tag:blogger.com,1999:blog-86334391545082252552018-03-14T04:25:38.227-07:00McPherson MathSharing resources and discussing issues about teaching and learning math.Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.comBlogger18125tag:blogger.com,1999:blog-8633439154508225255.post-10277951804634889222017-11-17T10:06:00.002-08:002017-11-17T10:06:57.606-08:00Understanding Standard DeviationLong story short, I was presented an opportunity to teach standard deviation and I jumped at the chance. There are times as a coach that I wish I were back in the classroom. I do miss working with students daily. Anyway, here's what I did.<div><br /></div><div><b><span style="color: blue;">Launch:</span></b></div><div>To start, I wanted students to think about the center of a data set. I asked them to individually consider each of the following and provide a typical number that best represents the situation.<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ijFCo0KVev8/Wg3nV5pppjI/AAAAAAAAAl0/PWRAAWG_EIgU3ioL5P_aVzw2DDWffMGMgCLcBGAs/s1600/Launch%2Bquestions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="442" data-original-width="647" height="218" src="https://4.bp.blogspot.com/-ijFCo0KVev8/Wg3nV5pppjI/AAAAAAAAAl0/PWRAAWG_EIgU3ioL5P_aVzw2DDWffMGMgCLcBGAs/s320/Launch%2Bquestions.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">I intentionally avoided using the words average, mean or center. Sometimes when I use a "mathy" word students hesitate. I'm not sure why, but I have my theories. I know I get better responses when I use the word typical so I went with that.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">As we discussed each one, I wanted to draw out some important characteristics: Minimum value? Maximum value? and Are the possible values close in range or spread out?</div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-rQV3xKCaw-I/Wg3roellxFI/AAAAAAAAAmM/Fh7_JNBjS5koO-ZbbFfagjQffA-CLdeEACLcBGAs/s1600/Launch%2BDiscussion.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="448" data-original-width="971" height="183" src="https://2.bp.blogspot.com/-rQV3xKCaw-I/Wg3roellxFI/AAAAAAAAAmM/Fh7_JNBjS5koO-ZbbFfagjQffA-CLdeEACLcBGAs/s400/Launch%2BDiscussion.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">I used the first situation as a contrast to the others to bring out variability. There are 24 hours in a day and this doesn't vary given you are talking about a day on Earth. The other situations have some variation in the data. That's what we are talking about today: variation in a set of data.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><b><span style="color: blue;">Going to the Movies</span></b></div><div class="separator" style="clear: both; text-align: left;">The <i>Justice League</i> movie is releasing next week and we are very excited about getting to see the movie. Imagine you are in the theater and settling in to watch the movie. There are people all around you and you begin to wonder...</div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>How old are they?</li><li>Are they all the same age? If not, what is the typical age?</li><li>How close are the other ages to that typical age?</li></ul>Here's what we got from that question.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Nxgt5xbwCFc/Wg3vE4d62TI/AAAAAAAAAmk/eAeJl6wF630nJTiCWgC78yZhoYVeWhWRwCLcBGAs/s1600/ages%2Bat%2Ba%2Bmovie.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="121" data-original-width="832" height="57" src="https://2.bp.blogspot.com/-Nxgt5xbwCFc/Wg3vE4d62TI/AAAAAAAAAmk/eAeJl6wF630nJTiCWgC78yZhoYVeWhWRwCLcBGAs/s400/ages%2Bat%2Ba%2Bmovie.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">The typical age is 26 years old with many people between the ages of 16 and 36. (Okay, I had to prompt them to think beyond the age of 30. At first it was 5 yrs old to 30 years old.)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Okay, so I've got them thinking about measure of center and variability. They've even put some values to that variability. Now, let's collect some real data.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="color: blue;"><b>Estimating Time</b></span></div><div class="separator" style="clear: both; text-align: left;">So here's why I wanted to teach this lesson. I really, really, really enjoy playing with <a href="https://nrich.maths.org/10629" target="_blank">this site</a> and like how its simplicity captures students' attention.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><i>Ever waited to go on vacation and it seems like time slows down but when you are on vacation the time goes by quickly? While the passing of time is constant, our estimation of how much time has passed varies. I wonder how close are we at estimating 10 seconds? Let's find out.