day continuing to pack and organize. The last day of school will be here soon! My school day ended with an "exit interview" conversation with the Headmaster, reflecting on what Moravian Academy does well and where Moravian Academy could improve. As I reflected on my twelve years of teaching at Moravian Academy, I realized that I grew as a teacher as a result of my desire to learn and continue to improve. It was not because of anything the school, administration or Board of Trustees did. The systems for faculty growth were missing when I started at MA and sadly, they are still absent now. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-oObNXgJcaHQ/Ww4jy4FfoEI/AAAAAAAAHSc/ZEPbFh1UkAIiS7524Gr7XiCLRmt4btssQCLcBGAs/s1600/Screen%2BShot%2B2018-05-30%2Bat%2B12.06.36%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="265" data-original-width="667" height="127" src="https://3.bp.blogspot.com/-oObNXgJcaHQ/Ww4jy4FfoEI/AAAAAAAAHSc/ZEPbFh1UkAIiS7524Gr7XiCLRmt4btssQCLcBGAs/s320/Screen%2BShot%2B2018-05-30%2Bat%2B12.06.36%2BAM.png" width="320" /></a></div><br />During my tenure at MA, I renewed my National Board Certification and helped three other teachers at Moravian (and a few at other schools) obtain their National Board Certificates. (The teachers taught middle school math, PE and Spanish.) I lead workshops on Desmos, Smartboard, plickers, Google Hangouts, flipping the classroom and twitter. Last year I visited the classroom of about a dozen teachers, many outside my department, to provide feedback to the teachers and consider how I might incorporate their teaching strategies in my own classroom. I tried to lead a group called "Faculty Fellows" where teachers would get together to discuss teaching strategies or articles we had read. When that failed (teachers did not want to give up their lunch time nor did they have time meet outside the school day), I tried to start a virtual book group on the book "Grade Smarter, Not Harder". Although twenty signed up for the group, only two have participated sporadically. I continued to grow and lead despite being in an environment that does not invest in growth.<br /><br />I am sure I will continue to lead in some capacity at my new school. However, I want to - no, need to - be surrounded by like-minded teachers who want to risk, share, and learn from each other daily. I hope that Kent Place School is a place where faculty professional growth is the norm and not the exception. I hope it is a school where faculty evaluation is primarily seen as an opportunity for professional growth, instead of a way to remove ineffective teachers. I need to be a part of an organization that values people as resources and invests in their continued journey to becoming better teachers, because I am not done learning or growing yet.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-38168390911139873272018-05-28T09:32:00.001-04:002018-05-28T09:32:15.445-04:00Teach 180: Exploring the Desmos CL (Day 169)Because I only had two classes on Friday and I was tired of sorting papers and packing up my classroom, I decided to spend some time playing around with the computational layer in Desmos. I had learned about the CL when I went to San Francisco last summer as a Desmos fellow Cohort #2. I had fully planned to dive into the CL when I returned from that weekend, but we still had two math teachers to hire with six weeks until the start of school. And when I go on vacation in Maine, we go to a place with limited WiFi.<br /><br />Thanks to @mrchowmath I was able to begin to dabble in the CL in an exploratory way through an Activity Builder he created. He posted his videos on his website <a href="https://www.blogger.com/">here<span id="goog_367410068"></span></a>. (I am now also noticing that he has some "<a href="http://www.mrchowmath.com/breakout.html">Breakout Room</a>" activities that he created with Desmos AB, too. I'll definitely need to look at these this summer.)<br /><br />What did I learn from Jay Chow's videos? First, that the computational layer is not as scary as the syntax makes it look. Second, that the built in de-bugging suggestions are very helpful. Jay created a google sheet for people to share their "homework" and now I have about a week and a half to try out one of the following.