tag:blogger.com,1999:blog-15917800872278398822024-03-14T04:13:42.910-07:00The Mathy MurkThe Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-1591780087227839882.post-2967911856867784272015-05-27T21:05:00.000-07:002015-05-28T08:44:00.170-07:00A Letter to My StudentsTomorrow will be the last day of my school year. It is always bittersweet to end a year with a group of students. There are those who have made us crazy, those who have made us proud, and sometimes those are even the same student. I wanted to express to my students how much I appreciate all of their hard work, their willingness to take risks, and for trusting me when I asked them to do math differently. This is the letter that I wrote to them. <br />
<br />
<br />
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Dear Students,<o:p></o:p></div>
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<br /></div>
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<span style="mso-tab-count: 1;"> </span>It is
always during this time of year that I do some reflecting on our time together
as teacher and student.<span style="mso-spacerun: yes;"> </span>What went
well?<span style="mso-spacerun: yes;"> </span>What could have been done better?<span style="mso-spacerun: yes;"> </span>What was fun?<span style="mso-spacerun: yes;">
</span>What was challenging?<span style="mso-spacerun: yes;"> </span>These are
questions that I always think about as the school year comes to an end.<o:p></o:p></div>
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<br /></div>
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<span style="mso-tab-count: 1;"> </span>When I
first met you in August, you were probably assuming that this was going to be
just another math class.<span style="mso-spacerun: yes;"> </span>A 95-minute
period, every other day, that would be filled with you sitting quietly as I
lectured, you sitting quietly taking notes, you sitting quietly as you practice
all that I’ve taught you on a worksheet with thirty problems.<span style="mso-spacerun: yes;"> </span>But, my hope is that when I first met you in
August, you knew right away that this math class was going to be different; you
were not going to sit quietly.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
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<span style="mso-tab-count: 1;"> </span>Many of you
have probably learned to hate math class for a variety of reasons.<span style="mso-spacerun: yes;"> </span>Being “good at math” has had a very narrow
definition for you and if you don’t fit that definition, why bother to try?<span style="mso-spacerun: yes;"> </span>As I look back on the last ten months, this
has been my single goal; to redefine for you what it means to be “good at math”.<span style="mso-spacerun: yes;"> </span>Math is not all about speed and accuracy,
memorizing formulas, and getting the right answer.<span style="mso-spacerun: yes;"> </span>Math is about communicating your thinking,
being able to solve a problem in a way that makes sense to you, and taking
ownership of your own learning.<span style="mso-spacerun: yes;"> </span>My hope
is that you experienced what it feels like to be good at math in these
capacities this year.<span style="mso-spacerun: yes;"> </span>For some of you,
you were hungry for the opportunity to feel this good about math, while others
needed a little more convincing that it was possible or real, but my belief is
that you all, in some way, got to feel good about yourself as a mathematics
student this year; and for that, you should be proud.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
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<span style="mso-tab-count: 1;"> </span>As you move
on to other math classes in the future, I will be honest, I have concerns.<span style="mso-spacerun: yes;"> </span>I’m concerned that you will go back to
feeling unsuccessful because the focus of the class will be on speed and
accuracy, memorization, and sitting quietly.<span style="mso-spacerun: yes;">
</span>Do not let this get in your way of finding success and knowing that you <i style="mso-bidi-font-style: normal;">are</i> good at math.<span style="mso-spacerun: yes;"> </span>You must believe that.<span style="mso-spacerun: yes;"> </span>When you eventually leave high school and
move on to a career or higher education, please understand that the skills that
I have tried to help you develop will be invaluable.<span style="mso-spacerun: yes;"> </span>You are all problem solvers, communicators,
and hard workers, but most of all, you are all curious.<span style="mso-spacerun: yes;"> </span>Do not let anyone kill your curiosity by
telling you that you have to solve a problem in a particular way.<span style="mso-spacerun: yes;"> </span>Find a way that works for you.<span style="mso-spacerun: yes;"> </span>I promise you, whether it be in math class or
in life, it is more rewarding that way.<o:p></o:p></div>
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<br /></div>
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<span style="mso-tab-count: 1;"> </span>As you
know, this has been an extremely challenging year for me as I try to navigate
my own plans for my future, but I want you to know that nothing has been more
rewarding than being your teacher and your friend.<span style="mso-spacerun: yes;"> </span>Please keep in touch with me and let me know
how you are doing.<span style="mso-spacerun: yes;"> </span>My favorite part of
being a teacher is the connections that I make with my students.<span style="mso-spacerun: yes;"> </span>You are all so incredibly interesting,
thoughtful, and lovable!<span style="mso-spacerun: yes;"> </span>I’m going to
miss you, but I will always remember how rewarding this year has been.<span style="mso-spacerun: yes;"> </span>Getting to see you succeed in ways you never
thought imaginable has been the greatest gift a student could give a
teacher.<span style="mso-spacerun: yes;"> </span>Thank you for all of your hard
work, for always making me laugh, and most of all, for proving me right when I
say to other teachers, “You <span style="font-family: Times, Times New Roman, serif;">know, you’d really be surprised by what students
can do, all you have to do is ask.”<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Times, Times New Roman, serif;">With love and respect,<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
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<span style="font-family: Times, Times New Roman, serif;">Ms. Balli</span><!--EndFragment-->
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I'd love to hear of ways that you end your school year. What works for you?The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com2tag:blogger.com,1999:blog-1591780087227839882.post-79814214421147855692015-04-27T14:29:00.002-07:002015-04-27T15:36:26.904-07:00Art & GeometryI recently was fortunate enough to attend the NCTM conference in Boston and while I was there, I checked out a really cool art museum in Providence, RI. I took these two pictures while I was there because I thought that they could connect to what we are learning about in our polygons unit.<br />
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When showing the pictures to my students, I projected each one and had them brainstorm some questions they were curious to know about. Of course, I encouraged them to be mathematically based questions, but I did still get, "What color is it?" Check out some of the questions that they came up with:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjL8KACZbKHKiKMDQ-mEGa9IgMg3NAiM-RLhZKivrfEIbWX3czb124yS1Vm6MlWhOEB3ipOOKu3W0bcg8X1GzGQoqYUtmC_LZFkWA8fjXpRXRNBnbV-_O4RqSp6eMM8qQjdbHCOmav7xHQ/s1600/IMG_3469.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjL8KACZbKHKiKMDQ-mEGa9IgMg3NAiM-RLhZKivrfEIbWX3czb124yS1Vm6MlWhOEB3ipOOKu3W0bcg8X1GzGQoqYUtmC_LZFkWA8fjXpRXRNBnbV-_O4RqSp6eMM8qQjdbHCOmav7xHQ/s1600/IMG_3469.JPG" height="320" width="240" /></a></div>
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<b>How many different squares do you see in the picture?</b></div>
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<b>What is the scale factor used if you dilate the center square to the outer square? What about from the outer square to the center square?</b></div>
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<b>How many lines are there?</b></div>
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I asked my students to explore 2 different questions. First, I said, "If I asked you to determine the number of 'spokes' in the picture, how might you do it?" Groups brainstormed ideas for how they might tackle the problem.</div>
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The general response was that students would break the picture down into smaller parts, count the number of spokes in that section and then multiply by the total number of those parts.</div>
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Once we had shared some ideas of how you might determine the number of spokes, I asked them to actually figure it out. They came up with 128 spokes.</div>
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The second question I had students explore was what would be the degree measure of the angle created by two adjacent spokes? </div>
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This was a little trickier, not because of the calculation they used (360/128), but they had a hard time defending their use of 360. Was it the Interior Angle Sum Theorem? The Exterior Angle Sum Theorem? Or the fact that there are 360 degrees around the center point where the vertex of the angle is?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAMwHtsz5DFatZOq_vJf6-z8Q27_LeLbz3GBIfrMRA2sPjO7XxLD1AayFlZ2-VioUcRC67feUxXeQ1_3fBBSQLzwEYtzMIi-sW_DriQ1GE6-APLgBYwA6RlIn3_hZgAu8WbK0v2Cq1cio/s1600/IMG_3470.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAMwHtsz5DFatZOq_vJf6-z8Q27_LeLbz3GBIfrMRA2sPjO7XxLD1AayFlZ2-VioUcRC67feUxXeQ1_3fBBSQLzwEYtzMIi-sW_DriQ1GE6-APLgBYwA6RlIn3_hZgAu8WbK0v2Cq1cio/s1600/IMG_3470.JPG" height="320" width="240" /></a></div>
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This was the second picture I shared with them. I really like this because it's deceiving. Is it a hexagon? An octagon? Neither...more of a hybrid of the two to form a heptagon. I find the inscribed circle to be really fun, too!<br />
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The question that I posed to the students was to find the measure of each of the seven interior angles. Lots of students found the sum of the interior angles (900 degrees) and then divided by 7, but soon realized that that would not work because not all angles are equal in measure. So, they revised and were able to make a lot of progress.<br />
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I find questions/problems like these to be so much more interesting than the way that I've taught this unit in the past. I feel like polygons have so much potential for interesting explorations, and I was happy to find some of those in art!<br />
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<b>What mathematical questions do you have about these pieces? </b>The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-53324951216089217732015-04-24T07:07:00.001-07:002015-04-24T07:07:39.543-07:00Breaking Bad...DefinitionsI posted a picture on Twitter the other day (see below) that included the <b>8 SMPs in "Kid Speak"</b>. Ashli Black (@Mythagon) pushed back on #6, saying that <b>precision is more than just numerical accuracy, but also precision of language</b>. I could not agree more!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyQhtVOh4daXUf2Bsk-Pw4NhPOGApkCMEn5r03oT5e9L_UEXggIddQCtI0fMSjMwDR834Fv5woDoTXmv3vLQyOfzCa5pctR0yGzKtmvO0CCqsdQaakRNnizkACyBDThZvhE1NuqTOfzqo/s1600/IMG_3515.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyQhtVOh4daXUf2Bsk-Pw4NhPOGApkCMEn5r03oT5e9L_UEXggIddQCtI0fMSjMwDR834Fv5woDoTXmv3vLQyOfzCa5pctR0yGzKtmvO0CCqsdQaakRNnizkACyBDThZvhE1NuqTOfzqo/s1600/IMG_3515.JPG" height="320" width="240" /></a></div>
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I have found that <b>necessitating the precision of language</b> is fairly easy in my classroom because my students do a ton of talking and writing. But I think the major difference is that they share their writing and verbal thoughts with their partners and the whole class, so I'm not the only audience member, which seems to be more important to the kids. (Remember, these are teenagers. <b>They care very much about what their peers think</b>.)<br />
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We were getting into our <b>unit on polygons</b> and I wanted to try out an activity that my colleague Dr. Patrick Callahan had shown me a few years ago. I was actually attending a PD session at UC Davis where he was presenting and he asked all of us (a room full of math educators) to <b>write down our definition of the word "polygon"</b>. I've adapted the activity slightly and I used it with my students just the other day.<br />
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We started with this handout:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgiuraWfKu5_RwuddnklT_YIOYg0gmImnHmEc5zdd3czJPKKATACuSY-Ovwh3oKm271VoCoaNV7sbvi67LnmQVk9Waod3oMu3t5f-HsA2pdAyrd1laZ7-EWwWdMEwY9cnGjrAjbQ1Dxgs/s1600/Screen+Shot+2015-04-23+at+6.29.12+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgiuraWfKu5_RwuddnklT_YIOYg0gmImnHmEc5zdd3czJPKKATACuSY-Ovwh3oKm271VoCoaNV7sbvi67LnmQVk9Waod3oMu3t5f-HsA2pdAyrd1laZ7-EWwWdMEwY9cnGjrAjbQ1Dxgs/s1600/Screen+Shot+2015-04-23+at+6.29.12+PM.png" height="320" width="232" /></a></div>
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The instructions read:<br />
<div class="MsoNormal" style="text-indent: .5in;">
"The purpose of this activity is to
work toward a mathematically viable definition of the term “polygon”.<span style="mso-spacerun: yes;"> </span>To begin, write a first draft of your
definition.<span style="mso-spacerun: yes;"> </span>Then, exchange papers with a
peer.<span style="mso-spacerun: yes;"> </span>When you read your peer’s
definition, consider if you can “break” it.<span style="mso-spacerun: yes;">
</span>To break a definition, draw a picture that <i style="mso-bidi-font-style: normal;">does</i> satisfy their definition, but <i style="mso-bidi-font-style: normal;">is not</i> a polygon.<span style="mso-spacerun: yes;"> </span>We will
have a mathematically precise definition of polygon when we cannot draw a
picture to break that definition. <o:p></o:p></div>
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<!--StartFragment-->
<span style="font-family: Cambria; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "MS 明朝"; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;"> For
example, if we were trying to define the term “vegetable”, my first draft might
be: A vegetable is a food. When I
exchange papers with my partner, they might break my definition by writing the
word “chicken”. This breaks my
definition because it <i>does</i> satisfy my
proposed definition, but it <i>is not</i> a
vegetable."</span><!--EndFragment--><br />
<span style="font-family: Cambria; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "MS 明朝"; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;"><br /></span>
<span style="font-family: Cambria; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "MS 明朝"; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">I should say that I was essentially looking for the following elements in an "<b>unbreakable</b>" definition of polygon:</span><br />
<br />
<ul>
<li><span style="font-family: Cambria;">Closed</span></li>
<li><span style="font-family: Cambria;">2-Dimensional</span></li>
<li><span style="font-family: Cambria;">Straight sides</span></li>
<li><span style="font-family: Cambria;">At least 3 sides/More than 2 sides</span></li>
<li><span style="font-family: Cambria;">Sides that intersect only at endpoints</span></li>
</ul>
<br />
<span style="font-family: Cambria; font-size: small;">I asked the students to begin writing down their initial ideas of what a polygon is. Many got right to it, others stalled to wait and see what others wrote, and some got flustered, </span><span style="font-family: Cambria;">panicked</span><span style="font-family: Cambria; font-size: small;">, and asked me "<b>What if I have no idea?</b>" I reassured them that we were <b>not going for the right answer at this stage</b> and I just wanted them to put down <u>ANYTHING</u> that they knew about the word. This seemed to help a bit.</span><br />
<span style="font-family: Cambria; font-size: small;"><br /></span>
<span style="font-family: Cambria; font-size: small;">I walked around and saw a variety of first drafts:</span><br />
<span style="font-family: Cambria; font-size: small;"><br /></span>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcQVnsJfVebuMt49avVruo9QKbOE-dvqKpaKrcsXVilIjvSsZTYFRx5m_BDuKKfYDTyqGj5fneCWkjKQasTKnYybgJgOcDwIAflx-MQSf73-AZcNkiUAtXmn2KQhS-bLS9KOPwCdaonz4/s1600/IMG_3484.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcQVnsJfVebuMt49avVruo9QKbOE-dvqKpaKrcsXVilIjvSsZTYFRx5m_BDuKKfYDTyqGj5fneCWkjKQasTKnYybgJgOcDwIAflx-MQSf73-AZcNkiUAtXmn2KQhS-bLS9KOPwCdaonz4/s1600/IMG_3484.jpg" height="240" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: xx-small;">polygon - a shape</span></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhyphenhyphenMrBWLOEi5ukq7WWpmEjE_mQrivB2tnSn7RFRIbxCwsgh13EJ6mlCOt9RU4kEK0PGdhbZcXPCpUB-Lsnv0KpegRQl_lcP2GHLtoIze2m_g2436qjlfC109pErLbnZyFs1Dg9aApSw_g/s1600/IMG_3487.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhyphenhyphenMrBWLOEi5ukq7WWpmEjE_mQrivB2tnSn7RFRIbxCwsgh13EJ6mlCOt9RU4kEK0PGdhbZcXPCpUB-Lsnv0KpegRQl_lcP2GHLtoIze2m_g2436qjlfC109pErLbnZyFs1Dg9aApSw_g/s1600/IMG_3487.jpg" height="240" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: xx-small;">A polygon is a shape or a figure</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuJRZD8KxHC4VihH0gH45GILuUcUMcxX6EPy_33uyeksoDYTGxSE8jpd4bPZBzEqVtmtVJppkFcHqHKQRTIWp1q7tbgVf6SRyIzjrgGs4DQYki_iLklAmumsSPOrgUwJux8fxDMEOzyb0/s1600/IMG_3509.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuJRZD8KxHC4VihH0gH45GILuUcUMcxX6EPy_33uyeksoDYTGxSE8jpd4bPZBzEqVtmtVJppkFcHqHKQRTIWp1q7tbgVf6SRyIzjrgGs4DQYki_iLklAmumsSPOrgUwJux8fxDMEOzyb0/s1600/IMG_3509.jpg" height="240" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: xx-small;">A closed shape with more than 2 sides</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS9exabvB_5FLP3Vb1oPJ-1TMxYeHcTu7dvDiPyGYEY5JCHdzHQTz8P0praO6VbWUFnlKFIGmOeHlXx1ZNtrOG-tURp0d-zrZgkXs2NjXfiNCZnsgw-Lge2TYw7bOIGbV23iczqDBQQO4/s1600/IMG_3510.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS9exabvB_5FLP3Vb1oPJ-1TMxYeHcTu7dvDiPyGYEY5JCHdzHQTz8P0praO6VbWUFnlKFIGmOeHlXx1ZNtrOG-tURp0d-zrZgkXs2NjXfiNCZnsgw-Lge2TYw7bOIGbV23iczqDBQQO4/s1600/IMG_3510.jpg" height="240" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: xx-small;">A polygon is a closed shape with no curved edges.</span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5c-pnnMlmOfmQ5SodjhTUr-XMIQSzLUCq98MsXbVWmnBBnqNIQTfDG3N2DTMRHQtJw0iBTjVnGqOjJnR3RtOjNIbVWprApzsydmOfX9kvnYR7Z1q7iAbcvn-2TGheCBZ0YU00KyV408w/s1600/IMG_3503.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5c-pnnMlmOfmQ5SodjhTUr-XMIQSzLUCq98MsXbVWmnBBnqNIQTfDG3N2DTMRHQtJw0iBTjVnGqOjJnR3RtOjNIbVWprApzsydmOfX9kvnYR7Z1q7iAbcvn-2TGheCBZ0YU00KyV408w/s1600/IMG_3503.jpg" height="240" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: xx-small;">A shape with more than one side</span></div>
<span style="font-family: Cambria; font-size: small;"><br /></span>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZzhP-btegVQprgGh1blqVhkHnGCTubL_ETdTAcHI2chESX2FcfE1GP9oZus8QLUxgBUu7MQElcNVZQuu1xFKSb3HCXYSmB7nRvaz_8n0QuBm2_igw7s7JP7_qtgfTIcanXkhMRNvzi6Y/s1600/IMG_3502.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZzhP-btegVQprgGh1blqVhkHnGCTubL_ETdTAcHI2chESX2FcfE1GP9oZus8QLUxgBUu7MQElcNVZQuu1xFKSb3HCXYSmB7nRvaz_8n0QuBm2_igw7s7JP7_qtgfTIcanXkhMRNvzi6Y/s1600/IMG_3502.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">A shape with 6 sides</span></div>
<br />
As papers started to move around for definitions to be broken, I found that this part was really challenging for some students. "<i>How am I supposed to break their definition if I don't even know what a polygon is?</i>"<br />
<br />
I grabbed a sample paper (see below) and put it under the <b>document camera</b>. The first draft read: "polygon - a shape". I asked the students how we might break this definition of polygon and one student said, "<i>You could draw a circle. Because even though it is a shape, it is not a polygon.</i>" This <b>started a lot of conversation</b> at each table and students were brainstorming ways that they could break the definition on the paper in front of them.<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqqopDVIwn9pTT8kff6RteLaCRI2LHivNYD_VMyZSQxWVygNVvO2AIZYY0nzCRu8_IZ-KSUFcZAgJk7zelBp3uIJlmv_5FiEFl_07n91oQuxp5RVowKxFW7Q3SIEWU4pWCIDh5Lc-1leg/s1600/IMG_3488.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqqopDVIwn9pTT8kff6RteLaCRI2LHivNYD_VMyZSQxWVygNVvO2AIZYY0nzCRu8_IZ-KSUFcZAgJk7zelBp3uIJlmv_5FiEFl_07n91oQuxp5RVowKxFW7Q3SIEWU4pWCIDh5Lc-1leg/s1600/IMG_3488.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">(as mentioned above)</span></div>
<br />
Here are some ways that students broke the first drafts: <br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrvLr70UU1gqep1wmVwIDgbtRB83MyWp4F568WyeVPhwQG6LwJ8pDEalbVYa778-VF3t-q86jOEjHrpqUTM0yDBwRTE3VqMqtv4fczj_d1MLaus1HTMN_1cJwEGHH6UpSI8FFFc8gsTPQ/s1600/IMG_3504.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrvLr70UU1gqep1wmVwIDgbtRB83MyWp4F568WyeVPhwQG6LwJ8pDEalbVYa778-VF3t-q86jOEjHrpqUTM0yDBwRTE3VqMqtv4fczj_d1MLaus1HTMN_1cJwEGHH6UpSI8FFFc8gsTPQ/s1600/IMG_3504.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">This student was calling out the fact that sides could not be curved</span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjE4in_NX0KQhi3KTXptq167bZMIaHELOdNYzdQ6xXW3Id9gFzthLFynIFYxIVkQyjvxqpPVVS2gzRr7N7mfaK8iheJjf-21v-VbMfTeyoIEs-UDLRCWRfjNZxw6aEMCXJntLgm9wZkdOA/s1600/IMG_3505.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjE4in_NX0KQhi3KTXptq167bZMIaHELOdNYzdQ6xXW3Id9gFzthLFynIFYxIVkQyjvxqpPVVS2gzRr7N7mfaK8iheJjf-21v-VbMfTeyoIEs-UDLRCWRfjNZxw6aEMCXJntLgm9wZkdOA/s1600/IMG_3505.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">This student was addressing the fact that a polygon could actually have less than 5 sides</span></div>
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<br /></div>
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Now, <b>not all students really got the way that the breaking process was working</b>. Here are some examples of pictures that did not <i>actually</i> break the proposed definition:</div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrecM5q9NW9wF4lk1fZC293KqfCVqevd9YmGFKDrZLslvHMqcdh234BU_INIpi4YhkWkzTRxErK-X40FRqCK3LPC3ePT7tmCidqSxRcsk6Kdz5aquHYqI-4T_xzSXRZRMfonyMRXYSkAc/s1600/IMG_3485.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrecM5q9NW9wF4lk1fZC293KqfCVqevd9YmGFKDrZLslvHMqcdh234BU_INIpi4YhkWkzTRxErK-X40FRqCK3LPC3ePT7tmCidqSxRcsk6Kdz5aquHYqI-4T_xzSXRZRMfonyMRXYSkAc/s1600/IMG_3485.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">"A polygon is a shape that has multiple sides and is not round" is not actually broken by a circle</span></div>
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUJY-ths1nf0gdjJhfvWL5utqg3vudO8dwJghDfKbbvPt0iqQQJrvwij4iMI-0YN4kWQdh2owT3oANq6fLHVh7RI2RXKVey7YqNy9IcU4YF16bnfeS0qAqtN8JunODbUzQcf5hJv4H4s8/s1600/IMG_3486.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUJY-ths1nf0gdjJhfvWL5utqg3vudO8dwJghDfKbbvPt0iqQQJrvwij4iMI-0YN4kWQdh2owT3oANq6fLHVh7RI2RXKVey7YqNy9IcU4YF16bnfeS0qAqtN8JunODbUzQcf5hJv4H4s8/s1600/IMG_3486.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">"A polygon is a geometric figure". And so is the shape drawn in the box.</span></div>
