tag:blogger.com,1999:blog-12678455589222470452017-07-23T01:57:47.043-07:00Math, etc.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.comBlogger22125tag:blogger.com,1999:blog-1267845558922247045.post-49340709743399263602014-05-28T08:37:00.004-07:002014-05-28T08:38:40.192-07:00Follow-Up: Letting Students' Curiosity Steer a LessonIn <a href="http://mtpberg.blogspot.com/2014/04/letting-students-curiosity-steer-lesson.html" target="_blank">an earlier post</a>, I wrote about exploring a student's question during class, and letting the subsequent questions of the whole group drive the remainder of the lesson. We were looking at critical points and inflection points with the first and second derivative, and my students had noticed that for several of our examples, the function's inflection point was the midpoint between the two critical points.<br /><div><br /></div><div>A couple of weeks after that lesson took place, I found a day for the class to just explore that question - when is the inflection point the midpoint of the critical points? We split the board into two sections and began recording functions that satisfied that condition and ones that didn't. My students set to work with their graphing calculators to create and explore functions. Here are some questions that arose during their work:</div><div><ul><li>What do you have to do to get two turning points?</li><li>Is it just true for all cubic functions?</li><li>Is it possible to find a quartic, or quintic function with an inflection point at the midpoint of two critical points?</li></ul><div>Eventually, most of the class gave up on quartic and higher degree functions, and the focus shifted mainly to cubics. Several times a cubic function was added to the "doesn't fit" column, but after double checking these we found that all of them had inflection points directly between their midpoints. So, by the end of the class period, we had a white board full of cubic functions that showed this critical-inflection point relationship, and no counterexamples. We were inclined to assume it was true for all cubic functions.</div></div><div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-i4B2Z-HIzh8/U3zB6gXijyI/AAAAAAAAARY/FNl0I_7XxTc/s1600/Critical+Inflection.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://1.bp.blogspot.com/-i4B2Z-HIzh8/U3zB6gXijyI/AAAAAAAAARY/FNl0I_7XxTc/s1600/Critical+Inflection.png" height="244" width="320" /></a></div><br /></div><div>In retrospect, it seems like a pretty obvious result, at least as far as the <i>x</i>-values are concerned. A cubic function's first derivative is quadratic, and the vertex of a parabola (where the zero of the second derivative will occur) is always halfway between its zeros. I wasn't certain about the <i>y</i>-values, though. To be sure, I wrote up a proof (<a href="https://drive.google.com/file/d/0B3QrlSzRpxztSmVtVjg1VVFhQVk/edit?usp=sharing" target="_blank"><span style="color: blue;">here's a link to that proof</span></a>). I began with a generic, standard form cubic function and carried out the differentiation to find the first and second derivatives, and then used algebra to find the zeros of those derivatives and the accompanying <i>y</i>-values for the function.</div><div><br /></div><div>Unfortunately, I didn't have a class period to spend working through a proof of this result with my class. The work is pretty accessible for students who have had Algebra II or more and some basic Calculus -- derivatives using the power rule, quadratic formula, and some binomial theorem for plugging numbers back in to the original cubic function. I made my proof available to my class for anyone who was interested. They were satisfied to know that their hypothesis was true at least for cubics, but they were still making conjectures about when the same relationship might be true in quartics and higher degree functions, which is great.</div><div><br /></div><div>Maybe I didn't word it well, but when I did a Google search to see if there was anything out there about this relationship between critical and inflection points, I didn't find much. I found at least one exploration worksheet that guided students to notice the relationship for some specific examples, but not much else. Does anyone out there know if this is a commonly known relationship? Maybe it's too "obvious" to be worthy of mention in most textbooks. I didn't notice it right away.</div><div><br /></div><div>I'm still curious, and so are my students, if this ever happens in higher degree functions. And if so, when? If anyone can point me in the direction of a resource that might help me learn more about it, I'd be grateful. Thanks!</div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-3884860234653660102014-04-19T13:23:00.001-07:002014-04-19T20:04:47.690-07:00Instructional Dialogue - Math AnxietyOne of our readings for Math 629 struck me as good material for my instructional dialogue. Jackson and Leffingwell's (1999) <i>The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College</i> left me asking myself whether or not I exhibited any of the anxiety producing behaviors their article listed. They wrote about <i>overt</i> and <i>covert</i> behaviors that teachers display that contribute to students' anxiety or otherwise send negative messages to students. While I could tell myself that I was innocent of many of the behaviors described, there were some that I had to think twice about, like relying on prerequisite knowledge, and even saying things like, "You have done this before in Algebra I." Even if that's not said with a condescending tone, it's disconcerting for any student who doesn't remember how to do whatever "this" is. Overall, the article left me concerned that I might, even if its usually unintentional, be doing or saying things that add to students' math anxiety, or make them feel worse about their ability level. After reading the article, I'm worried about an ill-timed sigh or furrowed brow might have a big impact on my students' comfort and confidence levels, or their willingness to ask questions.<br /><br />With these things in mind, I asked one of my fellow math teachers to observe a class period. Before the observation, I asked him to be mindful of this list of questions and concerns:<br /><br /><ul><li>Does Matt display any behaviors that might increase students' math anxiety? (Examples might include expressed frustration at repeated questions, gestures or mannerisms that suggest annoyance, avoidance of certain questions, lack of eye contact, etc.)</li><li>Does Matt do anything that might empower one gender or another to be more or less vocal and participatory during class. (Does he call on one gender more often? Does he not do enough to get responses from a variety of students, allowing two or three to dominate? Does one gender seem to dominate discussion?)</li><li>Do you see anything in general that Matt could improve upon or that he is doing well?</li></ul><br /><br />The lesson my friend observed was on graphing rational functions. We were taking generic graphs with asymptotes and intercepts plotted (no scale or numbers) and sketching in the shapes of the graphs following some graphing "rules" that we had discussed for rational functions (for example: the curve can only pass through the <i>x</i>-axis at an <i>x</i>-intercept; the curve cannot pass through a vertical asymptote, but instead must go to positive or negative infinity as it approaches one).<br /><br />The lesson went well, unusually so. A wider variety of students than usual were volunteering to answer questions than usual, and just about an even balance between male and female students. There was a lot of discussion, and even some arguing about the mathematics. Students were asking insightful questions about the behavior of the graphs and what causes it. It was a fun class period. I was reminded again the next day how well things had gone, because the class seemed kind of flat by comparison.<br /><br />My friend didn't notice any of the behaviors or tendencies I had listed in my questions for him. He kept a tally of responses from different genders, and again, it was almost an even split. He also noted that my questioning of all students was consistent in difficulty, and that I wasn't guiding with my questions. He said it appeared as if I have great rapport with my students and that they seemed very comfortable with me, asking questions and offering responses. Another thing that he made note of that he liked was that I took students' suggestions of how to sketch a portion of a graph, drew it that way on the board whether it was right or wrong, and then asked the class whether they agreed or disagreed. On some days my class might have gotten frustrated with me for not being direct about right and wrong answers, but on that day they seemed to embrace it and liked the added discussion.<br /><br />While I was relieved that my friend didn't notice any of the behaviors I was worried about, since things went so well I am left wondering why they did, in hopes that I could have more class discussions like that. I don't know that I can take any credit for how well it went - most of that is probably due to my students and their interest and willingness to discuss - but here are some things that I think may have helped:<br /><br /><ul><li>Having another teacher in the room, especially someone who's there just to watch you teach, really makes you bring your "A game", I think. I'm not sure exactly how that affected me, but it probably made me relish the good discussion that was happening, and be more thoughtful about the questions and answers I was offering.</li><li>Rational functions was a new topic for my class, one that they hadn't had much prior exposure to in earlier classes. I think this leveled the playing field a bit and contributed to a wide variety of students taking part in the discussion. The graphing rules that we had may have also empowered them to argue with each other a little more, too, rather than just waiting to see if I said the answers were right or wrong.</li><li>The list of questions I had given my friend to look for was fresh in my mind, and as such I was especially mindful of how I was asking and answering my questions, the tone of voice I was using, and the mix of students I was calling on. I was even thinking about my use of eye contact when I asked questions. I think you can sometimes draw out a response to a question from one student or one section of the room by directing your eye contact at them while you ask, and looking at them during the wait time. Or maybe it just makes them nervous.</li><li>It might have just been one of those days when things are going to go well, and I was lucky enough to have another teacher there to witness the good discussion.</li></ul><br />After all, I really appreciated the instructional dialogue process. I think just picking something for one of my peers to look at and thinking about that ahead of time improved my teaching a bit. Having someone else in the room to observe my teaching helped me to remember what my A game looks like, and left me challenged to try to put forth my best effort every day, whether another teacher is watching or not. Even though the lesson that my friend observed went well, his observation notes left me with more to think about and some things to keep working on and building on. I'm hoping I can do more instructional dialogues with my colleagues in the future.<br /><br />Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-34362926198976991702014-04-12T23:30:00.002-07:002014-04-23T10:46:01.038-07:00Math 629 Project Update - Improving TransferFor my Math 629 project, I am doing some work that will hopefully contribute to my master's project. What I am hoping to produce is a working draft of the literature review portion for my project. In my foundations and curriculum development courses at Grand Valley, I have done some work addressing the problem of poor transfer ability in mathematics students, and higher order thinking skills in general. I am planning to do the same for my master's project.<br /><br />To transfer learning is to take what you have learned and apply it to something new or different. I am often surprised at the difficulty my students have with math problems that are only slightly different than ones they have encountered previously. It's an issue that is very disconcerting to me, too, because I want their learning to have value. If they can't do anything with it beyond a narrow set of examples, it's not very valuable. Another immediate concern is the Smarter Balanced Assessment, which will (likely) replace the Michigan Merit Exam. It will demand a lot more flexibility and transfer ability from students.<br /><br />To address this problem in my master's project, I have in mind a capstone unit that would fit at the conclusion of an algebra II course. The unit would engage students in problem-based learning and cooperative group work, with a lot of metacognitive reasoning structured into the activities. I do think that all of these things would be best applied throughout a course, but for my project a capstone unit seemed appealing to me, since it offers more opportunities to build connections between a variety of algebra II topics.