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-babCkVnsjWo/WwwD4ZanyrI/AAAAAAAAHSQ/-1vv7V0R7MsBac-wLXq_pxUrQEflAfbYwCLcBGAs/s1600/Screen%2BShot%2B2018-05-28%2Bat%2B9.17.43%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="701" data-original-width="1027" height="218" src="https://4.bp.blogspot.com/-babCkVnsjWo/WwwD4ZanyrI/AAAAAAAAHSQ/-1vv7V0R7MsBac-wLXq_pxUrQEflAfbYwCLcBGAs/s320/Screen%2BShot%2B2018-05-28%2Bat%2B9.17.43%2BAM.png" width="320" /></a></div><br />I am thinking of trying #1 where a student enters a parabola in standard form and the vertex is reported back in a note along with the x-intercepts. It would be interesting to see if students notice that the x-coordinate of the vertex is the average of the x-intercepts. Right now my students don't notice this. It may be because I only have them focus on the ordered pairs or I only have them focus on the graph. Perhaps having these together will help them to see the relationship better.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-64419346774235877802018-05-25T08:44:00.001-04:002018-05-25T08:44:06.574-04:00Teach 180: Inverse Trig Functions (Day 168)Note: After flooding the lawnmower multiple times, doing ironing, making dinner and spending time with my daughter, I went to bed and realized that I had not written in my blog for the day. The events I am about to describe happened yesterday.<br /><br />Most of my day was spent copying final exams for PreCalculus and organizing files in my classroom. The one class I had today had only one student. (The other student in that class was at his sister's college graduation.) We spent about 45 minutes of class time reviewing inverse trig functions. We looked at the graph of f(x) = sin x and g(x) = sin<sup>-1</sup> x and realized that we would need to restrict f(x) so that it would be one-to-one. Desmos made this very easy to see and very easy to check the validity of our chosen restriction for the domain of f(x) = sin x. The graph of f(x) = sin x is in purple and the graph of its inverse is in black. The dotted line is the line y = x to show that f(x) and its inverse are reflections of each other over the line y = x. I added the points to the "ends" of the pieces to show that x and y coordinates are swapped when graphing a function and its inverse.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ops3VXmin5M/Wwf4-BDEEwI/AAAAAAAAHRs/EmR74aUN2zwsDN3RWGjc7JCKHmPBjBZKACLcBGAs/s1600/Screen%2BShot%2B2018-05-25%2Bat%2B7.46.11%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="571" data-original-width="858" height="212" src="https://2.bp.blogspot.com/-ops3VXmin5M/Wwf4-BDEEwI/AAAAAAAAHRs/EmR74aUN2zwsDN3RWGjc7JCKHmPBjBZKACLcBGAs/s320/Screen%2BShot%2B2018-05-25%2Bat%2B7.46.11%2BAM.png" width="320" /></a></div><br />What about f(x) = cos x and g(x) = cos<sup>-1</sup> x? Could we use the same restriction of -π/2 <u><</u> x <u><</u> π/2? We used Desmos to see what would happen if we did this.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Qtin0yBLDS4/WwgCeWjwcxI/AAAAAAAAHR4/_N-ECbvS5_sVai-l8yqwTlGO8vMaDSySQCLcBGAs/s1600/Screen%2BShot%2B2018-05-25%2Bat%2B7.55.12%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="441" data-original-width="637" height="221" src="https://4.bp.blogspot.com/-Qtin0yBLDS4/WwgCeWjwcxI/AAAAAAAAHR4/_N-ECbvS5_sVai-l8yqwTlGO8vMaDSySQCLcBGAs/s320/Screen%2BShot%2B2018-05-25%2Bat%2B7.55.12%2BAM.png" width="320" /></a></div><br />We could easily see that the restriction of -π/2 <u><</u> x <u><</u> π/2 was not correct. This was visible in two ways. First, the restricted cosine function did not pass the horizontal line test. Second, the red graph f(x) = cos x and the blue graph g(x) = cos<sup>-1</sup> x were not reflections of each other over the line y = x. We decided to change our restriction to 0 <u><</u> x <u><</u> π and it worked!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-YTiDVAXxJoE/WwgEOdKbF1I/AAAAAAAAHSE/Lv97AGOfRiQFW5YRQKyArAgUuB1hqhcAQCLcBGAs/s1600/Screen%2BShot%2B2018-05-25%2Bat%2B8.40.11%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="580" data-original-width="772" height="240" src="https://1.bp.blogspot.com/-YTiDVAXxJoE/WwgEOdKbF1I/AAAAAAAAHSE/Lv97AGOfRiQFW5YRQKyArAgUuB1hqhcAQCLcBGAs/s320/Screen%2BShot%2B2018-05-25%2Bat%2B8.40.11%2BAM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>I love how Desmos makes it easy to construct understanding and to play with the mathematics. If we had chosen to do this discover by hand or with a graphing calculator, we would have been bogged down by the computations or button pushing. Thank you Desmos for making discovery more accessible to everyone!