</i></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Each student was asked to estimate 10 seconds and record the amount of time to the nearest tenth on a sticky note. (Because this is a small school, I collected data earlier that morning from a variety of teachers and students. This allowed us to have enough data to analyze.) We built a histogram from the data. Great opportunity to assess student understanding of values within each interval.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-UoYX-oxupUs/Wg7ZY92kjzI/AAAAAAAAAnA/zlZ1QG0bmOMBTX7rDHdIZxtKYzeMnRbBwCLcBGAs/s1600/histogram.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="614" data-original-width="1600" height="122" src="https://4.bp.blogspot.com/-UoYX-oxupUs/Wg7ZY92kjzI/AAAAAAAAAnA/zlZ1QG0bmOMBTX7rDHdIZxtKYzeMnRbBwCLcBGAs/s320/histogram.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The conversation started broad by asking what they noticed about the data and if it helped to answer our question about how close we are at estimating 10 seconds. I directed the conversation towards getting closer to standard deviation (without saying standard deviation, yet).</div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-TjQCC7SDyAA/Wg8OX1nW_FI/AAAAAAAAAng/BqbbtPn-658fZtqdrz1g2yDZjLT4I_c_gCLcBGAs/s1600/student%2Bestimates.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="405" data-original-width="612" height="263" src="https://2.bp.blogspot.com/-TjQCC7SDyAA/Wg8OX1nW_FI/AAAAAAAAAng/BqbbtPn-658fZtqdrz1g2yDZjLT4I_c_gCLcBGAs/s400/student%2Bestimates.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">The students were developing a good sense of center and the values in relation to the center. Students talked about the estimates being 1.5 seconds less than 10 and 1.5 seconds more than 10.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><i>This 1.5 you are using to describe how far away the values are from the center was a number mathematicians wanted to calculate. It's not the full range of the data but a value that can tell us how far some of the data is from the center. This number is called the "standard deviation".</i></div><div class="separator" style="clear: both; text-align: left;"><i><br /></i></div><div class="separator" style="clear: both; text-align: left;"><span style="color: blue;"><b>Illustrative Math Task</b></span></div><div class="separator" style="clear: both; text-align: left;">Students were given the Illustrative Math Task <a href="https://www.illustrativemathematics.org/content-standards/tasks/1886" target="_blank">Understanding the Standard Deviation</a>. I used Part 3 as the formative assessment to gauge student understanding.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="color: blue;"><b>Up Next</b></span></div><div class="separator" style="clear: both; text-align: left;">The next lesson is to calculate the standard deviation and work on interpreting it. I like the EngageNY lesson Algebra 1 Module 2 Lesson 6 <a href="https://www.engageny.org/resource/algebra-i-module-2-topic-b-lesson-6" target="_blank">Interpreting the Standard Deviation</a>.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><i>In case you were wondering, because, let's face it, I was wondering, how close were the estimates for mean and standard deviation? The mean was 10.4 seconds and the standard deviation was 1.9 seconds.</i></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div></div>Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-9642741329913236432017-11-06T14:29:00.000-08:002017-11-07T04:56:11.441-08:00Reflections on NCCTM17The North Carolina Council of Teachers of Mathematics has their annual Leadership and Math Conferences this past week.It was a whirlwind three days with fabulous speakers including Jennifer Bay-Williams, Peg Smith, Juli Dixon, William McCallum, Joliegh Honey, Jennifer Wilson, .... The list goes on. Anyway, I always enjoy the conference and wanted to share a few highlights.<br /><div><br /></div><div><a href="https://3.bp.blogspot.com/-_VGM6o7RPZM/WgDZIAB0wKI/AAAAAAAAAkg/bOhHOzMu5io-Inig4lKRjpsLB94yrapZwCLcBGAs/s1600/Factoring%2Bwith%2BArea%2BModel.PNG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="440" data-original-width="780" height="179" src="https://3.bp.blogspot.com/-_VGM6o7RPZM/WgDZIAB0wKI/AAAAAAAAAkg/bOhHOzMu5io-Inig4lKRjpsLB94yrapZwCLcBGAs/s320/Factoring%2Bwith%2BArea%2BModel.PNG" width="320" /></a>First, I had the privilege to present with one of my teachers, Kim Clark. We presented <a href="http://www.mcphersonmath.com/p/presentations.html" target="_blank">Factoring Using the Area Model</a>. It is very rewarding to work with a teacher who pushes you to grow because she wants to be better. I have learned a lot from her over the years and was thrilled to get to present this session with her.