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It was at this stage that I realized that my description of "breaking" a definition was <b>not precise enough</b>. In my vegetable example, I had gone in the direction of not being specific enough in my definition, or as Dr. Callahan would say, I've "included too much". I hadn't considered the other direction of not including enough. In the example above where the student broke the definition by drawing a trapezoid, the original definition had left out figures with 3 and 4 sides, which had excluded too much. This was <b>an opportunity to revise my own definition</b> of "breaking". Obviously I always love a great opportunity to model what I am asking my students to do!</div>
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Also at this point, students got their papers back and started revising. You can see that <b>the precision of language was called out without me really having to say too much</b>. There was a sense of competition that students had with one another because they wanted to come up with an "unbreakable" definition before anyone else did. Take a look at their second drafts:</div>
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<i>(Please note that not all of these are great, but you can see that students were attending to the precision of their language, regardless of whether or not the picture actually broke their previous draft. Many students relied on the examples show to the whole class under the document camera or other papers that they saw to revise their definitions.)</i></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7xVdww-q29Ox1uwACpiPIG2TDAnsEK2kat0AE6-RRiSoEOO4Ns6di5BBjqjHCn1ZQnbfM1d5xDCs-VRrE7KCRfJACv_1TvmT0Yjp4lBw28pxXgG3hVElbSuQr3Wr2NEJfvJVP6nlNSR8/s1600/IMG_3490.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7xVdww-q29Ox1uwACpiPIG2TDAnsEK2kat0AE6-RRiSoEOO4Ns6di5BBjqjHCn1ZQnbfM1d5xDCs-VRrE7KCRfJACv_1TvmT0Yjp4lBw28pxXgG3hVElbSuQr3Wr2NEJfvJVP6nlNSR8/s1600/IMG_3490.jpg" height="240" width="320" /></a></div>
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From this point on, I didn't need to do a lot of modeling to the whole class, they were switching papers back and forth on their own. Students would call me over if they thought they had an unbreakable definition and I might tell them that there was still a way to break it, or I might pose it to the whole class to decide. <b>Could they break it?</b><br />
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Here are a couple examples of further revisions:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz3peRKwcz7lsPDNciIBSz373K2a_5EkNpalOsVHLYdTuJ_NbvyPkDDY8GxTfo4XjoPGApGpWbGIr_7qLrhHH7vMnLr-U_oHrRXfiFM3VDLCkfvUJBV5PVVf52PTSt6ZrdEe6sYWSfUuA/s1600/IMG_3495.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz3peRKwcz7lsPDNciIBSz373K2a_5EkNpalOsVHLYdTuJ_NbvyPkDDY8GxTfo4XjoPGApGpWbGIr_7qLrhHH7vMnLr-U_oHrRXfiFM3VDLCkfvUJBV5PVVf52PTSt6ZrdEe6sYWSfUuA/s1600/IMG_3495.jpg" height="320" width="240" /></a></div>
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Also, after a few rounds of revision, most students were good with their definitions including closed figure, straight sides, and at least 3 sides. <b>Now, it was time for me to push them</b> on the piece that could describe the requirement of sides only intersecting at endpoints.<br />
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I drew these images on the board:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiejKlmQM1sYbcjW0olgt00CnoML6HpDeHP6Blx61FgepZR7da5WeYTt78yZRHEcGZDbNOI1lipIqxBmxDEp249xNWKsE6Q5UJnrv_fn_DcVl28tRBpBveiwYE0y6Jegwf-RjDHB9UcccY/s1600/IMG_3498+(1).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiejKlmQM1sYbcjW0olgt00CnoML6HpDeHP6Blx61FgepZR7da5WeYTt78yZRHEcGZDbNOI1lipIqxBmxDEp249xNWKsE6Q5UJnrv_fn_DcVl28tRBpBveiwYE0y6Jegwf-RjDHB9UcccY/s1600/IMG_3498+(1).jpg" height="240" width="320" /></a></div>
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They knew that the first picture was unacceptable because it wasn't closed. They were fine with the middle image. But I told them that I could break a lot of their existing definitions with the third picture.</div>
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A lot of students said, "<i>Sides can't intersect!!!</i>" would be their revision. This is when <b>I pushed back</b> with, "Well, you can't have both. You can't say that it's a closed figure <b>AND</b> that sides don't intersect."</div>
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Silence.</div>
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I asked them to brainstorm with their group how they might describe the difference between the middle picture and the one on the right. <b>What made one acceptable, but not the other?</b> Check out what they came up with:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpgg-dT8Q_E7dnIxQLL9AZeCtrolBesL4gq5aFZB0e6WLoaVY61TrTdWsAr3N9_o86UfzPxEYe1vyXZGrVWSR2XfFXnRvNvIJKbJCRm8m-OQunrZPmWzUtSjO2-YK_-IxEOu_JEyfXRd4/s1600/IMG_3514.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpgg-dT8Q_E7dnIxQLL9AZeCtrolBesL4gq5aFZB0e6WLoaVY61TrTdWsAr3N9_o86UfzPxEYe1vyXZGrVWSR2XfFXnRvNvIJKbJCRm8m-OQunrZPmWzUtSjO2-YK_-IxEOu_JEyfXRd4/s1600/IMG_3514.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">She told me later that she meant "vertex" not "vertices"</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3ljtB0hJKpWsuMua25N3gKsuxIhkeieybmwq5GS5OyzXhOoQpSVR2LnvrDFsJLOO51p_o_Jo3G1Tg2erLCRgLUemzyi9n3PnM5hf5isuYzMCHP4brhCB6rbPv3jkEpNJ77nxKpyFK580/s1600/IMG_3513.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3ljtB0hJKpWsuMua25N3gKsuxIhkeieybmwq5GS5OyzXhOoQpSVR2LnvrDFsJLOO51p_o_Jo3G1Tg2erLCRgLUemzyi9n3PnM5hf5isuYzMCHP4brhCB6rbPv3jkEpNJ77nxKpyFK580/s1600/IMG_3513.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">This student actually came up with this language before I addressed it with the class</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxlRsR77u3BvWxonPDbxgeMURLvzsGhDYJ80mLzAK2q_NAPO5Y5IChrmnRPSREtwvNzPnAy5go3PoBbBmQIASMuj9_qXXCw6BsXbWAdRQVtbI0QDxLE9fOLvAV-rqvd8baynBEmksX40w/s1600/IMG_3496.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxlRsR77u3BvWxonPDbxgeMURLvzsGhDYJ80mLzAK2q_NAPO5Y5IChrmnRPSREtwvNzPnAy5go3PoBbBmQIASMuj9_qXXCw6BsXbWAdRQVtbI0QDxLE9fOLvAV-rqvd8baynBEmksX40w/s1600/IMG_3496.jpg" height="320" width="240" /></a></div>
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<span style="font-size: xx-small;">This is one of my top performing students. He had most of it down from the beginning.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnyyyrTg1IHUuvHYFZmT9nE-5j9al9wcrxdovtprM_ISCOP7B0KogdJhsMVPIzDHGoTRaCuGxrqYjB66e_qSyc92rhu-UHkKY2_pfnavbEqjfsogxztXscCjNYDXttLtWzh-VBBWJFfbE/s1600/IMG_3499.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnyyyrTg1IHUuvHYFZmT9nE-5j9al9wcrxdovtprM_ISCOP7B0KogdJhsMVPIzDHGoTRaCuGxrqYjB66e_qSyc92rhu-UHkKY2_pfnavbEqjfsogxztXscCjNYDXttLtWzh-VBBWJFfbE/s1600/IMG_3499.jpg" height="320" width="240" /></a></div>
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<span style="font-size: xx-small;">This last definition is one that the group came up with together</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOyvfkN4WVe4-kbc1Nhjj2T1hqTSmJ4LmDPngkJd8Y1WkbujjsGQ616okiKT7ygMomTdtIDeyx53lMg7jbRmPx9UHIv6y2t669hQ89TPJUomdTN77cdC7lo9qg5Vz4tLcHJqx9saGeLno/s1600/IMG_3500.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOyvfkN4WVe4-kbc1Nhjj2T1hqTSmJ4LmDPngkJd8Y1WkbujjsGQ616okiKT7ygMomTdtIDeyx53lMg7jbRmPx9UHIv6y2t669hQ89TPJUomdTN77cdC7lo9qg5Vz4tLcHJqx9saGeLno/s1600/IMG_3500.jpg" height="320" width="240" /></a></div>
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<span style="font-size: xx-small;">You can see this student's process and her final product below</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisIyHKtiXxJz9_yiG1OR1LMKkHH-l0252Rsg4d_mVBT47qXh2lUhMafMA0WUHGD-pY4kmTNBEHC-Y2ufrmbZtxyx9__w1V8e4IwfoXlVz73ME2-fgiLJ_f0By5iCOsnrhi-I83iMWZ60g/s1600/IMG_3501.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisIyHKtiXxJz9_yiG1OR1LMKkHH-l0252Rsg4d_mVBT47qXh2lUhMafMA0WUHGD-pY4kmTNBEHC-Y2ufrmbZtxyx9__w1V8e4IwfoXlVz73ME2-fgiLJ_f0By5iCOsnrhi-I83iMWZ60g/s1600/IMG_3501.jpg" height="320" width="240" /></a></div>
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<span style="font-size: xx-small;">See above</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWDDWd5OHTtgpJFMnJME_DA3dfXrCdvqI8XAcjtfdWS6np_5iKZxECGZnVmDn52qt9Wbbt2pf1ubTqHL010o3sqMULADuMU1iDhQte6uAEXaoqhruCDONjUt7KzQlNHSQvkDf31zJT4lo/s1600/IMG_3508.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWDDWd5OHTtgpJFMnJME_DA3dfXrCdvqI8XAcjtfdWS6np_5iKZxECGZnVmDn52qt9Wbbt2pf1ubTqHL010o3sqMULADuMU1iDhQte6uAEXaoqhruCDONjUt7KzQlNHSQvkDf31zJT4lo/s1600/IMG_3508.jpg" height="240" width="320" /></a></div>
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<span style="font-size: xx-small;">I loved this idea of the vertical angles connection</span></div>
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At the end of the day, of course, I was wanting my students to have <b>a solid understanding of what the term "polygon" means</b>, but this activity allowed for so much more than that. I know that I mentioned <b>the importance of the precision of language</b>, but another layer of this is the process of determining if something "belongs" or not. Furthermore, if I think it should belong, but my definition doesn't allow for it, I need to <b>revise my definition</b>. Or, if I don't think it belongs and my definition does allow for it, again, revision is needed. Dr. Callahan really made this clear for me when I debriefed this activity with him. It was something that I had not really considered, but was so powerful for students.<br />
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And just to wrap it all up, I had to share Dr. Callahan's creative side...<br />
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You're welcome.The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com4tag:blogger.com,1999:blog-1591780087227839882.post-88495733957391625472015-04-12T11:38:00.000-07:002015-04-12T11:38:07.552-07:00The Mathy Murk: An Impact SurveyHello Colleagues!<br />
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I'm wanting to gather some information about what impact my blog has on teaching practices. If you have 4 minutes, I would appreciate hearing from you! Thank you in advance for your feedback!<br />
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Please choose ONE of the versions below. You only need to answer once, as the questions are exactly the same. The only reason I created a Survey Monkey version is because I know that Google is blocked in some parts of the world. :)<br />
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Thanks!<br />
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Google Forms version: <a href="http://bit.ly/mathymurksurvey">http://bit.ly/mathymurksurvey</a><br />
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Survey Monkey version: <a href="https://www.surveymonkey.com/s/59WVQJJ">https://www.surveymonkey.com/s/59WVQJJ</a><br />
<br />The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-34589345922969265272015-04-10T14:36:00.000-07:002015-04-10T14:36:24.352-07:00One Hot Topic: Law of Sines & CosinesWhen I work with high school math teachers in my consulting life, I am always surprised by how often the topic of <b>Law of Sines and Law of Cosines</b> gets brought up. Now, this may seem random, but it is often in the context of what content standards we <b>must </b>teach students, yet we don't have time for. However, with the adoption of Common Core State Standards, you may notice that these are actually now "Plus Standards", which means that they will <b>not actually be assessed</b>. So, my take away from this feeling of panic from teachers, is that they are finding the transition to Common Core difficult, they are unsure of what they still need to teach from the previous standards, and what is it exactly that they can now take off of their plates.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM_gHY8Qtwn2M-9ALEn5Blw8ZhIlnw81yH5w4iF-kZLsG4LQ79LfKf5n4dIshA9BtFBdr3Oy-ADL4bKrdWcIJHEoWu3WpAPO4CnouVLU_pXlgH_L_zxvRKvMoDmPzMfngIms4bQeotJuQ/s1600/Screen+Shot+2015-04-10+at+9.06.16+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM_gHY8Qtwn2M-9ALEn5Blw8ZhIlnw81yH5w4iF-kZLsG4LQ79LfKf5n4dIshA9BtFBdr3Oy-ADL4bKrdWcIJHEoWu3WpAPO4CnouVLU_pXlgH_L_zxvRKvMoDmPzMfngIms4bQeotJuQ/s1600/Screen+Shot+2015-04-10+at+9.06.16+AM.png" height="149" width="320" /></a></div>
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<span style="font-size: xx-small;">http://www.corestandards.org/Math/Content/HSG/SRT/</span></div>
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This always sparks great discussion because the bigger conversation, in my opinion, is: "<i>Is it really that important that we teach students a formula to memorize?</i>" Law of Sines and Cosines are not the only ideas or concepts that fall into this category, and I was guilty for many years of spending my classroom time lecturing students on topics that involved some nasty formula to memorize. However, I have made a huge effort to change this culture in my classroom.<br />
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If you haven't seen <b>Diana Laufenberg's TED Talk</b>, you need to! (Find it <a href="https://www.ted.com/speakers/diana_laufenberg">here</a>!) I love the way she talks about the role of school changing as technology becomes more available to our students and they no longer need to rely on their teacher for all the information; they can find it elsewhere. I am not suggesting that there is no longer a place for the teacher in a child's education, it just looks different than it used to.<br />
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So, I <b>tried something a little different</b> this year. My students had just returned from a week long spring break and before the vacation we had finished up Right Triangle Trigonometry. So, as a review, I drew this picture on the board and asked them to find all of the missing parts:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJeAYoOZJFzf_dBSoObbJ2CYb8_a_r3SfGRQzfKcO3EfgYlg-rXC6M_zXLfVdjFHUsp2nRbWVQKuxksCoOkA6zMW1nAKS4xs98EsZ3mCMnb0Qn2S50eprcYnUgeaFQAAXxSYj-_9Rebsg/s1600/Screen+Shot+2015-04-10+at+11.05.08+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJeAYoOZJFzf_dBSoObbJ2CYb8_a_r3SfGRQzfKcO3EfgYlg-rXC6M_zXLfVdjFHUsp2nRbWVQKuxksCoOkA6zMW1nAKS4xs98EsZ3mCMnb0Qn2S50eprcYnUgeaFQAAXxSYj-_9Rebsg/s1600/Screen+Shot+2015-04-10+at+11.05.08+AM.png" height="320" width="250" /></a></div>
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The context was: you have a 12 ft ladder that is propped up against the side of your house and the bottom of the ladder is 5 ft away from your house.<br />
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Students proficiently used their <b>trig tables that we had generated together</b> (I blogged about that <a href="http://themathymurk.blogspot.com/2014/04/building-trig-tables.html">HERE</a>) to accurately solve for the missing pieces. All spirits were high and I was pleased to see that students comfortably fell back into the swing of school after a week away.<br />
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Then I threw them a <b>curve ball</b>...<br />
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Same situation, you've still got a 12 ft ladder that is 5 ft away from the base of the building, however, instead of this building being your house, it's the <b>Leaning Tower of Pisa</b>. Students realized immediately that this was no longer a right triangle and they were curious to know how they might solve it. "<i>Can we use trig? Can we use the Pythagorean Theorem</i>?" I told them that the angle in the bottom left was 60 degrees and that it was indeed NOT a right triangle, so our "traditional" methods for solving were unavailable. <br />
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I gave them a quick pep talk that essentially summarized Diana's TED Talk and let them know that <b>my purpose for giving them this problem was to see what they could discover and research on their own</b>. I did not want them to come back with the values of the missing side and angles, instead, I wanted them to <b>explore possible methods</b> for how they might find the answers. Also, I told them that I made up this problem, so don't bother Googling it directly.<br />
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My hopes were for them to think about:<br />
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<ul>
<li><i>What makes this problem different/the same as the first one? </i></li>
<li><i>What question should I Google/ask?</i></li>
<li><i>How do I decide if what I've found is valid?</i></li>
<li><i>Is this method that I've found really applicable/appropriate?</i></li>
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I'll admit, <b>I had no idea how the process would go</b> and was mentally preparing myself for this to be a total flop. But when they came back to class, <b>my mind was blown</b>...<br />
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When students returned to class, I reminded them that I didn't expect them to have found the answers for the missing values. I wanted them to come to class with an idea of <i>how</i> to solve for the missing values and have a discussion about these ideas with their peers.<br />
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Within seconds of discussion I was hearing students say, "<i>I found this thing called the Law of Sines that I think we can use here.</i>" Another student said, "<i>I found that, too</i>!" One student said, "<i>I found the Law of Sines, but I also found the Law of Cosines and I think that is the one we should use based on the information given</i>."<br />
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<b><i>Are you frickin' kidding me?!?!</i></b><br />
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Now, <b>not all students had this information</b>. Some hadn't looked at it at all, others used the Pythagorean theorem and right triangle trig properties, but there were a good handful of students who were on the right path and had hit the jackpot! I took a quick lap around the room and was able to capture some images of what students had written down on their papers. Check it out:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjj40fs3As0Frenyyyzye1cdCkWtxLkppQUBDERCeIA_154VDAcXIzisAwZ52kviF1kad2k7R_sTXRefgM8avk9x6D7wOWxEYpFtnilQRNKc0JHGU2q9sUPOvW5DMCyqIHfKK9_iVO81GY/s1600/Screen+Shot+2015-04-10+at+1.01.05+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjj40fs3As0Frenyyyzye1cdCkWtxLkppQUBDERCeIA_154VDAcXIzisAwZ52kviF1kad2k7R_sTXRefgM8avk9x6D7wOWxEYpFtnilQRNKc0JHGU2q9sUPOvW5DMCyqIHfKK9_iVO81GY/s1600/Screen+Shot+2015-04-10+at+1.01.05+PM.png" height="139" width="320" /></a></div>
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<span style="font-size: xx-small;">Trying to use some of our more "traditional" methods for solving non-right triangles</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg41qi3dp403o-0Hsxb-owCFp41Id9J89k3dMeLy5uQcRbjiXDbEXU8u51yWyewbclOlHbmtB83G1LxAdHyAB6NaFFHKKJUL1WaTrf_UWZun7tGigJV6KeFFoc9zF_xvr3pKNT_adDIl7Y/s1600/Screen+Shot+2015-04-10+at+1.01.17+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg41qi3dp403o-0Hsxb-owCFp41Id9J89k3dMeLy5uQcRbjiXDbEXU8u51yWyewbclOlHbmtB83G1LxAdHyAB6NaFFHKKJUL1WaTrf_UWZun7tGigJV6KeFFoc9zF_xvr3pKNT_adDIl7Y/s1600/Screen+Shot+2015-04-10+at+1.01.17+PM.png" height="199" width="320" /></a></div>
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<span style="font-size: xx-small;">I almost fell over when I saw this written down. Imagine how I felt when she started explaining it to her peers.</span></div>
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<span style="font-size: xx-small;">Note: when I came by to take the picture, she said, "Don't mind the raspberry juice that I spilled on my paper."</span></div>
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<span style="font-size: xx-small;">Perhaps she's onto something :)</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRNbhUa5wJEibALvlj7_uLGP7ikRireIbXnKgeCBZZlAEghRT-G713tFdB8liICilIlKMviOzHE2oelIUbL_q3gJjgadIjCwt-y0e412bjWt9hjq2ABTHr-cg-iyhWnyaqlZf1l9azbyU/s1600/Screen+Shot+2015-04-10+at+1.01.30+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRNbhUa5wJEibALvlj7_uLGP7ikRireIbXnKgeCBZZlAEghRT-G713tFdB8liICilIlKMviOzHE2oelIUbL_q3gJjgadIjCwt-y0e412bjWt9hjq2ABTHr-cg-iyhWnyaqlZf1l9azbyU/s1600/Screen+Shot+2015-04-10+at+1.01.30+PM.png" height="320" width="283" /></a></div>
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<span style="font-size: xx-small;">This student admitted that she had found the formula on Wolfram Alpha and got stuck</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfGDj7hbWR0vLSro8RTXhPQZNkicaX7kSatmZl7SdxMk_zsy4BAJOyzafm65FGqY-58AvEJbBheMMcQmpXlOgpI_nHEjg9wm2ui_nWcKnx7fP9w20ZmoknaiQxF4Yntl02KgRIinX9MjA/s1600/Screen+Shot+2015-04-10+at+11.35.01+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfGDj7hbWR0vLSro8RTXhPQZNkicaX7kSatmZl7SdxMk_zsy4BAJOyzafm65FGqY-58AvEJbBheMMcQmpXlOgpI_nHEjg9wm2ui_nWcKnx7fP9w20ZmoknaiQxF4Yntl02KgRIinX9MjA/s1600/Screen+Shot+2015-04-10+at+11.35.01+AM.png" height="299" width="320" /></a></div>
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<span style="font-size: xx-small;">I feel like the question mark says something about her understanding of the concept,</span></div>
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<span style="font-size: xx-small;">but she was able to plug in values correctly</span></div>
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Admittedly, <b>students really were unsure</b> about what the Law of Sines and Law of Cosines was all about, but with a little nudging, I was able to get them plugging things in, refining their research, and persevering in their problem solving. It was fun to bounce back and forth between groups of students to help push them further in the problem or answer some questions that had arisen since my last visit. It was also exciting to hear them <b>argue</b> with each other and ask such <b>thoughtful questions</b> of their peers.<br />
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But the most beautiful part was that students were <b>sincerely interested</b> in finding the values. The opportunity for them to <b>take ownership of their learning</b> and for them to <b>discover what was important</b> played a major role in <b>their willingness to try</b>.<br />
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My big <b>ah-ha moment</b> here was that students really <i>are</i> capable of finding things on their own, but it is still important for the teacher to be involved. My students still needed me to answer specific questions and to help explain things along the way, but <b>THEY</b> were a huge part of that conversation and process, which was magical and something that I hope they will remember. I know I certainly will.<br />
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I told them that my goal was for them to take away some bigger skills from this activity than just remembering a formula. I admitted to them that when I was a math major in college, it was <b>essential</b> that I knew how to work with others and ask appropriate questions when solving a problem. Without these skills, I would have sunk like a stone in many of my upper division math classes!<br />