<br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-44774980538890231062018-05-23T21:28:00.005-04:002018-05-23T21:28:58.728-04:00Teach 180: Hello, Anyone Out There? (Day 167)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-KCN-xVWbkEw/WwYPRBTCvCI/AAAAAAAAHQ4/kRtr9i_LGcwXfOiGQgus29nzDJQqqs-vQCLcBGAs/s1600/Screen%2BShot%2B2018-05-23%2Bat%2B9.02.11%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="252" data-original-width="821" height="121" src="https://4.bp.blogspot.com/-KCN-xVWbkEw/WwYPRBTCvCI/AAAAAAAAHQ4/kRtr9i_LGcwXfOiGQgus29nzDJQqqs-vQCLcBGAs/s400/Screen%2BShot%2B2018-05-23%2Bat%2B9.02.11%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: Times, Times New Roman, serif;">At the end of February, twenty faculty members signed up to be a member of the google classroom "Grading Smarter, Not Harder". I was excited by the possibility of sharing ideas with my colleagues around a topic that really interests me - assessment. My initial post said, "<span style="background-color: white; color: rgba(0, 0, 0, 0.87);">This is a virtual book group that is open to all upper school faculty. Discussion prompts will be based on the book "Grading Smarter, Not Harder" by Myron Dueck. Reading the book is not required to participate. Prompts will be posted weekly or bi-weekly." I have created prompts on each of the chapters for the past two months. However, </span>only nine posts have been made since the beginning of March and six of them were made by me. If a post happens in a discussion forum and no one responds to it, did the post happen? </span><br /><span style="font-family: Times, Times New Roman, serif;"><br /></span><span style="font-family: Times, Times New Roman, serif;">I know that my blog is essentially me talking to myself and I feel like that in the google classroom, too. But the purpose of a google classroom forum is to have interaction with other participants. Being a part of communities like #MTBoS, the Global Math Department and the 2018 AP Statistics Reader Facebook Group help me to grow as a <u>math</u> teacher. However, I want to have discussions about more general education topics that all teachers face. We don't have those conversations at lunch or in the faculty room. Discussions tend to focus on trying to understand the bell schedule, when grades are due, what to watch on Netflix and rating the school coffee. Conversations like this are a nice diversion. However, I want to learn from my math and non-math colleagues and grow in my profession. Setting up face-to-face meetings during lunch did not work last year (only 2 or 3 people attended). So, I thought this might work, because it would give teachers more flexibility with participation. But with 2/3 of the posts being written by me, it appears I am mostly talking to myself. Hello, anyone out there?</span><br /><span style="font-family: Times, Times New Roman, serif;"><br /></span><span style="font-family: Times, Times New Roman, serif;">Here is the post I did today about Chapter 4 - ReTesting. Maybe someone out there will read it and comment on it.</span><br /><span style="font-family: Times, Times New Roman, serif;"><br /></span><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-gPcueAbEayw/WwYUueqo56I/AAAAAAAAHRc/fjQUffAEx7Ybqhzhzj71R-boZtyZcUKawCLcBGAs/s1600/Screen%2BShot%2B2018-05-23%2Bat%2B9.25.52%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="403" data-original-width="755" height="341" src="https://4.bp.blogspot.com/-gPcueAbEayw/WwYUueqo56I/AAAAAAAAHRc/fjQUffAEx7Ybqhzhzj71R-boZtyZcUKawCLcBGAs/s640/Screen%2BShot%2B2018-05-23%2Bat%2B9.25.52%2BPM.png" width="640" /></a></div><span style="font-family: Times, Times New Roman, serif;"><br /></span><span style="font-family: Times, Times New Roman, serif;"><br /></span><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><span style="font-family: Times, Times New Roman, serif;"><br /></span>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-27516325868300075792018-05-22T16:47:00.000-04:002018-05-22T16:47:07.111-04:00Teach 180: Transforming Trig Graphs (Day 166)As students learn new concepts or skills, it is important for them to get immediate feedback. Saying "Just check your answer against the back of the book." is NOT feedback. Sure, a student can see if they got the right answer, but it doesn't let them know why they got it wrong. And if they get it wrong, the student really should have a chance to try it again.<br /><br />In Calculus today, we reviewed transforming graphs of trig functions. The students were able to get immediate feedback and targeted practice through DeltaMath. The short video below shows a little about how we used DeltaMath for this feedback and practice. (Note: It does not have any sound.