<br /><br />I have found a number of good primary research articles to support those three elements (problem-based learning, cooperative group work, and metacognitive reasoning) as a means of addressing the problem of low transfer ability. One in particular has shaped my approach to the problem more than any other one article. It addresses the problem, and ties together most of the ingredients I am applying in my proposed solution. Kramarski and Mevarech (2003) studied the effects of cooperative group work and metacognitive instruction on secondary students' mathematics reasoning and ability. Their study compared a control group to three other treatment groups, one of which underwent cooperative learning, another received metacognitive instruction, and another underwent a combination of the two. The results showed better outcomes in the group that received both treatments than in any other group. This group provided more correct explanations for their reasoning, and they outperformed their peers on tasks designed to assess transfer ability. In an earlier study, Kramarski, Mevarech, and Arami (2002) found that metacognitive instruction improved student performance on both standard mathematics tasks and authentic tasks.<br /><br />Both of these studies are well designed with decent sample sizes, and they look at secondary math students, which is my focus. Their main drawback with respect to my project is that they are Israeli studies, which raises the question of external validity. But they are more robust than most of the other studies I have found on cooperative learning and/or metacognitive instruction, and their control-group design also makes me more confident that I could extend their results to my own setting. As I look for and gather more sources, though, I am hopeful that I can find a few more domestic studies touch on the same topics.<br /><br /><br />Referenced Sources:<br /><br /><div class="MsoNormal" style="line-height: 200%;"><span style="font-family: "Times New Roman","serif";">Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effects of cooperative learning and metacognitive training. <i>American Educational Research Journal, 40</i>(1), 281-310.<o:p></o:p></span></div><div class="MsoNormal" style="line-height: 200%;"><span style="font-family: "Times New Roman","serif";"><br /></span></div><div class="MsoNormal" style="line-height: 200%;"><span style="font-family: "Times New Roman","serif";">Kramarski, B., Mevarech, Z.M., & Arami, M. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks. <i>Educational Studies in Mathematics, 49</i>, 225-</span><span style="font-family: 'Times New Roman', serif; line-height: 200%;">250.</span></div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com1tag:blogger.com,1999:blog-1267845558922247045.post-86005322248227833392014-04-11T19:59:00.001-07:002014-04-11T19:59:27.098-07:00Letting Students' Curiosity Steer a LessonA few weeks ago in my calculus class, I was introducing the use of the first and second derivatives to find a function's critical points and inflections points. For our first example, we took a look at <a href="http://1.bp.blogspot.com/-vPTwPB9XD4Q/U0Nre4nAgKI/AAAAAAAAAPw/pt9-6ukayi4/s1600/Blog+Pictures.png" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-vPTwPB9XD4Q/U0Nre4nAgKI/AAAAAAAAAPw/pt9-6ukayi4/s1600/Blog+Pictures.png" /></a>. We first graphed the function using <a href="http://www.desmos.com/" target="_blank">Desmos</a>, and then set about finding its derivatives. The first and second derivatives are <a href="http://1.bp.blogspot.com/-__ZZzKlZkhU/U0NsUZ1dbvI/AAAAAAAAAP4/miBAGHAS5Pc/s1600/Blog+Pictures.png" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-__ZZzKlZkhU/U0NsUZ1dbvI/AAAAAAAAAP4/miBAGHAS5Pc/s1600/Blog+Pictures.png" /></a>and <a href="http://3.bp.blogspot.com/-Pf-m2rBiITU/U0NsjOh8CYI/AAAAAAAAAQA/KbHWgC0QHKw/s1600/Blog+Pictures.png" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-Pf-m2rBiITU/U0NsjOh8CYI/AAAAAAAAAQA/KbHWgC0QHKw/s1600/Blog+Pictures.png" /></a>, which are pretty easy to work with when finding the zeros. The critical points of the function are (0, 4) and (2, 0), and the inflection point is (1, 2).<br /><div class="separator" style="clear: both; text-align: center;"><img border="0" src="http://4.bp.blogspot.com/-tgvy9pNEFKI/U0NudEWRYBI/AAAAAAAAAQM/weg8RBhSA2w/s1600/Blog+Pictures.png" height="208" width="320" /></div><div class="separator" style="clear: both; text-align: left;">I was about ready to move on to a second example, when one of my students asked, "Is the inflection point always going to be the midpoint of the critical points?" I acknowledged that it was a great question, but started to explain right away why I was pretty sure the answer was no, knowing that a higher odd degree function could be created to have two critical points with any variety of different behaviors in between.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">What was I thinking?! Here I had a perfect opportunity, a student-initiated reason to do and pay attention to more examples, and the first thing I did was to dismiss the possibility that the conjecture was true. Halfway through my explanation I realized that I was making a mistake, but fortunately for me, my students were undeterred. Instead of dropping the point, another student refined the conjecture, and suggested, "Maybe it only happens when the function displays its maximum number of possible turning points," like a cubic function with two turning points, or a quartic function with three turning points, etc.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I really didn't know whether that conjecture was true or not, but I had also finally realized how valuable this student inquiry was. We let the question be the drive for our next few examples. We took a look at another cubic function next, fairly similar to the first, and once again the inflection point was the midpoint between the two critical points. Since our class time was running short, we decided to move on to a quartic example. We used Desmos to set up a quartic graph with sliders for the coefficients, which made it easy to manipulate and create a function with three turning points.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The quartic function we looked at was <a href="http://3.bp.blogspot.com/-NxckR30TMrY/U0N4edaB2zI/AAAAAAAAAQc/64j9L6M8e1E/s1600/Blog+Pictures.png" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-NxckR30TMrY/U0N4edaB2zI/AAAAAAAAAQc/64j9L6M8e1E/s1600/Blog+Pictures.png" /></a>, which did turn out to be a counterexample for our class' conjecture. Class was just about over, so we didn't have time to look at more examples, but even as they were packing up, a few of my students were still throwing out possible modifiers on the conjecture.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">That class period reminded me, and demonstrated to me in new ways, how valuable it can be to run with students' questions. The question about the critical and inflection points' relationships to one another had most of the class curious. It provided an <b>"intellectual need"</b> for more examples. It even added some <b>suspense </b>to the remaining examples we covered. The whole class was <b>engaged </b>in exploring whether the conjecture was true, or otherwise when it might be true. This in turn meant they were <b>thinking critically</b> about the topic, and asking questions that got beyond the plain mechanics of how to find and plot critical points and inflection points.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Seeing the value in that experience, I have tried to be more open to exploring student questions, particularly with my calculus class. They have always asked more questions than most of my other groups, but it seems they have been asking even more, as we take more time to consider them in class. I am finding that I have to be selective again, though, about which bigger questions we take time for. There are some that have more potential to lead to enhanced learning than others, and sometimes the "others" need to be left for another time for the sake of covering a new concept. But maybe my priorities still need some adjusting?</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">As for the question of when inflection points coincide with the midpoint of two critical points, I still don't know the answer, but we have noticed it in many examples since that class period. Maybe it's a commonly known theorem, but I don't want to Google it yet. I'm hoping to find a day yet this year for my class to explore the question to see if we can find some commonalities in the functions that behave that way. It would be a good exploratory math experience for them.</div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com4tag:blogger.com,1999:blog-1267845558922247045.post-30756169077550618872014-03-05T10:40:00.001-08:002014-03-05T10:40:09.233-08:00Polar Functions in a Daily Desmos Challenge<span style="font-family: 'Times New Roman', serif;">I was intrigued by <a href="https://www.desmos.com/" target="_blank">Desmos</a> when it was introduced in class, so I was interested in spending a little more time with it in my math work. Desmos is an online graphing calculator with a lot of neat features.</span><span style="font-family: 'Times New Roman', serif;"> We took a look at <a href="http://dailydesmos.com/" target="_blank">dailydesmos.com</a> in class, which is a website that posts Desmos-generated graphs as challenge problems for other Desmos users to try. </span><span style="font-family: 'Times New Roman', serif;">On that particular night of class, </span><i style="font-family: 'Times New Roman', serif;">Daily Desmos #285</i><span style="font-family: 'Times New Roman', serif;"> was up on the website, and caught my eye as an interesting graph and a fun challenge.</span><br /><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><br /></span><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-ngCi2mQXWNI/Uxdhz6fFVQI/AAAAAAAAAM8/npUQy2zbzUY/s1600/desmos.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-ngCi2mQXWNI/Uxdhz6fFVQI/AAAAAAAAAM8/npUQy2zbzUY/s1600/desmos.png" height="205" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Daily Desmos</i> #285 (Advanced)</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"></div><span style="font-family: 'Times New Roman', serif;">B</span><span style="font-family: 'Times New Roman', serif;">y the look of it, I assumed that this was probably a polar function.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">In our precalculus class, we go over some of these – I think the ones we do are called cardioids and limaçon curves, and spirals as well – but I had never seen one that looked quite like this.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">I played around with some polar functions on Desmos to try to refresh my memory on how they work.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">I tried</span><a href="http://3.bp.blogspot.com/-cwlbvIJoFlg/UxditE4TjgI/AAAAAAAAANM/zygZUCwt2Q8/s1600/Untitled1.png" imageanchor="1" style="font-family: 'Times New Roman', serif; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-cwlbvIJoFlg/UxditE4TjgI/AAAAAAAAANM/zygZUCwt2Q8/s1600/Untitled1.png" /></a>a<span style="font-family: 'Times New Roman', serif;">nd then </span><a href="http://3.bp.blogspot.com/-RQvvyjgHvaQ/UxdjDFgDZrI/AAAAAAAAANY/w5Eo-egEyYE/s1600/Untitled2.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-RQvvyjgHvaQ/UxdjDFgDZrI/AAAAAAAAANY/w5Eo-egEyYE/s1600/Untitled2.png" /></a>a<span style="font-family: 'Times New Roman', serif;">nd then </span><a href="http://2.bp.blogspot.com/-yShzVra00Z0/UxdjDJrFgAI/AAAAAAAAANc/xpfLgSdmsuQ/s1600/Untitled3.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-yShzVra00Z0/UxdjDJrFgAI/AAAAAAAAANc/xpfLgSdmsuQ/s1600/Untitled3.png" /></a>,<span style="font-family: 'Times New Roman', serif;"> t</span><span style="font-family: 'Times New Roman', serif;">he last of which produces a circle centered at the origin, with radius 1.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">I decided this circle equation was the one to begin manipulating, since it does not pass through the origin, or pole.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">Having each of those terms squared seemed like it would be a key feature for this rule no matter how it turned out, since </span><i style="font-family: 'Times New Roman', serif;">r</i><span style="font-family: 'Times New Roman', serif;"> looks to stay positive throughout the graph. If <i>r</i> <i>did</i> ever change from positive to negative, the graph would have pass through the pole.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span style="font-family: "Times New Roman","serif";">The <i style="mso-bidi-font-style: normal;">Daily Desmos</i> graph begins at about</span><a href="http://3.bp.blogspot.com/-SCS6TZTlbrI/Uxdk_6KAyYI/AAAAAAAAAOY/TPfKxURwLjc/s1600/Untitled4.png" imageanchor="1" style="font-family: 'Times New Roman', serif; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-SCS6TZTlbrI/Uxdk_6KAyYI/AAAAAAAAAOY/TPfKxURwLjc/s1600/Untitled4.png" /></a><span style="font-family: "Times New Roman","serif";">.<span style="mso-spacerun: yes;"> </span>Since it goes out about that far, and just a little farther (maybe 2 units?), when </span><a href="http://1.bp.blogspot.com/-r1Ye98s9_3k/UxdlAPSox_I/AAAAAAAAAPQ/1B4JOqTP_wY/s1600/Untitled5.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-r1Ye98s9_3k/UxdlAPSox_I/AAAAAAAAAPQ/1B4JOqTP_wY/s1600/Untitled5.png" /></a>,<span style="font-family: "Times New Roman","serif";"> I figured I would not add a constant to my rule.<span style="mso-spacerun: yes;"> </span>I thought that would make the graph lop-sided, and while it's not actually symmetric, it's close (can you be <i>close</i> to symmetric?).<span style="mso-spacerun: yes;"> </span>Instead, I decided to try a multiplier of 17 on the </span><a href="http://4.bp.blogspot.com/-mqArGGTck6Y/UxdlAWNcCBI/AAAAAAAAAO8/G9dyFZ3fjWc/s1600/Untitled6.