tag:blogger.com,1999:blog-47847168147972780122018-03-20T00:11:28.003-04:00MATHEMATICAL 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.comBlogger155125tag:blogger.com,1999:blog-4784716814797278012.post-44877683838094025242018-03-11T22:40:00.000-04:002018-03-11T22:40:04.670-04:00Teach 180: I Want Proof (Day 121)A few days ago I was talking to a colleague about the fact that her Honors Geometry students weren't as curious this year. When she asked the students if they wanted to know why something was true or why it made sense, most of them said "No". These same students want their math teachers to tell them what strategy they should use or what the assignment is so they can do it and get it over with. These students say, "But how are we supposed to know how to do that problem on the test. We didn't have any like it on the homework."<br /><br />When I was in high school, I always wanted to know why something was true and my teachers were very good out outlining steps to a proof or leading us in reasoning why something was true. Just because some of my students don't share the same curiosity I did doesn't mean that I shouldn't expose them to mathematical arguments. In fact, one of the <a href="http://www.corestandards.org/Math/Practice/" target="_blank">Common Core State Standard for Mathematical Practices</a> states<span style="font-family: Times, Times New Roman, serif;"> that "Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments." In addition, "Students at all grades can listen to or read arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments." </span><br /><span style="color: #202020; font-family: "lato light"; font-size: 16.8px;"><br /></span>Most of my Calculus students are seniors and simply want to be told formulas for the derivative. Few of them care to understand why the derivatives of f(x) = a<sup>x</sup> and f(x) = log<sub>a</sub>x are what they are. The fact that they don't care, doesn't mean that I should not share the argument with them. In class today, I presented the following argument to show that the derivative of f(x) = a<sup>x</sup> is (ln a)(a<sup>x</sup>). Enjoy!<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/GeFt4UUb8vo/0.jpg" src="https://www.youtube.com/embed/GeFt4UUb8vo?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-46400589381733949962018-03-08T19:10:00.003-05:002018-03-08T19:10:23.155-05:00Teach 180: Project Based Learning (Day 120)At my current school, we meet about five times a year (approximately an hour each time) in cross-divisional groups to discuss something we ranked as being of interest to us. Unfortunately, I was one of several people who did not have one of his or her top 3 choices in the list of options. This meant that I had to choose among the following options: project-based learning, social emotional learning, intrinsic motivation, well-being or integrating play in learning. I was really interested in focusing on assessment as a tool for learning and the closest group that was related to assessment was Project-Based Learning. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-sx4VQGvr4Iw/WqHN73Y22FI/AAAAAAAAGjs/TM5qucWS9-gWh9qXefF2bAtV10i-3cfrwCLcBGAs/s1600/Screen%2BShot%2B2018-03-08%2Bat%2B6.51.17%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="407" data-original-width="287" height="320" src="https://3.bp.blogspot.com/-sx4VQGvr4Iw/WqHN73Y22FI/AAAAAAAAGjs/TM5qucWS9-gWh9qXefF2bAtV10i-3cfrwCLcBGAs/s320/Screen%2BShot%2B2018-03-08%2Bat%2B6.51.17%2BPM.png" width="225" /></a></div>My group consisted of a middle school math teacher, a middle school history teacher, a middle school English teacher, an upper school Spanish teacher and myself. We read the book "Setting the Standard for Project Based Learning". The practical advice offered was mainly to not rush, make sure all stakeholders buy-in and to allow for plenty of time for collaboration. Little was offered in the way of assessing projects or assessing group skills. The one project that was specifically given as an example for math would have required about two weeks to complete and it covered a topic that we typically spend two days on in class. Although the book said that PBL is better for students, especially for learning collaboration skills and for at-risk students, at no point did it offer any empirical evidence. All the evidence that was presented was anecdotal. At no point did it offer suggestions for what content should be removed from the high school mathematics curriculum to make time for PBL.<br /><br />Today was the last day our group met and we put our "book report" in google slides to share with our colleagues. Here was the slide I created.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-vOEM84L6eEE/WqHPOgZV-bI/AAAAAAAAGj4/MUXi24_QtEcjDUVOaK5jR6V-W3HZVoYewCLcBGAs/s1600/Screen%2BShot%2B2018-03-08%2Bat%2B7.02.19%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="393" data-original-width="844" height="185" src="https://2.bp.blogspot.com/-vOEM84L6eEE/WqHPOgZV-bI/AAAAAAAAGj4/MUXi24_QtEcjDUVOaK5jR6V-W3HZVoYewCLcBGAs/s400/Screen%2BShot%2B2018-03-08%2Bat%2B7.02.19%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;">Overall, our group was not impressed with this book and felt like it was a sales pitch for the Buck Institute for Education, one of the two main publishers of the book. Would PBL work in a large scale at my current school? With so many different initiatives happening and faculty being drained of time and energy, I would say it would be unwise to attempt a PBL initiative at this time.</div><br /><br /><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-61508686838801442532018-03-07T08:08:00.003-05:002018-03-07T08:08:32.249-05:00Teach 180: SolveMe Mobiles (Day 119)It's not a teaching day today, but I will still spend about 4 or 5 hours at home today working on school related things. Next week is spring break and I would love to spend a minimal amount of time on school related items over spring break. (I'd rather spend my spring break spoiling my niece and nephew.)<br /><br />One of the things I'll be doing today is playing at the site <a href="https://solveme.edc.org/Mobiles.html" target="_blank">SolveMe Mobiles</a>. Your progress can be saved and you can create your own puzzles. Here is puzzle #15 from the explorer collection and puzzle #71 from the puzzler collection.<br /><br /><table><tbody><tr><td><a href="https://1.bp.blogspot.com/-qPz67cdtnR4/Wp_fFjF0HUI/AAAAAAAAGig/tJGeb40zvdoiPSc6zU3gTBPLyqklGP7gQCEwYBhgL/s1600/Screen%2BShot%2B2018-03-07%2Bat%2B7.36.26%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="362" data-original-width="420" height="170" src="https://1.bp.blogspot.com/-qPz67cdtnR4/Wp_fFjF0HUI/AAAAAAAAGig/tJGeb40zvdoiPSc6zU3gTBPLyqklGP7gQCEwYBhgL/s200/Screen%2BShot%2B2018-03-07%2Bat%2B7.36.26%2BAM.png" width="200" /></a></td> <td><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-7A5RyQe-PyM/Wp_fGTmnhsI/AAAAAAAAGis/gQj7h7dxDvAUyKorNYoSDcwE4qKTYmdSQCEwYBhgL/s1600/Screen%2BShot%2B2018-03-07%2Bat%2B7.41.10%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="428" data-original-width="575" height="236" src="https://3.bp.blogspot.com/-7A5RyQe-PyM/Wp_fGTmnhsI/AAAAAAAAGis/gQj7h7dxDvAUyKorNYoSDcwE4qKTYmdSQCEwYBhgL/s320/Screen%2BShot%2B2018-03-07%2Bat%2B7.41.10%2BAM.png" width="320" /></a></div></td></tr></tbody></table><br />We know that to keep the mobile balanced what is on the left must equal what is on the right. As a matter of fact, if you click on the horizontal bars and drag them, you get systems of equations. And then if you "pull down" with your mouse on the right heart, it subtracts one heart from both sides of the equation.<br /><br /><table><tbody><tr><td><a href="https://1.bp.blogspot.com/-N3h-kAesT0Q/Wp_hWbB2jXI/AAAAAAAAGjM/veCCmT65_6MxohVDBDq7BXn1jzdn5NBTgCLcBGAs/s1600/Screen%2BShot%2B2018-03-07%2Bat%2B7.52.40%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="422" data-original-width="802" height="168" src="https://1.bp.blogspot.com/-N3h-kAesT0Q/Wp_hWbB2jXI/AAAAAAAAGjM/veCCmT65_6MxohVDBDq7BXn1jzdn5NBTgCLcBGAs/s320/Screen%2BShot%2B2018-03-07%2Bat%2B7.52.40%2BAM.png" width="320" /></a> </td> <td><a href="https://4.bp.blogspot.com/-zIIxkinKcFE/Wp_hWAE3i5I/AAAAAAAAGjI/gSxBoYv3fkUSoZu7OFdG9rge3sdkbd1qACLcBGAs/s1600/Screen%2BShot%2B2018-03-07%2Bat%2B7.52.47%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="408" data-original-width="760" height="168" src="https://4.bp.blogspot.com/-zIIxkinKcFE/Wp_hWAE3i5I/AAAAAAAAGjI/gSxBoYv3fkUSoZu7OFdG9rge3sdkbd1qACLcBGAs/s320/Screen%2BShot%2B2018-03-07%2Bat%2B7.52.47%2BAM.png" width="320" /></a><br /><br /> </td></tr></tbody></table>Finally, when you enter the values for each shape, you get instant feedback about your solution. By the way, the answer to this one is not trapezoid = 3, heart = 1 and square = 5.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-wmOM4pXToJQ/Wp_hWd4wynI/AAAAAAAAGjc/uhi5VcI5PQAIKXjNfb078bReBz5CDNGpQCEwYBhgL/s1600/Screen%2BShot%2B2018-03-07%2Bat%2B7.54.53%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="406" data-original-width="512" height="253" src="https://1.bp.blogspot.com/-wmOM4pXToJQ/Wp_hWd4wynI/AAAAAAAAGjc/uhi5VcI5PQAIKXjNfb078bReBz5CDNGpQCEwYBhgL/s320/Screen%2BShot%2B2018-03-07%2Bat%2B7.54.53%2BAM.png" width="320" /></a></div><br />Thanks to Kevin Smith for reminding me about this resource at his Global Math Department session last night. His session was called "Gamify the Math Classroom" and you can view the video of the webinar <a href="https://www.bigmarker.com/GlobalMathDept/Gamify-the-Math-Classroom" target="_blank">here</a>. Note that if you are not a member of the Global Math Department, you will be prompted to set up a free account. However, when you do that you'll be able to see many of the other wonderful free webinars that have been recorded over the past several years. I serve as host for many GMD sessions and we are always looking for quality presenters. If you are interested in presenting, send me an email at leighnataro@gmail.com.