</div><div><br /></div><div>This is a slide from our presentation. Did you know that the diagonals of the area model have the same product? Using that fact, we asked participants to find the possible area models. This question is the lynch pin of the investigation we did with students to help them develop a strategy for factoring when a is not 1. It's pretty cool and effective.</div><div><br /></div><div><a href="https://2.bp.blogspot.com/-OTjV4wQSOfo/WgDa3vBk8II/AAAAAAAAAks/j5enBrG0LScD1asezhs3_Jtrhl6X81-tACLcBGAs/s1600/J%2BBay-Williams%2BSketchnote.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="727" data-original-width="1096" height="212" src="https://2.bp.blogspot.com/-OTjV4wQSOfo/WgDa3vBk8II/AAAAAAAAAks/j5enBrG0LScD1asezhs3_Jtrhl6X81-tACLcBGAs/s320/J%2BBay-Williams%2BSketchnote.jpg" width="320" /></a>Second highlight was hearing Jennifer Bay-Williams talk about <i>Becoming Fluent in Developing Procedural Fluency</i>. This is an area of growth and learning for me. I am an avid supporter and believer that conceptual understanding is critical to students understanding and making sense of mathematics. What I am learning is that it takes specific and intentional instruction to help students develop that understanding into procedural fluency. I am also learning how to better define fluency as more than just quick.</div><div><br /></div><div>The last highlight I'd like to share was the session from Dr. Valerie Faulkner on <i>Opportunity, Equity & Agency: How do our grouping practices mediate student sense of mathematical identity? </i>She makes a very convincing argument about how grouping in education (high and low students) just doesn't make a lot of sense. The statement I have been using to summarize her talk is one she made.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-oTYbyQ9AX2o/WgDg1hyoTPI/AAAAAAAAAlU/ZxUsEFivqNcdguUEtQVKTgtT5vvQZQqSACLcBGAs/s1600/V%2BFaulkner%2BSketchnote.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1139" data-original-width="785" height="320" src="https://3.bp.blogspot.com/-oTYbyQ9AX2o/WgDg1hyoTPI/AAAAAAAAAlU/ZxUsEFivqNcdguUEtQVKTgtT5vvQZQqSACLcBGAs/s320/V%2BFaulkner%2BSketchnote.jpg" width="220" /></a></div><div><br /></div><div style="text-align: center;"><i><span style="color: blue;"><b>"We are surprisingly bad at evaluating what a person can and will do even if we have a vested interest in doing so."</b></span> </i></div><div><br /></div><div>We should stop trying to separate students using some abstract high and low characteristic of student ability. We must begin to talk about students in regards to what they know and when they need to work on.</div><div><br /></div><div>As I shared, the three days were a whirlwind. I am excited about new friendships developed and being able to connect with teachers from across the state. Maybe I'll even start blogging more often. Haaaa.....Haaaaa.....Haaaa. No promises but always a goal.</div>Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-77561549557744388762016-09-28T09:23:00.001-07:002016-09-28T09:23:12.096-07:00Floating Down the RiverIn an effort to support teachers across the state with the implementation of the revised NC Math 1, 2, and 3 standards, the K-12 Mathematics team are hosting weekly webinars. Each Thursday focuses on a different course: Math 1, Math 2, Math 3, and Math Leaders. <a href="http://maccss.ncdpi.wikispaces.net/Secondary+Webinars" target="_blank">Find out more here</a>.<br /><br />The sessions present a math task (if you register early you will get it in advance) and frames the discussion on standards, implementation, anticipating student misconceptions, and connections.<br /><br /><a href="https://3.bp.blogspot.com/-5wOPEIu0jWA/V-vsp9auytI/AAAAAAAAALI/exqs6TQIgb0td0CTJA2Ba1PuvzCkc5yfgCLcB/s1600/floating%2Bdown%2Briver.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="107" src="https://3.bp.blogspot.com/-5wOPEIu0jWA/V-vsp9auytI/AAAAAAAAALI/exqs6TQIgb0td0CTJA2Ba1PuvzCkc5yfgCLcB/s200/floating%2Bdown%2Briver.PNG" width="200" /></a>This past month I was able to participate in the Math 1 and Math 2 sessions. The Math 1 session was on Functions and we were given the <a href="https://drive.google.com/open?id=0B56ZJgN42AOXa0NtUm9zV21aeVE" target="_blank">Floating Down the River</a> task. (Which I tweaked a little. You can find the original version <a href="https://drive.google.com/open?id=0B50js0sjHc4DcFhTaGlzTTF4eEE" target="_blank">here</a>.) I really like this problem especially to discuss the key features of the functions and interpreting them in context.