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/kpwpL655CYU/0.jpg" src="https://www.youtube.com/embed/kpwpL655CYU?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Students worked for a full 30 minutes and completed about 25 problems each. I was able to help students, as they needed it, but students also were also able to figure out many of their errors for themselves.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-74842093175527128272018-05-21T15:15:00.001-04:002018-05-21T15:15:54.350-04:00Teach 180: The Tower of Hanoi (Day 165)<table><tbody><tr><td><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-UAM-7ABe2Co/WwMX9bDIi6I/AAAAAAAAHQs/hIn_Ta8e2KwZrWsaXRDKmPD4yrGSPyTIACEwYBhgL/s1600/Screen%2BShot%2B2018-05-21%2Bat%2B3.02.46%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="349" data-original-width="517" height="216" src="https://4.bp.blogspot.com/-UAM-7ABe2Co/WwMX9bDIi6I/AAAAAAAAHQs/hIn_Ta8e2KwZrWsaXRDKmPD4yrGSPyTIACEwYBhgL/s320/Screen%2BShot%2B2018-05-21%2Bat%2B3.02.46%2BPM.png" width="320" /></a></div><br /></td><td></td><td>There are certain math ideas or puzzles that I believe every student should experience. The Tower of Hanoi is one of those puzzles. After we reviewed the graphs of the six basic trig functions in Calculus today, we had some time to play with this puzzle. Our goal was to move the disks from the left peg to the right peg in as few moves as possible. The rule is that you can't place a larger disk on top of a smaller disk. Rather than starting with 7 disks, we started with 1 and then moved to 2 and then 3, looking for a pattern as we increased the number of disks. It only took about 10 minutes to see that the number of moves, M, based on the number of disks D, is given by the function M(D) = 2<sup>D</sup> - 1.<br /><br /></td></tr></tbody></table>If you are looking for an animation of the tower of Hanoi, there is a nice 2-D and 3-D simulation at <a href="http://towersofhanoi.info/Animate.aspx">Towers of Hanoi Animation</a>. I will probably show this tomorrow and we may even create <a href="http://www.jamesrahn.com/pages/geometry/pop_up_cards.htm">a pop-up card based on the towers of Hanoi</a>. <br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-87437802646473107792018-05-19T11:37:00.001-04:002018-05-19T11:37:22.379-04:00Teach 180: Dance Factor Dance (Day 164)We are continuing to look at patterns in PreCalculus. As students walked in the room today, I had the website <a href="http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/">www.datapointed.net/visualizations/math/factorization/animated-diagrams/</a> open. The short screencast here you through the first 30 seconds of what they saw. (Music is called "Sunglasses" courtesy of "Loyalty Freak Music".)<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/vJC9OKiZY8M/0.jpg" src="https://www.youtube.com/embed/vJC9OKiZY8M?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />I asked them "What's going on with the dots?" At first they noticed the pattern of colors. Something that I didn't even notice. But soon after that they noticed things like prime numbers forming a circle, factors of 3 forming triangles, factors of 4 forming squares, factors of 5 forming pentagons, etc. Although we weren't talking about factors explicitly today, it helped them to continue to think about patterns and was a nice way to get their brains thinking about math.<br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-28117235392073662202018-05-17T23:13:00.002-04:002018-05-17T23:13:50.710-04:00Teach 180: My Geeky Lightbulb Moment (Day 163)In PreCalculus today, we were doing some problems involving summation. One was the sum of n<sup>2 </sup> from n = 1 to 4. I knew Desmos could do this one and I showed my students how this worked in Desmos.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-UWk2_5R1wX8/Wv5BQZEwTdI/AAAAAAAAHPw/529UawjbsnsEqrZb4GAcCNSv_zErTycwwCLcBGAs/s1600/Screen%2BShot%2B2018-05-17%2Bat%2B10.45.26%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="180" data-original-width="418" height="136" src="https://3.bp.blogspot.com/-UWk2_5R1wX8/Wv5BQZEwTdI/AAAAAAAAHPw/529UawjbsnsEqrZb4GAcCNSv_zErTycwwCLcBGAs/s320/Screen%2BShot%2B2018-05-17%2Bat%2B10.45.26%2BPM.png" width="320" /></a></div><br />Then, we did the following "by hand".