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-18716847051898326152018-03-06T18:48:00.001-05:002018-03-06T18:48:16.314-05:00Teach 180: I Call Time Out!!! (Day 118)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-5arABebdRlY/Wp8gWapJrNI/AAAAAAAAGiM/TSiSY1uPE8o4MUl9W1VJlYEBxlYWR6n_ACLcBGAs/s1600/Screen%2BShot%2B2018-03-06%2Bat%2B6.11.43%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="479" data-original-width="642" height="147" src="https://4.bp.blogspot.com/-5arABebdRlY/Wp8gWapJrNI/AAAAAAAAGiM/TSiSY1uPE8o4MUl9W1VJlYEBxlYWR6n_ACLcBGAs/s200/Screen%2BShot%2B2018-03-06%2Bat%2B6.11.43%2BPM.png" width="200" /></a></div>Thanks to winter storm Riley the Friday evening performance of the musical had to be moved to Sunday evening. Students involved in the musical were allowed to miss school on Monday. (They were at school from about 9 AM to midnight, or later, on Sunday.) This meant that I missed 43% of my AP Statistics class on Monday. Initially we were going to review for a test on Chapter 9. It is an AP course and students who are absent are expected to learn things on their own when they are absent. But the students who missed class would have no clue what I was talking about. The class they missed was on one-sample t-tests for means <u>and</u> paired t-tests.<div><br /></div><div>Time out!!! Change of plans. I dismissed the 57% who had been in class on Monday and sent them to lunch. Then, I taught the lesson on one sample t-tests to the 43% that had missed class. When we reconvened from lunch all students had time to work on the problem set on one sample inference for means. I am fortunate that I work in a school that allows me the flexibility to work with the students in this way. However, with more snow in the forecast, I am nervous about students learning all the content needed before AP exams start. Right now, we have 5 days on the schedule to review for the exam. This is about half or one-third of the time I have had in previous years.</div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-62988590683619457792018-03-06T11:46:00.001-05:002018-03-06T11:46:03.031-05:00Teach 180: A Sequence of Logs (Day 117)Rarely do I have time to do something extra in my classes and that is especially the case this year. Even though we are about 2 weeks behind where we were last year, my class of PreCalculus students is pretty hardworking and inquisitive. They are ahead of the other PreCalculus classes and it looks like I'll end up doing something fun after spring break to allow the other PreCalculus classes time catch up.<br /><br />Today I was starting to purge my files at school and I ran across this gem of a question. I have some thoughts about how I will teach it. My students aren't familiar with the concepts of geometric sequences and arithmetic sequences, but they do understand exponential and linear functions. I'll have plenty of time between now and March 19th (the day back from spring break) to design a lesson around this question. One of my goals will be for the language of sequences (terms, initial term, geometric, arithmetic, common ratio and common difference) to develop naturally as a consequence of playing with this problem. Any comments or thoughts on this welcomed!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ohvTcoXjCWY/Wp7EmDa63fI/AAAAAAAAGh8/eiOZk739trUs7diOMRb8Xxf2K-TRxPoAACLcBGAs/s1600/Screen%2BShot%2B2018-03-06%2Bat%2B11.39.42%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="625" data-original-width="1343" height="297" src="https://4.bp.blogspot.com/-ohvTcoXjCWY/Wp7EmDa63fI/AAAAAAAAGh8/eiOZk739trUs7diOMRb8Xxf2K-TRxPoAACLcBGAs/s640/Screen%2BShot%2B2018-03-06%2Bat%2B11.39.42%2BAM.png" width="640" /></a></div><br /><br /><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-9281070327022781622018-03-02T13:28:00.000-05:002018-03-02T13:28:12.658-05:00Teach 180: You Fool. That's Wrong (Day 116)It's currently 1 PM and I am writing this blog entry from the comfort of my dining room table thanks to Winter Storm Riley. Assuming we don't lose power, I'll also be able to get my laundry done, clean the bathrooms and vacuum before 4 PM. I only taught the first two periods today and those classes met. This is very good news. It means I won't be having to revise my schedule and shuffle things around for classes that did not meet. This winter has been very harsh and it has made it challenging to have continuity. "Whenever it was we last met" has been a common phrase coming from my mouth this year.<br /><br />Today I showed the "<a href="https://www.youtube.com/watch?v=RN6qa3PxGU0" target="_blank">You Fool. That's Wrong.</a>" video in AP Statistics. If you aren't familiar with it and you teach AP Statistics, you should be. Although we have only been doing hypothesis testing for about a week, I showed the video to my students today and it was met with some chuckles. One student even said, "Now I finally understand p-value!" So, thanks to Steve Willott for bringing levity to my classroom and helping one of my students today. (Also, <a href="http://stevewillott.com/10-11%20ap%20stats%20notes%20for%20website/AP%20Stats%20hypothesis%20test%20mistakes.pdf" target="_blank">here is a link to a handout</a> that is related to the scenario in the video.)<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/RN6qa3PxGU0/0.jpg" src="https://www.youtube.com/embed/RN6qa3PxGU0?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-6519174124572507292018-03-01T22:59:00.001-05:002018-03-01T22:59:42.638-05:00Teach 180: Is the Coin Fair? (Day 115)<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-4HHa_pBXKRg/WpjH6bSAiwI/AAAAAAAAGhE/bMfWrdp-LsI-mg9_zOV4j91r3Ll26dEYwCEwYBhgL/s1600/Screen%2BShot%2B2018-03-01%2Bat%2B10.37.32%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="159" data-original-width="220" src="https://1.bp.blogspot.com/-4HHa_pBXKRg/WpjH6bSAiwI/AAAAAAAAGhE/bMfWrdp-LsI-mg9_zOV4j91r3Ll26dEYwCEwYBhgL/s1600/Screen%2BShot%2B2018-03-01%2Bat%2B10.37.32%2BPM.png" /></a></div>Although students can spout off what Type I and Type II errors are for hypothesis testing, they often have trouble understanding what would lead to making this type of error. Today I gave them information on two coins. Students had to determine if the coin was fair or not. The null hypothesis corresponded to the fair coin and the alternative hypothesis corresponded to the unfair coin. What would you conclude about Coin 1 - fair or unfair? What would you conclude about Coin 2 - fair or unfair?<br /><span style="text-align: center;"><br /></span><span style="text-align: center;"><br /></span><span style="text-align: center;"></span><br /><center><span style="text-align: center;"><b>FLIPPING RESULTS FOR COIN 1</b></span></center><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-JaZcIHo7w3o/WpjJnziVJyI/AAAAAAAAGhg/hwSBuXJxY9kPNA4rwFZXmkohRdcH1YO7QCEwYBhgL/s1600/Screen%2BShot%2B2018-03-01%2Bat%2B10.46.41%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="548" data-original-width="415" height="640" src="https://1.bp.blogspot.com/-JaZcIHo7w3o/WpjJnziVJyI/AAAAAAAAGhg/hwSBuXJxY9kPNA4rwFZXmkohRdcH1YO7QCEwYBhgL/s640/Screen%2BShot%2B2018-03-01%2Bat%2B10.46.41%2BPM.png" width="484" /></a></div><span style="text-align: center;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="text-align: center;"></span><br /><center><span style="text-align: center;"><b>FLIPPING RESULTS FOR COIN 2</b></span></center><center><span style="text-align: center;"><br /></span></center><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-YXdI1x404Fw/WpjJn5mDXOI/AAAAAAAAGho/wLdiEtEWX1ANP58AlUqU4IsrcMZMA17MACEwYBhgL/s1600/Screen%2BShot%2B2018-03-01%2Bat%2B10.47.17%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="514" data-original-width="398" height="640" src="https://3.bp.blogspot.com/-YXdI1x404Fw/WpjJn5mDXOI/AAAAAAAAGho/wLdiEtEWX1ANP58AlUqU4IsrcMZMA17MACEwYBhgL/s640/Screen%2BShot%2B2018-03-01%2Bat%2B10.47.17%2BPM.png" width="494" /></a></div><center><br /></center>I had students discuss this in groups and one group of students couldn't decide. They thought both coins were fair. So, I asked them to look at the 95% confidence intervals. If they could only choose one coin to use for the NFL Superbowl coin toss, which one would it be? They said Coin 2. The confidence interval contains .50 and the proportion of heads is closer to 50% than the other coin.<br /><br />The big reveal came when they flipped the papers over. Coin 1 was actually the fair coin and Coin 2 was actually the unfair coin. With Coin 1 they had made a Type I error and with Coin 2 they had made a Type 2 error. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-H-gake2vTNc/WpjJn3jeEKI/AAAAAAAAGhk/ThNnuYnYrGgzg8HDIjpgNJlD7iF54btaACEwYBhgL/s1600/Screen%2BShot%2B2018-03-01%2Bat%2B10.47.08%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="374" data-original-width="381" height="392" src="https://1.bp.blogspot.com/-H-gake2vTNc/WpjJn3jeEKI/AAAAAAAAGhk/ThNnuYnYrGgzg8HDIjpgNJlD7iF54btaACEwYBhgL/s400/Screen%2BShot%2B2018-03-01%2Bat%2B10.47.08%2BPM.png" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-lgub1D_MpYw/WpjJn33WT3I/AAAAAAAAGhs/Ls32n46Amvsu27GjD2VX0Wr3vd4lC2CPgCEwYBhgL/s1600/Screen%2BShot%2B2018-03-01%2Bat%2B10.47.39%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="395" data-original-width="478" height="330" src="https://4.bp.blogspot.com/-lgub1D_MpYw/WpjJn33WT3I/AAAAAAAAGhs/Ls32n46Amvsu27GjD2VX0Wr3vd4lC2CPgCEwYBhgL/s400/Screen%2BShot%2B2018-03-01%2Bat%2B10.47.39%2BPM.png" width="400" /></a></div><br />We'll see if this helps student understanding when they have their test next Friday - the day before Spring Break! (NOTE: I created this simulation in Fathom and kept re-randomizing until I made errors for both the fair and the unfair coin. If you would like a copy of this Fathom file, let me know in the comments below or send me an email to leighnataro@gmail.com.)Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-36602695367388548302018-02-28T12:48:00.001-05:002018-02-28T12:48:35.449-05:00Teach 180: The Golden Ratio (Day 114)<div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-ZN7F6v0EDWs/Wpbo2CpIy0I/AAAAAAAAGg4/YOuHPauOU_c-ADXJijofhcFoc3H0rZSmACEwYBhgL/s1600/Screen%2BShot%2B2018-02-28%2Bat%2B12.36.07%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="612" data-original-width="1023" height="238" src="https://1.bp.blogspot.com/-ZN7F6v0EDWs/Wpbo2CpIy0I/AAAAAAAAGg4/YOuHPauOU_c-ADXJijofhcFoc3H0rZSmACEwYBhgL/s400/Screen%2BShot%2B2018-02-28%2Bat%2B12.36.07%2BPM.png" width="400" /></a></div>Today I gave a chapel talk in the middle school called "The Golden Ratio: A Common Pattern". The bigger theme for the chapel was to look for common things we share with others rather than focusing on the differences. My talk focused on generating the Golden Ratio from the Fibonacci sequence. I then showed how the golden ratio is found in architecture, art, insects and the human body. Did you know that the ratio from the tip of the index finger to the wrist and the wrist to the elbow is in the golden ratio? Or that the width of the ear to the height of the ear is in the golden ratio? In fact, the ratio of the length of a minor groove to a major groove in DNA is the golden ratio!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Pk3CtWlWhyU/Wpbo3CzwuyI/AAAAAAAAGg0/7chOVy_jP2ghOKX2hsehBpJZKKLM9Q6CwCEwYBhgL/s1600/Screen%2BShot%2B2018-02-28%2Bat%2B12.36.38%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="777" height="99" src="https://3.bp.blogspot.com/-Pk3CtWlWhyU/Wpbo3CzwuyI/AAAAAAAAGg0/7chOVy_jP2ghOKX2hsehBpJZKKLM9Q6CwCEwYBhgL/s320/Screen%2BShot%2B2018-02-28%2Bat%2B12.36.38%2BPM.png" width="320" /></a></div><br />So, this ratio is seen many places. But is it really that common? Take any two integers and form a Fibonacci-like sequence from them. As you find ratios of consecutive terms, it approaches the golden ratio. The golden ratio truly is a common pattern.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-uMvrbdnLGaY/Wpbo3KZ7z7I/AAAAAAAAGg4/GD_F9fgWPToAqbbjwRsJC5UvJo-_pBatwCEwYBhgL/s1600/Screen%2BShot%2B2018-02-28%2Bat%2B12.36.53%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="636" data-original-width="1033" height="246" src="https://4.bp.blogspot.com/-uMvrbdnLGaY/Wpbo3KZ7z7I/AAAAAAAAGg4/GD_F9fgWPToAqbbjwRsJC5UvJo-_pBatwCEwYBhgL/s400/Screen%2BShot%2B2018-02-28%2Bat%2B12.36.53%2BPM.png" width="400" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-19066673817919140452018-02-27T23:15:00.001-05:002018-02-27T23:15:03.344-05:00Teach 180: A Visit to a French Market (Day 113)<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ZACgd24BdMM/WpYq9UdvchI/AAAAAAAAGgM/oQlzvfZ7DeMJYfOKH_8LtrX1NcxbXTy4wCLcBGAs/s1600/Screen%2BShot%2B2018-02-27%2Bat%2B11.06.38%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="202" data-original-width="306" height="209" src="https://2.bp.blogspot.com/-ZACgd24BdMM/WpYq9UdvchI/AAAAAAAAGgM/oQlzvfZ7DeMJYfOKH_8LtrX1NcxbXTy4wCLcBGAs/s320/Screen%2BShot%2B2018-02-27%2Bat%2B11.06.38%2BPM.png" width="320" /></a></div>Bonjour! Je m'appelle Leigh Nataro y je ne parle pas le français tres bien. But today I spent about 15 minutes visiting a French market in one of the French classrooms. The students had items for "sale" and were practicing their speaking skills relative to buying and selling food items at a market. Several of these students will be traveling to France over spring break and this activity helped them prepare for the trip. Faculty were invited to the classroom to participate in the activity.<br /><br />I said, "J'ai soif", and a student poured me a delicious (raspberry, I think) sparkling soda. Next I told one of my current AP Statistics students, "Je voudrai un meringue." It was a tasty cookie, but a bit overcooked. I also paid way too much for deux carottes. I think it was 2 Euros for the baby carrots?? Hmmm, perhaps next time I should study the Quizlet cards a bit more. In any event, I enjoyed seeing my students in a different setting and this was a fun way to show them that teachers can be students, too.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-92155311620002972672018-02-26T23:28:00.001-05:002018-02-26T23:28:09.666-05:00Teach 180: Teaching the Wrong Way (Day 112)Today I did a disservice to my period A class. I lectured for <b>the entire class period</b>, which was nearly an hour long. I could tell many of their brains were full after about thirty minutes, but I plowed ahead. So, why did I lecture for a full hour? The answer is the AP exam; it is on May 17th. This is the first time in fifteen or so years of teaching that the date of the AP exam has been a concern to me.<br /><br />In years past, I had plenty of time to review for the AP exam. I split my content up into bit-sized, easily digestible 40 or 50 minute chunks. Some days were entirely for students to work through problems. Other days were driven by a data collection activity. Some days included lecture for part of the class period. <br /><br />In years past, we had about three weeks to review prior to the start of AP exams. This year AP exams start about one week later and you would think that would allow me more time for my students to review for the AP exam. Unfortunately, that is not the case. Based on my estimates, we will have the equivalent of 5 one-hour classes to prepare for the AP exam. My AP Stat colleague from souther states will have 20-30 hours. I fear my five hours of review will be woefully insufficient.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Vae3BO_fjCg/WpTYGzwyt8I/AAAAAAAAGf8/2E5-U45PEbYBcD_1BxL2YJKfRDfFB-N0ACLcBGAs/s1600/Screen%2BShot%2B2018-02-26%2Bat%2B10.53.47%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="263" data-original-width="234" src="https://4.bp.blogspot.com/-Vae3BO_fjCg/WpTYGzwyt8I/AAAAAAAAGf8/2E5-U45PEbYBcD_1BxL2YJKfRDfFB-N0ACLcBGAs/s1600/Screen%2BShot%2B2018-02-26%2Bat%2B10.53.47%2BPM.png" /></a></div>In order to protect those five minimal review days, I chose to take two 30 minute lectures today and put them in one 60 minute lecture. It is true that 2 x 30 = 60 and technically, the timing equates. We talked about null and alternative hypotheses, p-values, the wording of conclusions, type I and type II errors, and quickly touched on power of a test. Basically, three big ideas that should have been separated into 2 or 3 separate class periods. Yes, it would have been better to separate these ideas, but spending an extra day on content now means one less review day in the future.<br /><br />Teaching in longer periods will still be my reality next year at my new job. So, what will I do differently? Mostly, I'll plan better. No time was allocated for teachers to do this at the beginning of the year or to pay them for this work over the summer. Had we been given time to work within our departments, we would not be scrambling to determine what content will be left out of our curricula for the remainder of this year. Or better yet, we could have reviewed the math curriculum that was over 12 years old to determine what content really matters the most.<br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-26226484050246444582018-02-25T17:00:00.001-05:002018-02-25T17:00:30.673-05:00Teach 180: Specific Student Feedback (Day 111)If you ever ask a teacher what they dislike the most about their job, it is likely that they will say grading. It never seems to end and when I taught in the public school, I could easily spend 10 hours a week or more grading tests, quizzes and problem solving tasks. One of the purposes of grading should be to provide students with feedback. I am not just talking about a percentage or number correct. I am talking about enough feedback that the student can understand where he or she fell short and how to improve.<br /><br />Feedback on assessments is usually quick, things like "be careful of your signs" or "don't forget the Such-and-Such property". In my mind the feedback is clear. But perhaps it isn't as clear as I think. Today a student came back and asked me about the following feedback I gave on his Calculus quiz.