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After about an hour of working on this problem, I told them that they could move on to the next activity that I had planned for the day. The only groups that moved on were those who had found the missing side and angles of the triangle. Other groups <b>persevered and continued working</b>. I brought to their attention that they had just spent an hour working on a challenging math problem and for them to reflect on this feeling of wanting to finish. I asked them to contrast this experience with approaching a challenging problem in their math classes of the past. Many students nodded and smiled; they knew that this was something special.<br />
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One student said to me, "<i>Yeah, I've never spent this much time on one single math problem and actually enjoyed it. What have you done to my brain, Ms. Balli</i>?"<br />
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I see it as this: I've helped them learn how to use their own brain and not rely on mine. Isn't that what we want for all of our students?The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com4tag:blogger.com,1999:blog-1591780087227839882.post-56100522618382995962015-04-04T14:58:00.002-07:002015-04-04T14:58:42.802-07:00International ConsultingI had the incredible opportunity to provide some math education professional development for an amazing group of K-12 teachers in China last week! Having lived in Asia for most of my life, I felt right at home, surrounded by my international people. :) <div>
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My colleague, Patrick Callahan, and I presented on Common Core Mathematics (an overview), the implications for international schools & students, mathematical modeling, as well as some teaching strategies that can help increase communication and critical thinking among students.</div>
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It was an incredible trip for many reasons, but one of my favorite parts was getting to read about the PD experience through the eyes of one of the teacher participants. Evan Weinberg is a high school math teacher in Hangzhou, China who blogged about his take-aways from the day <a href="http://evanweinberg.com/2015/03/27/before-a-break-ccss-math-bogram-problems-and-peer-feedback/">here</a>. Make sure to check out all the cool things that he's blogging about! He's an extremely passionate and innovative educator who has a ton of great ideas to share! His blog is: <a href="http://evanweinberg.com/">http://evanweinberg.com</a></div>
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I hope that this will be the first of many international consulting experiences that I can be a part of. The teachers were fabulous, and of course getting to see the world while doing what you love is a true gift! I can't wait to go back!</div>
The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com1tag:blogger.com,1999:blog-1591780087227839882.post-14752853565832239642015-03-22T08:33:00.002-07:002015-03-22T08:33:30.465-07:00Using Proportions: What If Our High School Was Our Geometry Class?With the big push for <b>proportional reasoning in Common Core</b>, I wanted to find a way that I could make the whole idea meaningful for my students. Teaching Geometry requires that I teach a unit on <b>similarity</b> (mostly related to triangles), but I wanted to do things a little differently. I started the unit by having students create a <b>scale drawing of the classroom</b>, so they had practice setting up and solving proportions with an application. It was cool to hear the issues that came up around converting units and the accuracy of their drawings. I felt that the <b>engagement was high</b> and that students were seeing the use of a proportion.<br />
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I was given a book a few years ago called <b>"If the World Were a Village"</b> and I thought this would be a great way to continue the application and help give context to similarity and proportional reasoning.<br />
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I started this section of the unit by reading the students the book and had them <b>brainstorm questions that they would be interested to know about the population of our high school</b>. Students worked in their table groups of four to generate questions and they came up with ones that I would have expected and others that surprised me. There were questions like, <i>How many people drive to school? and How many people play a sport?</i> And then there were also questions such as, <i>How many students own a pair of designer jeans? or How many students have lost their virginity?</i> I typed up all the questions as they read them to me and then I narrowed them down to a list of appropriate questions that were reasonable to put into a survey.<br />
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Here's the final survey that I used to collect data:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxdNXvZAnim0Xx7a8kUtwwLCPc9cdApCdV6vb5VAhL-7Gr9_LTk43jE4Nx9I397Yd1lbOe22cPyWOXmXNOeGSN7YBUWAH-DNH-U6CtJT53O7PTFPjIp9Fbtt98MbUh-wMglLZy_h8AAS0/s1600/Screen+Shot+2014-04-17+at+4.57.05+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxdNXvZAnim0Xx7a8kUtwwLCPc9cdApCdV6vb5VAhL-7Gr9_LTk43jE4Nx9I397Yd1lbOe22cPyWOXmXNOeGSN7YBUWAH-DNH-U6CtJT53O7PTFPjIp9Fbtt98MbUh-wMglLZy_h8AAS0/s1600/Screen+Shot+2014-04-17+at+4.57.05+PM.png" height="320" width="261" /></a></div>
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The way that our schedule is structured, all students are enrolled in an English class, so I asked the English teachers if they would be willing to survey the students. They were more than accommodating and within a couple of days we had our data.</div>
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Through all of the "number crunching" it became obvious that students struggled to keep their data organized, but they eventually had some totals that they could work with. <b>If I were to do this project again, I could see taking my students to the computer lab to work with Excel.</b> The big, overarching question here was, "What if our high school was our Geometry class?" There were roughly 1,300 students surveyed and we wanted that population scaled down to the size of our Geometry class of 32 students. This is, of course, where the proportional reasoning comes in. If 472 students (I made that up) said that they drive to school, students solved the equation 472/1300 = x/32.</div>
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As a final product, I wanted students to <b>create an info-graphic</b> that could show the relationship between the entire surveyed population and our own Geometry class. I had them choose 6 of the 19 survey questions that they liked and put the data into the info-graphic. Also, I asked all the students to represent the ratio of boys to girls in their info-graphics (that was the only required survey question to represent). I recommended that they use <a href="http://www.easel.ly/">easel.ly</a> or <a href="http://piktochart.com/">Pictochart</a> to generate their graphics. I've included some of my favorite info-graphics, please enjoy!<br />
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The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com10tag:blogger.com,1999:blog-1591780087227839882.post-11304279631907052552015-03-21T21:43:00.003-07:002015-03-21T21:43:34.705-07:00Transformations TestI work with a lot of Geometry teachers in my consulting work and the <b>question of assessment</b> comes up a lot. I have made a point of making my classroom a place where we <i>focus on understanding and reasoning as opposed to number-crunching and doing math without context</i>. <br />
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I wrote this assessment back in September and used it as my unit test on transformations, but I didn't realize how many teachers would find it useful. <b>So, here it is!!</b> By far, it is the one assessment that I am most proud of because I feel as though it is a genuine representation of what I find important: <i>balance</i>. You will notice that there are some procedurally driven questions, but also lots of questions that asses student understanding of the concepts.<br />
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In the past, when I would write assessments, I would do my best to come up with a clever extra credit question that was challenging, but doable, so long as students extended their understanding of the given topic. Oddly enough, I rarely had students even attempt the extra credit question because they thought it was too hard. That was NOT the case on this test. I would say that <b>80% of my students tried it</b> and of those who attempted it, more than 50% of them got it right! I could not have been happier!<br />
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So, take a look, let me know what you think. I'd love to hear your ideas for additional/other questions, what has worked for you, and any other nuggets of wisdom you have to share.<br />
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<a href="http://bit.ly/TransformationsTest">Link to the Test</a>The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-3649199217605816332014-11-07T06:52:00.003-08:002014-11-07T06:52:30.130-08:00Student CuriosityI have spent the last six weeks in my Geometry class developing definitions for rigid transformations (reflection, rotation, translation). Some might call me crazy for devoting that much time, but I would argue that in addition to learning the <i>mathematical content</i>, students have learned <i>mathematical habits of mind</i> that will carry them through the rest of the year, and hopefully, the rest of their lives. I'm a "<b>big picture</b>" kinda gal, and I want my students to learn skills that are transferrable to their experiences outside my classroom.<br />
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Toward the end of the unit, I gave my students a task that was written by Jade White (High Tech High) in order to sort of "bring it all together".<br />
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As you can probably guess, the last problem has students graph a letter "H", but it has been rotated 90 degrees and is therefore not oriented correctly. The task I gave my students was:<br />
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<b>Identify a single transformation that will orient this letter correctly and send it to Quadrant 1.</b><br />
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I was impressed to see that no students were trying translations. I even heard them saying to each other, "We can't use a translation because that would not change the orientation." Pretty awesome stuff!<br />
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Students grabbed patty paper and were <b>testing out their ideas</b>. They were rotating, trying to figure out where the center of rotation should be. They were reflecting, trying to find a line of reflection that would work. I was impressed to see their <b>level of commitment and engagement</b> in this problem. In addition to that, I was excited to see that they were <b>using their definitions</b> of transformations to look for their, what we call, "<i>key ingredients</i>" (line of reflection, center and degree of rotation).<br />
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Once students had arrived at an answer, I asked them to share a transformation that satisfied our needs (proper orientation and located in Quadrant 1). One student had identified a line of reflection of <i style="font-weight: bold;">y = -x</i>. We tested it, and it worked. We connected it back to our definition of reflection and students saw the connections. Students presented other lines of reflection.<br />
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Another student had found a <b><i>center of rotation</i></b> at the point <i style="font-weight: bold;">(-3, 3)</i> and rotated it <b><i>90 degrees clockwise</i></b>. This was also a viable transformation and again, students connected the fact that this "worked" back to their definition of rotation. Students presented other centers and degrees of rotation.<br />
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At this point in the discussion, we explored what constituted as "Quadrant 1". Students shared that they found lines of reflection or centers and degrees of rotation that had segments or points from "H" mapped <b>onto</b> the x- or y-axis. <b>Is that allowed?</b> Then a student asked, "Well, <b>what's the definition</b> of Quadrant 1?" After I picked my jaw up off the floor, I said that my understanding of Quadrant 1 was that the x-values and y-values had to be positive. Students decided for themselves if they wanted to "count" the x- and y-axis as Quadrant 1, but the fact that they even asked was exciting for me.<br />
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The question that inspired me to write this post came from a student who is always thoughtful in her learning. She does not take things at face-value, she wants to know and understand concepts deeply and as completely as possible. <br />
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"<b>I noticed that the center of rotation of (-3, 3) is on the line of reflection y = -x, and both satisfied our requirement of sending H to Quadrant 1 with the correct orientation. Is it always the case that the center of rotation will be on a line of reflection that works?</b>"<br />
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<i>I didn't know the answer</i>. I gave students some room to explore, which I guess was a way for me to buy myself some time, but honestly, not all of them were engaged or willing to persevere through the question, so we moved on. I have since explored this question using Geogebra and I have some ideas surrounding this question, but I still don't know the answer.<br />
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What are your thoughts? What if it's not an "H" we're talking about? Is this idea generalizable?<br />
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For me, this is exactly what I want to see in my classroom: <i>students being curious about the mathematics they are learning and feeling safe enough to ask about it</i>. This is the kind of magic that happens when we give students the <b>time and space to explore.</b>The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com1tag:blogger.com,1999:blog-1591780087227839882.post-15477703067124200262014-11-05T13:28:00.000-08:002014-11-05T13:34:01.214-08:00Mathematical Modeling: Counting TreesEstimation is something that my students have always struggled with. In my experience, some reasons for this are that <b>a.)</b> students are rarely asked to do it, <b>b.)</b> students are conditioned to not take risks in math class, and <b>c.)</b> there is a genuine fear of being wrong that students grapple with when engaging in mathematics. But take a second to consider <b>how often, in the course of a single day, do we make estimates?</b> Whether it be at the grocery store, driving in your car, or cooking a meal, I would argue that it is a skill that we use on a regular basis.<br />
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So, how do we get students to develop this skill? I heard Dan Meyer (<a href="http://blog.mrmeyer.com/">http://blog.mrmeyer.com</a>) speak a few months ago and as we were working through a series of problems, he asked us to write down an answer that we knew was too high and one we knew to be too low. I had never done this before, but this particular strategy helped lower my fear of being wrong because, in a way, he was asking me to be exactly that...<b>wrong</b>. Calling it out, and recognizing that this particular answer that I'm giving you is incorrect, provided the safe space to take risks and explore the solution more authentically.<br />
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I have made it a priority to provide my students with opportunities to engage in mathematical modeling tasks this year. I wrote about our first modeling experience back in August (<a href="http://themathymurk.blogspot.com/2014/08/what-do-you-mean-theres-no-right-answer.html">What Do You Mean, There's No Right Answer?!?</a>) and since then, I have noticed that students are more willing to take risks, try a variety of methods, and overall be less comfortable (that's when I know they're learning). Last week, I gave them a task called "<a href="http://map.mathshell.org/materials/download.php?fileid=1239">Counting Trees</a>" from the <a href="http://map.mathshell.org/materials/index.php">Mathematics Assessment Project</a> website. This particular task necessitates the use of <b>estimation</b> and <b>communicating reasoning</b>; two skills that I know my students need to work on.<br />
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When I handed this task to my students, there was a general feeling of <i><b>annoyance</b></i> and <i><b>irritation</b></i>. "Do we really have to count all of these trees? That's going to take forever!" I told my students that I couldn't agree more, how annoying would it be to have to count all of these trees? But then I directed them to the wording of the task, "Think of a method Tom could use to <i><b>estimate</b></i> the number of trees of each type." This seemed to help with the grumbling and students felt a little less anxiety about having to actually count the trees.<br />
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When I worked on this problem for myself, I approached my estimation as a sampling of the entire collection of trees. I broke it into smaller, more manageable pieces and then scaled up. I have done this task with teachers and this is a very common approach, so I anticipated seeing it a lot with my students, but that was not the case. <i>More about that later</i>.<br />
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In my experience, a modeling or open-ended task can be <b>extremely meaningful for students</b>, but to get the most "bang for my buck", I have learned that it is essential for me to have a <b>game plan</b>. When I first started doing these kinds of tasks, I would just give them to my students and see what happened. There were exciting moments, but I felt like the experience as a whole lacked coherence and a sense of culmination. I hadn't really answered that "<i>So what?</i>" question. For the Counting Trees task, I tried taking notes in a systematic way so that I could structure the conversation/debrief in a way that made sense as well as answered the "<i>So what?</i>" question.<br />
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The first picture is the document that I created for each class period. I was curious what methods I would see my students using, but I wanted to remember who used that particular method, and then I wanted to record how many old and new trees their method got them. The second picture is from all three of my classes. I wanted to get a record of what numbers student were coming up with. I then wrote the number of old and number of new trees on the board in order from least to greatest number of old (along with its corresponding number of new trees). See below<span style="color: white;"><span style="background-color: white;">.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7v_vUIFCaqSBigs0-Wy3H1VhP8fbzFGoYMwUZfMrduYxPb5YItdaitthHRUJ-rDukkU5Q75d7kCDZ5YqlQIRejVxrbU3HCsFvOWDHG3QemyetGo4gwcJt_xld3_F3WdxP-WvlmcA9JxM/s1600/IMG_2518.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7v_vUIFCaqSBigs0-Wy3H1VhP8fbzFGoYMwUZfMrduYxPb5YItdaitthHRUJ-rDukkU5Q75d7kCDZ5YqlQIRejVxrbU3HCsFvOWDHG3QemyetGo4gwcJt_xld3_F3WdxP-WvlmcA9JxM/s1600/IMG_2518.JPG" height="320" width="240" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhry3YfOd2VSZW2l4K-oYtprPfyB_uUd0RtWjg3xAE4boFPfi2nj51xX843IJx3oMj2lIHLDBvRIO_wVWsgpSrZoxs5WPtBUNtVLRKc4wmL7oYD6UE2laUIiziL_PmJ4LWD6pMjCVvAdww/s1600/IMG_2519.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhry3YfOd2VSZW2l4K-oYtprPfyB_uUd0RtWjg3xAE4boFPfi2nj51xX843IJx3oMj2lIHLDBvRIO_wVWsgpSrZoxs5WPtBUNtVLRKc4wmL7oYD6UE2laUIiziL_PmJ4LWD6pMjCVvAdww/s1600/IMG_2519.JPG" height="240" width="320" /></a></div>
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These records of student data (above) helped drive the conversation about how many old and new trees Tom could estimate to be in the plantation. I began by asking students to look at the data and give an argument as to why they disagreed with a particular data set, and why. For example, in the first picture, students were not comfortable with the 89, 37 numbers because they knew that there were way more than 126 trees in the picture. In the second picture, students were not comfortable with the 330, 110 numbers because that was a total of 440 trees and they knew that there was enough space for 2500 trees, which would have accounted for only 18% of the plantation. Looking at the picture, they knew that there needed to be more trees. Similarly, the 1500, 750 data was, for some students, too high of an estimate. They did not think that 90% of the plantation was covered with trees. There was not consensus on this point, however. Some students argued that this was reasonable.</div>
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As the discussion continued, I asked student to come to the document camera and <b>explain the method</b> they used to estimate the number of new and old trees in the plantation. I started with the most common approach, which was to count the number of new and old trees in a single row and the number of new and old trees in a single column and multiply the two numbers together. Some students described this as being similar to a multiplication chart, others thought of it as an area model.</div>
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<i>Please note that all of these student work samples come from different classes and are a sample of each approach to the problem.</i></div>
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<b>Row Times Column</b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs6jXBciFQk0lvxNruDCKE-R5qlFR_5KlY4ESl7UULcP3RPWx6-H-cuCaIXquActuCkNpYHFrdiP44r54hmBcwdMoJ0d2IWI9sa8TUbxkMSwB4QzR12ooYVDWdajlDX4lkjKouHi9nOKo/s1600/FullSizeRender.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs6jXBciFQk0lvxNruDCKE-R5qlFR_5KlY4ESl7UULcP3RPWx6-H-cuCaIXquActuCkNpYHFrdiP44r54hmBcwdMoJ0d2IWI9sa8TUbxkMSwB4QzR12ooYVDWdajlDX4lkjKouHi9nOKo/s1600/FullSizeRender.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCyPCaIxo1sW5bdDbOrcVsLGFDHxwruDP642Yd0fxJbRHgihFyogM_eaVEgjgpyUJ17O3_zYVei_NO8az-dL1tDRfjMpgVoc7XSmx0q0dktxZG0d66QtCbDrMJQdV1KiTM_onMczT7Qc4/s1600/FullSizeRender_1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCyPCaIxo1sW5bdDbOrcVsLGFDHxwruDP642Yd0fxJbRHgihFyogM_eaVEgjgpyUJ17O3_zYVei_NO8az-dL1tDRfjMpgVoc7XSmx0q0dktxZG0d66QtCbDrMJQdV1KiTM_onMczT7Qc4/s1600/FullSizeRender_1.jpg" height="240" width="320" /></a></div>
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I then built on this idea, but showed a slightly different approach where students didn't look at a single row and column, but instead at the perimeter of the plantation.</div>