<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-KblwIRa6a_I/Wv5CGviGjyI/AAAAAAAAHQE/LxmDfKZ8nF4dEfyMbOMZeG4F89IA_o10wCLcBGAs/s1600/Screen%2BShot%2B2018-05-17%2Bat%2B11.01.18%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="250" data-original-width="318" height="156" src="https://2.bp.blogspot.com/-KblwIRa6a_I/Wv5CGviGjyI/AAAAAAAAHQE/LxmDfKZ8nF4dEfyMbOMZeG4F89IA_o10wCLcBGAs/s200/Screen%2BShot%2B2018-05-17%2Bat%2B11.01.18%2BPM.png" width="200" /></a></div><br />I asked my students, "what would Desomos do with this problem". I wasn't really sure. When I entered the expression, it didn't give me "an answer", it gave me a graph. "Why is it doing that?" I wondered.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-FDcUQv8AlnU/Wv5Cl082z0I/AAAAAAAAHQQ/wUpmzX31N9o5Fvm9rclxofyKkP8VhBh_gCLcBGAs/s1600/Screen%2BShot%2B2018-05-17%2Bat%2B11.03.17%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="475" data-original-width="735" height="257" src="https://4.bp.blogspot.com/-FDcUQv8AlnU/Wv5Cl082z0I/AAAAAAAAHQQ/wUpmzX31N9o5Fvm9rclxofyKkP8VhBh_gCLcBGAs/s400/Screen%2BShot%2B2018-05-17%2Bat%2B11.03.17%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">And then it dawned on me! Desmos was graphing the polynomial that resulted from the summation. We confirmed this as a class by entering the expression 1 + x + x<sup>2</sup>/2 + x<sup>3</sup>/6 + x<sup>4</sup>/24 in line 2 and it graphed over the graph that was already there. As I was thinking aloud and having my lightbulb moment, my students asked me, "Wait, you mean you haven't done this in Desmos before?" They thought I was acting surprised and had done this before. I do that sometimes. But not this time, my surprise was real. About an hour later, I shared my discovery with the colleagues in my math department. They were also impressed. This is definitely something that can't happen on a TI-whatever.</div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-21638000577114856802018-05-16T09:30:00.002-04:002018-05-16T09:30:57.699-04:00Teach 180: The Law of Sines Video (Day 162)Today I am reviewing Law of Sines with my two students in Calculus. As I was looking through my folder, I came across this video I created using my smartboard software and Animoto. Since I made the video back in 2010, it is not the best quality. However, it is a quick explanation as to why the Law of Sines work. Perhaps I should work on one for Law of Cosines to add to my YouTube collection.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/gJRaJxEZHrE/0.jpg" src="https://www.youtube.com/embed/gJRaJxEZHrE?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-52766092120513477632018-05-15T19:56:00.002-04:002018-05-15T19:56:18.977-04:00Teach 180: Playing with Patterns (Day 161)In PreCalculus today we began a short unit on sequence and series. Today students played around with about twelve different sequences to discover various patterns. Next we looked at a graph of the sequence 20, 17, 14, 11...in Desmos. When I asked students what they noticed, they said it followed a line. When I asked what the slope of the line would be, they could quickly see that it was -3 and that the slope made sense relative to the pattern of subtract 3 from the previous number. However when we graphed the line y = -3x, it didn't go through the points. How could we get the correct y-intercept? Natalie said that the y-intercept would be 23, because she used the pattern backwards - adding 3 to 20. Without formally talking about arithmetic sequences and the formal notation of an arithmetic sequence, students were able to create a formula to generate the nth term.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-FnVeTQs0KAQ/Wvtv9VRPwqI/AAAAAAAAHO8/7qQ5pWGoYC0HwC4AC-3OdSnfVpBvbDzSACLcBGAs/s1600/Screen%2BShot%2B2018-05-15%2Bat%2B7.39.48%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="612" data-original-width="894" height="273" src="https://1.bp.blogspot.com/-FnVeTQs0KAQ/Wvtv9VRPwqI/AAAAAAAAHO8/7qQ5pWGoYC0HwC4AC-3OdSnfVpBvbDzSACLcBGAs/s400/Screen%2BShot%2B2018-05-15%2Bat%2B7.39.48%2BPM.png" width="400" /></a></div><br />Next students worked in groups to generate formulas for 1, 5, 25, 125... and 1, 3, 7, 15, 31,... After graphing them, the students realized that the pattern was curved and that an exponential equation would work better than a linear equation. With a little bit of trial and error in Desmos, students discovered the equation y = 5<sup>x-1</sup> would work for the first sequence and y = 2<sup>x</sup>-1 would work for the second sequence. We didn't formally talk about geometric sequences today, but the fact that students are already thinking about a pattern found by multiplying by the same number again and again, means that they will be ready for the formal introduction of a common ratio next week.