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-eyn2y_0dq1s/WpMwGlQGSXI/AAAAAAAAGfc/ZfuRbpN1jyUNjUJ3Si6_u-3WO0pj06sPACLcBGAs/s1600/Screen%2BShot%2B2018-02-25%2Bat%2B4.52.09%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="226" data-original-width="523" height="138" src="https://4.bp.blogspot.com/-eyn2y_0dq1s/WpMwGlQGSXI/AAAAAAAAGfc/ZfuRbpN1jyUNjUJ3Si6_u-3WO0pj06sPACLcBGAs/s320/Screen%2BShot%2B2018-02-25%2Bat%2B4.52.09%2BPM.png" width="320" /></a></div><br />He was wondering what my feedback was showing. What did he do wrong? Had he not asked me about it, my feedback to him would have been meaningless. Would writing sentences like these have helped? "If you don't have the parentheses, only the 4 is multiplied by 10e<sup>x</sup>. You need the parentheses, because we want the entire denominator to be multiplied by the derivative of the numerator." Maybe it would have been helpful. Or maybe not. It would certainly take me about 10 times longer to write that than what I wrote.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-BPylO3T5mdo/WpMxkgZTLoI/AAAAAAAAGfs/TwoY8B_4OKcJR_Ue-4nzF3NTjPL-z6CMQCLcBGAs/s1600/Screen%2BShot%2B2018-02-25%2Bat%2B4.57.39%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="312" data-original-width="243" src="https://3.bp.blogspot.com/-BPylO3T5mdo/WpMxkgZTLoI/AAAAAAAAGfs/TwoY8B_4OKcJR_Ue-4nzF3NTjPL-z6CMQCLcBGAs/s1600/Screen%2BShot%2B2018-02-25%2Bat%2B4.57.39%2BPM.png" /></a></div>So, how can I get students to understand my feedback and not use up the ink of a dozen pens? Would giving the students a copy of the answer key help? Perhaps. But I am guessing most students would see that their answer to a question as wrong, but not fully understand why. Would doing test or quiz corrections help? Perhaps. But that leads to more grading for me. Would having students consult with their neighbor about quiz or test errors work? Maybe. But student grades should be private. <br /><br />So, what is the answer to providing more specific student feedback AND not drowning in red ink? I am hoping to find the answer to this in the new book I starting to read "<a href="http://www.ascd.org/Publications/Books/Overview/Grading-Smarter-Not-Harder.aspx" target="_blank">Grading Smarter, Not Harder" by Myron Dueck</a>. Any ideas I try, I'll be sure to share them in my blog.<br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com2tag:blogger.com,1999:blog-4784716814797278012.post-52922196158013402052018-02-22T20:35:00.001-05:002018-02-22T20:35:23.030-05:00Teach 180: Smelling Parkinson's (Day 110)Today students were introduced to hypothesis testing with an <a href="https://www.youtube.com/watch?v=Ct-zg2eTu2o" target="_blank">activity developed by Doug Tyson</a>. The lesson begins by showing students a <a href="https://www.youtube.com/watch?v=XB0QgwVffvk" target="_blank">video about Joy Milne</a>, the woman who can smell Parkinson's disease. (If you don't like this particular video, there are many others online.) Students learn that Joy smelled t-shirts and correctly identified the shirt as being from a Parkinson's patient or not in 11 of 12 t-shirts. We then wonder how likely it is to get 11 of 12 identifications correct.<br /><br />We set up the null and alternative hypotheses (Ho: p = 0.50 and Ha: p > .5) and then did a simulation by hand to determine how likely it is for 11 of 12 identifications to be correct by chance alone. The pictures below show students smelling index cards that have either a P or NP on the back of the card. (It is quite fascinating how excited students get when the guess correctly.)<br /><br /><table><tbody><tr><td><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-2cBUBJprm_g/Wo9ooskFqBI/AAAAAAAAGfA/UHk35TpJZbgI1sQc4pdN3DRDuq22BExIQCEwYBhgL/s1600/IMG_5587.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="200" src="https://1.bp.blogspot.com/-2cBUBJprm_g/Wo9ooskFqBI/AAAAAAAAGfA/UHk35TpJZbgI1sQc4pdN3DRDuq22BExIQCEwYBhgL/s200/IMG_5587.JPG" width="150" /></a></div><br /></td><td><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-RqoJjmnoP2E/Wo9opcNxryI/AAAAAAAAGfE/OTVnsrr528gKXq3zccg3avVxnfTAFDXXQCEwYBhgL/s1600/IMG_5589.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="200" src="https://1.bp.blogspot.com/-RqoJjmnoP2E/Wo9opcNxryI/AAAAAAAAGfE/OTVnsrr528gKXq3zccg3avVxnfTAFDXXQCEwYBhgL/s200/IMG_5589.JPG" width="150" /></a></div><br /></td> <td><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-l1-kzJcWPvA/Wo9operRZmI/AAAAAAAAGfI/9i7WQafu9C8Rkhg_wjTHFxT17dRGEGW-ACEwYBhgL/s1600/IMG_5590.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="200" src="https://1.bp.blogspot.com/-l1-kzJcWPvA/Wo9operRZmI/AAAAAAAAGfI/9i7WQafu9C8Rkhg_wjTHFxT17dRGEGW-ACEwYBhgL/s200/IMG_5590.JPG" width="150" /></a></div></td></tr></tbody></table>We plotted the results of 40 simulations on a dotplot and got an estimated p-value of 0.025. (One student got 11 of 12 correct by guessing.) Finally, we did 10,000 runs of the simulation using <a href="http://www.lock5stat.com/StatKey/sampling_1_cat/sampling_1_cat.html" target="_blank">statkey</a>. Based on these simulated results, our p-value was 0.0045. This simulated p-value was definitely small enough for us to reject the null hypothesis.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-RDRcBKTQtug/Wo9oqpmyPFI/AAAAAAAAGfI/Mw40DhRGV4U4-hyrI5rPDPsfYpC3WSUswCEwYBhgL/s1600/Screen%2BShot%2B2018-02-22%2Bat%2B8.03.18%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="621" data-original-width="972" height="255" src="https://2.bp.blogspot.com/-RDRcBKTQtug/Wo9oqpmyPFI/AAAAAAAAGfI/Mw40DhRGV4U4-hyrI5rPDPsfYpC3WSUswCEwYBhgL/s400/Screen%2BShot%2B2018-02-22%2Bat%2B8.03.18%2BPM.png" width="400" /></a></div>Incidentally, one of the patients that Joy had identified as not having Parkinson's returned to the lab several months later to report that he, in fact, had Parkinson's. Joy was actually 100% accurate in all of her identifications!!<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-16616579402111537452018-02-21T23:06:00.001-05:002018-02-21T23:06:19.896-05:00Teach 180: The In-Service Day (Day 109)<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-kPxOJbZMoG8/Wo45vkhGjkI/AAAAAAAAGeM/YgkCsS63WK8J-AgPqzKXPSCuW6SgM5jVACLcBGAs/s1600/Screen%2BShot%2B2018-02-21%2Bat%2B10.31.36%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="329" data-original-width="243" height="200" src="https://1.bp.blogspot.com/-kPxOJbZMoG8/Wo45vkhGjkI/AAAAAAAAGeM/YgkCsS63WK8J-AgPqzKXPSCuW6SgM5jVACLcBGAs/s200/Screen%2BShot%2B2018-02-21%2Bat%2B10.31.36%2BPM.png" width="147" /></a></div>It's not a teaching day today. It's an in-service day. Thinking back to my in-service days over the past 25 years of teaching I would say that there is really only one that I remember with lasting importance. When I taught in Ames, Iowa, there was a 3-day cooperative learning seminar that the entire school was required to attend. We had to create sub plans and 1/3 of the school was out for 3 consecutive school days. By the end of day 9, we all had been trained by the authors of the book <a href="https://www.thriftbooks.com/w/learning-together-and-alone-cooperative-competitive-and-individualistic-learning_david-w-johnson/419616/?mkwid=sIxihvvkm%7cdc&pcrid=70112885352&pkw=&pmt=&plc=&gclid=EAIaIQobChMIspqxwbq42QIVx1YNCh11nQv6EAQYBCABEgJ0UfD_BwE#isbn=0205155758&idiq=2698011" target="_blank">Learning Together and Learning Alone</a>. Some teachers were definitely more into learning than others, but having a common language and understanding common strategies for cooperative learning was very beneficial. Language like "think-pair-share" and "jigsaw method" were quickly understood by the students, because so many teachers were implementing cooperative learning methods. We had follow-up meetings throughout the year to discuss successes and setbacks. I must have learned something about cooperative learning, because I still use some of the structures in my classes.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div>Today's in-service day was nothing like that. It was one day that was fairly fragmented. We had much to "cover". We had a presentation about a travel grant. Then, the Head of School shared the new school vision, and data on student engagement that had been collected last spring. Next, we met in cross-divisional groups to continue to work on group projects. We have been given 5 hours to complete a group project on a topic of interest and prepare a presentation or document to share in April. My group has come to the consensus that Project-Based Learning would not work well on a large scale at our school due to the lack of time needed to collaborate on a weekly basis.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-hAyLq6oA4WE/Wo5BZ1P8HcI/AAAAAAAAGec/XUBQAppyeacz1GW4bqSraRxEzZVxPabewCEwYBhgL/s1600/Screen%2BShot%2B2018-02-21%2Bat%2B11.04.27%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="452" data-original-width="373" height="200" src="https://1.bp.blogspot.com/-hAyLq6oA4WE/Wo5BZ1P8HcI/AAAAAAAAGec/XUBQAppyeacz1GW4bqSraRxEzZVxPabewCEwYBhgL/s200/Screen%2BShot%2B2018-02-21%2Bat%2B11.04.27%2BPM.png" width="165" /></a></div>Finally, math and science teachers met with a consultant for 3.5 hours to talk about concerns we have about the new schedule and its impact on the math and science curriculum. We touched on curriculum, assessment, and strategies for teaching in longer periods. We could have easily spent 3.5 hours or more on each of these topics individually. The consultant had a mathematics background and certainly understood our concerns, but we didn't really have enough time to make any substantive changes for the immediate future (say next week) or for the fall. We all agreed that more time was needed to make curricular changes and that we needed to investigate that as a group and not in isolation.<br /><br />The day ended with announcements relative to enrollment, finances and the school calendar for next year. We got through our agenda for the day and we certain did "cover" much. However, we did not accomplish much and what I learned today will unfortunately have little impact on what happens in my classroom.