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<b>Perimeter</b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdfcRDFtN-mSXNBgaqWZYKJZ3e5bN8pQsI-o_WESYxn_G9-XzJufVDxwWoNGeJrFsTsDNQlUBpotT-vqNUTxDojqQMzZtIXxOEMqjGsrLX1uAQXp4KyNiZ6zrWpIZcxHLnFYWrDgUIDkw/s1600/FullSizeRender.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdfcRDFtN-mSXNBgaqWZYKJZ3e5bN8pQsI-o_WESYxn_G9-XzJufVDxwWoNGeJrFsTsDNQlUBpotT-vqNUTxDojqQMzZtIXxOEMqjGsrLX1uAQXp4KyNiZ6zrWpIZcxHLnFYWrDgUIDkw/s1600/FullSizeRender.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDER-AxqbD4Iuir2o4IaUMUsHlTVhbFon9ldbq4ek8zuNVQbEM8zTW262IQ1yRi4EM02yUzwUoqYdHtot4oI7xaF6qdri5nBQVxr8CeBazVjlD57h44nvGOSymr1WPX-Y31FYeaS9if4s/s1600/FullSizeRender_1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDER-AxqbD4Iuir2o4IaUMUsHlTVhbFon9ldbq4ek8zuNVQbEM8zTW262IQ1yRi4EM02yUzwUoqYdHtot4oI7xaF6qdri5nBQVxr8CeBazVjlD57h44nvGOSymr1WPX-Y31FYeaS9if4s/s1600/FullSizeRender_1.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUqCEcURQgl-z_fxbo6LuaPJ9ShXKlsIziADfxCDXhnMHhgVwq9dbz9csGpEn7Ce6_GGJGB3mIr5H4WwFK3PmxFTiplemn4s9qi3CjqIN5T54-A6miXPG2CH96mdUBKLo_ZfX35trxVis/s1600/FullSizeRender_2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUqCEcURQgl-z_fxbo6LuaPJ9ShXKlsIziADfxCDXhnMHhgVwq9dbz9csGpEn7Ce6_GGJGB3mIr5H4WwFK3PmxFTiplemn4s9qi3CjqIN5T54-A6miXPG2CH96mdUBKLo_ZfX35trxVis/s1600/FullSizeRender_2.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPGmHeETyHV1nixUIkgvY1AcZzaoDOW8ggLB3_YbzvbxCaXZTPe1_jpiuq41qWzKVI5l5EATHGnJEugk7ppKVigPcL0oDb9qAyQ6nk5FivN1eAONY5WA_1Qrsy6vNjM7KC5QXAm5h3Xlc/s1600/FullSizeRender_3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPGmHeETyHV1nixUIkgvY1AcZzaoDOW8ggLB3_YbzvbxCaXZTPe1_jpiuq41qWzKVI5l5EATHGnJEugk7ppKVigPcL0oDb9qAyQ6nk5FivN1eAONY5WA_1Qrsy6vNjM7KC5QXAm5h3Xlc/s1600/FullSizeRender_3.jpg" height="240" width="320" /></a></div>
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Students discussed <i>similarities and differences</i> between the two methods, which they thought would be more accurate, and even how it was similar or different to their own method.</div>
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I then extended the conversation to include student work samples that showed using multiple rows in their sample (as opposed to a single row and column). Students found that this may have been slightly more accurate because it was a larger sample, and still manageable, so they didn't have to try and count all of the trees in the plantation.</div>
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<b>Using 4-5 Rows of Trees</b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPh5FhaHV96-HbAzFSf_5dgMV9D8o9AzfpZyW-KXl-jWgq-9w02mlkDuN3Sfg3nT6lDltQj0mp_NDr9mza-HFPWbJ0J2QvaaOW2_CYm-zXUEa_CxI32M4Qnf83XC4_k0YaREURL3_j1-4/s1600/FullSizeRender.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPh5FhaHV96-HbAzFSf_5dgMV9D8o9AzfpZyW-KXl-jWgq-9w02mlkDuN3Sfg3nT6lDltQj0mp_NDr9mza-HFPWbJ0J2QvaaOW2_CYm-zXUEa_CxI32M4Qnf83XC4_k0YaREURL3_j1-4/s1600/FullSizeRender.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYSwZnKYe-JR4JIBgDPMM5eRiz8tTLNG7-L0U8-V-n8nhuNFgAwgU3YYCQ-6yrsI-gn9po1D-uTBVcR2bfV89BUHQDdF57edclRjUDqB93oAcJN9Dl51prsQvtwj73_SGLzRhw2WwVJNU/s1600/FullSizeRender_1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYSwZnKYe-JR4JIBgDPMM5eRiz8tTLNG7-L0U8-V-n8nhuNFgAwgU3YYCQ-6yrsI-gn9po1D-uTBVcR2bfV89BUHQDdF57edclRjUDqB93oAcJN9Dl51prsQvtwj73_SGLzRhw2WwVJNU/s1600/FullSizeRender_1.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirfBjGGpVT8vfCmnDaCJLEQUvOy9nKjDypZ0swgeNPiWh7SkijFcluDnLjtiwQyWucmSqvhaCjBFN61E4pDH9vuOj2XMzgvnTSNF4nJ_Z2zhUYec00gLE8IEU4utEWCCaYPguEsAg4iTk/s1600/FullSizeRender_2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirfBjGGpVT8vfCmnDaCJLEQUvOy9nKjDypZ0swgeNPiWh7SkijFcluDnLjtiwQyWucmSqvhaCjBFN61E4pDH9vuOj2XMzgvnTSNF4nJ_Z2zhUYec00gLE8IEU4utEWCCaYPguEsAg4iTk/s1600/FullSizeRender_2.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVG_1DCdNQeM_oTI1XGtm_KHoUemKc7jvvKZeHP5oAhcmID0nGPv51zcYYtIGk7uCiPewumz9OcJSQ0Q2ov4CR3TnEigsTJCRbL2AEQqrygxIzsqTBUypYnXu8KExRgX8bxoSaVDb12do/s1600/FullSizeRender_3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVG_1DCdNQeM_oTI1XGtm_KHoUemKc7jvvKZeHP5oAhcmID0nGPv51zcYYtIGk7uCiPewumz9OcJSQ0Q2ov4CR3TnEigsTJCRbL2AEQqrygxIzsqTBUypYnXu8KExRgX8bxoSaVDb12do/s1600/FullSizeRender_3.jpg" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhunBCa9xMf_CRBQFEWJDftz5jBo1wO_nH1Ap1mVTpWy62vPDk2C-aXB1J0JudpTDhTwyRbuxZJx_Okl3iUkUuYyvKeVHwma9-sxXbnJPxuF4vq8DrRj_71k63qseWD3_vu1-jJt1ByUxU/s1600/FullSizeRender_4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhunBCa9xMf_CRBQFEWJDftz5jBo1wO_nH1Ap1mVTpWy62vPDk2C-aXB1J0JudpTDhTwyRbuxZJx_Okl3iUkUuYyvKeVHwma9-sxXbnJPxuF4vq8DrRj_71k63qseWD3_vu1-jJt1ByUxU/s1600/FullSizeRender_4.jpg" height="240" width="320" /></a></div>
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The next set of student work shows the estimation approach of breaking up the plantation in a variety of ways. As I mentioned above, this was the most common approach I saw teachers use, but this was not true for students. The single row and column approach (above) was the most common for students.</div>
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<b>Breaking it into Smaller Regions</b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6LQjYmQ7CxpBjOZVtbpyi2rUBQI8oRBGQFfeJa9d1HwEYuB0Y9g18EPSvlchB62L4J9HymoIBa-n68TMBAjR0kV1I7FxcbZovL9un7eMy_AIUj9qOu2-AcjU7c8NAdtD1karfAhcupA4/s1600/IMG_2644.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6LQjYmQ7CxpBjOZVtbpyi2rUBQI8oRBGQFfeJa9d1HwEYuB0Y9g18EPSvlchB62L4J9HymoIBa-n68TMBAjR0kV1I7FxcbZovL9un7eMy_AIUj9qOu2-AcjU7c8NAdtD1karfAhcupA4/s1600/IMG_2644.JPG" height="320" width="240" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgc7wv5-b_uwb4MY1nQiKGFdcFc3QhP1Nelv2lvF0qi8AvkzgD_K1LiUZpu-z3twZHJayp__T2UTdegR7-SRnir4_o9kEGX8Tj6HYAaPwmI34x5Amalqo1x0xWAD0nJ-4NKNuwnaxQSDq0/s1600/IMG_2645.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgc7wv5-b_uwb4MY1nQiKGFdcFc3QhP1Nelv2lvF0qi8AvkzgD_K1LiUZpu-z3twZHJayp__T2UTdegR7-SRnir4_o9kEGX8Tj6HYAaPwmI34x5Amalqo1x0xWAD0nJ-4NKNuwnaxQSDq0/s1600/IMG_2645.JPG" height="320" width="240" /></a></div>
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After looking at these approaches, several students thought that these would be the most accurate. It was interesting to talk about why different ways of breaking up the plantation may have yielded different estimations. A lot of students said they preferred this method to others they saw or used themselves.</div>
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Something that came up in our discussion was and <b>interesting use of division</b>. Some students calculated the number of total trees and then divided by two. I put this back on the students: "<i>What are they assuming by dividing the total number of trees by two?</i>" They were spot on with their response: "<i>They're assuming that there are and equal number of new and old trees.</i>" We talked about whether or not that was reasonable. Most students felt that there were not an equal number and that there were more old than new.</div>
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<b>Equal Numbers</b>:</div>
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Another interesting idea that I saw students using was counting the number of blank spaces and using that to guide their estimate. Essentially, there are three things to consider: number of new trees, number of old trees, and number of blank spaces. Most students only focused on the number of trees, while others thought it was important to also include the number of blanks in their calculations.</div>
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<b>Using the Blank Spaces</b>:</div>
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Lastly, the approach that I saw that rounded out the conversation dealt with <b>ratios</b> of trees. As you can see, some students were not able to clearly articulate how they intended to use ratios, but I was impressed to see that students were able to make a practical connection to the use of ratios to help guide their estimate. As I said, this was the last approach I showed because for most students, it was the most sophisticated. </div>
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<b>Ratios</b>: </div>
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In reflecting back on this experience, I think that I could improve it by making stronger connections to the similarities between the various methods. I did my best to bring it back to the mathematics that the students were using, but also, what were the assumptions that they made? Students are learning that just because not everyone arrives at the same answer, they're more or less correct. They are learning that as long as you state your assumptions and use sound mathematics to answer the question, your solution will make sense. </div>
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I feel that this was a great task to use for students in developing their writing skills. I would like to place more emphasis on the precision of language next time because I feel like students were a little too vague in their descriptions of their approaches. However, overall, I feel that they were successful and it was an authentic, meaningful experience.</div>
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And I know what you're all thinking...<b>Did anyone just count the number of trees?</b> </div>
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Yes...yes, they did. :)</div>
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The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com2tag:blogger.com,1999:blog-1591780087227839882.post-29436206319513192452014-09-27T17:48:00.002-07:002014-09-27T17:48:55.963-07:00Building Definitions, Bongard StyleIn Geometry, I always find it difficult to get students to communicate the "<b>Why?</b>" or the "<b>How do you know?</b>" part of their solution. At a very basic level, I believe that this is why traditional two-column proofs are so challenging for students; they are rarely asked to defend their reasoning in a math class. As I mentioned in a previous post, I've been doing a lot of things surrounding vocabulary and definitions this year, and I have been pleased with the results. <b>Students are starting to defend their solutions</b> with statements like, "<i>By the definition of midpoint, I know that segment AB is congruent to segment BC</i>." I believe that this transition is greatly attributed to the fact that they were given the opportunity to build on their own understanding and definition of a term, I just helped solidify it and give it a more academic spin.<br />
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<span style="font-size: large;">What is a Widget?</span><br />
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My colleague and friend Amy Zimmer (<a href="http://zicker63.blogspot.com/">http://zicker63.blogspot.com</a>) shared an image with me at the beginning of the year that comes from the Discovering Geometry textbook (see below). The goal is for students to develop a definition of a Widget based on the pictures provided. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic7EzdFXM0E59n-WxII3RYkRi3shgK6SyFrg6FFj1idbZeNA6C6s1Ar0yST7PfT4q1S3L67ysvMiS1s6Juu08f9Fx3r_wk60aEbZhVfQ4EItcvXoC_YDXGSK98hYVYuL4ormcx45LdkL0/s1600/Widget.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic7EzdFXM0E59n-WxII3RYkRi3shgK6SyFrg6FFj1idbZeNA6C6s1Ar0yST7PfT4q1S3L67ysvMiS1s6Juu08f9Fx3r_wk60aEbZhVfQ4EItcvXoC_YDXGSK98hYVYuL4ormcx45LdkL0/s1600/Widget.JPG" height="163" width="320" /></a></div>
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I displayed the picture on the board and asked students to quietly observe what they thought a Widget might be. I asked them to be mindful of the characteristics that they were looking for to establish their definition. After a minute of solo think time, I had students turn and talk to a partner about what they were noticing and what their initial ideas were for what it meant to be a Widget. I called on a few students to share the characteristics they were considering, not to share their definition just yet (see list below).<br />
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Once the class agreed that we had listed all of the characteristics that were important, or even ones that didn't end up being relevant, we started to use those to refine our definition. I had students turn and talk to their partner again to formalize their definition and I asked them to share it out with the rest of the class. I wrote notes on the board and our definition was born.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtCjz4yF-QSmJYrMnD0EzhAdxnf2hH7tIv50L91OXin-ZkrhG4RShq7Swgo2jNvLIEMzqdyo5rUAJNijPQyBA7olNvjSo_avNqQJqk_enS7gRR9TzTlMzgl9k40NSZXY64RfDm2GQe6tU/s1600/photo-27.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtCjz4yF-QSmJYrMnD0EzhAdxnf2hH7tIv50L91OXin-ZkrhG4RShq7Swgo2jNvLIEMzqdyo5rUAJNijPQyBA7olNvjSo_avNqQJqk_enS7gRR9TzTlMzgl9k40NSZXY64RfDm2GQe6tU/s1600/photo-27.JPG" height="240" width="320" /></a></div>
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Once we had our definition, we tested it on the "Widgets" and "Not Widgets" from the picture to make sure it held true and that we had not overlooked anything. The last step of the process was to see which of the four images (A, B, C, and D) were Widgets. Without having to ask, students were able to say Yes or No and provide a reason why. I asked, "Is A a Widget?" and students responded with, "Yes, by our definition, A is a Widget." When I asked, "Is B a Widget?" students said, "No, because it does not have exactly one eye and one claw." It was obvious that they actually appreciated the concrete set of rules that <i>they</i> had developed in defending their answer.<br />
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In two of my classes, the rule about the eye and claw was not immediately as precise as "Exactly one eye and one claw". Students had said something about they had to have eyes and claws. When we looked at picture B, it required us to go back and revise our definition, which was so wonderful to show them that <b>our first draft did not need to be our final draft</b>.<br />
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It is my guess that for most students, this kind of abstract reasoning is not happening very often. When students think about definitions, they probably think about their English class and having to look up words in a dictionary. I wanted to show them that <b>using definitions is an acceptable way to build an argument</b> and that those definitions and arguments could be even stronger if they were given the chance to build them based on their own understanding.<br />
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<span style="font-size: large;">Bongard Problems</span><br />
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Dr. Patrick Callahan (Co-Director of the California Math Project) had shown me these problems a while ago and I was recently re-inspired when I saw him share them with a group of teachers in Fresno during a professional development session. Here is a slide from Dr. Callahan's Power Point:<br />
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A Bongard Problem is a collection of 12 images where the 6 on the left hand side all share a characteristic (or characteristics), and that characteristic is not present in any of the 6 images on the right hand side. The task for the reader (or student in my case) is to determine a rule that works for <b>all</b> images on the left, and <b>none</b> on the right. Here is an example (also taken from Dr. Callahan's PPT):</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEij-qcwWHjzq6xg5UtQos0ITmfT8Zbg631igOrvQUrxpWAOiSdsikdrZhvm9JolOJN55RtlVV9VjA6p1ZSOQ790DGoGeNsYfE8s29XMXtDHa9rJR23BB2DVXOS6-DOzIlharjIfGIiEgrw/s1600/Screen+Shot+2014-09-27+at+12.33.08+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEij-qcwWHjzq6xg5UtQos0ITmfT8Zbg631igOrvQUrxpWAOiSdsikdrZhvm9JolOJN55RtlVV9VjA6p1ZSOQ790DGoGeNsYfE8s29XMXtDHa9rJR23BB2DVXOS6-DOzIlharjIfGIiEgrw/s1600/Screen+Shot+2014-09-27+at+12.33.08+PM.png" height="208" width="320" /></a></div>
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Can you figure out the rule? This one is fairly easy. <b>Spoiler Alert!!!</b> All images on the left hand side are triangles and none of the figures on the right hand side are triangles. Try this one:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs9XeYYm0D7Bggz0nzgFNIr-8SqEBh0HETJzKpxj4VmmzxQ46L7x8DLmatxfwZ32Jrh3mlldw10qy5LyzlMQed8ffhi-i-I-ZslMOFCWJILAm8F1seEu3PpaqMpPNRKlhTghHQeCG6e-w/s1600/Screen+Shot+2014-09-27+at+12.34.51+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs9XeYYm0D7Bggz0nzgFNIr-8SqEBh0HETJzKpxj4VmmzxQ46L7x8DLmatxfwZ32Jrh3mlldw10qy5LyzlMQed8ffhi-i-I-ZslMOFCWJILAm8F1seEu3PpaqMpPNRKlhTghHQeCG6e-w/s1600/Screen+Shot+2014-09-27+at+12.34.51+PM.png" height="209" width="320" /></a></div>
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<a href="http://www.foundalis.com/res/bps/bpidx.htm" style="text-align: start;"><span style="font-size: xx-small;">http://www.foundalis.com/res/bps/bpidx.htm</span></a></div>
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Here's what I love about this particular problem (it's #4 on the website) -- students might not know the words concave or convex, but what so many of them pick up on is that the images on the right hand side are all "sunken in", or "have a bend in them", or even that they have a "part missing". They notice that all of the images on the left hand side are "pushed out" or have "nothing missing". This is exciting because it's an <b>opportunity to build on what they already know</b> and can observe about a figure to then <b>introduce the more formal term for these ideas</b> (concave and convex).<br />
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When I first introduced these to my students, I shared these two, along with some others that were more challenging. What I found interesting was that <b>there wasn't a sense that I was the one who held the answer</b>. Students had already picked up on the fact that they could check to see if their rule held true just by looking at the images on the left and the images on the right. I know that some teachers struggle with the idea of not being the authority in the classroom in terms of knowledge, or having the answer, but this happened so naturally that I actually addressed it with my students. I wanted them to acknowledge that they had the skills to be able to answer the question "<i>Does that work?</i>" or "<i>Is that right?</i>" <b>They learned very quickly that the responsibility of coming up with a rule fell on them, not on me.</b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivOA8RI7vTi9aDLDcT_gQajWURrf2gc7lRkMUeC8qGNNt-NqSKWbBX2DvEMJ6033a5WQ5Cpyr5huoj2gMKZSZbmRb_KTHTRHWaWump2_cAXRqX26Qij-7SGh_wmHur0-WXMfE_8w6-q2g/s1600/photo-28.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivOA8RI7vTi9aDLDcT_gQajWURrf2gc7lRkMUeC8qGNNt-NqSKWbBX2DvEMJ6033a5WQ5Cpyr5huoj2gMKZSZbmRb_KTHTRHWaWump2_cAXRqX26Qij-7SGh_wmHur0-WXMfE_8w6-q2g/s1600/photo-28.JPG" height="240" width="320" /></a></div>
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For homework that night, I had students <b>create their very own Bongard Problem</b>. I wanted to see what they would come up with and how creative they would be. These are some of my favorites. Which ones can you figure out the rule for?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYFRiUsrYYsNZHMP_zcg80hYUOAGjBLa4V4as1gLSxnPwfSzXfzYm9UvO3kO5BJZcsU3QlFxbQ7hZu02h4K5JCQtyAsxmvVQlprcmK7n61jm12dEnBvQw3UYaptU58WRcMfWU_aOBC5Rw/s1600/photo+1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYFRiUsrYYsNZHMP_zcg80hYUOAGjBLa4V4as1gLSxnPwfSzXfzYm9UvO3kO5BJZcsU3QlFxbQ7hZu02h4K5JCQtyAsxmvVQlprcmK7n61jm12dEnBvQw3UYaptU58WRcMfWU_aOBC5Rw/s1600/photo+1.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR54esxvcYClKHinWTxyemAXdYxovxKIMUFORd804KiXkn3eNiztWXiC6GRLv6SvIrk7uiqLrPWcm7Ww8j1J1_11ruB8L-96w-hB3QjIORbpmh2_tdoTScXi22kGUF5aInzkhzKxKmpJk/s1600/photo+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR54esxvcYClKHinWTxyemAXdYxovxKIMUFORd804KiXkn3eNiztWXiC6GRLv6SvIrk7uiqLrPWcm7Ww8j1J1_11ruB8L-96w-hB3QjIORbpmh2_tdoTScXi22kGUF5aInzkhzKxKmpJk/s1600/photo+2.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYWwmBf5TUc43izxJJO5H1zzQecsNNdfPqHOXDf0_MCyIciduJDqRLM3MdHc-xqUW2KyFOUmf_z4ZdA2qGfus3YASUHVZ_DccLT32hqlCMstmNLwpER6r3ftVhB2OghCEqLVLnUOUumgk/s1600/photo+3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYWwmBf5TUc43izxJJO5H1zzQecsNNdfPqHOXDf0_MCyIciduJDqRLM3MdHc-xqUW2KyFOUmf_z4ZdA2qGfus3YASUHVZ_DccLT32hqlCMstmNLwpER6r3ftVhB2OghCEqLVLnUOUumgk/s1600/photo+3.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP04tUF0IY3Ku4kui9O6VAmv-r_Gy964WPKNwg8Bz8BXYlwXisW8TUxB_Wi4RnnTqN8lNeRNVmstqj3DBgG7YQBAiTPVgzxSeuYFfaeVJg-3NNBysSunu-J66ppOYT26uOFR4EkW4jO4I/s1600/photo+4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP04tUF0IY3Ku4kui9O6VAmv-r_Gy964WPKNwg8Bz8BXYlwXisW8TUxB_Wi4RnnTqN8lNeRNVmstqj3DBgG7YQBAiTPVgzxSeuYFfaeVJg-3NNBysSunu-J66ppOYT26uOFR4EkW4jO4I/s1600/photo+4.JPG" height="240" width="320" /></a></div>
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<span style="font-size: large;">And Now What?</span><br />
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I wanted to continue with this work because I feel that the use of definitions is something that we can rely on throughout the school year. But, I wanted to formalize it and see if I could help students organize their thoughts with a template or structure that allowed them to articulate their thoughts more clearly. This is what I came up with:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizKAUiB3YlQvQoEp1YPgRHId0l1oCbkR92Ew0Mv6jsZyiSCL88vcj4UCMYvOrvZDUvm9guv2Am2KsIlEKN7_a55ejLzyniROOHfzIsuL7e_m2IG129ALuDHm58YWE2PXobIBGXpPbY1N4/s1600/Screen+Shot+2014-09-27+at+4.55.02+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizKAUiB3YlQvQoEp1YPgRHId0l1oCbkR92Ew0Mv6jsZyiSCL88vcj4UCMYvOrvZDUvm9guv2Am2KsIlEKN7_a55ejLzyniROOHfzIsuL7e_m2IG129ALuDHm58YWE2PXobIBGXpPbY1N4/s1600/Screen+Shot+2014-09-27+at+4.55.02+PM.png" height="400" width="312" /></a></div>
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When I introduced the template, it was clear that students were a little <b>apprehensive to write anything down</b> because it seemed so final. But with some practice, they realized that it was a work in progress and that they could take the risk of not having it perfectly stated the first time.</div>
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I asked a random student to pick a number between 1 and 100 (the number of problems that Bongard wrote) and in my first period class, the chosen student selected 94. So, we looked at Bongard Problem #94:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZRL05KNNeX3RV1H3BvDA-eh-S0TAXVCk0fbK27kH7YdyTuuue6AMQRMJlSp4wEgCie3XVWCh80tjU3OwVp0Mm3fGqC5mqtoklb7aZ2YXWbEii6nLlz8p-HtHJ3rAhksbiTG6VmCRfTwg/s1600/Screen+Shot+2014-09-27+at+5.04.41+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZRL05KNNeX3RV1H3BvDA-eh-S0TAXVCk0fbK27kH7YdyTuuue6AMQRMJlSp4wEgCie3XVWCh80tjU3OwVp0Mm3fGqC5mqtoklb7aZ2YXWbEii6nLlz8p-HtHJ3rAhksbiTG6VmCRfTwg/s1600/Screen+Shot+2014-09-27+at+5.04.41+PM.png" height="207" width="320" /></a></div>
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This ended up being a great one to address <b>precision of language</b>. Students realized that a subtle difference in wording could actually mean something that they did not intend to say. With the template, students were given individual think time to develop their own rule and write ideas down, then they could turn and talk to a partner to refine their rule and perhaps see things they had not initially observed. After about 2 minutes, I selected a group at random to share their rule with the class. I gave them one minute to come to consensus as to what I would write on the board. <b>Once I had their first draft of a rule written on the board, it was the job of the rest of the class to brainstorm feedback or clarifying questions for this particular group. It was not an opportunity for a new group to share the rule that they come up with for us to compare. I said that we were committed to helping this particular group come up with the best version of their rule as we could.</b></div>