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-kSSvOwN4UnA/V-vsM3VzwyI/AAAAAAAAALE/e78wHxUt1oMhSdCm6rAoGMs0AZf9zUVnACLcB/s1600/function%2Bfamilies.PNG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="118" src="https://3.bp.blogspot.com/-kSSvOwN4UnA/V-vsM3VzwyI/AAAAAAAAALE/e78wHxUt1oMhSdCm6rAoGMs0AZf9zUVnACLcB/s320/function%2Bfamilies.PNG" width="320" /></a></div>This is one of my SOAP BOX concerns: the difference between <a href="http://www.corestandards.org/Math/Content/HSF/IF/B/4/" target="_blank">F-IF.4</a> and <a href="http://www.corestandards.org/Math/Content/HSF/IF/#CCSS.Math.Content.HSF.IF.C.7" target="_blank">F-IF.7</a>. When we discuss functions it quickly becomes about the "families of functions." You know the ones I am talking about - linear, quadratic, exponential, etc.). With these functions we are able to use the symbolic representation and determine key features. For example, rewrite a quadratic into vertex form and identify the vertex. This is F-IF.7.<br /><br /><a href="https://1.bp.blogspot.com/-xrmYjYaVCBU/V-vs9Ar55BI/AAAAAAAAALM/HqIFFND0PQgBMf_wRzekHx3BV0jKEnkKQCLcB/s1600/functions%2Btell%2Bstory.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="118" src="https://1.bp.blogspot.com/-xrmYjYaVCBU/V-vs9Ar55BI/AAAAAAAAALM/HqIFFND0PQgBMf_wRzekHx3BV0jKEnkKQCLcB/s200/functions%2Btell%2Bstory.gif" width="200" /></a>So what is F.IF.4? It doesn't seem like it should be the same thing. IMHO - it is not. While F-IF.7 focuses on those classical function families, F-IF.4 is broader. It includes <b>all</b> functions. This includes functions I refer to as <i>functions that tell a story</i>. These functions may be represented symbolically, often by a piecewise function, which is well beyond the focus of Math 1. However, it is not unreasonable for students to reason with and interpret the key features using a table or a graph.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-gGWHxBS0A5U/V-vtyJVTOpI/AAAAAAAAALY/uHAV7n1dtbc-puB6jbnCVRZDzvWm82dhwCLcB/s1600/function%2Bstudent%2Bwork.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="257" src="https://1.bp.blogspot.com/-gGWHxBS0A5U/V-vtyJVTOpI/AAAAAAAAALY/uHAV7n1dtbc-puB6jbnCVRZDzvWm82dhwCLcB/s320/function%2Bstudent%2Bwork.png" width="320" /></a></div>That's why I like the <a href="https://drive.google.com/open?id=0B56ZJgN42AOXa0NtUm9zV21aeVE" target="_blank">Floating Down the River</a> task. Students are given multiple tables of values and a simple question is posed. I would expect students to do what we did during the session and graph the values. (Further commentary could be given on whether depth is positive or negative but I'll leave that up to you to decide.) Now, there are opportunities for students to discuss intervals of increase/decrease, maximum/minimums, average rate of change, and intercepts. They also have to compare and connect the events. Such as, when the water is shallow the speed increases. Can students also recognize that the distance function becomes steeper? Will they recognize why that would be occur?<br /><br />There is great potential in the task which is why I am encouraging my teachers and sharing with you. Try it and see what students do with it. I guarantee learning will take place. Also, consider joining me and others at the next Math 1 webinar.Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-25328995260566056672016-08-26T14:00:00.001-07:002016-08-26T14:00:25.607-07:00Starting a New School Year!!!The 2016-2017 school year begins next week and we are busy getting ready. North Carolina revised the standards for Math 1, 2 and 3 so the past few weeks have been focused on getting the curriculum materials aligned. Math 1 has some changes but Math 2 and 3 had a lot.<br /><br /><a href="https://4.bp.blogspot.com/-4FhOA9AEZ7A/V8CtHvsuiGI/AAAAAAAAAIY/-12k-h5v4BkOq9S0XDSwB4KTme_32r2ZwCLcB/s1600/transition.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="112" src="https://4.bp.blogspot.com/-4FhOA9AEZ7A/V8CtHvsuiGI/AAAAAAAAAIY/-12k-h5v4BkOq9S0XDSwB4KTme_32r2ZwCLcB/s200/transition.png" width="200" /></a>I've been getting requests for our pacing guides and outcomes. They have been updated on the respective pages (Math 1 and Math 2).<br /><br />For Math 3, a major concern is to make sure students do not experience gaps in concepts due to the movement of some standards out of Math 3 and into Math 2. For example, complex solutions to quadratic equations is now in Math 2. This year is a transition year in Math 3 and won't look the same as last year or next year. Therefore, we are not revising anything as much as we are "tweaking" some things. If you have questions about Math 3 please contact me and I'll share what we are doing. If there is enough interest I may post it here.