<br /><br />Tomorrow we will begin class by looking at some patterns found at <a href="http://www.visualpatterns.org/">Visual Patterns by Fawn Nguyen</a>. Here are two that I plan to use. Can you figure out how many stars and footballs are in step 43?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ab7-g8D0YI0/WvtzKOHg5-I/AAAAAAAAHPU/BDdmP8MjpCocPikY85FSxq2QrhOMrF6MgCLcBGAs/s1600/Screen%2BShot%2B2018-05-15%2Bat%2B7.47.45%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="175" data-original-width="570" height="98" src="https://2.bp.blogspot.com/-ab7-g8D0YI0/WvtzKOHg5-I/AAAAAAAAHPU/BDdmP8MjpCocPikY85FSxq2QrhOMrF6MgCLcBGAs/s320/Screen%2BShot%2B2018-05-15%2Bat%2B7.47.45%2BPM.png" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-if-t3eTlMgs/WvtzKLBg4qI/AAAAAAAAHPQ/CwFEI6k-UmoxzO31mNoQLrjXHGu6ti7lwCLcBGAs/s1600/Screen%2BShot%2B2018-05-15%2Bat%2B7.53.28%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="167" data-original-width="473" height="112" src="https://3.bp.blogspot.com/-if-t3eTlMgs/WvtzKLBg4qI/AAAAAAAAHPQ/CwFEI6k-UmoxzO31mNoQLrjXHGu6ti7lwCLcBGAs/s320/Screen%2BShot%2B2018-05-15%2Bat%2B7.53.28%2BPM.png" width="320" /></a></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-58980228058070512522018-05-14T13:13:00.000-04:002018-05-14T13:13:16.695-04:00Teach 180: Happy Monday (Day 160)Luckily I did not have too many copies to make this morning and could just print the 12 pages or so on the printer. Why could I not make copies?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-TtlsBwfpb_E/WvnDQrUkvqI/AAAAAAAAHOc/8Kcbd34okc8OVD4lP4XdcgiDRoC6-g-DwCLcBGAs/s1600/Screen%2BShot%2B2018-05-14%2Bat%2B1.11.02%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="443" data-original-width="486" height="291" src="https://3.bp.blogspot.com/-TtlsBwfpb_E/WvnDQrUkvqI/AAAAAAAAHOc/8Kcbd34okc8OVD4lP4XdcgiDRoC6-g-DwCLcBGAs/s320/Screen%2BShot%2B2018-05-14%2Bat%2B1.11.02%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br />Thankfully I only have I have two classes scheduled today - first period and last period. For the first period class, I only had one Calculus student. The other student is taking the AP Bio exam. The seniors in Calculus are done with classes and don't need to attend class. What did I do with just one student? We reviewed the unit circle together and then the student worked on an assignment in DeltaMath.<br /><br />For my last period class, the students are not required to come to class either. They are all seniors. In fact, some of them may not be able to come to class, because they have a senior exam scheduled. However, I am hoping that several students show up to ask some more last minute questions before the AP Exam on Thursday. Time will tell. Happy Monday, everyone!!Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-54308607556332896932018-05-13T15:55:00.001-04:002018-05-13T15:55:23.940-04:00Teach 180: The AP Statistics Murder Mystery (Day 159)Friday was the last official day of classes for seniors and I felt that a fun review day for AP Statistics was needed. Although I had seen the AP Statistics Murder Mystery in the AP Statistics Reader Facebook group, I wasn't sure if I would do it. Then, I read several posts reporting how much fun students had with the activity. This lead me to read the questions and I concluded that it was a pretty comprehensive review in a format that the kids would find fun. I created envelopes for each of the six worksheets. Each worksheet led to either a weapon, location or suspect being eliminated.<br /><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-WAAUMHU392U/WviWTtccGUI/AAAAAAAAHOI/obgk8hSdPEINb8YQ_f5SRlu2vGwHG_lvACLcBGAs/s1600/IMG_5808.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://1.bp.blogspot.com/-WAAUMHU392U/WviWTtccGUI/AAAAAAAAHOI/obgk8hSdPEINb8YQ_f5SRlu2vGwHG_lvACLcBGAs/s320/IMG_5808.JPG" width="240" /></a></div><div><br /><div>Students worked in groups of 3 - 4 to solve the mystery. My first class took about 40 minutes to complete the activity and my second class whipped through it in about 30 minutes! Not only did they review many of the AP Statistics topics, students reviewed some of the features on their calculator, too. As always, there will be a few things I will tweak in future years with this activity, specifically the directions on a few of the handouts required some clarification. A special thank you to Celia Rowland for sharing this activity.