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-47935884071172485292018-02-18T22:00:00.003-05:002018-02-18T22:01:08.424-05:00Teach 180: Log Properties by Discovery (Day 108)When I was in high school, I was content with being told certain rules in math. I could either see that they were true for myself or I willingly accepted them as true. We rarely, if ever, discovered rules or were asked to make and test conjectures. Discovery, reasoning, and sense-making is more common in my classroom now than when I first started teaching, but it can be easy to fall back into pulling notes out of a file cabinet and just working through examples of concepts with the students. If discovery is better for learning, why not just create lessons based on noticing, wondering and discovery? Two main reasons: time and collaboration. With inadequate time to plan and lacking collaboration with a colleague, the chances of moving closer to discovery are greatly reduced.<br /><br />This year I decided to make the time to work with a colleague and change our lesson on logarithm properties. I had reached out via twitter asking for ideas and got a response back from Ralph Pantozzi (a.k.a.@mathillustrated). He gave my colleague and me something he had used with his students and that was where we started.<br /><br />These were the objectives for the lesson:<br />1) Students will make observations about numbers from a table of values for x and f(x), where f(x) was log<sub>2</sub>x.<br />2) Students will use the observations to create the product, quotient and power property for logarithms.<br />3) Students will practice using the properties to combine and expand logarithms.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-jzEq-rA9zK4/WonbPCIU5LI/AAAAAAAAGdQ/f3wG50nyqIomZiswiTSG3gRVNArYyWXmACLcBGAs/s1600/Screen%2BShot%2B2018-02-18%2Bat%2B2.02.34%2BPM.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="298" data-original-width="644" height="185" src="https://4.bp.blogspot.com/-jzEq-rA9zK4/WonbPCIU5LI/AAAAAAAAGdQ/f3wG50nyqIomZiswiTSG3gRVNArYyWXmACLcBGAs/s400/Screen%2BShot%2B2018-02-18%2Bat%2B2.02.34%2BPM.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Page 1</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-UrLOSRW2IiU/WonbPGb5uwI/AAAAAAAAGdM/drNxqoXMqtU5qCnCGZpA5f1jAd2TZbnXwCLcBGAs/s1600/Screen%2BShot%2B2018-02-18%2Bat%2B2.02.44%2BPM.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="462" data-original-width="588" height="313" src="https://4.bp.blogspot.com/-UrLOSRW2IiU/WonbPGb5uwI/AAAAAAAAGdM/drNxqoXMqtU5qCnCGZpA5f1jAd2TZbnXwCLcBGAs/s400/Screen%2BShot%2B2018-02-18%2Bat%2B2.02.44%2BPM.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Page 2<br /><br /></td></tr></tbody></table>Page one shows part of the full table of values we gave our students. Page two was what we had planned to give to our students, if they had trouble coming up with some ideas. But they had so many ideas!!! We didn't need to give them the page with the questions. They just took off on their own and our full class discussion led to observations about each of the log properties. Photos of two of the boards from our discussion are shown below.<br /><br /><a href="https://2.bp.blogspot.com/-zlc_Ms3k22Q/WoncBcvindI/AAAAAAAAGdg/ye0E4YPqNBoVIvVUFTvFmCs1Iw2OQ1Q7ACLcBGAs/s1600/IMG_5578.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://2.bp.blogspot.com/-zlc_Ms3k22Q/WoncBcvindI/AAAAAAAAGdg/ye0E4YPqNBoVIvVUFTvFmCs1Iw2OQ1Q7ACLcBGAs/s320/IMG_5578.JPG" width="240" /></a><a href="https://2.bp.blogspot.com/-GUyUpL1g1YQ/WoncBG2CpBI/AAAAAAAAGdc/2TjDMOvrQ1ImiLNYhEkWj5LXbzdNykv2QCLcBGAs/s1600/IMG_5579.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://2.bp.blogspot.com/-GUyUpL1g1YQ/WoncBG2CpBI/AAAAAAAAGdc/2TjDMOvrQ1ImiLNYhEkWj5LXbzdNykv2QCLcBGAs/s320/IMG_5579.JPG" width="240" /></a><br /><br />At one point, my students asked if what we were doing would work for all bases. For some reason, I didn't anticipate that the students would ask this. I didn't have a spreadsheet set up where we could change the base to be a different value. Was there a way to test to see if what we were doing with the sum and product would work for all bases?<br /><br />Desmos to the rescue! I instructed students to type in two expressions in Desmos: log<sub>b</sub>M + log<sub>b</sub>N and log<sub>b</sub>MN, where b was a base of their choosing and M and N where numbers of their choosing. Students could see that the results of both calculations were the same! I made expressions with sliders for M and N to show that changing values for M and N produced the same result. The screenshot here shows something similar with the quotient property.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-JC8N3zTABOo/Wonfx4-Z7pI/AAAAAAAAGd0/OPRO1SlLzAkPt4V2ySNL-WTJ9wM5xAShACLcBGAs/s1600/Screen%2BShot%2B2018-02-18%2Bat%2B3.03.57%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="434" data-original-width="463" height="297" src="https://3.bp.blogspot.com/-JC8N3zTABOo/Wonfx4-Z7pI/AAAAAAAAGd0/OPRO1SlLzAkPt4V2ySNL-WTJ9wM5xAShACLcBGAs/s320/Screen%2BShot%2B2018-02-18%2Bat%2B3.03.57%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Based on our results from the product property, we derived the power property. We also reviewed the values of log<sub>b</sub>b and log<sub>b</sub>1. Next, I had the students type f(x) = b<sup>log<sub>b</sub>x</sup> into desmos. Students were instructed to choose a base for b and a value for x. Why did it make sense that the f(x) always equal to x? Why was the input value the same as the output value? Does this happen for all values of base b? I created a table of values and a slider and when the function b<sup>log<sub>b</sub>x</sup> was plotted, it resulted in a line. (See the short screencast below.) This led to a discussion of the fact that b to a power and log base b were inverse functions. So, it made sense for them to undo each other and for f(x) to equal x.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowFullScreen='true' webkitallowfullscreen='true' mozallowfullscreen='true' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dyRfsim7n_RNSmcc1aFZum1GoHXnY5_WjdAtpRYgl9eH7_O2wznUCfgVLjjLdL6soJ4DRQFOMwReBmhfBkFOg' class='b-hbp-video b-uploaded' FRAMEBORDER='0' /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br />So, what lessons have I learned about moving closer to having students learn by discovery, by noticing, by wondering? Two things: First, it's ok to not plan for everything. I didn't think that my students would ask about other bases, but we were able to quickly examine a multitude of bases. Second, don't be afraid to let the students take control of the lesson. They have the ability to notice and wonder, if you let them. Their insights might surprise you.<br /><br />Discovery can come naturally. Learning logarithm properties by discovery made the lesson more fun for me to teach and it made it more interesting for them to learn.<br /><br /><br /><div><br /></div><div><br /></div><div><br /></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-30327767053873716462018-02-15T14:56:00.001-05:002018-02-15T14:56:37.257-05:00Teach 180: Contest Day! (Day 107)<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-kXjtsRmgO88/WoXlGPECmaI/AAAAAAAAGc8/O1BKnWhY4DY6OvTE3OPAed9flEvXLPKewCLcBGAs/s1600/Screen%2BShot%2B2018-02-15%2Bat%2B2.47.29%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="658" data-original-width="352" height="320" src="https://2.bp.blogspot.com/-kXjtsRmgO88/WoXlGPECmaI/AAAAAAAAGc8/O1BKnWhY4DY6OvTE3OPAed9flEvXLPKewCLcBGAs/s320/Screen%2BShot%2B2018-02-15%2Bat%2B2.47.29%2BPM.png" width="171" /></a></div>Today our contingency plan went into effect. We had a snow day the previous week and that meant we had to give the <a href="https://www.maa.org/math-competitions" target="_blank">American Mathematics Competition (AMC)</a> Form B contest, instead of the Form A contest. This year we held the contest during an assembly period and students only missed half of one class period. (In previous years students missed two class periods. However, I think it is important for students to be in class, because you can't replicate the learning that happens as students interact with each other.)<br /><br />The contest lasts 75 minutes and consists of 25 multiple choice questions. Students cannot use a calculator and the questions are quite challenging. After the contest was over, the room was buzzing with chatter, such as "How did you solve question #8?" and "How many did you actually put down an answer for?" Tomorrow students can get their test books back from me and I look forward to discussing some of the problems with them.<br /><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-10740876414494047632018-02-14T17:47:00.003-05:002018-02-14T17:47:39.908-05:00Teach 180: Mathograms (Day 106)There are times when the school day is over and I just don't feel like working. Sure I could plan for future lessons or grade the test I just gave. I could even organize the desktop of my computer. Instead I chose to get lost in some math. Yesterday I saw a tweet about Desmos Mathograms. I found the tweet again today and opened up the link <a href="http://mathogram.desmos.com/">http://mathogram.desmos.com/</a> Then I sent Mathogram Valentines to my daughter and husband. (My husband responded about a minute later with "Geek. Luv ya too.")