<br /><br />Okay, time to get started with the new year!!! Thanks for checking in.Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-2221116617769456562016-05-19T09:42:00.000-07:002016-06-01T11:33:18.828-07:00NCFE Review SolutionsThere have been multiple requests for answer keys for the NCFE review sets. Well, I have finally had some time to sit and work on them. I have posted the solution sets for <a href="https://drive.google.com/open?id=0B56ZJgN42AOXUHN3UlRGLTZTUU0" target="_blank">Math 2</a> and <a href="https://drive.google.com/open?id=0B56ZJgN42AOXYXE4T1o3MEU0Qlk" target="_blank">Math 3</a> right now. <strike>My goal is to work on Math 1 in the next few days and also get it posted.</strike> Yeah!! <a href="https://drive.google.com/open?id=0B56ZJgN42AOXTDMwNmtmM3gtbmc" target="_blank">Math 1</a> is also done!<br /><a href="https://2.bp.blogspot.com/-HcWXQ-dmdLg/Vz3sHBxq9hI/AAAAAAAAAH8/vVrI598i9XU7h3ET44ab7Q0DNa13Cc9PQCLcB/s1600/answer%2Bkey.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-HcWXQ-dmdLg/Vz3sHBxq9hI/AAAAAAAAAH8/vVrI598i9XU7h3ET44ab7Q0DNa13Cc9PQCLcB/s200/answer%2Bkey.jpg" width="200" /></a><br /><span style="color: red;"><b>***Warning***</b></span><br />There may be mistakes. Actually, I would be surprised if there weren't any mistakes. So, if you find one (or two or three or....), please let me know.<br /><br /><b>Mistake #1</b><br />Math 1: Review 4: Problem 2 -- the area of the triangle should be 13 unit squares not 26.Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-1431729872912162612016-05-14T23:09:00.000-07:002016-05-14T23:09:20.838-07:00Math 1 EOC Review MaterialsYour students have spent all semester or possibly all year studying the Math 1 standards. Now they must take the Math 1 EOC. In an effort to provide review material for teachers, my colleague and I put together some resources.<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-id4zFsgZvkQ/VzgRxNIyBnI/AAAAAAAAAHs/38jIIuamrq0zyqQJ6Yxq08jYkSbgsB_AQCLcB/s1600/Standardized%2Btest.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="129" src="https://4.bp.blogspot.com/-id4zFsgZvkQ/VzgRxNIyBnI/AAAAAAAAAHs/38jIIuamrq0zyqQJ6Yxq08jYkSbgsB_AQCLcB/s200/Standardized%2Btest.jpg" width="200" /></a></div><div><br /><div><div class="separator" style="clear: both; text-align: left;">We designed the resources around the conceptual categories: Number & Quantity, Algebra, Functions, Geometry, and Statistics. We also did a Linear & Exponential folder. In each folder you will find Resource Description. In this document we have listed the resources along with the alignment to the standards, type of resource, recommendation about calculator use, and some instructional suggestions. Our intent was to provide some variety for review along with opportunities for class discussion and strategies for solving problems.</div><div><br /></div><div>There are two other folders. One is General Resources which provides information about the specifics of the Math 1 EOC. The other is Comprehensive Resources. This folder has some calculator inactive tasks, midterms we have used in our district, the released EOC and the mini quiz review sheets.</div><div><br /></div><div>I hope you find some of this beneficial and I wish your students the very best on the exam.</div><div><br /></div><div>Here is the link to the entire <a href="https://drive.google.com/folderview?id=0B56ZJgN42AOXcWExMl8wSXlWOWc&usp=sharing" target="_blank">Math 1 EOC Review Resources</a>. I will also post on the Math 1 page.</div></div></div>Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-1464079681385963822016-03-01T11:42:00.001-08:002016-03-01T11:42:22.212-08:00Updates<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-91t_Vlu2NC0/VtXwPWX6JVI/AAAAAAAAAGo/VZ59cFJn3j4/s1600/update.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://1.bp.blogspot.com/-91t_Vlu2NC0/VtXwPWX6JVI/AAAAAAAAAGo/VZ59cFJn3j4/s200/update.jpg" width="200" /></a></div><span style="color: blue;"><b>Contact Information</b></span><br />Wow!!! The traffic to this website has exploded overnight. More and more people are visiting every day. To help in communication, I have placed a contact form on the left hand side. This will allow you to email me directly with comments or requests.<br /><br /><span style="color: blue;"><b>EOC and NCFE Reviews</b></span><br />Several of you have requested answer keys to the review resources. I'm not ignoring your request. I just don't have them. However, I am working on them. I hope to have them posted and revised before the testing in May/June.