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://4.bp.blogspot.com/-mqArGGTck6Y/UxdlAWNcCBI/AAAAAAAAAO8/G9dyFZ3fjWc/s1600/Untitled6.png" /></a><span style="font-family: "Times New Roman","serif";">term of my rule.<span style="mso-spacerun: yes;"> </span>Made a similar decision for the </span><a href="http://1.bp.blogspot.com/-BrbR1kV6QVE/UxdlA1p8B-I/AAAAAAAAAO4/hogNkHk0sT8/s1600/Untitled7.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-BrbR1kV6QVE/UxdlA1p8B-I/AAAAAAAAAO4/hogNkHk0sT8/s1600/Untitled7.png" /></a><span style="font-family: "Times New Roman","serif";">term of my rule.<span style="mso-spacerun: yes;"> </span>Since the </span><a href="http://3.bp.blogspot.com/-vP43yO2FAlU/UxdlBeD-qLI/AAAAAAAAAPI/kDNxU3EP88c/s1600/Untitled8.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-vP43yO2FAlU/UxdlBeD-qLI/AAAAAAAAAPI/kDNxU3EP88c/s1600/Untitled8.png" /></a><span style="font-family: "Times New Roman","serif";">term would drop out to 0 on the <i style="mso-bidi-font-style: normal;">y-</i>axis whenever </span><a href="http://1.bp.blogspot.com/-_oMqiAKT7qg/UxdlBzEWvVI/AAAAAAAAAPU/m9LokGBUXUs/s1600/Untitled9.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-_oMqiAKT7qg/UxdlBzEWvVI/AAAAAAAAAPU/m9LokGBUXUs/s1600/Untitled9.png" /></a><span style="font-family: "Times New Roman","serif";">is a multiple of </span><a href="http://2.bp.blogspot.com/-qLlWnLO3aUI/Uxdk-Bu_s9I/AAAAAAAAAN0/Goy_7UMWb4g/s1600/Untitled10.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-qLlWnLO3aUI/Uxdk-Bu_s9I/AAAAAAAAAN0/Goy_7UMWb4g/s1600/Untitled10.png" /></a><span style="font-family: "Times New Roman","serif";">, the </span><a href="http://3.bp.blogspot.com/-Qa6YETBWeww/Uxdk-AIPZRI/AAAAAAAAAN4/bBCZUsnB_ew/s1600/Untitled11.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-Qa6YETBWeww/Uxdk-AIPZRI/AAAAAAAAAN4/bBCZUsnB_ew/s1600/Untitled11.png" /></a><span style="font-family: "Times New Roman","serif";">would be the main player at those points.<span style="mso-spacerun: yes;"> </span>The first of these points on <i>Daily Desmos #285</i> is about </span><a href="http://2.bp.blogspot.com/--3FVRzE2F8M/UxdlASP4G3I/AAAAAAAAAOs/Vqh6SOqdA0k/s1600/Untitled12.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/--3FVRzE2F8M/UxdlASP4G3I/AAAAAAAAAOs/Vqh6SOqdA0k/s1600/Untitled12.png" /></a><span style="font-family: "Times New Roman","serif";">, but I decided that the multiplier on </span><a href="http://1.bp.blogspot.com/-GdX0vJRNX8Q/Uxdk-fF_wnI/AAAAAAAAAOA/D-1ZoLQ_7FE/s1600/Untitled13.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-GdX0vJRNX8Q/Uxdk-fF_wnI/AAAAAAAAAOA/D-1ZoLQ_7FE/s1600/Untitled13.png" /></a><span style="font-family: "Times New Roman","serif";">should be just a little bit lower than 4, due to the spiral effect going on in this graph.</span></div><div class="MsoNormal"><br /><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">From teaching precalc, I knew that a function of the form </span><a href="http://2.bp.blogspot.com/-qqcu5TNRnIs/Uxdk-vBXMtI/AAAAAAAAAOQ/s-UKW7a-1to/s1600/Untitled14.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-qqcu5TNRnIs/Uxdk-vBXMtI/AAAAAAAAAOQ/s-UKW7a-1to/s1600/Untitled14.png" /></a><span style="font-family: "Times New Roman","serif";">would produce a spiral, so I knew I would need a term like this as well.<span style="mso-spacerun: yes;"> </span>Looking at the successive passes of the curve on one axis at a time gave me the idea that this term should be increasing <i style="mso-bidi-font-style: normal;">r</i> values by a little bit less than 5 for each full rotation.<span style="mso-spacerun: yes;"> </span>I decided to try 4.6 as my first estimate.<span style="mso-spacerun: yes;"> </span>To figure out what <i style="mso-bidi-font-style: normal;">k</i> should be for this term, I solved</span><a href="http://2.bp.blogspot.com/-jnVZhqJAgMc/Uxdk-7lh-NI/AAAAAAAAAOI/bG5aR3TmXv4/s1600/Untitled15.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-jnVZhqJAgMc/Uxdk-7lh-NI/AAAAAAAAAOI/bG5aR3TmXv4/s1600/Untitled15.png" /></a><span style="font-family: "Times New Roman","serif";">for <i style="mso-bidi-font-style: normal;">k</i> to get </span><a href="http://1.bp.blogspot.com/-Imja4viv9m4/Uxdk_GXH3dI/AAAAAAAAAOU/K4micswJRM8/s1600/Untitled16.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-Imja4viv9m4/Uxdk_GXH3dI/AAAAAAAAAOU/K4micswJRM8/s1600/Untitled16.png" /></a><span style="font-family: "Times New Roman","serif";">.</span><br /><span style="font-family: "Times New Roman","serif";"><br /></span><span style="font-family: "Times New Roman","serif";"><br /></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">With these guesses and estimates, I put together my first attempt at really matching the graph:<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-viVnYkLZuFg/Uxdk_iEE90I/AAAAAAAAAOo/P8n1QW0xITE/s1600/Untitled17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-viVnYkLZuFg/Uxdk_iEE90I/AAAAAAAAAOo/P8n1QW0xITE/s1600/Untitled17.png" /></a></div><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><span style="mso-text-raise: -14.0pt; position: relative; top: 14.0pt;"></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"></div><span style="font-family: 'Times New Roman', serif;">Here’s the result:</span></div><div class="MsoNormal"><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-EN2Z9_wiRak/Uxdhz6jLgHI/AAAAAAAAANA/QtTVMmN6Uy0/s1600/my+desmos+attempt.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><table 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://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2F3.bp.blogspot.com%2F-EN2Z9_wiRak%2FUxdhz6jLgHI%2FAAAAAAAAANA%2FQtTVMmN6Uy0%2Fs1600%2Fmy%2Bdesmos%2Battempt.png&container=blogger&gadget=a&rewriteMime=image%2F*" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-EN2Z9_wiRak/Uxdhz6jLgHI/AAAAAAAAANA/QtTVMmN6Uy0/s1600/my+desmos+attempt.png" height="198" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">My Attempt</td></tr></tbody></table></a><a href="https://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2F3.bp.blogspot.com%2F-EN2Z9_wiRak%2FUxdhz6jLgHI%2FAAAAAAAAANA%2FQtTVMmN6Uy0%2Fs1600%2Fmy%2Bdesmos%2Battempt.png&container=blogger&gadget=a&rewriteMime=image%2F*" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: left;"></a></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="background-color: transparent; float: left; margin-right: 1em; text-align: left;"></table><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;"><img border="0" src="http://4.bp.blogspot.com/-ngCi2mQXWNI/Uxdhz6fFVQI/AAAAAAAAANE/-WUgPOx82eE/s1600/desmos.png" height="205" style="margin-left: auto; margin-right: auto;" width="320" /></td></tr><tr><td class="tr-caption" style="text-align: center;">The <i>Daily Desmos</i> Challenge</td></tr></tbody></table><br /><span style="height: 186px; margin-left: 0px; margin-top: 10px; mso-ignore: vglayout; position: absolute; width: 301px; z-index: 251663355;"><br /></span><span style="height: 190px; margin-left: 323px; margin-top: 6px; mso-ignore: vglayout; position: absolute; width: 296px; z-index: 251662331;"></span><span style="font-family: "Times New Roman","serif";"></span></div><div class="MsoNormal"><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1267845558922247045" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: 'Times New Roman', serif;">I was pretty satisfied that I had at least discovered the right form for my polar function.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">All it would take to get things to match more accurately would be to adjust my constants a little bit.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><br /></span><span style="font-family: "Times New Roman","serif";">In retrospect, I didn’t take advantage of Desmos as an exploratory tool very well. If you type in a function in Desmos with letters in place of numbers, it can automatically create sliders for the would-be constants. It might have made more sense to set up a rule with sliders for the constants, once I had decided on a form for the rule. That is, I would have entered </span><a href="http://2.bp.blogspot.com/-z_OlMVutqHo/UxdpowhqxJI/AAAAAAAAAPc/DyPm-yzuoCs/s1600/Untitled18.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Times New Roman', serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-z_OlMVutqHo/UxdpowhqxJI/AAAAAAAAAPc/DyPm-yzuoCs/s1600/Untitled18.png" /></a><span style="font-family: "Times New Roman","serif";">into Desmos. That would have made it easier to investigate different values for each constant. At the same time, it was good to do some analysis of the points and make educated guesses.</span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><br /></span><br /><span style="font-family: Times New Roman, serif;">This was a fun way for me to review some polar functions concepts, but it would also be a great activity for my precalculus students in the future. Providing students with laptops, or going to a computer lab, and then presenting the with graphs like this would be a great way for them to test and expand their understanding at the close of a polar functions unit. Something similar could certainly be done in other classes and units as well, with other types of functions. If computers aren't readily available, this could be done with graphing calculators as well. Taking the time to do this activity deepened and reinforced my understanding of polar functions and how they work, and got me thinking about the features of several types of polar graphs. I think it could do the same for my students.</span></div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com1tag:blogger.com,1999:blog-1267845558922247045.post-30282675387301612952014-02-26T02:55:00.003-08:002014-02-26T02:55:41.432-08:00Using "The Letter Game" to Introduce Deductive Systems and Proof in GeometryA new trimester will begin in just a couple of weeks, and I will have a new bunch of geometry students. Before long, we'll be digging in to geometric proof, which is a concept/activity/process that we will develop and use throughout most of the course. It is always a tough concept to develop with students, and I am always wondering how I can make it easier for them to grasp.<br /><br />One really nice, fairly accessible activity that we have used with students at my school is "The Letter Game", which was published by Don Gernes in his article <i>The Rules of the Game </i>(full citation below). The game amounts to a very simple deductive system, in which students are asked to "prove" various statements. It gives students an opportunity to practice working within the bounds a deductive system, and to get a sense for the structure of proofs and the process of proving something. This comes before they have to confront the added complexity of geometry, which is the big deductive system they ultimately will be learning to work within.<br /><br />Before getting to the "The Letter Game", Gernes suggests having students think about games that they are already familiar with: monopoly, basektball, soccer, etc. Each of these games has <i style="font-weight: bold;">undefined terms</i>, <i style="font-weight: bold;">defined terms</i>, <i style="font-weight: bold;">postulates</i>, and maybe even some <i style="font-weight: bold;">theorems</i>. These are the makings of a deductive system. Gernes presents basketball as an example, and here's what he lists in each of these categories:<br />______________<br /><b>Undefined Terms</b><br />Ball<br />Player<br />Court<br />Baskets<br /><br /><b>Defined Terms</b><br />Field Goal<br />Foul<br />Free Throw<br />Traveling<br /><br /><b>Postulates</b><br />If a player is fouled, then the player gets to shoot a free throw.<br />If a player travels, then the other team gets possession of the ball.<br />If a player makes a field goal, then the player's team gets two points.<br /><br /><b>Theorem</b><br /><u>The referee objectively applies the rules of the game to each play.</u><br /><br />This obviously doesn't represent a complete list of all of the terms and rules that make up the game of basketball, but it is enough to have a discussion about what the undefined terms, etc. are, and how each of them contributes to the structure of a deductive system.<br /><br />The Letter Game is where students get a chance to try their hand at some proofs. Gernes keeps the system simple, so students don't have too much to keep track of as they begin proving "theorems". Here's the deductive system he establishes:<br />______________<br /><b>Undefined Terms</b><br />Letter M, I, and U<br /><br /><b>Definition</b><br />x means any string of I's and U's.<br /><br /><b>Postulates</b><br />1. If a string of letters ends in I, you may add a U at the end.<br />2. If you have Mx, then you may add x to get Mxx<br />3. If 3 I's occur, that is, III, then you may substitute U in their place.<br /><u>4. If UU occurs, you drop it.</u><br /><br />With these four postulates, students take given strings of letters and then "prove" another string of letters, using one postulate at a time to manipulate the string. For example - Given: MIII Prove: M.<br /><br />Even though many of my students end up groaning about geometric proofs, most of them end up enjoying the letter game. It is accessible enough that most of them are able to make some headway, and after a quick example or two, most of the class is able to work independently. I usually don't get any students reverting to "shut down" mode because they can't get it. Sometimes I even have students finished early who create their own new letter "theorems" and challenging one another to prove them. While the activity doesn't make them masters of geometric proof, it does reinforce the concept of a deductive system, and what it is to work in and prove new theorems in a deductive system. Every step that is taken or statement that is made needs to be backed up by a definition, postulate, or theorem.