</div></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-85723899923306392342018-05-10T21:02:00.003-04:002018-05-10T21:02:57.272-04:00Teach 180: Marbleslide Challenges (Day 158)Today was my last day for seniors in Calculus and I wanted them to spend their final day of math in high school just playing with mathematics. Thanks to Sean Sweeny's Marbleslide Challenge in Desmos Activity Builder we did just that! One pair of girls completed eleven of the challenges in about an hour. The short video below shows the one that I think they were most proud of.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/muRy1SNY13E/0.jpg" src="https://www.youtube.com/embed/muRy1SNY13E?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Next year I am thinking of doing this as <a href="http://sweeneymath.blogspot.com/2017/08/a-full-year-set-of-36-weekly.html">Sean describes in his blogpost</a>, using teacher pacing and releasing one Marbleslides challenge a week. However, I may want to reorder the slides for this activity builder based on what functions we are studying in class. Looks like it might be time to start my summer list of "Ideas for School This Fall".</div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-13063575510151145262018-05-09T22:16:00.000-04:002018-05-09T22:16:36.235-04:00Teach 180: The Five Pointed Star (Day 157)The seniors last day of classes is this Friday and we essentially finished the content in Calculus last Friday. With only 4 classes remaining, I spent the last two days showing students the documentary called "<a href="http://www.pbs.org/independentlens/between-the-folds/">Between the Folds</a>". It does a great job looking at how origami blends with math and science. It also shows that math can be learned for the sake of learning math and that applications of the math often come later. Here is the trailer for the video:<div><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/tE4lqYzS2m0/0.jpg" src="https://www.youtube.com/embed/tE4lqYzS2m0?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">In the video mathematics professor Eric Dermaine shows that paper can be folded and one cut made to make various shapes. In the video, Eric folds a piece of paper and makes one cut to reveal a swan. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">After the video, I worked with students to fold paper and make one cut to form a five pointed star. The same fold and cut trick that Betsy Ross supposedly showed George Washington when the American flag was being designed. <a href="http://www.ushistory.org/betsy/flagstar.html">Here are the directions we used</a>. My five pointed star is seen below. We also explored mobius strips and what happens when you cut a mobius strip 1/3 from the edge of the strip and in the center of the strip. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-wuVplOyNgCI/WvOpffxYdEI/AAAAAAAAHN4/GZdQN9bDKWktZujA0t_XWl6RQbaXJDR2wCLcBGAs/s1600/IMG_5806.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://2.bp.blogspot.com/-wuVplOyNgCI/WvOpffxYdEI/AAAAAAAAHN4/GZdQN9bDKWktZujA0t_XWl6RQbaXJDR2wCLcBGAs/s320/IMG_5806.JPG" width="240" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-44162619702674863242018-05-08T18:37:00.001-04:002018-05-08T18:37:37.841-04:00Teach 180: What's Going on in This Graph? (Day 156)Today I let my students experience the new New York Times feature called <a href="https://www.nytimes.com/interactive/2018/05/03/learning/08WGOITGraphLN.html?rref=collection%2Fcolumn%2Fwhats-going-on-in-this-graph&action=click&contentCollection=learning&region=stream&module=stream_unit&version=latest&contentPlacement=1&pgtype=collection">What's going on in this graph?</a> The students worked in groups to study the graph and answered the following three questions. 1) What do you notice? 2) What do you wonder? and 3) What might be going on in this graph? After they discussed it together, we created a class post in the comment section. Here are two screenshots of the graph, as it changed over time.