<br /><br /><span style="color: blue;"><b>Curriculum Maps</b></span><br />These are coming. I added two more in Math 1 today and should add more to Math 2 soon. As the maps are added so are the resources noted in the maps.Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-88631062810857037492016-02-18T07:18:00.000-08:002016-02-18T07:18:21.671-08:00PtA Series #2: Implement Tasks the Promote Reasoning and Problem Solving<div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">Let's begin this discussion with a question. What is a task? I once asked a teacher what task students were going to do in class that day. The response was "No task. They are just going to complete a worksheet and then start on an investigation." Hmmmm..... aren't those tasks? My guess is that for some teachers a task is a rich math problem that stretches your mind. Like the one below:</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-N6ezHLa5tu4/VsXdBxD-gEI/AAAAAAAAAEU/D2QQ1sEzJf4/s1600/Cory%2Bthe%2BCamel.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><img border="0" height="151" src="https://2.bp.blogspot.com/-N6ezHLa5tu4/VsXdBxD-gEI/AAAAAAAAAEU/D2QQ1sEzJf4/s320/Cory%2Bthe%2BCamel.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">For me a task is any activity given to students to do. That includes worksheets, quizzes, writings, discussions, etc. Suffice it to say -- tasks come in all shapes and sizes. That may be a little cliche but the emphasis is important. Teachers chose every day what tasks to give students. This choice is extremely important and one that bears deeper discussion.<o:p></o:p></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><br /></div><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><b><i>"Effective mathematics teaching uses tasks as one way to motivate student learning and help build new mathematical knowledge through problem solving." </i>(p.17)</b><o:p></o:p></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><br /></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">What should a teacher consider when choosing a task? The decision is about the opportunities afforded students to make sense of problems and explore solution methods connected to the established goal of the lesson. (<a href="http://www.mcphersonmath.com/2015/04/pta-series-1-establish-goals-to-focus.html" target="_blank"><span style="color: blue;">see practice 1</span></a>) So what do I think about the task we used in the <a href="https://drive.google.com/open?id=0B56ZJgN42AOXc2pkMVYzQklnSE0" target="_blank"><span style="color: blue;">Leaping Lizard Lesson</span></a>?<o:p></o:p></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><br /></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">Using the <a href="https://drive.google.com/open?id=0B56ZJgN42AOXT0hjUWZsQm5laXc" target="_blank"><span style="color: blue;">taxonomy</span></a> designed by Stein and Smith, I think that the <a href="http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod6_ccp_se_71713.pdf" target="_blank"><span style="color: blue;">Leaping Lizard</span></a> task falls within the category of a higher-level demand. Students are encouraged "to engage in active inquiry and exploration" (PtA p. 19) I lean more towards <i>procedures with connections</i> as suggestions are provided to students for ways to look at the transformations (i.e. connect image and pre-image points). However, students are required to expend some cognitive effort to develop a deeper understanding of the definitions of the transformations.<o:p></o:p></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><span style="color: blue; text-decoration: none;"><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1026" type="#_x0000_t75" alt="https://4.bp.blogspot.com/-RAX9rlvMdQ4/VsXQLgfy_WI/AAAAAAAAADs/6bs3k2YAWFM/s200/garmin-gps.jpg" href="https://4.bp.blogspot.com/-RAX9rlvMdQ4/VsXQLgfy_WI/AAAAAAAAADs/6bs3k2YAWFM/s1600/garmin-gps.jpg" style='width:150pt;height:117pt;visibility:visible;mso-wrap-style:square' o:button="t"> <v:imagedata src="file:///C:\Users\628465~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.jpg" o:title="garmin-gps"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><o:p></o:p></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><a href="https://3.bp.blogspot.com/-P4rQEqTpHts/VsXdPr7k9dI/AAAAAAAAAEQ/39speyc1sO0/s1600/garmin-gps.