<br /><br />Opening up the much more broad deductive system of geometry, with all of its definitions, postulates, and theorems, always makes geometric proof more difficult for my students. The Letter Game hasn't eliminated those difficulties for my geometry students, but it has been a good first experience with proof for them. Gernes' activities build a nice segue from concepts that students are familiar and comfortable with to the way geometry works as a system. His article is definitely worth a look:<br /><br />Gernes, D. (1999). The rules of the game. <i>The Mathematics Teacher</i>, <i>92</i>(5), 424-429.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com1tag:blogger.com,1999:blog-1267845558922247045.post-28066234926723073442014-02-05T10:22:00.002-08:002014-02-05T10:22:50.225-08:00An Introductory GeoGebra Activity for Calculus StudentsMy calculus students had never used <a href="http://www.geogebra.org/" target="_blank">GeoGebra</a> before this activity, so I wanted to do something fairly simple with them. This activity takes about five steps on GeoGebra, so it wasn't too demanding for first time users. At the same time, it provided some nice visual support for the concepts we had been learning in class, and provided them with a tool to explore those concepts further. I've tried to make the instructions below friendly for first time GeoGebra users, and I'm a novice anyway.<br /><br /><h3><b><i>Setting Within the Course</i></b></h3>When I brought my class to the computer lab for this activity, my calculus students had already been using the limit definition of a derivative for three days. We had also been discussing the derivative at a point as the function's instantaneous rate of change, and the slope of a line tangent to the function at that point. When we had opportunities to do so, we were taking special notice of times when the derivative was zero and the tangent line was horizontal. We had also spent just a little bit of time more generally discussing the relationships between graphs of functions and the graphs of their derivatives, including a few mentions of concavity. Furthermore, after working with the limit definition for several days, my students had noticed and we had informally discussed the power rule.<br /><br /><h3><b><i>The Activity</i></b></h3>With these concepts in mind, we did the following together on GeoGebra:<br /><br />1) First, we entered a function in the <b>Input </b>bar at the bottom of the window. The function we started with was <b><i>y</i> = <i>x</i>^3+3<i>x</i>^2-1</b>. In the algebra pane, GeoGebra relabels this as <i>f</i>(<i>x</i>). This function provided opportunities to talk about horizontal tangent lines (critical points) as well as changes in concavity (inflection points). At this point I also showed my students how to use the Move Graphics tool to center their graph in the window, and to adjust the scale on each axis to get a good picture of the function.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-jOx5BJQpQus/UvJOcz-7MwI/AAAAAAAAALA/UDYlhY_NYBQ/s1600/5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-jOx5BJQpQus/UvJOcz-7MwI/AAAAAAAAALA/UDYlhY_NYBQ/s1600/5.png" height="297" width="400" /></a></div><br />2) With our function graphed, we added a new point on the curve by selecting the <b>Point</b> tool and clicking on the curve. GeoGebra automatically labels this point <b>A</b>.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Pi3JhnCfygM/UvJOGl5YA2I/AAAAAAAAAK4/BT2JEOxJGfI/s1600/4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-Pi3JhnCfygM/UvJOGl5YA2I/AAAAAAAAAK4/BT2JEOxJGfI/s1600/4.png" height="297" width="400" /></a></div><br />3) We then put a tangent line on the curve at point A. GeoGebra has a tool for this under the fourth box from the left. Once the <b>Tangents</b> tool is selected, click on the curve and click on the point of tangency (point <b>A</b> for us), in no particular order. This is a good time to show students how to use the <b>Move</b> tool if they haven't discovered it already. It's the cursor at the upper left of the tool bar (or pressing the Esc key will select this tool). If students click and hold point A with the move tool, they can slide it along the curve and watch the tangent line change.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-EDvhoOTUJC4/UvJNgTtIpRI/AAAAAAAAAKg/PhDywjee20g/s1600/2.5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-EDvhoOTUJC4/UvJNgTtIpRI/AAAAAAAAAKg/PhDywjee20g/s1600/2.5.png" height="297" width="400" /></a></div><br />4) Now we'll have GeoGebra measure the slope of the tangent line (the value of which is that of the derivative). GeoGebra has a variety of measuring tools. Once the <b>Slope</b> tool is selected, click on the the tangent line and GeoGebra will add a rise/run triangle and display the slope. The value of the slope gets added to the algebra pane as <b>m</b>.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-RMycUlw7De8/UvJO-19bOMI/AAAAAAAAALM/2vGRmC4QNp4/s1600/6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-RMycUlw7De8/UvJO-19bOMI/AAAAAAAAALM/2vGRmC4QNp4/s1600/6.png" height="297" width="400" /></a></div><br />5) We finally add one more point to the sketch, using the <b>Input</b> bar again. Type in <b>(x(A),m)</b> to define this last point, which GeoGebra will call point <b>B</b>. This tells GeoGebra to use the <i>x</i>-value from point A (the point on the curve) and to plot the slope of the curve as the <i>y</i>-value. This point displays the value of the graph's derivative at any <i>x</i>-value. By right-clicking on point <b>B</b>, <b>Trace On</b> can be selected. <br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-MAxvm81QCVg/UvJQHuPoDJI/AAAAAAAAALg/ghnUZTZ2kA4/s1600/8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-MAxvm81QCVg/UvJQHuPoDJI/AAAAAAAAALg/ghnUZTZ2kA4/s1600/8.png" height="297" width="400" /></a></div><br />With trace turned on, the path of this point will be traced out on the sketch as point <b>A</b> is moved along the curve, revealing the shape of the derivative function. (I also changed the color of point <b>B</b> using the pull-downs at the top of the graphics view.)<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-XYH0RC9BdZI/UvJVcrJZhdI/AAAAAAAAAMM/UY_ZNZVniIY/s1600/10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-XYH0RC9BdZI/UvJVcrJZhdI/AAAAAAAAAMM/UY_ZNZVniIY/s1600/10.png" height="297" width="400" /></a></div><br /><h3><b><i>Discussion</i></b></h3><div>The completed sketch opens up a lot of good talking points. Students should be able to see that the local maximum and local minimum of <i>f</i>(<i>x</i>) line up with the zeros of the derivative function. If inflection points have been discussed, it's also nice to see that this point lines up with the vertex of the derivative. As point <b>A</b> is moved back and forth across the inflection point, the tangent line makes a nice rocking motion that shows the slope is beginning to increase/decrease. In general, the tangent line in the sketch provides a good visual connection between the curve of <i>f</i>(<i>x</i>) and the sign of the derivative.</div><div><br /></div><div>Of course, students can also try to predict the shape of the derivative function, or find the derivative function algebraically and plot it on the sketch as well. Wherever possible, it seems good to keep students' visual and algebraic understandings closely intertwined. (Check out Aspinwall and Shaw's (2002) article, <i>When Visualization is a Barrier to Mathematical Understanding</i>, in Mathematics Teacher vol. 95, no. 9)</div><div><br /></div><div>This GeoGebra sketch is also a good tool for students to explore derivatives for functions that they don't yet know how to differentiate algebraically. Helping my students work through this activity took less time than I had anticipated, so they had about ten minutes of class time in the computer lab after we were done with the first function. After I explained that they could change <i>f</i>(<i>x</i>) by double-clicking it in the algebra pane without altering any of the other features of the sketch, most of them started exploring other functions. Students were discovering on their own that cosine is the derivative of sine, and one of them was looking at the graph of tangent, and noticing similarities (but important differences) between the sketched derivative and the secant function. Having a tool available to explore derivatives increased their curiosity. I may pull up the same sketch again in class as we learn about the derivatives of specific types of functions.</div><div><br /></div><div>If you would like to go to the completed sketch, here is the <a href="http://www.geogebratube.org/material/show/id/83330" target="_blank">GeoGebra Tube link</a>.</div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com2tag:blogger.com,1999:blog-1267845558922247045.post-10787519797017254922014-01-20T21:59:00.000-08:002014-01-20T21:59:44.980-08:00My Teaching SituationI am currently teaching Geometry, Algebra II, and Calculus (non-AP). Aside from my Calculus class, my classes are all fairly sizable, at 26, 28, and 30 students. Calculus, on the other hand, has only 10 students in it, so I get to know those students especially well. There were more signed up at the beginning of the year, but I think many of those students made the jump to AP Calc before the term began.<br /><br />We are on trimesters, with 5 class periods a day (we call them "macros"). Each class period is 72 minutes long. We have a great building and a lot of resources available. I have a document camera and projector in my room. We do not have one-to-one technology, but there are laptop carts and computer labs available when I have a <a href="http://www.geogebra.org/" target="_blank">GeoGebra</a> activity to do with my students.<br /><br />One aspect of my teaching situation that has been a little bit challenging at times is the distribution of our math department throughout the building. There are two other math teachers in my wing of the building, but the other eight of them are spread out in other locations. When I am teaching the same class with those other teachers, I have to be deliberate about checking in with them and collaborating with them. There would be a lot more impromptu collaboration if the departments in our building were arranged by wing. I know others in Math 629 have mentioned much more real isolation, though, so this is only a minor challenge.<br /><br />Another challenge, although I know it's not the least bit unique, has always been time. I feel this particularly in Geometry. Both halves of our Geometry course fit very tightly within a trimester.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-91798053299476651782013-12-09T14:38:00.002-08:002013-12-09T14:38:42.147-08:00Euclid's Elements - Book IIIAt this point, I am finished with my project - a translation of Book III of the elements with links to GeoGebra manipulatives (no proofs at this point, however). Please take a look, and feel free to leave feedback.<div><br /></div><div>You can use the link to the right under Pages, or <a href="http://mtpberg.blogspot.com/p/euclids-elements-book-iii.html" target="_blank">click here</a>.</div><div><br /></div><div>If you'd like to just take a look at the collection of my GeoGebra worksheets for the project, <a href="http://www.geogebratube.org/collection/show/id/6993" target="_blank">click here</a>.</div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-22445534212632985802013-12-09T14:32:00.006-08:002013-12-09T14:32:58.350-08:00Big Take-Aways from Math 641As the semester wraps up, here are some of the big things I will take away from Math 641, Modern Geometry:<br /><br /><i>Get students asking questions.</i><br />As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.<br /><br /><i>GeoGebra is an excellent tool and resource</i>.<br />I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.<br /><br /><i>There are <b>far </b>more ways to present and explore mathematical ideas than I am currently aware of.</i><br />That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like <i><a href="http://www.nctm.org/publications/toc.aspx?jrnl=mt" target="_blank">Mathematics Teacher</a></i>, resources like the <a href="http://mathtwitterblogosphere.weebly.com/" target="_blank">mathtwitterblogosphere</a> are another great place to find new ideas.<br /><br /><i>Geometry and mathematics in general are far more broad than most students get a chance to see</i>.<br />Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing <i>this </i>with my students, they would be way more engaged, and a few more of them might even decide they like math. I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical. Maybe I'll have to keep up a good bulletin board for once.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-23283231428998538412013-12-09T08:49:00.001-08:002013-12-09T08:49:18.639-08:00Learning GeoGebra - Part 2Working on <a href="http://mtpberg.blogspot.com/2013/12/project-update.html" target="_blank">my project for Math 641</a> has forced me to learn a lot more about <a href="http://www.geogebra.org/" target="_blank">GeoGebra</a>. Some of the manipulatives for the propositions in Book III were quite simple, but others had a lot going on. Here are some of the new things that I learned:<br /><br /><u>How to define a figure made up of two or more other objects</u><br />I first ran into this issue when I was trying to make a picture for a circular segment. I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color. After a few Google searches, I found instructions for how to create a <i>list</i>. In the input line at the bottom of the GeoGebra window I used set braces to enter the following, for example<br /><br /> {Segment[A,B], Arc[c,B,A]}<br /><br />This would define the figure bounded by line segment AB and the arc on circle c between B and A. It showed up as a <i>list</i> in the algebra pane, which I could select and give a color to for emphasis. Of course, different types of figures could be included in a list, but I used these mainly to create segments, since my work was with circles.