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-SaMdyfDfVnU/WvIkhyDPHvI/AAAAAAAAHNI/tXIO9AXSX2cMUeCoytVC2d_5OUnQYY4pwCLcBGAs/s1600/Screen%2BShot%2B2018-05-08%2Bat%2B11.21.18%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="414" data-original-width="1064" height="248" src="https://4.bp.blogspot.com/-SaMdyfDfVnU/WvIkhyDPHvI/AAAAAAAAHNI/tXIO9AXSX2cMUeCoytVC2d_5OUnQYY4pwCLcBGAs/s640/Screen%2BShot%2B2018-05-08%2Bat%2B11.21.18%2BAM.png" width="640" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Va2QPv1oQtw/WvIkh-pmifI/AAAAAAAAHNM/kHj8ulIY0Z86jTTaoHw8Qh8piGflV9H2wCLcBGAs/s1600/Screen%2BShot%2B2018-05-08%2Bat%2B11.21.24%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="418" data-original-width="1030" height="257" src="https://1.bp.blogspot.com/-Va2QPv1oQtw/WvIkh-pmifI/AAAAAAAAHNM/kHj8ulIY0Z86jTTaoHw8Qh8piGflV9H2wCLcBGAs/s640/Screen%2BShot%2B2018-05-08%2Bat%2B11.21.24%2BAM.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Here is what we composed as a class response to the three questions. Our observations received a response from the moderator and a response from someone else. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ZZo2lDpE0Lk/WvIlA-1A3vI/AAAAAAAAHNg/mutGfObYKpopl14HUObH3FIlnZg6D_PMgCLcBGAs/s1600/Screen%2BShot%2B2018-05-08%2Bat%2B6.25.44%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="500" data-original-width="796" height="401" src="https://2.bp.blogspot.com/-ZZo2lDpE0Lk/WvIlA-1A3vI/AAAAAAAAHNg/mutGfObYKpopl14HUObH3FIlnZg6D_PMgCLcBGAs/s640/Screen%2BShot%2B2018-05-08%2Bat%2B6.25.44%2BPM.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-49_wOFXtjps/WvIk5d8ypyI/AAAAAAAAHNc/3adZOp8X9boVJqtOkYqO0WBYfrtBbUlmgCLcBGAs/s1600/Screen%2BShot%2B2018-05-08%2Bat%2B6.25.57%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="413" data-original-width="794" height="332" src="https://1.bp.blogspot.com/-49_wOFXtjps/WvIk5d8ypyI/AAAAAAAAHNc/3adZOp8X9boVJqtOkYqO0WBYfrtBbUlmgCLcBGAs/s640/Screen%2BShot%2B2018-05-08%2Bat%2B6.25.57%2BPM.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-FE3oxk4gCy0/WvIk5Y54FYI/AAAAAAAAHNY/m--9iG_3wWYIYmo68CTpv60DiC7wB3pNwCLcBGAs/s1600/Screen%2BShot%2B2018-05-08%2Bat%2B6.26.13%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="149" data-original-width="792" height="120" src="https://1.bp.blogspot.com/-FE3oxk4gCy0/WvIk5Y54FYI/AAAAAAAAHNY/m--9iG_3wWYIYmo68CTpv60DiC7wB3pNwCLcBGAs/s640/Screen%2BShot%2B2018-05-08%2Bat%2B6.26.13%2BPM.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">I'll be definitely taking these comments back to my students. Once we look at the article at the bottom of the page, it will probably help students understand why including something, or leaving something off, a graph can help the graph tell a story.</div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-27259209829385070152018-05-07T22:45:00.000-04:002018-05-07T22:45:42.419-04:00Teach 180: Which Inference Procedure? (Day 155)Today we decided to mix it up a bit in AP Stat by using a Kahoot! to review which inference procedure should be done. Thanks to Kahoot user david.kish for giving me something to use/modify. After this activity, several students recognized that they may need to do a little work in this area. So, I sent them this <a href="http://www.ltcconline.net/greenl/java/Statistics/catStatProb/categorizingStatProblems.html">link in an email</a>. Notice the variety of concepts that students can review!!<br /><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-7ySrkmHQMPQ/WvCVX-13m-I/AAAAAAAAHM0/l4pCFtPPtTgigKCOjssvcSwBNEMu-lm7QCLcBGAs/s1600/Screen%2BShot%2B2018-05-07%2Bat%2B12.44.15%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="632" data-original-width="525" height="640" src="https://3.bp.blogspot.com/-7ySrkmHQMPQ/WvCVX-13m-I/AAAAAAAAHM0/l4pCFtPPtTgigKCOjssvcSwBNEMu-lm7QCLcBGAs/s640/Screen%2BShot%2B2018-05-07%2Bat%2B12.44.15%2BPM.png" width="529" /></a></div><div><div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">As we continue to review for the AP exam, I think it is important for my students to figure out what they don't know well. I give them way more practice problems and resources than they need. The idea is that they need to start to target their areas of weakness.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Today we also started an 8 day take-home final exam. The students are getting one question sent to them at 8 AM for the next 8 school days. Their solution is due at 8 AM on the following school day, at which time a new question and the solution to the previous question is released. It is about 11 PM and already about half of the students have turned in their work. In fact, several sent me their work before leaving school for the day. Be sure to read a future blog about this form of assessment and how I think it impacted my results on the AP Statistics Exam.</div></div></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0