<span style="text-align: center;"> </span><br /><br /><div style="text-align: center;"><a href="https://goo.gl/ubnmNC" target="_blank">This is the one I sent my husband. </a></div><br /><div style="text-align: center;"><a href="https://goo.gl/ubnmNC" target="_blank"><img border="0" data-original-height="385" data-original-width="498" height="308" src="https://3.bp.blogspot.com/-iCMlJpfqXzs/WoS3ce4hkvI/AAAAAAAAGcw/GfjV1-kiVh0fu5vTZrNvMBn7HTIIh2f-gCEwYBhgL/s400/Screen%2BShot%2B2018-02-14%2Bat%2B5.25.18%2BPM.png" width="400" /></a></div><br /><br /><div style="text-align: center;"><a href="https://goo.gl/J623Gi" target="_blank">This is the one I sent to my daughter.</a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-xXKym9AvpD4/WoS3zUYVXgI/AAAAAAAAGcw/_jtbQKO4cGkWxUGhKwI33IVLpZmTjsXJgCEwYBhgL/s1600/Screen%2BShot%2B2018-02-14%2Bat%2B5.26.41%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="380" data-original-width="513" height="296" src="https://4.bp.blogspot.com/-xXKym9AvpD4/WoS3zUYVXgI/AAAAAAAAGcw/_jtbQKO4cGkWxUGhKwI33IVLpZmTjsXJgCEwYBhgL/s400/Screen%2BShot%2B2018-02-14%2Bat%2B5.26.41%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">One of the things I love about both of these is that Desmos lets you see the "Behind the Scenes" of each graph. Seeing the flower transform to a heart, I was instantly curious how it happened. And I was surprised to see that it happened in just 9 lines of code! The Sierpinski triangle, just 4 lines of code!! Now I need to take some time to break it down and see just how it works. I wonder if my tests will get graded tonight.</div> <br /><br /><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-32907105812268334702018-02-13T23:46:00.001-05:002018-02-13T23:46:39.923-05:00Teach 180: Helping Students (Day 105)One of the aspects of my job that I enjoy the most is working with students individually. When I taught in the public school, there was little time to give students the additional help they needed. If they were free, I would usually have a class. Or if I was free, they would usually have a class. This would mean coming to school earlier than the start of the school day or sticking around after school was over to offer individual help. Either before school or after school was an impossibility for many students due to bussing, part time jobs, or caring for siblings.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-BjWLXhzzV7Y/WoO82rUCknI/AAAAAAAAGcY/n5T9gwJMNDQpRFk8g4ca5NgGYBlbmq71QCLcBGAs/s1600/Screen%2BShot%2B2018-02-13%2Bat%2B11.36.25%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="187" data-original-width="256" height="145" src="https://4.bp.blogspot.com/-BjWLXhzzV7Y/WoO82rUCknI/AAAAAAAAGcY/n5T9gwJMNDQpRFk8g4ca5NgGYBlbmq71QCLcBGAs/s200/Screen%2BShot%2B2018-02-13%2Bat%2B11.36.25%2BPM.png" width="200" /></a></div><div style="text-align: left;"></div>At my current job, I teach 4 classes and this allows me to have 3 periods where I am available to work with colleagues and meet with students. Some days I don't work with any students and other days I might work with five different students. Today I met with two students. One had been absent several days in AP Stat and was trying to get caught up. She had the basic understanding of what she was doing relative to confidence intervals, but needed some encouragement. She had contacted me via email to arrange for a meeting time.<br /><br />The second student had done poorly on a quiz and realized that she really didn't understand the basic properties of logarithmic and exponential functions. She sought me out and found me talking to a colleague about the recent Calculus test. This student wants to do well, but she often misplaces items, like her homework. Even though I have time to meet with students during the school day, there are still times I need to meet with students before school or after school. I have been meeting with one student once or twice a week since the beginning of October. He recognizes that without this accountability he probably wouldn't get his work done for PreCalculus.<br /><br />Some may think that these students should be able to figure math out on their own. After all, the students I have described are seniors. Doesn't giving the students help make them weaker? First, the fact that they are willing to ask for help is a sign of strength. It can be challenging to admit that you need help and are struggling. If the first time a student asks for academic help is in college, it is likely to be even more challenging. It is good to learn this skill now. Second, I ask the students that I work with questions that they should be asking themselves. How is this like the problem we just did? How is it different? Does your answer seem reasonable, and why or why not? What do we know and what are we trying to figure out? Modeling the thinking and questioning that students should be doing will ultimately help the student to help themselves.<br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-49813852551452571072018-02-12T19:34:00.002-05:002018-02-12T19:34:25.066-05:00Teach 180: The Derivative of e^x (Day 104)I had to have a substitute cover my last period class today and I never count on a sub to know math, especially Calculus. Less damage control needs to be done the next day, if I go with that assumption. Today's lesson was finding the derivative of f(x) = e<sup>x</sup> and using the chain rule to find the derivative of g(x) = e<sup>f(x) </sup>. In years past, I had students use their graphing calculators to find the value of the derivative of f(x) = e<sup>x</sup> at various values of x. Next, we would plot points such as (0, 1), (1, 2.718), (2, 7.4) and so on, and students would see that these points lined up on the function f(x) = e<sup>x</sup>.<br /><br />Since I wasn't going to be in class, I couldn't easily lead the students through this activity, and I had no expectation that my sub would be able to do this. So, I created a <a href="https://www.youtube.com/watch?v=ZAiR6Le75jc&list=PLNEJJgypdNblGDxagLF_JDMX04z-lWfLt&index=7" target="_blank">short 8 minute video</a> instead and posted it to my <a href="http://www.youtube.com/mathteacher24" target="_blank">YouTube channel</a> to show students why the derivative of f(x) = e<sup>x</sup> is f(x) = e<sup>x</sup> by using the limit definition of the derivative and a little help from desmos. With desmos we could easily see the value of the following limit was 1.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-zfz6ohhpVi0/WoIt-_sNeII/AAAAAAAAGbo/tHwI6wyMdLg0HhztEOr21GOQRwof7JBSACLcBGAs/s1600/Screen%2BShot%2B2018-02-11%2Bat%2B10.54.51%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="236" data-original-width="416" height="181" src="https://2.bp.blogspot.com/-zfz6ohhpVi0/WoIt-_sNeII/AAAAAAAAGbo/tHwI6wyMdLg0HhztEOr21GOQRwof7JBSACLcBGAs/s320/Screen%2BShot%2B2018-02-11%2Bat%2B10.54.51%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">To convince the students further, I used desmos to create a table of values for f(x) = e<sup>x</sup></div>and g(x) = f ' (x). Behold, they have the exact same values! Why?!? Because they are the exact same function!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-BFpdue8AqXo/WoIwtn7elHI/AAAAAAAAGb0/udIs5T2FNyoTo-HiD0jxfRNNtBWMYcxMgCLcBGAs/s1600/Screen%2BShot%2B2018-02-11%2Bat%2B10.55.02%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="467" data-original-width="457" height="320" src="https://1.bp.blogspot.com/-BFpdue8AqXo/WoIwtn7elHI/AAAAAAAAGb0/udIs5T2FNyoTo-HiD0jxfRNNtBWMYcxMgCLcBGAs/s320/Screen%2BShot%2B2018-02-11%2Bat%2B10.55.02%2BPM.png" width="313" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">It's hard to believe that I started my YouTube channel back in November of 2008, almost 10 years ago. Prior to Desmos, YouTube and <a href="https://screencast-o-matic.com/" target="_blank">Screencast-o-matic</a> my sub plans would have consisted of students reading the book and working through examples. Technology has certainly made my math lessons richer and has allowed for me to help students draw more connections among various representations. </div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-81508539129556781212018-02-11T19:28:00.002-05:002018-02-11T19:28:29.184-05:00Teach 180: The Calculus Test (Day 103)Today I gave a calculus test on two sections - applications of the derivative, such as maximizing revenue or minimizing amount of material to make a box, and implicit differentiation, which included related rates. As students took the test, I became worried. A few of them approached my desk to ask a question or two. Several of them were confusing surface area and volume. Others were are trying to solve the problems without using <u>any</u> Calculus! The title of the class is Calculus and we have been taking derivatives for the past 4 months!!!!<br /><br />(Fast forward to Saturday) I graded the tests and there was 1 A, 1 B, 2 Cs, 6 Ds and 2 Fs. I think the lack of continuity over the past few weeks is partly to blame. Our bell schedule is new this year and we don't meet each day. Normally, we meet 6 out of every 8 school days. But here is how things have worked out for the past 3 weeks. (Note that classes are 1 hour in length.) If the students had been doing some Calculus each of the days we didn't have class, they would have had reviewed for the test for the equivalent of five school days. As a point of reference, we only reviewed for a comprehensive midterm for three consecutive school days in January.