<br /><br /><br /><u>More varied ways of assigning conditions to show objects</u><br />In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures. For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same. I learned how to put measures other than a slider value in the conditions to show object line in object properties. I used the distance between points,<br /><br /> Distance[A,B]<br /><br />frequently. I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles. If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example<br /><br /> abs(a-b)<0.1<br /><br />As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added. Most of the time, I had constructed completely separate segments with the different colors or marks. It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!). However, it was still nice to be able to change line styles and thicknesses in addition to color.<br /><br /><br />I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now. They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.<br /><br />If you know of easier ways to do the things described above, I'd love to know! Please leave a comment.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-54808362445226590692013-12-09T06:59:00.004-08:002013-12-09T06:59:48.233-08:00An Interesting IdeaThe basic idea is this: How to motivate students to help one another improve their understanding.<br /><br />In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes. Here are some basics about the structure of our homework that I need to mention before I describe the system further:<br /><br />- We have two-day homework. That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.<br />- Actual homework (what is done out of the book) is not part of students' calculated grade. On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn. They may use their homework as a reference on daily assessments.<br />- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points. They have to complete some additional practice problems first. We call the retakes "reworks".<br /><br />At the start of a unit, my colleague creates groups of 3 to 4 students. He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together. On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson. While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups. So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.<br /><br />The especially interesting part of this group system is the incentive my colleague attaches to it. He calculates each group's overall average grade at the start of the unit, and again at the end of the unit. If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.<br /><br />The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty. After all, the students with the lowest grades are the ones that have the most room for improvement. Alternatively, it's hard to bring a 98% up a whole lot. Helping a struggling group member is the most promising way of earning those extra credit points. Teaching is the great way to learn, so there's a benefit to the excelling students also. My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.<br /><br />I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores. In some form or another, though, this sounds like a great idea. Does anyone else out there have a good method for getting students to invest in each others' learning?Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-27243694999413999882013-12-03T11:43:00.003-08:002013-12-03T11:43:43.010-08:00Project UpdateFor my Math 641 project, I am working on a "translation" of Book III of Euclid's Elements. As my source for the Elements, I am using <a href="http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/bookIII.html">Joyce's Elements webpage</a>. It is obviously in English already, so by translate I mean that I am trying to use more modern terminology for the things described in Book III. It has been a challenge though, particularly for some of the longer, multi-faceted propositions. It is difficult to know how to state some of those more clearly, without getting much too wordy (as I'm prone to do).<br /><br />For visuals to accompany the definitions, and for manipulatives to accompany each proposition, I am using GeoGebra. I am still a rookie when it comes to GeoGebra, so it is probably taking me longer to develop these things than most, but I have been surprised at how time consuming making these manipulatives has been. I've been learning a lot about how to set conditions to show objects, and having to think carefully about what conditions I can set in a diagram to emphasize each proposition. I had originally imagined that I would make an illustrated step-by-step GeoGebra file for the proofs of each proposition as well, but that might be an ongoing task for some time in the future.<br /><br />To organize and display my work, I am adding an additional page to this blog per Dr. Golden's suggestion. I had considered using Google Sites, but after looking that over it seemed just as well to lay out the definitions and propositions in the blog, with links to the GeoGebra pages for each manipulative.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-8066048013648674002013-11-26T13:56:00.003-08:002013-11-26T13:56:34.587-08:00Compass and Straight-Edge Constructions<div class="separator" style="clear: both; text-align: left;">Compass and straight-edge constructions are a part of my school's curriculum for Geometry A. The constructions that we teach and assess include:</div><div class="separator" style="clear: both; text-align: left;">- copying a segment</div><div class="separator" style="clear: both; text-align: left;">- copying an angle</div><div class="separator" style="clear: both; text-align: left;">- perpendicular bisector of a segment</div><div class="separator" style="clear: both; text-align: left;">- bisecting an angle</div><div class="separator" style="clear: both; text-align: left;">- constructing a line perpendicular to a given line through a point not on the line</div><div class="separator" style="clear: both; text-align: left;">- constructing a line parallel to a given line through a given point</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I personally enjoy doing and teaching these constructions. It's hands on, and I enjoy seeing how they work out. I'm intrigued by all that can be done with just a compass and straight-edge, especially when it comes to things like <a href="http://www.geogebratube.org/material/show/id/49815">quadrature</a> and more complicated constructions. I like the history of it, too, and it seems amazing what early geometers and mathematicians were able to figure out with these tools.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I wonder sometimes what value these constructions have for my students, though. Some of the basic constructions can be used to draw other things that we study, like a triangle with certain lengths or angles, or the incenter of a triangle, although we usually have them explore concepts with difficult constructions like that on GeoGebra. I'm not sure whether these types of constructions really enhance students' visual-spatial reasoning relative to the things they are constructing either. I know I could do a better job emphasizing the fixed distance feature of arcs, but even if I did that, I'm afraid most of my students would still learn these as mechanical processes.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Eric Pandisico, in his article <i>Alternative Geometric Constructions: Promoting Mathematical Reasoning</i> suggests that students can take the most away from constructions when they have the chance to try them with a variety of tools. Rather than stick primarily to compass and straight-edge constructions, Pandisico promotes the use of several tools to do the same constructions, including Miras, a simple right angle source, like a notecard, and even a two-edged straight-edge. He suggests that because each tool has the potential to emphasize a different feature or property of each construction, doing one construction three different times with different tools can give students a deeper and more full understanding of the geometry underlying each construction.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">We only teach these constructions with a compass and straight-edge. We don't use any other methods for creating these things. After trying these constructions and others with patty paper in Math 641, I could see the value in using more than one method. The patty paper made the symmetry of the constructions much easier to see, and even though the results were the same, the thinking I was doing about each construction was much different, and for some of them more intuitive.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Classmate Trevor Kuzee did a nice job of comparing and contrasting patty paper and compass/straight-edge constructions in <a href="http://kuzeemath.blogspot.com/2013/10/patty-paper-vs-compassstraight-edge.html">this blog post</a>. I'm not necessarily drawn toward one method over another, but I will try to give my students a chance and using more than one method the next time I teach constructions.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">To help my students review the compass and straight-edge constructions that they had learned, I made the <a href="http://www.geogebratube.org/material/show/id/50690">GeoGebra worksheet</a> pictured below. This didn't represent much new GeoGebra learning for me. It was a lot of assigning values to show objects again, and there were lots of hidden circles underneath those arcs. I did set points to which I fixed the text boxes for the instructions, which allowed me to keep those well aligned even between constructions.</div><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;"><a href="http://4.bp.blogspot.com/-_dF0--jonDw/Uo5HcyVXqxI/AAAAAAAAAHU/nZK2QZnribM/s1600/Geometry+A+-+Constructions+Review.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-_dF0--jonDw/Uo5HcyVXqxI/AAAAAAAAAHU/nZK2QZnribM/s1600/Geometry+A+-+Constructions+Review.gif" height="322" width="640" /></a></div><br /><div>Reference:<br />Padisico, Eric A. (2002) Alternative geometric constructions: Promoting mathematical reasoning. <i>The Mathematics Teacher, 95</i>(1), 32-36.</div><div><br /></div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-9165731911562643242013-11-20T13:40:00.001-08:002013-11-20T13:40:17.093-08:00Reflective Properties of Conic SectionsOne of the things I find most interesting about conic sections is their reflective properties. Here are those properties, as I understand them (for the sake of a common description, I'll suppose a "beam" is bouncing off each conic section):<br /><br />1) Any beam parallel to the axis of a parabola will be reflected to its focus. Or, any beam leaving the focus of a parabola will be reflected to a path that is parallel to the axis.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-8KRIG8gqgT0/Uo0qIuyy3SI/AAAAAAAAAG0/QdDBez7uZWM/s1600/Parabola.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-8KRIG8gqgT0/Uo0qIuyy3SI/AAAAAAAAAG0/QdDBez7uZWM/s1600/Parabola.png" height="135" width="200" /></a></div><br />2) Any beam that passes through or leaves one focus of an ellipse will be reflected to pass through the other focus.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-g-m7S7frPWc/Uo0qIGgiICI/AAAAAAAAAGs/_96b44MU6XE/s1600/Ellipse.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-g-m7S7frPWc/Uo0qIGgiICI/AAAAAAAAAGs/_96b44MU6XE/s1600/Ellipse.png" height="138" width="200" /></a></div><br /><br />3) Any beam directed toward one focus of a hyperbola will be reflected toward the other focus.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-uxTLlpCrkYs/Uo0qIDQ6oYI/AAAAAAAAAGw/44kOlbUeTn8/s1600/Hyperbola.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-uxTLlpCrkYs/Uo0qIDQ6oYI/AAAAAAAAAGw/44kOlbUeTn8/s1600/Hyperbola.png" height="200" width="162" /></a></div><br />These are a nice talking point when teaching students about conic sections, since they provide some interesting reasons for locating the foci or finding equations for sections with specific foci or focal distances. Flashlights and satellite dishes are familiar objects for kids, and it's always nice to have an answer to the question "Who uses this stuff?"<br /><br />One of these days I'd like to team up with the engineering teacher at my school and make a model that demonstrates these properties. I imagine setting up pieces of conics with shared foci, as sketched out below. Our technical education department has a computerized lathe, so I think we could get the appropriate grooves cut into a piece of particle board and then put a thin strip of reflective material in each one. There would be a track for a laser pointer that runs perpendicular to the axis of the parabola, so the beam could move back and forth and always be parallel to the axis. If it were set up correctly, the beam would always hit the last focus (the second focus of the hyperbola) even as the laser pointer is moved back and forth. I'm not sure how tight the precision would have to be, but I guess that last focus could just be as big as necessary.