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Students provided feedback on things that surrounded the precision of language. <b>They referenced the actual picture to defend why they thought their feedback was important.</b> If the first draft of the rule said "<i>The black dot is not at the end of the string of dots</i>", students might ask for more specifics of the black dot's location in regards to the white dots. Or, if the first draft said "<i>The black dot is between 2 white dots</i>", students might ask about the image on the left hand side that is on the right in the middle row. This elicited more precision of language such as "<i>between at least white dots</i>", etc. </div>
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Here are some student work samples of how they worked through the process of writing a rule:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSyeRQzmp9Ro2-SvjxffV56je7wxOs57Yrw-e-hbBo30tiyC_vbWCfLIF6bnKE4aPuvExsS7Qg-62Q_F4I9r9iPHliJAB3VkGL4WoyeZXoBT-EEx7d2ErG3Q9pJdKt1ub1KHGtO14pXYQ/s1600/photo+1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSyeRQzmp9Ro2-SvjxffV56je7wxOs57Yrw-e-hbBo30tiyC_vbWCfLIF6bnKE4aPuvExsS7Qg-62Q_F4I9r9iPHliJAB3VkGL4WoyeZXoBT-EEx7d2ErG3Q9pJdKt1ub1KHGtO14pXYQ/s1600/photo+1.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz1Js7rhvexeHRhJbXRUP8y4dv_zNkb8E0h9BS63UlwkVPi2JO_wRg9b4m2qbVzhfFwT7ec3ksMkaOXN9U8GCldYzMeR10985bkOzTUCi9neA7QQCLYeXp7FilE9kXL2k7vrxgDdcydME/s1600/photo+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz1Js7rhvexeHRhJbXRUP8y4dv_zNkb8E0h9BS63UlwkVPi2JO_wRg9b4m2qbVzhfFwT7ec3ksMkaOXN9U8GCldYzMeR10985bkOzTUCi9neA7QQCLYeXp7FilE9kXL2k7vrxgDdcydME/s1600/photo+2.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBSeoPHysVEyYnPLBj4j_LkpUr0QQY2vIJP7MjTRts7L5hhEtvMkSILP07yOvZF9SXD-IniUjAyCpRrxNbP3bPENjlkP8Po465C3_qkUPiLYg1X1rJm8_4oOPJb4mfDC2CDGAi-xKpBYM/s1600/photo+3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBSeoPHysVEyYnPLBj4j_LkpUr0QQY2vIJP7MjTRts7L5hhEtvMkSILP07yOvZF9SXD-IniUjAyCpRrxNbP3bPENjlkP8Po465C3_qkUPiLYg1X1rJm8_4oOPJb4mfDC2CDGAi-xKpBYM/s1600/photo+3.JPG" height="240" width="320" /></a></div>
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We did five of these Bongard Problems as a class over the course of about 3 weeks. I would start class with them, almost like a warm-up. I especially like how this practice went along with the first unit in our textbook, which is very heavy with new terminology and vocabulary. It was nice to have an example separate from the textbook to use that showed <b>the importance of developing a definition to build an argument</b>. I am excited to see where this takes us in our next unit on transformations!</div>
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Lastly, I wanted to share some pictures from the last Bongard Problem that we did as a class. These come from the three sections of Geometry that I teach, and <b>I wanted to capture the original rule and the result of giving feedback and modeling revision</b> that I used with my students.</div>
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The Bongard Problem was #58:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9GDc4AoA4HSX5Xy5asIvfWoy6iEz3IUhPyUOQ7IsuPykRbA4AL4arsbBK487K56SK0G_PsJmbelV92hmLP1XwCArQBkDEqJuFaL66qCi8xOSc1_AZisNF-xijXsYIA-r37-KFAdcN7OU/s1600/Screen+Shot+2014-09-27+at+5.38.43+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9GDc4AoA4HSX5Xy5asIvfWoy6iEz3IUhPyUOQ7IsuPykRbA4AL4arsbBK487K56SK0G_PsJmbelV92hmLP1XwCArQBkDEqJuFaL66qCi8xOSc1_AZisNF-xijXsYIA-r37-KFAdcN7OU/s1600/Screen+Shot+2014-09-27+at+5.38.43+PM.png" height="205" width="320" /></a></div>
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And here is how students developed their rule:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIqF2Xccvu3SY2_Il3j5GrVJre6czZCUmfVsb_GezQUvFeYpwiaEmBYRK3WnWtNxvCLbodhVuAaO6_xwF5OcmTN14XbgM06kgnPbWBEmV7-W0_Hj1yG3IQf7HM0SRWCxpObGurbhAjF9E/s1600/photo+1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIqF2Xccvu3SY2_Il3j5GrVJre6czZCUmfVsb_GezQUvFeYpwiaEmBYRK3WnWtNxvCLbodhVuAaO6_xwF5OcmTN14XbgM06kgnPbWBEmV7-W0_Hj1yG3IQf7HM0SRWCxpObGurbhAjF9E/s1600/photo+1.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnEgeK2lsY6vOJBCbqlKr12fuKSRH05GTyKSmojzVOKhckvh8uMhU5wgDtG2QFNzr29va6ziwaDfIJvQkjfcvcZSBpJz5rN2YjA-w-BEf8R3h7Zo5blKr0wn4G9L5FH6sJB9i-kJEj00Y/s1600/photo+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnEgeK2lsY6vOJBCbqlKr12fuKSRH05GTyKSmojzVOKhckvh8uMhU5wgDtG2QFNzr29va6ziwaDfIJvQkjfcvcZSBpJz5rN2YjA-w-BEf8R3h7Zo5blKr0wn4G9L5FH6sJB9i-kJEj00Y/s1600/photo+2.JPG" height="240" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtPCxux_nERj3i1AbZ4_QKpqUJlJWKye2OwZt-HlyczBeQlN2Pza517PtOnGWCNk0ARw-Drd957VCG9eIkHVtVTvNRdJJ6-N4cUN9lPrPSmiXtrR-2IRg42xQgU0T242M6Sj41d9qYRfI/s1600/photo+3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtPCxux_nERj3i1AbZ4_QKpqUJlJWKye2OwZt-HlyczBeQlN2Pza517PtOnGWCNk0ARw-Drd957VCG9eIkHVtVTvNRdJJ6-N4cUN9lPrPSmiXtrR-2IRg42xQgU0T242M6Sj41d9qYRfI/s1600/photo+3.JPG" height="240" width="320" /></a></div>
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I would encourage teachers to use Bongard Problems in their classes to help students build a foundation for using definitions and understanding the importance of precision of language. There are some Bongard Problems that are quite challenging, but for the most part, I have found that students enjoy the puzzle element of finding the rule and are willing to take the risk of being wrong the first time. I have seen growth in their reasoning skills and I am encouraged by their perseverance and overall willingness to try and work hard.</div>
The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com8tag:blogger.com,1999:blog-1591780087227839882.post-55269415332917362622014-09-10T17:08:00.001-07:002014-09-10T17:08:57.097-07:00Redefining Geometry VocabularyIt's no secret that a high school Geometry course is riddled with wordy, complex definitions for terms that students inherently know something about based on life experiences. The struggle for a teacher is how to access that prior knowledge and make connections to the Geometric concept? In the past, I have been the one to define the terms for the students, simply using the definitions from the textbook, thinking that those would be the most accessible to students. Surprisingly, this was largely unsuccessful.<br />
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A few weeks ago, I read a blog post by Lisa Bejarano (<a href="http://crazymathteacherlady.wordpress.com/2014/08/23/starting-geometry-with-definitions-synthesizing-stratagies/">Check it out here</a>) where she spoke about similar struggles to provide access to students in the realm of Geometry terms and definitions. The element of her blogpost that was the most inspiring to me was her use of examples and non-examples for each term. I was also inspired by a colleague from High Tech High in San Diego who had created an activity that included an element of revision that I found very interesting (see sample below).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi59qLVqNqOpKBHjCRVJziD8wrUthbnZ3q97_4sdGRm9LsuiyLr4QSMcS8QodVU6AmnY0p-ECsZ0o_K-WEZkUarZIETVQbI0kwcPDIx0CqPSKT6dyEcSsX-q-TKq-t52LZ9b22VCCrpnEQ/s1600/Screen+Shot+2014-09-10+at+8.16.43+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi59qLVqNqOpKBHjCRVJziD8wrUthbnZ3q97_4sdGRm9LsuiyLr4QSMcS8QodVU6AmnY0p-ECsZ0o_K-WEZkUarZIETVQbI0kwcPDIx0CqPSKT6dyEcSsX-q-TKq-t52LZ9b22VCCrpnEQ/s1600/Screen+Shot+2014-09-10+at+8.16.43+AM.png" height="91" width="400" /></a></div>
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<i>Headings: Word, Student Generated Definitions (words & images), </i></div>
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<i>Exact Definitions (words & images)</i></div>
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Together, Lisa and Jade inspired me to develop an authentic way for students to acknowledge what they already knew about a term or concept, but then also have the perseverance to learn more and build understanding.</div>
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I began with the first section in my textbook, and I listed all of the key vocabulary terms. The list included point, line, plane, collinear, coplanar, segment, endpoint, and ray. (The old me just fell asleep thinking about having to teach these terms) :) I created a chart that was similar to Jade's, but also brought in elements of Lisa's activity by addressing examples and non-examples (see sample below).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT3KJUzeQXmpUEZYP1znLOoBnKZOsW88vrnaHRe1y3vnZj_xoiuk57IksJh-90H6vyCQNsx-C-pXOxp3mK6V1seCj2B2Dh9v3Xj7Iu5Ypyqk-wPRxmwqGqlDtiuLP73-qD06Ax_criWLw/s1600/Screen+Shot+2014-09-10+at+8.23.29+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT3KJUzeQXmpUEZYP1znLOoBnKZOsW88vrnaHRe1y3vnZj_xoiuk57IksJh-90H6vyCQNsx-C-pXOxp3mK6V1seCj2B2Dh9v3Xj7Iu5Ypyqk-wPRxmwqGqlDtiuLP73-qD06Ax_criWLw/s1600/Screen+Shot+2014-09-10+at+8.23.29+AM.png" height="235" width="400" /></a></div>
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<i>Headings: Word, Student Generated Definition (words/images), </i></div>
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<i>More Precise Definition (words), Picture, Non-Example</i></div>
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I gave each student a copy of the vocabulary chart and asked them to read through the list of terms, checking in with themselves about what they already knew about the word or concept. They had the choice to write down their thoughts or not, but if they wanted to, they would do so in the "Student Generated Definition" section. Next, students turned to their peers (at their table groups) and shared what they already knew about each term, again they had the choice to write down what they wished. <br />
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This is what I had written on my whiteboard as a reference after they had time to think independently:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbGOWAiMUM_7YTboUZyzkHyYoRLuvL41KPaUTGK-xT183e1yom1fIli6Dm7T0bwYXUFAz5tq4jWdQTR3tS0TeQxWWArQJEFUP_7X7cBImRj9FUALGpUfwdJkTxG3Xc2goCViLdiZTG7vc/s1600/IMG_2282.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbGOWAiMUM_7YTboUZyzkHyYoRLuvL41KPaUTGK-xT183e1yom1fIli6Dm7T0bwYXUFAz5tq4jWdQTR3tS0TeQxWWArQJEFUP_7X7cBImRj9FUALGpUfwdJkTxG3Xc2goCViLdiZTG7vc/s1600/IMG_2282.JPG" height="240" width="320" /></a></div>
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The next stage of the activity involved students getting up and moving between tables where I had distributed strips of paper with examples and non-examples for each term (see sample below -- THANK YOU LISA!!!). Students were asked to study the pictures in the two categories and record what they needed in their vocabulary chart. If they needed to draw all of the pictures, that was fine, if they saw some were redundant, I left it up to them to choose what to include.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTzhg_mfTUVWvWNo5AoXuy6FaS1ceJ7aq7f28lTI2AVXcZt2nYSXXsGT-zmWPOLp2NE3lRVdBVI9ItCxDNkQjgqsLdbsE5GFtSnKBnh3xyHD7MFq0Oeu-_D46ae-9ppa78eMWe9d507A4/s1600/Screen+Shot+2014-09-10+at+1.55.01+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTzhg_mfTUVWvWNo5AoXuy6FaS1ceJ7aq7f28lTI2AVXcZt2nYSXXsGT-zmWPOLp2NE3lRVdBVI9ItCxDNkQjgqsLdbsE5GFtSnKBnh3xyHD7MFq0Oeu-_D46ae-9ppa78eMWe9d507A4/s1600/Screen+Shot+2014-09-10+at+1.55.01+PM.png" height="245" width="320" /></a></div>
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<i>Please note that these pictures are computer generated, but the ones I created for my students were drawn by hand, so they looked slightly different. For example, for the line pictures, I had more about labeling and I included points on the lines, etc.</i></div>
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What I found during this piece was that there was not a lot of discussion happening. Students were traveling around the room simply copying the pictures as if that was the most important part. So, what I did was I actually asked them to move around quietly as they observed and copied pictures; no talking. I don't know why I thought to do it this way, it just seemed like a battle that I was not going to win if I kept asking them to talk, talk, talk. It just seemed so forced.<br />
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So, when they were done checking out the pictures, I asked them to return to their seats and told them that they were not allowed to have their pen or pencil in their hand, that drawing and writing time was over for a little while and that this was now time for discussion. <b>I think by defining the time more clearly, it allowed for students to focus on one element of the activity at a time.</b> Sure enough, students began brainstorming what they thought was an appropriate way to define these terms. They were building on each other's ideas, they were referencing the pictures, and they were accessing their background knowledge.<br />
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Once discussions began to die down, I invited them to write their updated versions for each term, still in the "Student Generated Definition" section. I was pleased to see that students who chose to write something right off the bat were not afraid to revise and make their definitions better. Students were open to learning from each other and building on each other's ideas. <b>This, however, is an area where I believe they can still grow.</b> <br />
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Often times when I ask my students to discuss an idea and I leave it open ended, it's not much of a two-way conversation, but more of a "I did this..." and "I did this..." from student A to student B and vice versa. They are still learning to engage in a conversation, not just telling each other what they did.<br />
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At this point, I was out of time for the period, so had to make a decision about what to assign for homework. Any Geometry teacher will tell you that the first section of the book is often times an exercise in learning, recalling, and using the Geometric vocabulary that we have discussed above, so I thought, Why not see how and what they do with it? So, I assigned 12 problems from the first section that involved everything from drawing pictures that used these terms, using the terms in ways to describe a picture, identifying these terms from a picture, and so on. I was hopeful that things would all sort of "come together" for them when they were given an opportunity to practice. <b>I knew that I was risking the possibility of students solidifying an incorrect definition of a term</b>, but I was confident, based on my observations, the opportunities to work with their peers, and access to the formal definitions in the book, that there was sufficient support for them to be successful.<br />
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When students returned to class the next time, I was pleasantly surprised by how well they were able to use the vocabulary terms in an appropriate context and when checking the homework, there were very few questions that were raised. After reviewing the assignment, I did take the time to provide students with a "More Precise Definition" that was a balance of the textbook definition, items involving notation, and an approachable way to think about the term. But, before I provided them with the more precise definition, I asked for them to share with me (and the class) some of the things they wrote down. It was fun to hear how they described certain things: "a ray is half a line", "collinear means together on the same line", because <b>it was obvious that they had made sense of the term, but perhaps didn't have an "academic" way of defining it</b>. That's where I could help extend their learning, not stifle what they authentically know.<br />
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It's been about a week since I did this activity and we have since moved on to discussing things like the Segment Addition Postulate and the construction of copying a segment For the first time in my 9 years of teaching, I have found that students are willing to and comfortable using the Geometry vocabulary from the activity. We are looking at problems that involve the use of the vocabulary and students are using them accurately.<br />
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Here's a perfect example:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzvFj736lXUnFSPM424EDJtoRXXTTkiHQygHxHjQlF3Nm9uHHPAduw3596uChpPAhnh9BReJOEhRkVTcAh4C89vrmp3KhCzQ6V02eLZrJm-K2PBgOQMgnB4tTMZqYvw8TebAX1MU5LDmk/s1600/Screen+Shot+2014-09-09+at+4.09.11+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzvFj736lXUnFSPM424EDJtoRXXTTkiHQygHxHjQlF3Nm9uHHPAduw3596uChpPAhnh9BReJOEhRkVTcAh4C89vrmp3KhCzQ6V02eLZrJm-K2PBgOQMgnB4tTMZqYvw8TebAX1MU5LDmk/s1600/Screen+Shot+2014-09-09+at+4.09.11+PM.png" height="234" width="320" /></a></div>
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<i>Holt Geometry</i></div>
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After posing this question to my students, some of them asked clarifying questions like "Are these points collinear?", "Can we think of this freeway as a segment? Or is it more of a line?", "Which towns are the endpoints of the segment?"</div>
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The last thing that I will say about this set up for teaching and learning vocabulary is that I'm finding that if a term comes up in class that they are still not quite sure what the definition is, they are so comfortable with opening up their binder and taking out their chart. In the past, when I have taught the definitions of these terms as a part of their notes, they never return to them again. I think that this is another strength of the chart and activity as a whole. <b>Students have interacted with the words and they hold on to that experience in a way that I have not seen in the past</b>. </div>
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I can't wait to hear about your ideas for how to make this activity even better! And again, thank you to Lisa Bejarano and Jade White for the inspiration!</div>
The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com4tag:blogger.com,1999:blog-1591780087227839882.post-43844545398076670512014-09-01T19:02:00.002-07:002014-09-01T19:03:01.087-07:00"Where Do I Put P?" An Introduction to Peer Feedback"Construct Viable Arguments and Critique the Reasoning of Others" is a new expectation for students that is proving to be challenging for them, and for me. How can I offer students opportunities to do this without just asking them to "give feedback"? They don't necessarily know how to give and/or receive feedback, so what structures can I put in place to help them be more successful?<br />
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I have used this problem from MDTP (<a href="http://mdtp.ucsd.edu/">http://mdtp.ucsd.edu</a>) in the past as a whole class activity. Students think about the problem independently, then discuss as a table group, and finally share with the whole class where they would put P. I would draw the number line on the board and a group member would come up and place a magnet on the line that represented the position of P. (Forgive the formatting - for the life of me I could not get the points to rest ON the line)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdmq5tZk03ZX_uD9jWfx3EwnMbrXwELOxWRixD6ERc3zg0oP1DK2T0az_RlQjUTAB1hq3tPmnDuUSfUTIjghMJKGff8ooFXEbtF0zDbhs7klegSOLJWuFF1TnBImhHeC0grDAPT4OXE7E/s1600/Screen+Shot+2014-08-31+at+9.26.41+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdmq5tZk03ZX_uD9jWfx3EwnMbrXwELOxWRixD6ERc3zg0oP1DK2T0az_RlQjUTAB1hq3tPmnDuUSfUTIjghMJKGff8ooFXEbtF0zDbhs7klegSOLJWuFF1TnBImhHeC0grDAPT4OXE7E/s1600/Screen+Shot+2014-08-31+at+9.26.41+AM.png" height="102" width="400" /></a></div>
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I chose to introduce this problem a little differently this year. I gave this task to the students using a template that Dr. Patrick Callahan (Co-Director of the California Math Project) had shown me. The template is designed to offer students a structure and process for providing and receiving feedback.<br />
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I knew that if I just gave this paper to students and said "GO", they would have felt very unsuccessful and discouraged, so I knew I needed more structure. I have found that the more explicit and transparent that I can be with my students, the more likely they are to try and persevere. Here is what I had written on the whiteboard:<br />
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We acknowledged that this was a new process for everyone, but that it was acceptable and expected to not have it perfect on the first time (notice what is written in orange). I read these expectations to my students at least 4-5 times throughout the lesson, reminding them to try hard and stick with it.<br />
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So, that's how I was building the culture and expectations of providing feedback, but I felt that they would also benefit from some specific examples, so here's what I wrote:<br />
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And with that, we were on our way. Here's how the lesson flowed...<br />
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<b>Step 1</b>: Students write their own individual argument in the "First Draft of Justification" box.<br />
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<b>Step 2</b>: I had students turn to their neighbor and read their argument to see if it made sense when read out loud, also this was the first chance that students had to share their argument (practice, practice, practice). The partner was required to provide some sort of verbal feedback to their peer (sentence starters on the board) and the student who just read their argument was given the opportunity to edit their first draft if they wanted to.<br />
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Here's an example of a pair of arguments from students who were working together:<br />
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<b>Step 3</b>: The other partner reads their argument, just like the first student. Then, as a pair, they decided whose argument would be shared with another pair to receive feedback. I did not give any suggestions as to what criteria should be considered when selecting whose argument should be selected, it was totally up to them. The reason that I had one argument from each pair "travel" to another group is because I wanted students to practice giving written feedback as pairs, not as individuals.<br />
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<b>Step 4</b>: I chose a random piece of student work (avoiding the disclosure of the name) to put under the document camera. As a class, we read the argument and brainstormed possible feedback that would be useful. How could we help this student make their argument clearer, more viable, or more logical?<br />
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Here are some that I chose to start the discussion:<br />
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<b>Step 5</b>: Students held up the paper that was to be sent to another pair, and I came by to grab them. I distributed one new paper to each pair and students began reading the arguments. Students consulted the whiteboard to see what sentence starters would be appropriate for providing written feedback on this particular argument. Together as pairs, students provided feedback on the first draft of the argument.<br />