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-gZhdpTCc7kU/Wn44zslt5bI/AAAAAAAAGbU/9_5U944An_Eaj84rMCMojItvdgx9jCtmgCLcBGAs/s1600/Screen%2BShot%2B2018-02-09%2Bat%2B12.20.00%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="262" data-original-width="720" height="232" src="https://3.bp.blogspot.com/-gZhdpTCc7kU/Wn44zslt5bI/AAAAAAAAGbU/9_5U944An_Eaj84rMCMojItvdgx9jCtmgCLcBGAs/s640/Screen%2BShot%2B2018-02-09%2Bat%2B12.20.00%2BPM.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">So, how do I rectify this situation? Do I simply curve the test and move on? Clearly that would be the easiest thing to do. But that would mean that the curved grades would not be a representation of the true level of understanding of the material.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">It looks like re-teaching and re-testing is in order. I am not quite sure how this will be accomplished yet, because I need to consult with the other Calculus teacher. Luckily, this class is an elective class and we are not bound by AP testing dates. We don't have to push the students through the course. </div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-86293456538630866142018-02-08T23:10:00.000-05:002018-02-08T23:10:04.014-05:00Teach 180: Students Lack of Sleep (Day 102)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-sEgb3jC6piU/Wn0eAZzHiCI/AAAAAAAAGa4/t3HPcpgs_kMbZMNJ42V5hA2Uj249_EqtwCLcBGAs/s1600/Screen%2BShot%2B2018-02-08%2Bat%2B11.04.43%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="131" data-original-width="126" src="https://2.bp.blogspot.com/-sEgb3jC6piU/Wn0eAZzHiCI/AAAAAAAAGa4/t3HPcpgs_kMbZMNJ42V5hA2Uj249_EqtwCLcBGAs/s1600/Screen%2BShot%2B2018-02-08%2Bat%2B11.04.43%2BPM.png" /></a></div>Students are naturally more engaged when the topic is of interest to them. Today's topic in AP Statistics was a review of confidence intervals for a population proportion and our data set was the number of hours of sleep my students got on the first night of school. I had gathered data from my students on the first day of school using a google form. I saved the sleep hours column as a .csv file and imported the data into <a href="http://fathom.concord.org/" target="_blank">Fathom</a> for us to review.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-sEHFYFBJe80/Wn0YDQzYLHI/AAAAAAAAGac/r4WsHHoZcXcfFe0vxv1uYHO-GWKZwM4swCLcBGAs/s1600/Screen%2BShot%2B2018-02-08%2Bat%2B10.22.57%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="108" data-original-width="405" height="84" src="https://3.bp.blogspot.com/-sEHFYFBJe80/Wn0YDQzYLHI/AAAAAAAAGac/r4WsHHoZcXcfFe0vxv1uYHO-GWKZwM4swCLcBGAs/s320/Screen%2BShot%2B2018-02-08%2Bat%2B10.22.57%2BPM.png" width="320" /></a></div>Next we created a column called "Less_Than_8_Hours" and created the formula at the left to change the quantitative variable to a categorical variable. If the variable <span style="color: magenta;">Sleep</span> was less than 8 hours, a "Y" was displayed. Otherwise an "N" was displayed. So, what is the point estimate for the proportion of the 47 students surveyed slept less than 8 hours? And what is the confidence interval? After we calculated it by hand, we used Fathom to confirm our results. Our conclusion is that we are 95% confident that the interval from 0.6702 to 0.9042 contains the true proportion of students who slept less than 8 hours the night before school started. It's crazy to think that students were lacking in sleep even before the school year started!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-R2wlRDBsntU/Wn0edN3XwDI/AAAAAAAAGa8/TIzBK8hwfPEseozW-aw7WdFvpow_7b4LQCLcBGAs/s1600/Screen%2BShot%2B2018-02-08%2Bat%2B11.06.23%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="179" data-original-width="449" height="157" src="https://1.bp.blogspot.com/-R2wlRDBsntU/Wn0edN3XwDI/AAAAAAAAGa8/TIzBK8hwfPEseozW-aw7WdFvpow_7b4LQCLcBGAs/s400/Screen%2BShot%2B2018-02-08%2Bat%2B11.06.23%2BPM.png" width="400" /></a></div><br /><br /><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-80434247068068458602018-02-07T08:51:00.002-05:002018-02-07T08:51:59.873-05:00Teach 180: Teaching on Snow Day #4 (Day 101)Although it is a snow day, we must keep moving ahead in AP Statistics. Other schools in the south start their review on April 1. Because our school year starts later and we typically have snow days, we are lucky to have review begin by April 20th. Losing one hour of clas today means one less hour of review for the AP exam. Due to our new bell schedule this year, my one AP Statistics class has missed 5 one-hour classes due to snow, delays and early releases. I believe my other AP Statistics section has only missed one class.<br /><br />To try to keep some momentum going, my AP Statistics students will be greeted with an email from me at some point today. (It's still before 9 AM and I am pretty sure they are still sleeping.) In the email there is a link to one of two videos. Period A will be learning the lesson that Period F did yesterday - <a href="https://www.youtube.com/watch?v=C7HNJQ8-H3o" target="_blank">determining sample size and confidence level for a given margin of error</a> for a confidence interval for proportions. And Period F will be learning about the <a href="https://www.youtube.com/watch?v=GjRw5YBQv-8" target="_blank">t-distribution and constructing confidence intervals for a mean</a>. Luckily for me, I had these videos ready to go on my YouTube channel thanks to snow events in 2014 and 2015.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-kgJHHSWoyrs/WnsEZc_pM_I/AAAAAAAAGaI/FOxZrMfP4JAGECYyAIwWH82KNKU00C8CACLcBGAs/s1600/Screen%2BShot%2B2018-02-07%2Bat%2B8.50.39%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="431" data-original-width="533" height="258" src="https://3.bp.blogspot.com/-kgJHHSWoyrs/WnsEZc_pM_I/AAAAAAAAGaI/FOxZrMfP4JAGECYyAIwWH82KNKU00C8CACLcBGAs/s320/Screen%2BShot%2B2018-02-07%2Bat%2B8.50.39%2BAM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div style="text-align: center;">Happy Snow Day everyone!</div><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-62985894526660553052018-02-06T20:23:00.001-05:002018-02-06T20:23:57.416-05:00Teach 180: You Know You are Winning When...(Day 100)<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-HD0-XsBUxsc/WnpR7WH8M2I/AAAAAAAAGZ4/y2TWE1NNVIAYxwOudyx43cEW8Ns2aiU4gCLcBGAs/s1600/Screen%2BShot%2B2018-02-06%2Bat%2B8.09.40%2BPM.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="610" data-original-width="741" height="263" src="https://4.bp.blogspot.com/-HD0-XsBUxsc/WnpR7WH8M2I/AAAAAAAAGZ4/y2TWE1NNVIAYxwOudyx43cEW8Ns2aiU4gCLcBGAs/s320/Screen%2BShot%2B2018-02-06%2Bat%2B8.09.40%2BPM.png" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">https://chehockey.files.wordpress.com/2014/06/winner2.jpg</td></tr></tbody></table>At the beginning of the school year, you quickly identify those students who you think will struggle. They often have weak backgrounds in a previous math class and have a low self-esteem relative to mathematics. (At my school, the students often cite "Hot Pencils", a timed math facts activity, as a pivotal moment that changed their view of mathematics.) Sometimes these students will even leave questions blank on a test or quiz rather than risk putting something down that they think is wrong and in their minds, makes them look stupid. <br /><br />At the beginning of the year, I identified a few students like this in my PreCalculus class. Today, I knew I had reached some of them when two of these students, independent of each other, asked me, "Why did log(-4) on my calculator give me an error message?" It was at the end of class and it was totally unsolicited. It's a question I had planned to ask my class that day, but we ran short on time. The bell had rung and these two students stayed after the bell to ask me their question. The fact that they had the curiosity to ask that question and not dart like Pavlov's dog at the sound of the passing bell means that I have won. It means that they, too, are now winning.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-52641882827950905562018-02-05T15:54:00.001-05:002018-02-05T15:54:19.360-05:00Teach 180: Planning to Discover Log Properties (Day 99)<div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Today we had our third (or maybe it was our fourth?) two-hour delay. We were scheduled to originally have periods A, B, C, D and E. I teach periods A, B, F and G. With the two-hour delay, I had <u>no</u> classes today. You may wonder why I even came into school. Why not call in sick? If you know me, you know that is not my style. I had a meeting with a student planned and I also had a meeting with a colleague planned to discuss PreCalculus.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-sYIfiviaI9k/Wni-Ii3YP-I/AAAAAAAAGZo/VpRvDqL_v2Q3ohkdBVvrm2sezJaIpx8egCLcBGAs/s1600/Screen%2BShot%2B2018-02-05%2Bat%2B3.26.11%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="515" data-original-width="1140" height="180" src="https://1.bp.blogspot.com/-sYIfiviaI9k/Wni-Ii3YP-I/AAAAAAAAGZo/VpRvDqL_v2Q3ohkdBVvrm2sezJaIpx8egCLcBGAs/s400/Screen%2BShot%2B2018-02-05%2Bat%2B3.26.11%2BPM.png" width="400" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-TPVZXG0YAro/Wni3_pmOZ_I/AAAAAAAAGZY/THl6ZdOJa8sIc91AB_bEOVy99zI8syn0wCLcBGAs/s1600/Screen%2BShot%2B2018-02-05%2Bat%2B3.00.09%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="440" data-original-width="175" height="320" src="https://3.bp.blogspot.com/-TPVZXG0YAro/Wni3_pmOZ_I/AAAAAAAAGZY/THl6ZdOJa8sIc91AB_bEOVy99zI8syn0wCLcBGAs/s320/Screen%2BShot%2B2018-02-05%2Bat%2B3.00.09%2BPM.png" width="127" /></a></div>When I met with my colleague, we talked about the upcoming unit on logarithms in PreCalculus. We wanted students to discover the properties for themselves, but we weren't quite sure how to do it. For example, we want student to discover that log<sub>b</sub>(MN) = log<sub>b</sub>M + log<sub>b</sub>N. We could have them create a table of values like the one seen at the left on a spreadsheet. But would they be able to see that log(2) + log(3) = log(6)? I felt like there were too many numbers for the students to see that relationship. Perhaps if we showed the values rounded to the nearest hundredth, it would be easier for the students to see that relationship. Would it work?