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-SRNo6GHBkbo/Uo0qIMEt9BI/AAAAAAAAAHE/qjLYo7XOVgw/s1600/Diagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-SRNo6GHBkbo/Uo0qIMEt9BI/AAAAAAAAAHE/qjLYo7XOVgw/s1600/Diagram.png" height="315" width="400" /></a></div><br />Maybe students wouldn't get as excited about this as I would, but I think it would be a pretty cool demonstration for class. Here's <a href="http://www.geogebratube.org/material/show/id/58734">my attempt at creating the same in GeoGebra</a>, but I'm not sure how to plot only part of a conic section yet.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-18100160810292941492013-11-16T22:55:00.001-08:002013-11-16T23:04:09.332-08:00Teaching ProofsAs I mentioned in <a href="http://mtpberg.blogspot.com/2013/09/van-hieles-levels-of-geometric-reasoning.html">my first blog post</a>, on Van Hiele levels, I have some misgivings about the way proofs are presented in the curriculum from which my geometry classes are taught. The work that we have been doing in Modern Geometry this semester has opened my eyes to the value of <a href="http://mtpberg.blogspot.com/2013/10/conjecturing.html">conjecturing</a>. Ideally, proof in my students' geometry classes would develop naturally out of discussion and defense of their own conjectures. Instruction on how to construct proofs more formally could follow.<br /><br />As it is, my students face formal geometric proof head on. They start by writing "algebraic proofs," which basically entails setting up and solving equations with justifications (maybe a theorem or two, mostly properties of equality) for each step. Then they practice proofs that are more based in geometry concepts. Early on most of their work is filling in blanks in two-column proofs (which is the format we stick to, for the most part), providing missing statements and reasons. The next chapter in the course includes parallel and perpendicular lines, and proofs are included again. The big finish for proofs in Geometry A (although there are more in Geometry B) is triangle congruence proofs.<br /><br />Proofs are always a struggle, for the majority of my students anyway. To help them along, I try to point out common patterns in the structure of the proofs we write in class, such as "<i>the final statement is whatever you were asked to prove</i>" or "<i>to change a word into an equation or an equation into a word, use its definition.</i>" I try to remind them to keep the end result in mind, and sometimes we even start at the end of a proof and work backwards a few steps. A lot of these things that I do to help students "get through" proofs don't really get at the meaning involved, and to a large extent I often feel I'm just teaching them the rules of a game.<br /><br />It was hard not to be somewhat encouraged though, when by the end of our chapter on triangle congruence, many of my students were able to put together a pretty solid triangle congruence proof on their own. It made me wonder if teaching formal proof head on might have some value. The big question, of course, was whether or not my students could transfer any of the skills they were demonstrating to more varied proofs. I got a partial answer to that question in my class' response to a bonus question on one of their unit tests.<br /><br />The bonus problem asked them to write a proof of the Triangle Sum Theorem (the sum of the angle measures is 180 degrees). Not every student attempted the proof. Among those who did, only two gave fairly conceptually solid proofs, and theirs weren't entirely rock solid as far as our formal standards from class went. Below is a copy of the problem on the test (I think more could have been said of what is "Given:"), and the two best proofs from my class.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Y9BH1yYKAbo/UohiqKYueOI/AAAAAAAAAGU/-smB_B15DwQ/s1600/pic+1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-Y9BH1yYKAbo/UohiqKYueOI/AAAAAAAAAGU/-smB_B15DwQ/s1600/pic+1.png" height="175" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-m8uCbcU8Et8/UohiqGa0kWI/AAAAAAAAAGM/2lomKzwJxSE/s1600/pic+2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-m8uCbcU8Et8/UohiqGa0kWI/AAAAAAAAAGM/2lomKzwJxSE/s1600/pic+2.png" height="296" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-65rPxiynLvM/UohiqORCYBI/AAAAAAAAAGQ/OYMhESDqx7w/s1600/pic+3.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-65rPxiynLvM/UohiqORCYBI/AAAAAAAAAGQ/OYMhESDqx7w/s1600/pic+3.png" height="168" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">For only two students to come this close took some wind out of my sails again. But at the same time, it is still the slightest bit encouraging that a couple of them were able to go "off pattern" and prove something fairly different from the triangle congruence cases we had been working with (maybe my hopes for students' proof fluency are way too low).</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So I still think that a heavily conjecture-based lead in to proof would serve more of my students better. But having gone through the usual proof sequence with my students once again, I don't believe what they've done will have <i>no </i>value for them as they continue in mathematics. If they aren't able to transfer their proof writing to more novel proofs now, might some ground work still have been laid for future proof writing? It seems like writing proofs formally will always be somewhat awkward the first time, even if it is preceded by lots of practice with informal arguments.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I will have many of the same students again next trimester in Geometry B, and the first unit is on quadrilaterals and their properties. We are planning to give students a few days to hopefully conjecture some of these properties on their own and then try to prove them with what they already know. This will be another opportunity to see how much they were able to take away from their education in proofs this trimester.</div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-85754861666280936232013-10-29T12:22:00.002-07:002013-10-29T12:22:47.887-07:00Review of "Selecting and Creating Mathematical Tasks: From Research to Practice"Schwan Smith, M., and Stein, M.K. (1998). Selecting and creating mathematical tasks: From research to practice<i>.</i> <i>Mathematics Teaching in the Middle School, 3</i>(5), 344-350.<br /><br />This article stresses the importance of thoughtfully selecting/assigning/creating mathematical tasks for students. The tasks presented to students should facilitate the intended learning, and to be sure that happens, teachers need to pay close attention to the level of cognitive demand tasks will potentially place on students. Low demand tasks may help students learn math facts and develop procedural fluency, but tasks with greater complexity and even procedural ambiguity have the potential to engage students in high level reasoning and connection-making.<br /><br />It would be very interesting to have the "Task Sort" discussing with my own math department.<br /><br /><br /><br />- Here's an example of a low-level task from my precalculus class:<br /><br />Rewrite | pi - sqrt(5) | without the absolute value symbol.<br /><br /><br />- Here's an example of a high-level task from precalculus from one of our problem solving days:<br /><i><br /></i>A line <i>tangent</i> to a circle is a line that intersects a circle at one point and is perpendicular to the radius at the point of intersection.<br />The line 2x-3y=-11 is tangent to a circle at (2, 5).<br />The line 3x+2y=29 is tangent to the same circle at (7, 4).<br />Find the standard form equation of the circle.Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-29189437503347419692013-10-27T20:19:00.000-07:002013-10-27T20:19:51.717-07:00Making an AmbigramOne of our assignments for class this week was to do some math art, and one of the suggestions was to try making an <a href="http://en.wikipedia.org/wiki/Ambigram">ambigram</a>. Ambigrams are words or phrases that can be read in more than one direction, or from different points of view. In class, we had looked at several examples by <a href="http://www.scottkim.com/inversions">Scott Kim</a>. Having never made an ambigram before, I decided to try what seems to be the simplest type - one that reads forward and backward (upside down).<br /><br />After thinking about the geometry involved for this particular type of ambigram, I understood that I was aiming to make a word that would look the same under a 180 degree rotation. With this in mind, I tried to think of words whose first and last letters might bear some similarities under rotation. Obviously, every letter would have to have its rotated partner, but I had to start (and end) somewhere. <i>Geometry</i> seemed like it could work well. It wasn't too much of a stretch to imagine a capital <i>G</i> that looked something like a <i>y</i>.<br /><br />Having settled on a word to try, I listed out the pairs of letters that would have to be rotations of one another. In retrospect, I suppose the best ambigrams don't necessarily match one letter with one other letter, but carefully weave shapes and features even between "letters" to achieve the desired effect. To keep it simple for myself, though, I tried to work with one letter pair at a time: <i>G</i> and <i>y</i>, <i>e</i> and <i>r</i>, <i>o</i> and <i>t</i>, <i>m</i> and <i>e</i>.<br /><br />I filled up the margins of my notepad with sketches of my attempts at making symbols for these pairs. I basically tried to write each letter with the other in mind, and then I would rotate my notebook around 180 degrees and add more marks to the letter to make it look like the other one. There were lots of attempts that I didn't use. The challenge felt like walking a fine line between ambiguity and specificity in what letter(s) each symbol represented.<br /><br />I had the hardest time with <i>e</i> and <i>r</i>. I was tempted to just borrow Scott Kim's solution to this one in his <a href="http://www.scottkim.com/inversions/gallery/superteacher.html">"SuperTeacher"</a> inversion, but I kept trying and came up with something else that worked. My solution to <i>m</i> and <i>e</i>, right in the middle of the ambigram, was to go with a basic lowercase <i>m</i> and turn it 45 degrees. Not fancy, but again, it worked. Once I had each symbol figured out (four of them altogether, since it was two letters for each), I set about drawing a large copy of the ambigram.<br /><br /><a href="https://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2F4.bp.blogspot.com%2F-0S2KSICw04g%2FUm3SbfnFLOI%2FAAAAAAAAAFc%2FoCcWxXLfsRk%2Fs1600%2FDSC_0253.JPG&container=blogger&gadget=a&rewriteMime=image%2F*" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-0S2KSICw04g/Um3SbfnFLOI/AAAAAAAAAFc/oCcWxXLfsRk/s1600/DSC_0253.JPG" height="209" width="320" /></a><a href="http://4.bp.blogspot.com/-tbSIFKzBjD0/Um3SaQI33TI/AAAAAAAAAFU/gstl3R_71Z4/s1600/DSC_0251.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-tbSIFKzBjD0/Um3SaQI33TI/AAAAAAAAAFU/gstl3R_71Z4/s1600/DSC_0251.JPG" height="130" width="200" /></a>To draw it accurately, I decided to mark the center of a piece of paper and then to draw out half of the word (the first, and the last half :) ) on one side of this center. Then I used an architect's scale to make a connect-the-dots copy of this half as a reflection through the center point. The resulting point symmetry is the same thing as 180 degree rotation symmetry. This was tedious, keeping the ruler lined up through the center and mirroring distances for lots of points. While I worked I kept thinking of more efficient ways to do this, like scanning the image and rotating it on the computer, or photocopying it, or even just tracing it on another piece of paper and attaching the two. But for the sake of having my finished product completely hand drawn, and on one, intact piece of paper, I persisted.<br /><br />The finished product turned out fairly well. Here are two photographs of the completed ambigram. I lined up the corners of the paper with the ruler and the pencil the same way in each, but the paper does get rotated 180 degrees. Fairly similar?<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-swrPa5QGLi8/Um3Sj62vwPI/AAAAAAAAAF0/qU5uEJNklds/s1600/DSC_0255.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-swrPa5QGLi8/Um3Sj62vwPI/AAAAAAAAAF0/qU5uEJNklds/s1600/DSC_0255.JPG" height="262" width="400" /></a></div><a href="http://4.bp.blogspot.com/-1ub04Cu8B0M/Um3ShrD9WOI/AAAAAAAAAFs/4JldRn8eYAY/s1600/DSC_0254.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-1ub04Cu8B0M/Um3ShrD9WOI/AAAAAAAAAFs/4JldRn8eYAY/s1600/DSC_0254.JPG" height="263" width="400" /></a><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com1tag:blogger.com,1999:blog-1267845558922247045.post-37471157106933845812013-10-15T13:20:00.001-07:002013-10-15T13:20:01.539-07:00Geometry at ArtprizeMy wife's family is very serious about <a href="http://www.artprize.org/">Artprize</a>. Her parents and sister volunteer several times during the event. My wife and daughter and I usually make two to three, if not four, trips downtown during Artprize. The first day we went, I took our camera along, in hopes of finding some geometrically interesting art. I had hoped to notice some geometry that was very subtle yet deeply tied to the art. Most of what I photographed, though, was pretty obvious in its geometry. Nothing too profound here. And this is only from one visit, so I'm sure there were many more "geometric" entries that I never ran across. But here are a few things that I saw.