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<b>Step 6</b>: Again, I chose a piece of student work at random to put under the document camera so that we could discuss the feedback and how it related to the "Feedback Expectations" I had written on the board. This time we considered if the feedback was useful, we were not as worried about the first draft of the argument.<br />
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Here are some samples of feedback that we looked at:<br />
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<b>Step 7</b>: Essentially, we repeated Step 5. This time, when students got a new paper they were reading the first draft of the argument and the first round of feedback to see how they could build on the story. What new feedback could they provide? I was very clear that it was unacceptable to just write "ditto" or "what they said". I pushed them to think of new feedback or a suggestion that would build on the existing feedback.<br />
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**When I have shared this template with other teachers, they have suggested that I reformat it so that it goes First Draft, Feedback #1, Second Draft (meaning the owner of the paper gets it back after the first round of feedback to revise their argument), Feedback #2, and finally Final Draft/Revised Version. I have not tried this yet, but it would potentially help with the "ditto"/"what they said" issue.<br />
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<b>Step 8</b>: I repeated Step 6 except now, we were looking at the whole "story" as it unfolded to consider what worked and what didn't. What were some good examples of meaningful and useful feedback, not so great examples, etc.<br />
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<b>Step 9</b>: I returned the papers to their original authors to read through the feedback and revise their arguments. Because I only had one paper per pair receive feedback, I had students work as pairs to brainstorm the best way to revise the original argument. Then, as individuals, students filled in the last part of the template on their own paper. Even if their argument did not receive any feedback, I thought that students had seen enough examples that would help them revise what they had originally written.<br />
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These are the final products of the two I shared above in Step 2 (notice that the paper that was not passed did not have written feedback):<br />
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<b>Closing</b>: There were essentially three main methods that students used to answer this question. The first, and most common solution was to assign values to X and Y and then add those values together to determine the value of P. The other two, which were very rare in each of my three classes, were first, identifying the distance (or length of the segment) between 0 and X and then adding that distance on to Y (in the positive direction), and second, speaking in more general terms about X and Y. For example, a student might say that X and Y are both greater than 0, therefore their sum would also be greater than 0. Furthermore, students might say that X is approximately one-half, and Y is greater than X, but less than 1, therefore their sum will be greater than 1, but less than 2. All methods involve very different thinking, and I LOVE it!<br />
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<b>Here are some samples of the various methods</b>:<br />
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Assigning Values<br />
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Distance<br />
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Location<br />
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I took the time at the end of the lesson to show my students these three approaches, because I felt like there was not a lot of variety in the arguments that they got to read. This was probably the greatest down side of this particular problem, because as a result, there was not a lot of rich discussion about alternate methods or solutions. However, once I showed my students the less common methods, it was obvious that they appreciated the fact that there were other ways to do it.<br />
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Lastly, I wrote four words on the board, "Successes, Challenges, Liked, Learned". I asked students to flip their papers over (the back of the template was blank) and write about any or all of these four topics. What I realized was that they really valued the open ended nature of this activity, they admitted that giving feedback was difficult, but they hoped that we would do it again. And oddly enough, when asked to write a reflection, nobody asked things like, "how long does it have to be?"<br />
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<b>Here are some of my favorite reflections</b>: Sorry for the light writing<br />
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Overall, I thought this was a really great way to start the year and it really set the tone for my expectations and my style of teaching. If I were to do it all over again, the only thing I would really change is the task I chose. I would prefer to use a task that elicited a wider variety of solutions from my students from the beginning. That being said, this particular task has two other parts, one where students need to place points on the number line that represent Y - X and X - Y, and also X times Y. Perhaps this could be a good follow-up task to try next week or later in the semester?The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com5tag:blogger.com,1999:blog-1591780087227839882.post-77267683467862330272014-08-29T18:32:00.000-07:002014-08-29T18:32:25.099-07:00Problem Solving with NPRA friend of mine introduced me to the free app "Stitcher" this summer (thanks, Solana!), which essentially is a way to organize podcasts that you enjoy listening to. I wasn't much of a "podcaster", but with the help of the app, I've learned a TON about a lot of really cool stuff! One of my favorite podcasts is the NPR Series: Sunday Puzzle. Considering I'm a math teacher, that should be no surprise to you!<br />
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The way the Sunday Puzzle works is they start the podcast with a recap of last week's puzzle and they reveal the answer. Then, they contact a random winner (someone who answered the puzzle correctly online) and have them on the show (via phone) to go through a series of smaller puzzles/riddles. These puzzles are not necessarily related to the Sunday puzzle. Finally, they introduce the new Sunday puzzle and give you until the following Thursday to submit your answer.<br />
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On August 17, the title of the show was, "Is There An Echo In Here?" and the Sunday Puzzle was one from Sam Lloyd. Here it is...<br />
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<b>"You have a target with six rings bearing the numbers 16, 17, 23, 24, 39, and 40. How can you score exactly 100 points by shooting at the target?"</b><br />
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I thought this would be a great puzzle for my students to try. After introducing it, they had some clarifying questions, which I thought were great!<br />
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"How many times do you shoot at the target?" I told them that it did not specify, so they could decide.<br />
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"Can you hit a number more than once?" I told them, again, that it did not specify, so they could decided.<br />
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I gave them about 3 minutes to work independently, trying it on their own, and it was completely silent. Every student was engaged. <br />
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After this independent work time, I wanted to tap into the minds of my students, so I just asked for their initial strategies.<br />
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Lots of students said that they were just randomly adding numbers to see what happened. Others said that they added all 6 numbers to get 159 and then tried to see if they could combine any of the numbers to get 59 that could be taken away. Students found that they could get close to 100, either just over or just under. Students also explored finding a factor of 100 and then repeating that process. Lots of good stuff here!<br />
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Students who arrived at a solution found that they could get to 50 by adding 16 + 17 + 17 and then they doubled it. Another approach that was shared was very sophisticated and inspiring. A student said that she added 16 six times and found that she got 96, which was 4 short of 100. She knew that 4 was not an option for a ring, but thought there might be a way to gain 4 by adding in a 17. She tested it by adding 16 five times and then 17 once to get 97. Her strategy was working...she was getting closer to 100. Ultimately, she found that four 17s and two 16s was the combination she needed. I loved it!<br />
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As I described earlier, the 5 minute podcast starts with the answer from the previous week's puzzle, so I shared the solution with my students (Ben Parks, August 24, "A Puzzle Hokey Pokey, That's What It's All About"). They were delighted to know that they had arrived at the (only) correct solution.<br />
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I decided to play the rest of the podcast for them, which they loved! The riddles involved 2-word phrases (first word has 5 letters, the second word 4 letters) that were generated by dropping the last letter of the first word, then reading the remaining 4 letters backwards to get the second word of the phrase. (Try this one...Where Peruvian pack animals shop: Llama Mall) :)<br />
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Finally, the moment came to hear the new puzzle for the week. Students were excited to get in on the action. Here's the latest Sunday Puzzle from Jason Zuffranieri:<br />
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<b>"Name a world leader of the 1960's, 2 words, change the last letter of the 2nd word, then switch the order of the words, that is putting the 2nd word in front, and the result will name a hit song of the 1990's. Who's the leader and what's the song?"</b><br />
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I didn't know what to expect, but I sent my students home with the challenge of figuring this out. I offered extra credit for getting a correct solution, but they had to somehow defend their process, so I would know that Google didn't deserve the points. :)<br />
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I returned to class today (Friday) after introducing this on Wednesday and I was pleasantly surprised that two of my students in my 3rd period class had figured it out! One young man said that he had his cell phone open to Google looking up world leaders, while his mom's phone was open to iTunes looking up hit songs of the 90's. He started piecing things together and arrived at his answer.<br />
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The other student said that she started with song titles from the 90's, looking for ones that included names (Mr. Jones, Mrs. Robinson, Jane, etc.). Then she looked at a list of leaders and found a last name that could easily be changed to a common word and pieced it together.<br />
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Here's what they came up with...<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQWGC32BayKyyqk7MIih21YlmIfIz3pCbaQdDBlBBGk3KnqeQx46zxTVAyFo52dlc0OJua4piDP3uWWqRix1iT2PmgjSC1fmCxjSh9qrAT7TbRhK08LkxuoClrYQCI0A8cD0Mgi0oMlBs/s1600/photo-25.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQWGC32BayKyyqk7MIih21YlmIfIz3pCbaQdDBlBBGk3KnqeQx46zxTVAyFo52dlc0OJua4piDP3uWWqRix1iT2PmgjSC1fmCxjSh9qrAT7TbRhK08LkxuoClrYQCI0A8cD0Mgi0oMlBs/s1600/photo-25.JPG" height="240" width="320" /></a></div>
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SO proud! I can't wait to incorporate this into my classroom on a more regular basis and even encourage my students to submit their answers online! It was obvious that these two students were not motivated by the extra credit, they were just enthusiastic puzzle solvers. What more could I ask for from a math student?The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-82356302070820785992014-08-24T08:49:00.000-07:002014-08-24T08:49:30.157-07:00"What Do You Mean, There's No Right Answer?!?" Wanting to start the school year off with a bang, I decided to introduce my students to my classroom culture through a mathematical modeling problem that I did not have an answer to. The <a href="https://www.illustrativemathematics.org/illustrations/1559">Laptop Battery Task</a> from Illustrative Mathematics (www.illustrativemathematics.org) is one that I've used with teachers during professional development, as well as with students, and I like that it makes everyone uncomfortable. Teachers and students alike have something to gain from engaging in this problem, and other problems of its kind. Students are uncomfortable because this is possibly their first encounter with a math problem where their teacher doesn't know the answer, and teachers are uncomfortable with presenting a problem that is so open ended. It's scary stuff!!!<br />
On the first day of school, I thought I would take a risk and see how my students would handle The Laptop Battery Task. I went over several things on the first day, laying groundwork for classroom culture, and with 15 minutes left in the 90 minute period, I put this task in front of them. I asked them to work independently and brainstorm a solution for when Jerry might have a fully charged battery. Students exited the class without discussing their work and their homework was to continue working on the task.<br />
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The next day, as students entered, I asked them to take out their work, being thoughtful to not use words like "answer" because I wanted to emphasize that this was a work in progress and their work could be revised. Students began bouncing ideas off of one another, comparing strategies and initial solutions. It became immediately clear that no two students arrived at the same answer, or if they did, they got there in different ways. <br />
As I circulated the room, I recorded times that students predicted Jerry's battery to be fully charged, as well as what methods students used to arrive at their solution. I then wrote the times on the whiteboard and students saw that there was a large range of solutions from 10:28 to 11:23. As far as a teacher move, at this point I wanted to make sure that students saw the two main approaches that students took in solving this problem (graphing and average rate of change), but that we also looked at student work from an individual on the 10:28 end, one on the 11:23 end, and one in the middle somewhere.<br />
I selected students whose work was to be shown to the rest of the class, but it was the task of their partner to articulate their reasoning and process. I took this as an opportunity to build collaboration and a culture of being responsible for understanding the reasoning, not just a passive audience member. It seemed to work very well!<br />
As the first two students presented, the class saw that graphing would have been a useful option/tool to solve this problem (only one or two students in each class chose to graph), but also that even though their own solution did not match exactly, their reasoning was very similar to those of their peers. The final student presentation was of the solution of 11:23. The time of 11:23 didn't sit well with the class, but they had a difficult time articulating why. They knew in their gut that 11:23 was too late (even the students who had this solution knew something was off), but I loved seeing the perseverance and the unwillingness to back down, as opposed to admitting defeat in arriving at an "incorrect" answer. What came next was simply magical.<br />
The partner of a student who had a solution of 11:23 came to the document camera and displayed the work for us to interpret. Students were instantly engaged and curious to know how their peer arrived at 11:23. The student work showed proportions, which many students had used, so there was a certain timidness for students to question the solution. Essentially, this is what the student work showed:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_42aUFl3G_crme6SaPeg6cztVTLUY7dk1Af7KolmiGnX-JaQuWI8jJFejC-bmcikfZszJ923pJaVLSD_iA3__xvj7MmY9ErwVfCcXDqw5BsnsajP_lc2dqxo6GMN2BR_2x_g5m9pEw9I/s1600/Screen+Shot+2014-08-24+at+8.13.58+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_42aUFl3G_crme6SaPeg6cztVTLUY7dk1Af7KolmiGnX-JaQuWI8jJFejC-bmcikfZszJ923pJaVLSD_iA3__xvj7MmY9ErwVfCcXDqw5BsnsajP_lc2dqxo6GMN2BR_2x_g5m9pEw9I/s1600/Screen+Shot+2014-08-24+at+8.13.58+AM.png" height="108" width="320" /></a></div>
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Students were able to make sense of the proportion and found the setup to be quite useful. They understood the meaning of the 132 minutes, but they still weren't buying the 11:23 final charge time. Finally, one student raised their hand and asked the question, "But why are they adding the 132 minutes to 9:11?" Silence. Nobody answered...they just looked at me, waiting for me to answer. I took this opportunity to have them turn to each other and brainstorm their answer to that question.</div>
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Ultimately, we came to the understanding that THIS was the issue with the 11:23 charge time. The 132 minutes of total time made sense to them, but they grappled with where that fit in the context of this problem.</div>
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A student asked if they could come to the board to draw something that they thought might help make sense of this particular issue. After picking my jaw up off the ground, I said of course and handed over the whiteboard marker. This is what she drew:</div>
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Without me saying anything, students looked at the image and began digesting its meaning. Students were arguing, the discussion was getting interesting. Finally, I asked if someone could articulate their connection to the diagram and how it helped them understand the problem? The original student who arrived at a solution of 11:23 raised her hand and gave a thoughtful response to how she had added the 132 minutes to 9:11, which would have meant that the battery charge was at 0% at that time, which was not true. That was the crux of her issue, and she was able to work through it without me saying a thing. Other students articulated their thoughts of where the other 66 minutes should go or even proposing another proportion we could set up from here. They were making sense of the argument of a peer to help build their own understanding.</div>
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Once the excitement had died down a bit, I brought closure to the discussion by highlighting some of the elements of this lesson. I even asked students to share with me some of their observations of the process as a whole, and they said things like, "You didn't lecture on this topic before giving us this problem", or "You placed a lot of emphasis on how we got an answer, not on the answer itself". Bingo.</div>
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Of course, it would have been too good to be true if it ended there. Before excusing them for the day, I had a student raise their hand and ask, "So, what's the answer?" With all eyes on me, a quick shrug of my shoulders communicated to them that that was not my priority, and I was sticking to it (and, oh, by the way, I have no idea what time it will be fully charged). Some students left irritated, but overall, I think the students understood that this was not going to be a typical math class. </div>
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Mission accomplished.</div>
<br />The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-63711963224754709382014-08-10T09:30:00.000-07:002014-08-10T09:41:24.857-07:00Memes on Day One Much like many of my friends and family, I am getting ready to start another year of teaching. As always, summer has gone by way too fast, my brain is filled with ideas of how I am going to make this year the best one yet, and I'm excited about the potential that this year holds. I wanted to do something a little bit different this year for the first day of school. <br />
Like many teachers, the first day of my class involves reviewing my syllabus, discussing expectations, setting boundaries, and other things that students will most likely not remember in a few hours. I thought I would mix things up this time around. One thing that always guarantees a laugh from me is a good meme. My personal favorites are the "Most Interesting Man In the World" memes that are in the form "I don't always...but when I do..." They get me every time. <br />
I've compiled a few memes in a powerpoint to share with my students on the first day (I did not include any of the most interesting man). My hope is that this presentation will accomplish two things; 1) they will learn and remember what my expectations are for the class, and 2) they will begin to understand my teaching style; a balance of humor, hard work, and a belief that they can be successful, regardless of their previous experiences in a math class.<br />
Stay thirsty my friends<br />
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<iframe frameborder="0" height="327" scrolling="no" src="https://onedrive.live.com/embed?cid=92825231238595F6&resid=92825231238595F6%21107&authkey=ADJDCiq9i99vTEg&em=2" width="402"></iframe>The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-49739731883718805342014-05-13T20:57:00.001-07:002014-05-13T20:57:31.877-07:00Beyond Can'tI was fortunate enough to receive a $15,000 grant in 2012 (Thank you CDE and PacTIN) that would essentially be used to pay for professional development on my journey to implementing Common Core. I had set out to use the money to hire outside experts to help my team learn about increasing the opportunities for our students to construct viable arguments and critique the reasoning of others (SMP 3), but what I discovered was that we weren't ready.<br />
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I felt that I could be successful in executing our original plan, however, some of my department members and colleagues were finding it difficult to even get started. So, instead of trying to fight it, I decided that the money would be better spent on hiring the outside experts, but shifting our focus to Common Core in general (Thank you, Patrick Callahan). <br />
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This is a video that I made that is my "digital story" of my journey with this grant. I chose to focus on how the grant impacted me personally and the growth that I experienced from receiving these funds. I believe that because of this grant I am a better teacher, colleague, and person. This is my story...<br />
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<a href="https://www.wevideo.com/hub/#media/ci/191744970">Beyond Can't</a></div>