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><img border="0" height="156" src="https://3.bp.blogspot.com/-P4rQEqTpHts/VsXdPr7k9dI/AAAAAAAAAEQ/39speyc1sO0/s200/garmin-gps.jpg" width="200" /></span></a><span style="font-family: Times, Times New Roman, serif; font-size: large;">While selection of the task is important so is its implementation. In an effort to help students that are struggling, a teacher can inadvertently GPS the task by "taking over the thinking" for students. This lowers the demand and effectiveness of the task. I've witnessed this happening with the Leaping Lizards task. The teacher told students upfront what they would discover and consequently students were no longer invested in doing the task. A teacher has to mindful to "support students without taking over their thinking" (p.24).<o:p></o:p></span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><div style="text-align: left;"><a href="https://www.nctm.org/store/Products/5--Practices-for-Orchestrating-Productive-Mathematics-Discussions/" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><img border="0" height="200" src="https://2.bp.blogspot.com/-4kGPEFOAptY/VsXdCGxr3NI/AAAAAAAAAEM/OcB3cyRz3Rk/s200/5%2BPractices%2BBook.PNG" width="141" /></span></a></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">Knowing not to GPS and not doing it are two different things. It takes practice and intentional planning to change and prevent oneself from falling into that habit. I recommend the book <i>5 Practices for Orchestrating Productive Mathematics Discussions </i>by Margaret Smith and Mary Kay Stein. A number of years ago I was introduced to this book during a lesson study. It has been my guide ever since in implementing tasks in the classroom. We will revisit this again in part 4 of this series when we talk about student discourse.</span></div><div class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family: Times, 'Times New Roman', serif; font-size: large; line-height: 107%;">Next time, the practice is use and connect mathematical representations and I will share one of my favorite tools when studying functions.</span></div><div class="MsoNormal"><o:p></o:p></div><div class="separator" style="clear: both; text-align: center;"></div>Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-10734143575632811062015-08-24T11:32:00.001-07:002015-08-24T11:32:51.820-07:00New Resources for 2015-2016I am so excited to start the new school year. I spent June working with a great group of teachers writing and preparing the Curriculum Maps. You can find these on the resource pages for Math 1, Math 2, and Math 3. These have been a goal of mine for a few years and finally we have them.<br /><br /><a href="http://2.bp.blogspot.com/-xyTtvrtkKRg/VdtcF0bV64I/AAAAAAAAABE/959JaU7bWkw/s1600/Karen%2Bfishing.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="132" src="http://2.bp.blogspot.com/-xyTtvrtkKRg/VdtcF0bV64I/AAAAAAAAABE/959JaU7bWkw/s200/Karen%2Bfishing.jpg" width="200" /></a>July was all about family. We took a road trip to Canada and spent a week by the lake. The Northern Pike are aggressive and a lot of fun to catch. I was super excited to catch a Lake Trout.<br /><br />August has me back at work where I continue as a high school math coach. My school assignments have changed a little so I have new relationships to build. There have also been a lot of turnover so there are several new teachers to get to know.<br /><br />I have posted new resources on the Math 1, 2, and 3 pages. Make sure you check them out. I used feedback from teachers and have revamped some of the investigations. You will also find some new resources inspired by NCSM and NCTM conferences I attended in Boston.<br /><br />Stay tuned for the return of the series on the Principles to Actions, a post about <i>Conversation with a Wrestling Coach</i>, and the upcoming NCCTM conference.Karen McPhersonhttp://www.blogger.com/profile/05349678274034275302noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-81526580175255730262015-04-12T19:06:00.002-07:002015-04-12T19:06:41.787-07:00PtA Series #1: Establish goals to focus learning<a href="http://newsroom.unl.edu/announce/files/file46818.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://newsroom.unl.edu/announce/files/file46818.jpg" height="200" id="irc_mi" style="margin-top: 0px;" width="137" /></a>I've been interested in the <i>Principles to Actions</i> publication by NCTM since it came out a year ago; however, I haven't found the time to read it. So, how do you accomplish something you want to do? You set a goal. My goal is to read the <i>Principles to Actions</i> (PtA) book and I'm holding myself accountable by making a commitment to document my learning here through illustrations and reflections.