<br /><br />My colleague had a suggestion for using index cards and placing log(M) on one side and the decimal approximation on the other side. Perhaps this might work. We could tell students to look at the side with the decimal values and find three cards such the sum of two cards equals the third card. Then, they flip the cards over and see if they can form a relationship with the numbers on the other sides of the cards. We could also do this to also allow them to discover that log<sub>b</sub>(M/N) = log<sub>b</sub>M - log<sub>b</sub>M. Clearly, we are still in the planning stages on this. But that's ok. This lesson is for NEXT Monday, assuming no more two-hour delays or snow days between now and then. If you have a way for students to discover log properties, let me know in the comments below.<br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-15509545591722933412018-02-02T16:22:00.002-05:002018-02-02T16:22:44.851-05:00Teach 180: Flipping Kisses (Day 98)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-sMwSMkXCI8k/WnTUm5zCpKI/AAAAAAAAGZE/YlEmB1NVepoFu2YTIQ2i4qX0XjpNDFg-gCEwYBhgL/s1600/IMG_5554.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://4.bp.blogspot.com/-sMwSMkXCI8k/WnTUm5zCpKI/AAAAAAAAGZE/YlEmB1NVepoFu2YTIQ2i4qX0XjpNDFg-gCEwYBhgL/s320/IMG_5554.JPG" width="240" /></a></div>Today AP Statistics students constructed their very first confidence interval based on data they collected. The data came from tossing or flipping Hershey kisses. Students flipped 10 kisses out of a cup and repeated that 5 times to have a sample of n = 50 flips. Students counted the number of times they landed bottom side down and recorded that as a sample proportion. From there, we looked at the general "recipe" for creating a confidence interval for their specific set of data. Intervals were plotted on chart paper and their calculations confirmed with their graphing calculators. <br /><br />It's unfortunate that half of the class was absent. Although learning can happen by watching a video or reading a book, nothing can replace a hands-on activity and face-to-face discussions to build understanding of a concept. Being able to eat the chocolate is, of course, an added bonus!<br /><br /><br /><br /><table><tbody><tr><td><br /><a href="https://2.bp.blogspot.com/-Eb_G0dAwV40/WnTUm3k82DI/AAAAAAAAGY8/GVDCAqwxDPo8pMfnMjIB0YZ857U-TrZHQCLcBGAs/s1600/IMG_5555.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://2.bp.blogspot.com/-Eb_G0dAwV40/WnTUm3k82DI/AAAAAAAAGY8/GVDCAqwxDPo8pMfnMjIB0YZ857U-TrZHQCLcBGAs/s320/IMG_5555.JPG" width="240" /></a></td> <td><a href="https://2.bp.blogspot.com/-7AE_FN0TsVw/WnTUm0nDbSI/AAAAAAAAGZA/CCS0jcoJE38YSyUf5SrrxnmTB8Y32nynACLcBGAs/s1600/IMG_5558.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" data-original-height="1600" data-original-width="1200" height="400" src="https://2.bp.blogspot.com/-7AE_FN0TsVw/WnTUm0nDbSI/AAAAAAAAGZA/CCS0jcoJE38YSyUf5SrrxnmTB8Y32nynACLcBGAs/s400/IMG_5558.JPG" width="300" /></a><br /><br /><br /><br /></td></tr></tbody></table><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/-Eb_G0dAwV40/WnTUm3k82DI/AAAAAAAAGY8/GVDCAqwxDPo8pMfnMjIB0YZ857U-TrZHQCLcBGAs/s1600/IMG_5555.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><br /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-4539896065337075632018-02-01T11:27:00.003-05:002018-02-01T11:27:25.889-05:00Teach 180: The Confidence Interval (Day 97)In AP Statistics today, we looked at how changing confidence level and sample size impacts the width of a confidence interval. We also saw how the confidence level is related to the long run capture rate of confidence intervals. The website that we used to investigate this was http://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html, but rather than having my students type this in, I told them to go to <a href="http://bit.ly/ConfidenceIntervalBasics">bit.ly/ConfidenceIntervalBasics</a>. (Creating a shorter, custom link is easy to do at <a href="http://bit.ly/">bit.ly</a> or <a href="http://tinyurl.com/">tinyurl.com</a> and makes it easier for the students to correctly enter the web address in their browser.)<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-R8zyJYmqHqA/WnMkCdJuS8I/AAAAAAAAGYM/darCQTn_9Q0t9M_aC0OcwKw8uBrEz6fzgCLcBGAs/s1600/Screen%2BShot%2B2018-02-01%2Bat%2B9.26.00%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="369" data-original-width="605" height="195" src="https://2.bp.blogspot.com/-R8zyJYmqHqA/WnMkCdJuS8I/AAAAAAAAGYM/darCQTn_9Q0t9M_aC0OcwKw8uBrEz6fzgCLcBGAs/s320/Screen%2BShot%2B2018-02-01%2Bat%2B9.26.00%2BAM.png" width="320" /></a></div><br />First, I asked students what they noticed about the two distributions that were shown on the screenshot above. They commented on the fact that they had the same normal distribution shape and same mean, but that the sampling distribution was less variable than the population distribution. Then, we used the applet to take several samples of size n = 20 and created 95% confidence intervals from each sample. I asked them what they noticed about the intervals. Students commented on the symmetry of the intervals, but weren't entirely sure if the dot in the center was the median or the mean. They also noted that the one interval was red. Why was that? Noticing the sample under the interval gave them a sense that this sample had many values that were above the mean and that the <u>entire interval</u> was bigger than the population mean. Seeing this in action with several intervals helped the students to begin to understand that the probability that <u>an individual interval</u> captures the population parameter is 0 or 1, and not 95% (or whatever the confidence level is). The 95% level refers to the long run capture rate of all possible intervals constructed by this method.<br /><br />Next, I had half the room keep the confidence level at 95% and investigate the change of sample size on the width of the interval. The other half of the room kept the sample size the same and investigated what happened as the confidence level changed. If I had students change both variables at the same time, they may not have fully understood the relationship between confidence level and interval width and sample size and interval width. Focusing on just one variable at a time was an important part of their discovery of the relationships. Below are screenshots of the applet and what students discovered. <br /><br />Changing Sample Size, Keeping Confidence Level Constant : Larger Sample, Narrower Intervals<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-YQx7bReoPR8/WnMkBsXGRNI/AAAAAAAAGYU/WQ_iGBMPAegvojmXpxjQF3Tr78ovRCuJwCEwYBhgL/s1600/Screen%2BShot%2B2018-02-01%2Bat%2B9.24.57%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="452" data-original-width="905" height="316" src="https://3.bp.blogspot.com/-YQx7bReoPR8/WnMkBsXGRNI/AAAAAAAAGYU/WQ_iGBMPAegvojmXpxjQF3Tr78ovRCuJwCEwYBhgL/s640/Screen%2BShot%2B2018-02-01%2Bat%2B9.24.57%2BAM.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-rEiyMmzxiQY/WnMly0mDq1I/AAAAAAAAGYg/mLPOh1vJmGMBF3RoCdQLLbL0V2tIvlgowCLcBGAs/s1600/Screen%2BShot%2B2018-02-01%2Bat%2B9.35.16%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="437" data-original-width="908" height="307" src="https://4.bp.blogspot.com/-rEiyMmzxiQY/WnMly0mDq1I/AAAAAAAAGYg/mLPOh1vJmGMBF3RoCdQLLbL0V2tIvlgowCLcBGAs/s640/Screen%2BShot%2B2018-02-01%2Bat%2B9.35.16%2BAM.png" width="640" /></a></div><br /><br />Changing Confidence Level, Keeping Sample Size Constant: Larger Confidence Level, Wider Intervals<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-xABaqA3uuLg/WnMmeDzGkkI/AAAAAAAAGYo/H1EeOV3a_nQzYImdnxC8GLJKNjCKAN4BgCLcBGAs/s1600/Screen%2BShot%2B2018-02-01%2Bat%2B9.38.19%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="441" data-original-width="906" height="310" src="https://3.bp.blogspot.com/-xABaqA3uuLg/WnMmeDzGkkI/AAAAAAAAGYo/H1EeOV3a_nQzYImdnxC8GLJKNjCKAN4BgCLcBGAs/s640/Screen%2BShot%2B2018-02-01%2Bat%2B9.38.19%2BAM.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-H_4uGij_5Tk/WnMkBnIg7sI/AAAAAAAAGYQ/mHifGpPurW4A05hSC-QJj9blRnZ-DwcQwCEwYBhgL/s1600/Screen%2BShot%2B2018-02-01%2Bat%2B9.24.24%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="440" data-original-width="873" height="322" src="https://2.bp.blogspot.com/-H_4uGij_5Tk/WnMkBnIg7sI/AAAAAAAAGYQ/mHifGpPurW4A05hSC-QJj9blRnZ-DwcQwCEwYBhgL/s640/Screen%2BShot%2B2018-02-01%2Bat%2B9.24.24%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;">I believe that building an informal understanding without any formulas leads to a better understanding of how the formulas work to create wider or narrower intervals. Tomorrow we toss Hershey Kisses to create our first intervals for the proportion of kisses that would land bottom side down when tossed.</div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0