The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-59921147861921285232014-04-15T19:09:00.000-07:002014-04-17T09:47:10.071-07:00"In the ______": Students Finding Trig in Their EnvironmentSo, as I mentioned in a previous post, I've had my students build their own trig-tables this year and I must say that they are doing extremely well! I was inspired by my amazing colleague, Amy Zimmer (<a href="http://zicker63.blogspot.com/">http://zicker63.blogspot.com</a>) to do this fun activity with my students. The original document involved a scene set in a park (see below, original document found at: <a href="http://www.funmaths.com/worksheets/downloads/view.htm?ws0012_1.gif">http://www.funmaths.com/worksheets/downloads/view.htm?ws0012_1.gif</a>) and Amy thought that it would be cool to switch it up and have the students come up with their own scene. In the woods, at the beach, in the kitchen, etc.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRZC4SHHy1-9r2PZB-cU96HJ_Zx8g7Fo0cT7A3ZCfIBy1BfA7fn4yMLKUg1vct4hq6Bwv4JEDBYgWweHEM_-xpEy9tExLYNtjvxwn-Zj0yl23LqsKomm5x0uKaG9dT6p_aQPrCHsuFcqQ/s1600/Screen+Shot+2014-04-15+at+3.04.33+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRZC4SHHy1-9r2PZB-cU96HJ_Zx8g7Fo0cT7A3ZCfIBy1BfA7fn4yMLKUg1vct4hq6Bwv4JEDBYgWweHEM_-xpEy9tExLYNtjvxwn-Zj0yl23LqsKomm5x0uKaG9dT6p_aQPrCHsuFcqQ/s1600/Screen+Shot+2014-04-15+at+3.04.33+PM.png" height="197" width="320" /></a></div>
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The requirement for the assignment was that they needed to create some sort of scene or scenario where they were setting up right triangles that they could solve for missing sides and angles. Another requirement was that they needed to set up one problem that involved using the ratio altitude/base, one using altitude/hypotenuse, and one using base/hypotenuse, all given one acute angle and one of the side lengths in the ratio. The fourth type of triangle they had to set up should have had both acute angles missing and given two side lengths, basically using inverse trig. The fifth triangle did not have a requirement, they could repeat any of the other types of triangles (the wildcard).<br />
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I enjoyed seeing their work and seeing how creative they got. The creation of the poster was essentially phase one of the activity and phase two was actually solving for all missing values of the triangle. Students jumped right in to using their trig charts and did surprisingly well with the triangles that were missing their acute angles. I really enjoyed seeing them do this because I think even high school students need to show their inner artist and get a little crafty sometimes. I had them staple their work to the back of the poster and any student that worked on the poster had to submit a page of work. I've included some of my favorites. Thanks, Amy for the inspiration! Enjoy!<br />
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From Amy Zimmer's Class:<br />
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<br />The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com5tag:blogger.com,1999:blog-1591780087227839882.post-18019308032256505382014-04-08T07:38:00.000-07:002014-04-08T07:38:08.289-07:00Building Trig Tables I'm sure I'm not the only Geometry teacher out there who gets frustrated with how their textbook approaches a subject. Right Triangle Trigonometry, for me, is one of those subjects. The textbook that my school uses just dives right in to introducing sine, cosine, and tangent with little to no context, and certainly no mention of similar right triangles. So, this year, I decided to do things a little differently.<br />
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A few months ago I was doing some work with the incredible Kate Nowak and this exact topic came up. She shared with us that when she was a classroom teacher, she had her students build their own trig tables and she ended the lesson with a Romeo & Juliet problem where Romeo needed to find how high the balcony was so that he could climb to see his star-crossed lover. (Check out Kate's blog post: http://function-of-time.blogspot.com/2009/04/introducing-right-triangle-trig.html) I thought I would give it a shot with my students.<br />
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I am teaching 6 sections of Geometry this year, so I thought that this would be a great opportunity to collect lots of data to calculate some reasonable values for sine, cosine, and tangent, even though they would not be called that until later.<br />
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Much like Kate's blog suggests, I had my students start by drawing 5 right triangles with a specified acute angle on an 8.5 by 11 piece of printer paper. This was challenging for some because a lot of my students do not know how to operate a protractor. Once we got over that hurdle, they were on their way.<br />
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After drawing their 5 right triangles, I set them up to measure the three side lengths of each triangle, using Kate's handout, students recorded their measurements. The third step was to calculate the ratios of the side lengths (aka: sine, cosine, tangent) :) of their 5 triangles. Lightbulbs began to go on around the room because students were realizing that their ratios for the 5 different triangles were all the same (or at least pretty close). This was a great opportunity to take a look at some data and discuss potential outliers or mistakes. Here's a sample of student data:<br />
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What I found was that students were not yet ready to be the "critical consumers" that I was wanting them to be. This sample doesn't have any glaring errors, but nevertheless, they were willing to accept the data at face value. I asked them to look for patterns and initially I was hearing things like "It's going up by 0.3". Well, that may be true for the "altitude" column from row 1 to row 2, but they didn't know how to look for any other type of pattern. Finally someone said, "They're all increasing," which was wonderful to hear! We talked about why that was a good sign and why it made sense. Then another student pointed out that the hypotenuse values were always the largest in the set of 3 data points and we discussed why that was a good thing. Next, we practiced finding the ratios to make sure that they were dividing correctly. It was great to discuss why all the numbers should be less than 1 or why a number should be bigger than another; conversations that I don't think they had ever had. Lastly, we practiced finding the averages of the 5 different ratio entries. I have to say, this was a bit of a train wreck! For example, in the first column (a/b), students told me that the average was 1.4 or something like that. They didn't understand why that was not possible. After some discussion, they were convinced that the average needed to be closer to 0.2-something.<br />
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Because I have so many sections of Geometry and so many students in each section, I thought this would be a great opportunity to collect lots of data, but also calculate averages. Each student was working with a specific acute angle, but so was their partner, so once they got their ratio averages, I had them average their averages with their neighbor. That's what we wrote in the yellow charts on the board (see below).<br />
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Each class generated this chart and now it was time to compare the charts from the 6 different classes. I asked students to help me calculate averages from the yellow chart (if there was more than one entry) as well as help me understand if any of the numbers just did not make sense. They picked up on things like increasing and decreasing patterns, and if there were two entries in a cell they were able to compare two numbers and see if they were reasonable or not. The next step involved me writing up the numbers from each class into a fairly large chart on the white board (see below).</div>
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The last step in this process was to look at all this data and determine if the numbers were reasonable from all 6 classes. I told the students that if they were comfortable with all the entries, they should calculate the average, if they felt that there was an outlier (or more than one), they should be able to defend why, throw it out, and then find the average of the remaining numbers. This was an interesting process because some students argued to keep specific numbers while others fought to get rid of them. The final product for each class was their own trig table (see below), which was a lot of work, but now we were ready to do some trig! :)<br />
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So, now it was time to actually do some work and I had a moment of panic (I actually lost some sleep over this) because I wasn't sure how I was going to bring this all together. Kate's worksheet had some great questions, so I started there. First of all, I wanted to take some time to marvel at the fact that all of this data collection happened independently (no two classes worked together), so how was it that their data was so close? To my surprise, it seemed that students had already considered this and very quickly responded with "Because the triangles are similar!" We discussed this further and it was clear that they understood the idea of similar triangles by AA. I felt they were ready to put this stuff to good use.<br />
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The last problem on Kate's handout involved Romeo & Juliet, and I thought we could give it a shot. I tried a sample using a 30 degree angle and a base of 50 feet. I asked students what I was solving for, they knew it was altitude, and then I asked, do I know anything about the relationship between altitude and base of a right triangle with a 30 degree angle? And that's when the lightbulbs REALLY started to go on! What an amazing moment for the students to realize that all of their hard work had lead them to this place where they could solve a problem without me telling them what to do. They were off to the races and solving for all different angles.<br />
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The last thing that I did was said, "Now, Romeo really wants to impress Juliet, so he is fashioning a zip-line for her to ride down and meet him. If he is 70 feet away, how long with the line need to be for your given angle?" Some started to set up the same ratio as before, but quickly realized that this relationship was now base and hypotenuse. After realizing that, they were able to set up the correct equation, but I have to say, solving it proved to be a bit more difficult. :)<br />
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Even though I spent a whole week doing this with my students (3 periods on block schedule), I feel that it was totally worth the time and effort that the students put in. They worked with data, they were critical, precise, and thoughtful, which are things that don't usually happen for them in a math classroom. Overall, I feel that this was a huge success, but I am already waiting for them to get mad at me when I tell them that they can just use a calculator to do some of this stuff. :)The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-29180044855185502352014-03-27T14:44:00.002-07:002014-03-27T14:44:32.926-07:00Mathematical Modeling: Best Night's SleepI got a Jawbone band for Christmas this year and I have loved seeing how well I do in achieving my sleep goal and steps goal. I am finding that I am actually a really good sleeper and that my attempt to take 10,000 steps a day is hit or miss. As a teacher, I do better with my steps on the weekend and days when my students are testing are my worst. I thought that my students could benefit from taking a look at some real-life data and apply what they've learned from our studies of ratio, similarity, and proportions.<br />
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With almost 200 students this year, I did not want to grade that many tests for the end of the chapter, so I made this a group assessment. My students sit in groups of four as it is, so the groups were already chosen. As you can see from the write-up below, each student had a different responsibility in the group. Students dove in to the data before choosing roles that spoke to them. I was impressed to see how many different methods there were that students latched on to. Some chose percentages, some chose to represent their data in bar graphs, other pie charts; it was interesting to see what they were drawn to.<br />
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Here are the three pieces of data that I showed the students:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7XfulPkWgDM_a34w8bwu9a6e0_G2gU9JzZzOntJqAUGCoNDKt-89b1ZxTkHqrrSlOzZH_mtOWJdaX1X0UKjBtZHwhBzaeM7C1mDWZRk5mp_i_5eOsv2VtIUn7miG0IYuTtzYV6yxPb9o/s3200/Screen+Shot+2014-03-27+at+11.27.25+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7XfulPkWgDM_a34w8bwu9a6e0_G2gU9JzZzOntJqAUGCoNDKt-89b1ZxTkHqrrSlOzZH_mtOWJdaX1X0UKjBtZHwhBzaeM7C1mDWZRk5mp_i_5eOsv2VtIUn7miG0IYuTtzYV6yxPb9o/s3200/Screen+Shot+2014-03-27+at+11.27.25+AM.png" height="200" width="155" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaHUoVZHw35ivXgM83OGv288894m08YnX_zR9sXSFbIsB6X7NbwoD_CEl4sN-3K2an0HWN3UTQCpokf259WnjIIOHp5vaUwpouZ56HyYN8l98wEPIv1Xv458qOCnwz0VKMrCF-bU4xOc8/s3200/Screen+Shot+2014-03-27+at+11.28.35+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaHUoVZHw35ivXgM83OGv288894m08YnX_zR9sXSFbIsB6X7NbwoD_CEl4sN-3K2an0HWN3UTQCpokf259WnjIIOHp5vaUwpouZ56HyYN8l98wEPIv1Xv458qOCnwz0VKMrCF-bU4xOc8/s3200/Screen+Shot+2014-03-27+at+11.28.35+AM.png" height="200" width="155" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTKZC6X1pUmDmUQHnQWHeYamOQGsix4L2qFdlSQf9nRlFs3sUmwv_3OVl6IyGDIeLIBwWvby10MHnZoyp1wrgGy0XXLZchbSV3icw6lFu2HbH45CciYTjCfiNplypmbOLusNL_j0dOgVc/s3200/Screen+Shot+2014-03-27+at+11.26.35+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTKZC6X1pUmDmUQHnQWHeYamOQGsix4L2qFdlSQf9nRlFs3sUmwv_3OVl6IyGDIeLIBwWvby10MHnZoyp1wrgGy0XXLZchbSV3icw6lFu2HbH45CciYTjCfiNplypmbOLusNL_j0dOgVc/s3200/Screen+Shot+2014-03-27+at+11.26.35+AM.png" height="200" width="151" /></a></div>
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I actually cut off the bottom portions that showed the amount of sound sleep, deep sleep, how long it took to fall asleep, and how long each person was in bed for. Students were given the bar graphs as well as the amount of time each person slept and what percent of their goal was met.</div>
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I started class by showing the three graphs through the projector so that they could see them with color. Students determined what the data showed them, what additional information they needed to know, and asked other questions that came up. I chose to answer some and leave some for them to answer because the data was available to do so.</div>
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At this time, I gave students the following handout for them to read and decide what role they were interested in. </div>
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Name:
____________________________________________<o:p></o:p></div>
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Geometry 2013-2014<o:p></o:p></div>
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Date:
_____________________________ Block: ________<o:p></o:p></div>
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<span style="font-size: 16.0pt;">Chapter 7 Similarity Test: Who Got the Best Night's Sleep?<o:p></o:p></span></div>
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<span style="mso-tab-count: 1;"> </span>This is a group test, but <b style="mso-bidi-font-weight: normal;">each student is responsible for submitting
his or her own portion of the test</b>.<span style="mso-spacerun: yes;"> </span>Please
put a check next to the role that you had in your group in the list below.<span style="mso-spacerun: yes;"> </span>If you are a group of 4, each student should
take on one role.<span style="mso-spacerun: yes;"> </span>If you are a group of
3, each student will take on their own role and then share the role of
“Processor”.<span style="mso-spacerun: yes;"> </span>When your group is
finished, please staple all of the parts together with this page as a “cover
page” for each section.<span style="mso-spacerun: yes;"> </span>Please staple
the pages in the order as they appear below.<o:p></o:p></div>
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Group Member 1: Graphics <o:p></o:p></div>
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<span style="mso-tab-count: 1;"> </span>When considering who got the best
night's sleep, it’s important to compare “apples to apples”.<span style="mso-spacerun: yes;"> </span>You may notice that in the three sets of data
that I provided you, all three of them use different scales because they went
to bed and woke up at different times.<span style="mso-spacerun: yes;">
</span>The job of the Graphic team member is to create a visual that uses the
same scale for all three sets of data and create this graphic in a way that
helps support your argument for who got the best night sleep.<span style="mso-spacerun: yes;"> </span>You will be graded on accuracy, neatness, and
quality of your graphic.<o:p></o:p></div>
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Group Member 2: Data <o:p></o:p></div>
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<span style="mso-tab-count: 1;"> </span>Looking at the three sets of data,
you can see that there is some “number crunching” to do.<span style="mso-spacerun: yes;"> </span>In order to compare the three sets of data
accurately, you need to use the same scale or the same units of measure.<span style="mso-spacerun: yes;"> </span>It is your responsibility to help your group
make sense of the numbers so that you can compare them accurately.<span style="mso-spacerun: yes;"> </span>You will be graded on the accuracy, neatness
and quality of your calculations.<o:p></o:p></div>
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________
Group Member 3: Argument Writer<o:p></o:p></div>
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<span style="mso-tab-count: 1;"> </span>I purposefully chose data that
didn’t have a “clear winner” so that you would have to defend your answer to
“Who got the best night's sleep?”<span style="mso-spacerun: yes;"> </span>Think
about what you value in a “good” sleep.<span style="mso-spacerun: yes;">
</span>Is it length of time spent sleeping?<span style="mso-spacerun: yes;">
</span>Most deep sleep?<span style="mso-spacerun: yes;"> </span>There is no
right answer here, so you need to convince me that your answer is correct.<span style="mso-spacerun: yes;"> </span>Imagine that you are convincing a skeptic, or
someone who got a different answer than you.<span style="mso-spacerun: yes;">
</span>You will be graded on the clarity of your argument, how well you connect
it to your data and graphic, and the quality of your writing.<o:p></o:p></div>
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________
Group Member 4: Processor<o:p></o:p></div>
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<span style="mso-tab-count: 1;"> </span>I may have some questions as to how
you arrived at your final answer, but that’s why we have the Processor.<span style="mso-spacerun: yes;"> </span>I would like to know how your group worked
together to come to your final answer.<span style="mso-spacerun: yes;">
</span>This should be a descriptive account of what your group did, but more
importantly, WHY?<span style="mso-spacerun: yes;"> </span>Because this is a
chapter test on similarity, please make sure to mention HOW you used
similarity, ratios, and/or proportions.<span style="mso-spacerun: yes;">
</span>You will be graded on the clarity of your process description, how well
you connect it to the other three group members, and the quality of your
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I found that students really struggled to understand how they could translate this data into a common measurement. Lots of students used percent, which I thought was great, but if they didn't think to use percentage, they struggled to come up with an alternate way.<br />
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I have to say that I was please with the level of engagement and focus that my students exhibited. They were interested to know who got the most sleep and what the data meant. I felt that this was a good exercise in mathematical modeling because students did not feel the pressure of arriving at the "right" answer and they were rewarded for their process as opposed to their answer. Also, the use of real-world data was beneficial and useful because it made things more applicable and gave the activity context...everybody has slept before. :)The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com2tag:blogger.com,1999:blog-1591780087227839882.post-29099718896100648132014-03-03T09:46:00.003-08:002014-03-03T09:46:51.215-08:00Transitioning to Transformations<span style="font-family: Tahoma, sans-serif; font-size: x-small;">Last week I gave my students a task from Illustrative Mathematics called <a href="http://www.illustrativemathematics.org/illustrations/1422">Similar Triangles</a> as an introduction to the concept of Angle-Angle Similarity. Here's how it played out...</span><br />
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A brief disclaimer is that although I have been teaching Geometry all year, I have not made the complete transition to teaching it through Transformations. I have done a few things here and there, but more than anything it was to try tasks and strategies out for myself and to challenge my students. They have had some experience with transformations and each time we do something with them, the students seem to grasp the concepts quite well. When doing the AA task, I encouraged my students to use patty paper as a tool for doing transformations. Also, this task was their first introduction to dilations.</div>
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I started by drawing 2 equiangular triangles on the board (see picture below) and asked the students if the 2 triangles were similar. A lot of them said NO because there wasn't a "nice" number to multiply 3 by to get 8. :) So, I wrote the equations from the picture and we came to the agreement that 8/3 or 3/8 would have done the trick, depending on which way we were scaling/dilating. They seemed to find this pretty magical...this should be an indication of the level of number sense my students are working with. :)</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5zo4_wGICjejxK3JIPz6szZLQFZfxmDpYj4Y5_qEQGYtYKS9j-GOHIxBLil7EsCfJdHy2Gth58jEPoDCWcBDRC3gIS038c_fzqRKx5fYzGyVFycWZ8vDpPZiDiaZia1eoPSE3U5TiPgs/s1600/IMG_1224.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5zo4_wGICjejxK3JIPz6szZLQFZfxmDpYj4Y5_qEQGYtYKS9j-GOHIxBLil7EsCfJdHy2Gth58jEPoDCWcBDRC3gIS038c_fzqRKx5fYzGyVFycWZ8vDpPZiDiaZia1eoPSE3U5TiPgs/s1600/IMG_1224.JPG" height="240" width="320" /></a></div>
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So, as they worked on the AA task, they did great with the translating and rotating. I wrote up a list of requirements (see picture below) and they were able to articulate them quite well. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcIN-hueVtBtD3IPTrRWcm5FRqyU2J0iJZMF-qQxaQQx8i03ezC3nSuDAcoI8hkvaYFTvW5QTcu6jsJ5nFQtBB34XKyGoJzpScFYrs5sD-ttH4YNs5jDV7l9tZgp6snQWQ6CRctQ42I-s/s1600/IMG_1226.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcIN-hueVtBtD3IPTrRWcm5FRqyU2J0iJZMF-qQxaQQx8i03ezC3nSuDAcoI8hkvaYFTvW5QTcu6jsJ5nFQtBB34XKyGoJzpScFYrs5sD-ttH4YNs5jDV7l9tZgp6snQWQ6CRctQ42I-s/s1600/IMG_1226.jpg" height="320" width="240" /></a></div>