<div><br /></div><div><a href="http://4.bp.blogspot.com/-DPQuZ1L3z38/Ul2I-yjt3zI/AAAAAAAAADA/kZrP_U4zXbE/s1600/Artprize+1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-DPQuZ1L3z38/Ul2I-yjt3zI/AAAAAAAAADA/kZrP_U4zXbE/s1600/Artprize+1.png" height="200" width="186" /></a>In the Gerald R. Ford Museum, one of the entries was <i>Reflection</i>, by Josemiguel Perera. Aptly named, I thought, since it has reflection symmetry across a horizontal. I added these circles to the picture to emphasize the concentric circles in each fan. The little circles are tangent to one another. Of course, then the larger circles with the same centers intersect. I thought it was interesting that connecting the points of intersection on the larger circles created a segment that passed through the point of tangency of the smaller ones. I suppose this would be the case any time two congruent larger circles are concentric with two smaller, congruent, externally tangent circles. That's a fairly specific situation, though. </div><div><br /></div><div><a href="http://2.bp.blogspot.com/-LHrMz4k4zSA/Ul2LIn_safI/AAAAAAAAADM/N8eXFWDnTaM/s1600/DSC_0100.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://2.bp.blogspot.com/-LHrMz4k4zSA/Ul2LIn_safI/AAAAAAAAADM/N8eXFWDnTaM/s1600/DSC_0100.JPG" height="131" width="200" /></a></div><div><a href="http://2.bp.blogspot.com/-JPw8UqyCKCE/Ul2XD-bFwjI/AAAAAAAAADo/5nf5ZAWQrEM/s1600/Sign+Fig+2.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://2.bp.blogspot.com/-JPw8UqyCKCE/Ul2XD-bFwjI/AAAAAAAAADo/5nf5ZAWQrEM/s1600/Sign+Fig+2.png" height="152" width="200" /></a>I also found some of the Artprize signs interesting. They suggest geometry at a pretty basic level, so this was no great discovery on my part - all circles, triangles, and squares. From the right side of the sign, this pair of shapes stood out to me (right end, middle row of figures). It's a quarter circle with an isosceles right triangle attached to the side, with legs equal to the radius of the circle. It's divided up differently than that, though, with a long diagonal across the middle. My drawing from GeoGebra shows three triangles, and all of them have the same area. Each of them has the radius of the sector as its height, and half of the radius as the base (the vertical radius is bisected by the diagonal - the two line segments are diagonals of a parallelogram). Each of the three triangles has an area of (pi*r^2)/4. The light blue portion of the this figure (as seen in the Artprize sign) has the same area as that of a full quarter sector of the circle with this radius. These are somewhat random musings, but it seems there is probably more in this picture that could be uncovered and might be interesting.</div><div><br /></div><div><a href="http://1.bp.blogspot.com/-ANxNl28XB5g/Ul2cxyVUkLI/AAAAAAAAAD4/9HYUperD0Vo/s1600/DSC_0129.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-ANxNl28XB5g/Ul2cxyVUkLI/AAAAAAAAAD4/9HYUperD0Vo/s1600/DSC_0129.JPG" height="131" width="200" /></a>At another venue, someone had put together a bunch of square Artprize signs into an interesting design. This got me thinking about squares inscribed inside one another, with vertices tangent to the midpoints of the outer squares' sides. It's related to the special, isosceles right triangles that show up when these midpoints are connected, and square roots of 2 canceling one another out, but there was an interesting relationship in these, too. As more inscribed squares were added, every second square would have its side length halved. In other words, the second inscribed square would have side lengths half as large as the outermost square. The side lengths between were, according to the special right triangle relationship, the outer square's side length times square root 2. I thought it was interesting that every second square had half the side length (and one fourth the area).</div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-A9ib8SEmv5o/Ul2eRz4erAI/AAAAAAAAAEE/6luexuZ8bEY/s1600/Sqrs+Tangent+at+Midpoints.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-A9ib8SEmv5o/Ul2eRz4erAI/AAAAAAAAAEE/6luexuZ8bEY/s1600/Sqrs+Tangent+at+Midpoints.png" height="305" width="400" /></a></div><div><a href="http://4.bp.blogspot.com/-a_NjsGjfpvo/Ul2fr5VE8UI/AAAAAAAAAEQ/O42ssPyDxJY/s1600/DSC_0121.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-a_NjsGjfpvo/Ul2fr5VE8UI/AAAAAAAAAEQ/O42ssPyDxJY/s1600/DSC_0121.JPG" height="211" width="320" /></a>There were some other geometric aspects of entries that caught my eye. <i>Silkwaves in the Grand</i>, by Al and Laurie Roberts consisted of flags in the Grand River. The flag poles were connected near their bases, which made it apparent that they were arranged in star formations - pentagons with triangles attached to each side. I wonder if there was a particular aesthetic reason this arrangement was chosen.</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><a href="http://1.bp.blogspot.com/-0-xdjpXPSz0/Ul2g-nHE7JI/AAAAAAAAAEc/reZpoa6lJeI/s1600/DSC_0125.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://1.bp.blogspot.com/-0-xdjpXPSz0/Ul2g-nHE7JI/AAAAAAAAAEc/reZpoa6lJeI/s1600/DSC_0125.JPG" height="131" width="200" /></a>At the B.O.B., <i>Through the Iris</i>, by Armin Mersmann was my favorite entry. It was impressive to me because I thought the pictures were photographs, but they were pencil drawings! I usually take for granted the perfect (or near perfect, considering my astigmatism) circular geometry of human eyes. It's pretty amazing, and it was nice to be reminded of it.</div><div><br /></div><div><br /></div><div>Most of my geometric noticing was fairly mundane, but it was fun to spend the day being attentive to the subject in my surroundings. Here are a few other Artprize entries that stood out as being more geometric than others:</div><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="http://4.bp.blogspot.com/-ape2NA17kOQ/Ul2h-OpxdoI/AAAAAAAAAEo/g0OBJuFmAt4/s1600/DSC_0102.JPG" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-ape2NA17kOQ/Ul2h-OpxdoI/AAAAAAAAAEo/g0OBJuFmAt4/s1600/DSC_0102.JPG" height="209" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Sacrifice</i>, by Tom Panei</td></tr></tbody></table><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-FgKv_Zbp7oY/Ul2iAVftHYI/AAAAAAAAAEw/NgA5fvYUWYg/s1600/DSC_0110.JPG" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-FgKv_Zbp7oY/Ul2iAVftHYI/AAAAAAAAAEw/NgA5fvYUWYg/s1600/DSC_0110.JPG" height="211" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Facing Al Aquaba</i>, by Maurice Jacobsen</td></tr></tbody></table><br /><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="http://1.bp.blogspot.com/-D3MxFiTH8LA/Ul2iDMiRmsI/AAAAAAAAAE4/pFjyQjJYCbQ/s1600/DSC_0131.JPG" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-D3MxFiTH8LA/Ul2iDMiRmsI/AAAAAAAAAE4/pFjyQjJYCbQ/s1600/DSC_0131.JPG" height="211" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Hilo - Sacred Geometry</i>, by Kimberly Toogood</td></tr></tbody></table><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-29EsdzAGP-Y/Ul2iGYdmFRI/AAAAAAAAAFA/UC3WCaazW7o/s1600/DSC_0127.JPG" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-29EsdzAGP-Y/Ul2iGYdmFRI/AAAAAAAAAFA/UC3WCaazW7o/s1600/DSC_0127.JPG" height="211" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">I failed to write down the exhibit name and artist for this one.<br />Nice to see pi in my search for geometry, but unfortunately<br />it was backward in every place it appeared in this exhibit.</td></tr></tbody></table><div><br /></div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com3tag:blogger.com,1999:blog-1267845558922247045.post-31372180492785877592013-10-08T13:55:00.003-07:002013-10-08T13:55:25.202-07:00Conjecturing<div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; line-height: 1.15; white-space: pre-wrap;">Our in class activities in Modern Geometry as well as several of the assigned readings have heightened my awareness of the absence of conjecturing in the geometry curriculum that I present to my students. In general, my students read or are presented with the postulates and theorems that are important to a topic, and then we work with them. Very rarely are they presented with a relationship that they are asked to uncover on their own.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In their article </span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: italic; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Geometry and Proof</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">, Battista and Clements (1995) suggest that students own conjectures are an ideal segue to the process of developing proofs. Providing students with an opportunity to discover a geometric relationship, and then leading them to defend their discoveries through class discussion and debate can demonstrate the importance of proof to students before they are asked to engage in it. The process of developing and testing a conjecture before proving it is also more true to real mathematical development. The article left me wanting to completely restructure the way proofs are presented in my geometry course, with ample time for exploration leading into informal proofs, and eventually some practice with formal deductive proofs. A progression like this would be more in line with <a href="http://mtpberg.blogspot.com/2013/09/van-hieles-levels-of-geometric-reasoning.html">Van Hiele‘s levels of geometric</a> reasoning as well. Letting conjecture be the impetus for proof would make the learning of proof more intrinsic for students, and it would also make their learning a more genuine mathematical experience.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The value of conjecture in geometry courses stood out to me again in Diemente’s (2000) article about the Euler line and its relationship to the sides of a triangle. The mathematics presented in the article are certainly interesting, but I was most struck by the fact that the content of the article was all prompted by students’ questions and conjectures. Taking the time to explore a students’ question about whether the Euler line can be parallel to a side of a triangle, Diemente and his class covered a wide swath of mathematical ground. Their exploration and proof included midpoints, slopes, systems of linear equations, and even conic sections. What a great example of how valuable students’ curiosity can be! Not only were these and other concepts used, reviewed, and learned, but they were shown in connection with one another, and moreover, with a real purpose! While an exploration like that of Diemente’s honors geometry class might be over the ability levels of my geometry students, it might be a great study for students in my precalculus class.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">So now I have a heightened sense of the value of providing students opportunities to make and investigate conjectures, and at the same time a greater awareness of the absence of such opportunities for my students. So now I am looking for places to create these opportunities. There are a couple on the horizon.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Tomorrow we will begin our unit on triangle congruence. Our department has an activity designed to help students discover the “shortcut” congruence theorems, like Side-Angle-Side, etc. It’s called “Triangle in a Bag”. Students can ask for one side length or angle measure of a certain triangle at a time, and the goal is to construct a triangle congruent to the one “in the bag” with as few given pieces as possible. The hope is that after doing this for several triangles, they might start to think critically about what key measurements are necessary to guarantee the same triangle, and develop the shortcut theorems. It’s a nice activity, but at the same time, I’m not sure if there’s much </span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: italic; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">learning</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> in it. A good class discussion might need to follow it, with some time for students to defend their conjectures as to why each set of three measures is enough. Any suggestions out there to make this activity better?</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The chapter following this one includes the centers of triangles (incenter, circumcenter, centroid, etc.). For the first time, our department is planning to get students working with this with the aid of <a href="http://www.geogebra.org/">GeoGebra</a>. I am looking forward to seeing how students respond, and I hope some of them will get excited about what they’re learning. Depending on how much time is available, it would be great to have students or groups of students share what they find about the relationships between these centers, or just the centers themselves. This might also be a good opportunity for some informal/formal proof writing.</span></div><br /><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"></span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">I took my class to the computer lab for the first time this past week just to do some basic GeoGebra constructions and find their way around the program. They liked it, and more than once I ran across students experimenting with lines and figures on the program - being curious. I hope I can find more opportunities to let them be curious about geometry and learn through their curiosity. For that matter, I hope I can do that for all of my math students, and not just my geometry students.