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They did great with the "vagueness" that I had been struggling with. So, when it came to dilating Triangle ABC (I didn't worry too much about the prime notation for this task), they knew that they had to make it bigger, but they didn't necessarily know by how much. This is where I wrote the new equation from the picture below (AB x ? = DE) and they were actually able to connect it back to the 3 and 8 sided triangles. The magic (or madness for some of them) continued. Ultimately, they were able to say that you needed to multiply it by DE/AB in order to dilate the smaller triangle to the larger triangle, and I was pleasantly surprised.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgH1qVDLNC580YPx_Tj5i9YxaY7y-TGV-gB9_RLBOw_CwCMfrnnYDA3dKZCEb8lsvlx9UmpC5FWpYUWN2NGPuhyphenhyphenqx6h6xQJk92j__uitq7_XM4wV_J-5NDkEGfoPmQ1WFHjVKHhUe3vL8c/s1600/IMG_1225.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgH1qVDLNC580YPx_Tj5i9YxaY7y-TGV-gB9_RLBOw_CwCMfrnnYDA3dKZCEb8lsvlx9UmpC5FWpYUWN2NGPuhyphenhyphenqx6h6xQJk92j__uitq7_XM4wV_J-5NDkEGfoPmQ1WFHjVKHhUe3vL8c/s1600/IMG_1225.jpg" height="320" width="240" /></a></div>
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Another issue that I'm having is, Where do I go from here? I guess I'm still having difficulty wrapping my head around the fact that THIS is the new definition of similarity, but I'm getting there. Pulling it all together and not seeing these tasks as individual exercises is something that I'm still working on. I appreciate your feedback. Please forgive my vagueness or lack of precision in notation, but I think that mostly comes from trying to meet my students where they are, and let's be honest...where I am, too.</div>
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Thanks for reading!</div>
The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com2tag:blogger.com,1999:blog-1591780087227839882.post-85285142668475045712013-12-13T22:26:00.000-08:002013-12-13T22:26:03.022-08:00Asilomar Mathematical Modeling Presentation<span style="font-family: Times, Times New Roman, serif;">This past weekend I had the pleasure of attending the annual CMC North, Asilomar Math Conference. I have attended this conference before, but this was the first year that I was going to be a presenter at the conference. The title of my presentation was "Embedding Authentic Modeling Problems in MS/HS Curriculum" and we featured two of Dan Meyer's 3-Act Video problems (<a href="http://blog.mrmeyer.com/">http://blog.mrmeyer.com</a>). The audience had a choice to work on one of the two problems that we included and it seemed to go very well. I've included the presentation below. Please enjoy!</span><br />
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<iframe frameborder="0" height="327" scrolling="no" src="https://skydrive.live.com/embed?cid=D8DE62D119811AA8&resid=D8DE62D119811AA8%21120&authkey=AGta_EgoYlCN-84&em=2" width="402"></iframe>The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-31542274495061205512013-12-13T21:43:00.003-08:002013-12-13T21:43:36.518-08:00Re-Engagment LessonThis lesson actually took place at the beginning of the year, but I am just getting around to it now...<br />
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<br />
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Through
the last eight years that I have been a classroom teacher, one common thread
that stands out to me is the amount of professional development that I have
participated in. I have learned about
everything from strategies to engage English Language Learners, to how to
incorporate technology into my classroom, to how to mentor a first-year beginning
teacher. With all of the different
themes and directions that my professional growth has taken on, none of them
have been as meaningful to me as those centered on Common Core and the
Standards for Mathematical Practices.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
When I was asked to participate in
the CPEC grant, I honestly had no idea what to expect, nor did I really know
anyone who was involved. I was anxious
and excited and could not wait to see what was happening in my region with
regards to Common Core. The impact that
this two-year grant has had on my teaching practice is invaluable and perhaps
even immeasurable. I continue to take
risks and try new things in my classroom, as well as persevere when I am not
successful the first or second, or even third time. One of my favorite risks that I have taken
recently was a re-engagement lesson in my Geometry classes.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
I have found in my eight years of
teaching that the first chapter test is often the hardest for students. I’m not sure if this has more to do with
brushing off the dust from summer vacation, or the fact that Geometry is quite
different than Algebra. Nevertheless, my
current Geometry students played into my theory and were not overly successful
on their Chapter One test. There were
errors in notation, reasoning, and even setting up a correct equation to model
a Geometric principle; all of which are areas that need to be addressed and
remedied in order for a student to be successful in later chapters.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
After grading the Chapter One tests,
it was obvious that I was faced with the age-old problem of the “haves” and the
“have-nots” in the sense that some students "have" the understanding and others don't. There was a large
population of students who were struggling with basic concepts on the test, but
there was a considerable amount of them who were successful and were ready to
move on to new applications and ideas.
So, as their teacher, what do I do?
Do I move on and hope that the “have-nots” will catch on
eventually? Or do I start over and cause
boredom and frustration amongst the “haves”?
My solution was to try a re-engagement lesson.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
I chose four areas/questions that I
wanted my students to focus on. The
first three questions on the test related to Geometric notation and basic
concepts such as where two planes intersect (many said a point). The second area of concern for me was on an
error-analysis question. The majority of
students were able to articulate which student in the scenario was correct
and/or incorrect, however their explanations and justifications were weak and
unconvincing. The third and forth questions
that I chose for my re-engagement both had Algebraic applications that stemmed
from Geometry. First, an Angle
Addition Postulate problem that involved four adjacent angles (two of which
were congruent) that formed a straight angle, and asked to find the value of
the variable. Second, an angle
bisector scenario where students were given the measure of one of the smaller
angles and the larger angle, both in the forms of an Algebraic expression, and
asked to find the measure of the smaller angle.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
My current class schedule is laid
out such that I teach three, 90-minute periods of Geometry every day. Every other day I see the same group of
students, so I see six periods over the span of two days. The way that this particular test panned out
was that I graded my first three classes during the first night and then graded
my second set the next night. However,
once I graded the first set of tests, I was familiar with common errors and
misconceptions, so I did not grade my second set of tests right away. Instead, I went to the photocopy machine and
made copies of student work from the second set of tests. This allowed me to copy student work that was
ungraded, so the students were unaware of the accuracy of the responses. I selected student work that represented the
common errors as well as a few correct solutions, some using a non-traditional
approach. <o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4eVkBdy9aUX1QNtXE7i7tQ9fC4WjccwjJwStvqo9k9rFGV6o0UxLmayTlrowcRS9u6GEsQfmOB_XqfPYennV2pC8cMXYWXK2L8a2gBgT_F_VAsy7o3OcVoLeOrI_sxwcHges95_sZHmM/s1600/IMG_0863.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div>
The next time that I saw each group
of students, this was also the day after their test, I told them that they
would get their tests back at the end of the period, but they had some work to
do first. I knew that if they received
their tests first, they would not buy in to the re-engagement activity. On the white board I had written prompts that
corresponded with each page of student work – there were four pages in total.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4eVkBdy9aUX1QNtXE7i7tQ9fC4WjccwjJwStvqo9k9rFGV6o0UxLmayTlrowcRS9u6GEsQfmOB_XqfPYennV2pC8cMXYWXK2L8a2gBgT_F_VAsy7o3OcVoLeOrI_sxwcHges95_sZHmM/s1600/IMG_0863.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4eVkBdy9aUX1QNtXE7i7tQ9fC4WjccwjJwStvqo9k9rFGV6o0UxLmayTlrowcRS9u6GEsQfmOB_XqfPYennV2pC8cMXYWXK2L8a2gBgT_F_VAsy7o3OcVoLeOrI_sxwcHges95_sZHmM/s320/IMG_0863.jpg" width="240" /></a></div>
<br />
For the first page, titled
“Notation”, students were asked to find the error in each student’s notation
and make appropriate corrections.
Because I photocopied the first three questions from the test and used
that for each anonymous student representative, there were some pieces of each
student work that were possibly correct.
If that was the case, students did not need to comment.<br />
<br /></div>
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The second page was called “True
Statements About Angles” and required students to a. make sense of the student
work sample, b. determine if the work was correct, c. if the student was
correct, they needed to describe the approach, and d. if the work was
incorrect, explain why.<o:p></o:p><br />
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<br />
The directions for the third page,
“Between vs. Midpoint”, were to first read the student explanation, and
secondly write a question or a comment that could make the argument better.<br />
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Finally, students worked on a page
of sample work called, “Angle Bisectors”, which asked them to a. look at the
student work and make sense of it, b. if the work was correct, state why, and c.
if the work was not correct, determine where the student made their mistake and
explain why it’s incorrect.<o:p></o:p><br />
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<br />
As you can see, what was required of
each student was a little different from page to page. I did not want them to do the same thing for
all, partly because the four pages dealt with four different mathematical ideas
that I wanted to showcase. My purpose
was for them to become re-introduced to the problems from the test, and to use
a critical eye to determine how they might make each piece of work better if it
was incorrect, or expand their knowledge if it was an unfamiliar approach.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
As students began to engage in this
activity, I was blown away by their interest and motivation to make sense of
the mathematics. Without asking them to
do so, they started taking out their notes to use as a reference, they were
communicating with their partner to see what they thought, and there were very
few questions of me. This was the polar
opposite of what I normally see in my classroom. It is a struggle for me to get them to
interact with their notes and textbook, they usually turn right to me for the
answers, and they don’t rely on their peers as resources or experts.<o:p></o:p></div>
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As I observed my students working, I
heard them discussing why a particular answer was true or false, they were
critiquing the arguments that had been provided by the selected students, and
they were working together to draw a conclusion. I even heard comments like, “I’m going to be
much more careful on my next test so that Ms. Balli doesn’t have to grade all
of this crap!” or “Oh no, I don’t think this is my work, but I know that this
is the way I did this problem, and now I see that it’s wrong.”<o:p></o:p></div>
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There was one student work sample
that I included because it was an example of an equation that had been set up
incorrectly, but yielded the correct answer.
This was an interesting discussion starter because students asked me,
“Would they get full credit? They got
the right answer!” Eventually students
began to realize that the importance did not rest in the correct answer, but
rather the correct method. It was
obvious that, to them, math is all about getting the correct answer, but they
were able to slowly move away from this because they saw evidence of getting
the right answer through an invalid process.
I believe that this was the start of a cultural shift in my classroom –
as their teacher, a correct answer is nice, but that is certainly not the only
thing that I am looking for.<o:p></o:p></div>
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When it came time to return the
tests, I did a short debrief with them about the whole re-engagement
experience. Several students stated that
they had never done anything like this before, and found it very useful. The majority of my students said that they
learned something from this lesson, and whether they scored an A or an F on the
first chapter test, they felt strongly about these particular problems that they
had examined and felt that they understood them better. Regardless of his or her grade on the test, I
did feel that each student was able to walk away from this experience with a
deeper understanding of the mathematics.<o:p></o:p></div>
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Just before handing back the tests,
I shared with my students some of the research that has been done with
relationship to feedback. The three
categories that were examined in this research study were teachers providing
only a grade (no feedback), teachers providing both written feedback and a
grade, and teachers providing only feedback (no grade). The results showed that the greatest amount
of student learning was from the population of students who only received
feedback, no formal grade. I had a good
laugh with my students, because despite the findings from this research, I had
provided them with both a formal grade and feedback. I did this because I knew that I wanted to
have my students participate in a re-engagement lesson prior to getting their
tests back, so I felt that this would be a vehicle for student learning,
regardless of me providing them with a grade.<o:p></o:p></div>
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As the research predicted, once
students received their tests with both written feedback and a formal grade,
they appeared to be ready to move on to the next topic of discussion, as there
was little interest in examining their mistakes. However, I will say that they were able to
connect mistakes that they made to mistakes that they saw from the student work
samples. The work samples may not have
been a photocopy of their work, but the same mistake was made, and I do feel
that there was something learned from making this connection.<o:p></o:p></div>
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An observation that I made about my
six Geometry classes was that, in some instances, there was not as much buy-in
from my B-Day students (second day). My
B-Day classes have fewer Honors students, as the Honors classes (English,
History, and Science) are offered more on B-Days, so I do find more Honors
students in my A-Day classes. However, I
am not sure that the schedule of classes is the only culprit. When I made photocopies of student work, I
only made copies from my B-Day classes, so I am wondering if they felt
discouraged because they recognized some of their work. Are the students in these classes not as
confident? I am not sure what the
reasoning is for the lack of connection, but I am interested to see that if I
include work from both days, would there be an increase in buy-in?<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
Finally, I want to take a moment to
reflect on the two successful re-engagement lessons that I have undertaken. The first re-engagement lesson that I taught
successfully was using a MARS task that I had given my students as a formative
assessment. I was teaching a unit on
surface area and volume and wanted to see how much they were able to apply to a
composite three-dimensional figure. The
task involved finding the volume of a wine-shaped glass that was part cylinder,
part hemisphere. Students struggled with
the composition of volume; forgetting to divide the volume of the sphere by
two, or did not remember the four-thirds from the sphere formula. There was a wide range of issues.<o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;">
When I did the re-engagement lesson,
I provided students with six samples of student work, which they were asked to
make sense of and then determine which student had the correct answer. Without exception, every student in my three
different classes was able to articulate why “Student B” was correct. It was an amazing sight to see and I was
confident that my students had really made a connection to the mathematics from
the task. <o:p></o:p></div>
<div class="MsoNormal" style="text-align: justify; text-indent: .5in; text-justify: inter-ideograph;">
Fast-forward a week and I am now giving them their summative
assessment. The end of unit test
included a problem asking them to find the volume of a hemisphere, and I was
devastated to find that less than a quarter of my students were successful with
this task. This made me question the
impact of the re-engagement lesson. Was
it not all that it was cracked up to be?
Was it really that effective? Or
in that moment during the re-engagement lesson did they actually connect to the
mathematics and were unable to make the connection to this new problem? Is the issue in their ability to see how
these concepts work together?<o:p></o:p></div>
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I am finding that this year I am eager to assess my students,
simply so that I can take them through the process of re-engagement. I am also excited to explore the applications
of re-engagement outside of assessment, as well as outside of mathematics. Would a re-engagement lesson be applicable or
effective in Science or Spanish? What
about English or History? As I gain
confidence in my teaching, I am open to sharing my thoughts and experiences
with others, because I know that there are so many things that I have yet to
learn. Not only have I learned from
professional development seminars, which are lead and populated by adults, but
even more so I have learned from the people that are inside the walls of my
classroom everyday; my students.</div>
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<!--EndFragment-->The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0tag:blogger.com,1999:blog-1591780087227839882.post-43694666216557064662013-10-05T12:24:00.002-07:002013-10-05T12:24:28.851-07:00Animoto and GeometryI always want my students to make connections between math and their surroundings, especially when it comes to Geometry. We are just now wrapping up the chapter on parallel lines cut by transversals, and I think there are so many examples of some of the Geometric ideas that we cover in their everyday environment. So, I had them download the free app Animoto and it was great because more than half of each class had the capacity to do so, which made groups small and doable. <br />
<br />
I had students get into groups of two and walk around campus looking for various Geometric elements. I had a list of twelve, which they had to find a minimum of ten. The list included:<br />
<br />
<ul>
<li>Parallel Lines</li>
<li>Perpendicular Lines</li>
<li>Alternate Interior Angles</li>
<li>Alternate Exterior Angles</li>
<li>Same-Side Interior Angles</li>
<li>Corresponding Angles</li>
<li>Intersecting Planes</li>
<li>Supplementary Angles</li>
<li>Complementary Angles</li>
<li>Vertical Angles</li>
<li>Parallel Planes</li>
<li>Skew Lines/Segments</li>
</ul>
<br />
<br />
Overall, I found that the students took the activity seriously. They loved being outside and making connections to what they were learning in class. They still relied on me for things like, "Mrs. Murk, would this be an example of Parallel Planes?" They still struggle with self-starting, but for the most part, they took the initiative. <br />
<br />
Something that I would change for next time is I would be stricter, or more descriptive, as to what the pictures are of. I don't want students to find two lines that are intersected by a transversal and use that picture for three or four different Geometric idea. I just find this boring and not pushing the students very hard to be creative.<br />
<br />
I've included some of my favorite videos below.<br />
<br />
<a href="http://animoto.com/play/1Ak5NYrDnHPTuuqHN008GQ" target="_blank">Geometry shtuff</a><br />
<a href="http://animoto.com/play/CJDzifWpKjIFRDk6b5BnAg" target="_blank">Thomas and Ben's swag master geometry video</a><br />
<a href="http://animoto.com/play/Nx9JUbBlV1bGDaVso0gvfQ" target="_blank">My Video</a><br />
<br />
<br />The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com1tag:blogger.com,1999:blog-1591780087227839882.post-24789689888734219752013-07-11T14:18:00.002-07:002013-07-11T14:19:15.172-07:00New Things on the HorizonSo, I know that as teachers we live for our summer holiday and the time off for relaxation and rejuvenation; but I can't seem to turn off my work mind. I love getting online through my lap top or my iPad and checking in to communities like Edmodo or Facebook to see what's new. I think that a big reason for this is because of the transition to Common Core that will be taking place this year and next in my classroom. I say classroom because I am 100% committed to making changes in my instruction, but I don't know what the goals area for my department, school, or district are as of yet.<br />
<br />
One thing that I have been working on is planning Common Core Math units that will be implemented this year. I brought one into my curriculum this year and it was successful in many ways. I'm anxious about all of the work that will need to go into the other units to follow; especially because of the lack of support from my district. I feel like the only one doing this kind of work, so it's a lonely road.<br />
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Another element of my teaching that I have been revamping is my use of technology in my classroom. I have been using apps and online communities to try to keep my students connected and interested in their learning. The challenge for me will be to sustain the use of these items in my work. For example, I have created an assignment on Edmodo that my students need to complete, but it's more just as a trial because I don't know how it all works. I want to see if it's a useful tool, and if it is, I want to continue to use it.<br />
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I want to stay positive and continue to think outside the box when it comes to my teaching, but at the same time, I do not want to constantly be trying new things for the sake of trying new things. I would like to find something that I think is useful, effective, and meaningful and make it work for me and my students.The Mathy Murkhttp://www.blogger.com/profile/09518226935498308758noreply@blogger.com0