</span><br /><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"><br /></span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"><u>References</u></span><br /><span style="font-family: Arial;"><span style="font-size: 15px; white-space: pre-wrap;">Battista, M.T., & Clements, D.H. (1995). Geometry and proof. <i>The Mathematics Teacher, 88</i>(1), 48-54.</span></span><br /><span style="font-family: Arial;"><span style="font-size: 15px; white-space: pre-wrap;"><br /></span></span><span style="font-family: Arial;"><span style="font-size: 15px; white-space: pre-wrap;">Diamente, D. (2000). Algebra in the service of geometry: Can Euler's line be parallel to a side of a triangle? <i>The Mathematics Teacher</i>, <i>93</i>(5) 428-431.</span></span>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com1tag:blogger.com,1999:blog-1267845558922247045.post-35808884092865910342013-09-22T12:58:00.000-07:002013-09-22T13:05:11.474-07:00Learning GeoGebraI am finally reaching the point at which I can put <a href="http://www.geogebra.org/" target="_blank">GeoGebra </a>to good use. I am getting to the steeper portion of the learning curve. I am sure I have much more to learn, still. But I have learned enough to make GeoGebra a useful tool for exploring geometry, and also for creating learning activities for my students.<br /><br />My first week of trying to use GeoGebra was frustrating. I was having a hard time making shapes that were adjustable and movable, or dynamic. At a basic level, I did not understand the <i>dependency </i>of points in Geogebra. Points can be free (completely movable), semi-dependent (only able to move along the object they are placed upon), and fully dependent (not movable, unless the object(s) they lie on move). I didn't realize that if I placed a point at the intersection of two lines, I wouldn't then be able to move those lines by moving that point. The point needed to come first. In my early attempts at figuring this out, I tried to make a rectangle with adjustable dimensions. I went back and forth between creating rectangles that were entirely rigid, to trying to use four free points and creating a quadrilateral that didn't maintain the properties of a rectangle.<br /><br />A Google search for how to make a dynamic rectangle in GeoGebra led me to a <a href="http://www.youtube.com/watch?v=kDTeEmc-lTQ" target="_blank"><span style="color: red;">YouTube video</span></a> presenting a challenge to do just that. The rectangle in the video was dynamic in three points, that is, three points could be moved and the shape would remain a rectangle. I had imagined that four points could be adjustable, so seeing that only three could move gave some more direction to my attempts, of which there were many. However, I still wasn't able to figure it out until I watched the<span style="color: red;"> <a href="http://www.youtube.com/watch?v=daR6tq63K_A" target="_blank"><span style="color: red;">longer challenge video</span></a></span>, which provided a key hint: one diagonal of the rectangle must lie on the diameter of a circle. Then I was able to figure out that the vertices on either end of that diagonal are the only free points on the rectangle. Another vertex needs to lie on the circle, and a right angle is guaranteed at that vertex, according to <a href="http://en.wikipedia.org/wiki/Thales%27_theorem" target="_blank"><span style="color: red;">Thales' Theorem</span></a>. Here's a link to the <span style="color: red;"><a href="http://www.geogebratube.org/material/show/id/49501" target="_blank"><span style="color: red;">rectangle</span></a> </span>on <a href="http://www.geogebratube.org/" target="_blank"><span style="color: red;">GeoGebra Tube</span></a>.<br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-4USYb7cI-Gc/Uj88jq_ckYI/AAAAAAAAACQ/Kwkz9Mauc9k/s1600/3pts+Dynamic+Rectangle.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-4USYb7cI-Gc/Uj88jq_ckYI/AAAAAAAAACQ/Kwkz9Mauc9k/s1600/3pts+Dynamic+Rectangle.png" height="234" width="320" /></a></td></tr><tr><td class="tr-caption" style="font-size: 13px; text-align: center;">My dynamic rectangle with semicircle shown.<br /><i>A</i> and <i>C</i> are freepoints, <i>B</i> is semi-dependent,<br />and <i>D</i> is fully dependent.</td></tr></tbody></table><br /><div>Working through the rectangle challenge problem helped me to understand how objects and figures behave and relate to each other in GeoGebra, and how to carefully put a sketch together to allow adjustment of certain dimensions or positions. Another thing that I particularly wanted to understand, especially for the sake of creating activities for my students, was how to show different steps in a GeoGebra worksheet.</div><div><br /></div><div>More Google searches on this topic helped me fumble my way through using <i>check boxes</i>. Several objects on a worksheet can be selected to be shown or hidden when a box is checked or unchecked. Here's a <a href="http://www.geogebratube.org/material/show/id/49497" target="_blank"><span style="color: red;">GeoGebra worksheet</span></a> that I created using check boxes. With the help of Professor John Golden, I learned how to use a <i>slider </i>to achieve the same results, but with more control of what is being shown or hidden. With a slider set to integer values, <i>conditions to show objects</i> could be assigned in the properties for each part of a drawing, according to the values on the slider. This also made it easier to have objects, like text, appear for only one step, or for several steps, without the need for unchecking a box again.</div><div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em;"><tbody><tr><td style="text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-shPPbz398rU/Uj9BGyAXkNI/AAAAAAAAACc/Uh5icruDgdw/s1600/Quadrature+Bk+II+Prop+14.png" height="103" style="margin-left: auto; margin-right: auto;" width="200" /></td></tr><tr><td class="tr-caption" style="font-size: 13px; text-align: center;">A worksheet with check boxes.</td></tr></tbody></table><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-QjueqCYH_IU/Uj9DiUTzp1I/AAAAAAAAACo/YSoo51eL6NA/s1600/Book+II+Prop+14+with+Proof.gif" imageanchor="1" style="clear: right; display: inline !important; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-QjueqCYH_IU/Uj9DiUTzp1I/AAAAAAAAACo/YSoo51eL6NA/s1600/Book+II+Prop+14+with+Proof.gif" height="187" width="320" /></a></td></tr><tr><td class="tr-caption" style="font-size: 13px; text-align: center;">A <a href="http://www.geogebratube.org/material/show/id/49815" target="_blank"><span style="color: red;">worksheet with a slider</span></a> to show steps. Plus, the<br />rectangle's length and width can be adjusted!</td></tr></tbody></table></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div>Being able to use a slider will be key to making worksheets for students to use that contain several different examples. In the near future I am hoping to make a GeoGebra worksheet to help my geometry students discover the four triangle congruence theorems (<i>SSS</i>, <i>SAS</i>, <i>ASA</i>, and <i>AAS</i>). In a similar vein, I think a GeoGebra worksheet could be very helpful for my precalculus students learning about the Law of Sines and the <i>SSA </i>case, and when two triangles are possible.</div>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com0tag:blogger.com,1999:blog-1267845558922247045.post-44043518740746486592013-09-11T15:00:00.002-07:002013-09-11T15:00:55.257-07:00Van Hiele's Levels of Geometric Reasoning<div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">“School geometry that is presented in a [Euclidean] axiomatic fashion assumes that students think on a formal deductive level. However, that is usually not the case…” </span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"> </span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">(Van Hiele, 1999)</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">This excerpt from Van Hiele’s article, </span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: italic; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Developing Geometric Thinking through Activities that Begin with Play</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">, immediately made me think of one topic in particular, the bane of (most) high school geometry students’ existence: proofs. The type of formal reasoning demanded by deductive proofs is seldom easy for students to attain, even in high school. If students have one complaint about geometry, it is usually the proofs.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">As such, I am always looking for ways to make proofs easier for students to understand. Before we get into geometric proofs, we always play a “proof game” with simple rules for manipulating strings of letters, to get them used to justifying every statement they make. I also try to emphasize the sequence and structure that is common among many proofs, like using definitions to change words into equations, after which properties of equality can be used to show a range of things. We also try to scaffold heavily as students learn how to write proofs, slowly bringing them along from filling in blanks to eventually writing complete proofs on their own.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">But Van Hiele’s article led me to think about students’ struggles with proof in a different light, and question whether my students have been given or are given enough opportunities to progress through Van Hiele’s levels of geometric reasoning. Perhaps some of their difficulty with proof stems from having too little time to engage in </span><span style="font-family: Arial; font-size: 15px; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">informal reasoning</span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; text-decoration: underline; vertical-align: baseline; white-space: pre-wrap;">Van Hiele’s Levels of Geometric Reasoning</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">1 - Visual</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">2 - Descriptive</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">3 - Informal Deductive</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">4 - Formal Deductive</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">5 - Axiomatic</span></div><b style="font-weight: normal;"><br /></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">I know that my students do not have many opportunities to do this, to take their time and consider what they know about geometric figures, and to discover out of their own knowledge further relationships and geometric truths. On the first day of class, they are introduced to four postulates, and at least one or two postulates or theorems follows in each subsequent lesson. They do come in a logical sequence, but still they are presented for students to take at face value most of the time, unless we take the time to prove them. It would be better if my students could be presented with new ideas along the way, and allowed to explore and struggle to find new relationships on their own, before having the formalized theorems presented to them. And they could practice justifying their discoveries, without immediately being tied to structure of a proof. I think they would much more thoroughly internalize important concepts than they do now.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Formal proof writing will always be an adjustment, though. It is a new style of writing for students. But if they have had more time to explore their way through new concepts, they might be better at making the connections that tie given information to a desired conclusion.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Van Hiele’s mosaic and the activities he describes to go with it are impressive. Its careful design provides a basis for a wide range of discussions and student discoveries. The example of the mosaic makes me want to find more exploratory activities that can apply to the concepts my high school geometry students will work with this year. I will certainly look for these, or try to come up with some, and look for opportune places to tie them into our curriculum.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Time might be the biggest challenge to integrating more exploratory activities. It is at such a premium already, with all that is supposed to be covered in a term. Giving students a day here or there to explore, rather than presenting material directly and moving on is not the most efficient way to go, time-wise. Our math department already has moved to include days dedicated to problem solving group work near the end of each unit. I wonder now if in geometry it might be better to use some of those days for exploratory activities at the start of a unit. That is a suggesting I will make to our geometry team. At the very least, I will be more apt to provide more time within lessons to have students discuss key ideas with each other.</span></div><b style="font-weight: normal;"><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; vertical-align: baseline; white-space: pre-wrap;"><u>Reference</u></span></div><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">Van Hiele, P.M. (1999). Developing geometric thinking through activities that begin with play. </span><br /><span style="font-family: Arial; font-size: 15px; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Teaching Children Mathematics</span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">, </span><span style="font-family: Arial; font-size: 15px; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">5</span><span style="font-family: Arial; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;">(6), 310-316.</span>Matthttp://www.blogger.com/profile/06565106214943299968noreply@blogger.com1