<br /><br />NCTM has a website specifically dedicated to PtA (check it out <a href="http://www.nctm.org/PtA/" target="_blank">here</a>). I watched the <a href="http://www.nctm.org/Conferences-and-Professional-Development/Webinars-and-Webcasts/2014-NCTM-Annual-Meeting---Principles-to-Actions_-Defining-Core-Practices-of-Teaching-Mathematics/" target="_blank">video</a> of the presentation by Dr. DeAnn Huinker at last year's NCTM Annual Conference in New Orleans. She describes and illustrates the 8 Mathematical Teaching Practices.<br /><br /><a 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" 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" style="margin-top: 0px;" width="200" /></a>At the beginning of the presentation, she asks "What is the best math lesson you ever taught?" The <span style="color: #38761d;">Leaping Lizard!</span> lesson came to mind and I thought it would be a worthwhile activity to reflect on that lesson within the framework of the 8 Mathematical Teaching Practices.<br /><br />One of my favorite things about coaching is working with teachers to develop lessons and teach. The <span style="color: #38761d;">Leaping Lizard! </span>lesson <span style="color: black;">is a collaboration between myself and another teacher. I will use this lesson to think about the 8 Mathematical Teaching Practices. Let's look at #1.</span><br /><ol></ol><span style="color: black;"><span style="color: blue;">1. Establish mathematics goals to focus learning. </span>In full disclosure,</span><span style="color: #38761d;"><span style="color: black;"> we didn't start here. We actually chose the task first; but in thinking about what we wanted students to learn from doing the task, we needed to talk about the goal. So, we may not have started here, but we did get here quickly. The standards we were addressing were</span></span><i><span style="color: #666666;"><b> </b></span></i><br /><br /><i><span style="color: #666666;"><b>G-CO.2 </b>Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).</span></i><span style="color: #666666;"><i><b> </b></i></span><br /><br /><span style="color: #666666;"><i><b>G-CO.4</b> Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.</i></span><br /><br />We wondered what students may already know about transformations. In 8th grade, students verify the properties of the transformations (8.G.1). They also describe a sequence that exhibits a transformation (8.G.2). We anticipated students would be able to recognize the transformations and use general descriptions such as flip, turn, or slide to define them. Our goal was for students to use lines, angles, and circles to verify a transformation. We also wanted students to represent a given transformation <i>(a skill we would continue to work on for the next few lessons)</i>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-IsZ4Dpfg4Rc/VSsfqYrr0lI/AAAAAAAABmA/mB1WybOK6X0/s1600/Leaping%2BLizard%2Bcollage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-IsZ4Dpfg4Rc/VSsfqYrr0lI/AAAAAAAABmA/mB1WybOK6X0/s1600/Leaping%2BLizard%2Bcollage.png" height="327" width="400" /></a></div><br />How does this measure with the actions outlined on p. 16?<br /><blockquote class="tr_bq"><ul><li>The goals are clear and articulate the mathematics that students are learning. <i>We knew we wanted students to take the transformations beyond just a simple movement. We wanted them to understand the relationship between the image and preimage points.</i></li><li>The goals fit within the learning progression. <i>This was the first lesson in the unit. We had to build upon the 8th grade standards.</i></li><li>The goals were explicitly stated throughout the lesson to focus student work. <i>During the lesson, we had to emphasize the goal multiple times. This helped to clarify with students how we expected their understanding of transformations to extend beyond the movement. Students were to explore the properties and develop definitions.</i></li><li>The goals guided the planning and decisions made during instruction. <i>This task could be used to accomplish a variety of goals. It was critical that we have a clear understanding of student learning.</i></li></ul></blockquote>There are a number of challenges with this practice. First and foremost is time. Teachers need time to research the vertical alignment of the standard and time to discuss and refine goal. In our district this means leveraging the PLC time to focus on goals of lessons. There is also a need for quality tasks. That is Practice #2 which I will address in the next entry in this series.Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com2tag:blogger.com,1999:blog-8633439154508225255.post-83267209384072863952015-01-06T11:22:00.000-08:002016-05-23T05:30:31.715-07:00Review Material for NC Testing<div style="text-align: center;"><b><span style="color: blue; font-size: large;">Happy New Year!!!</span></b></div><br /><a href="http://www.buncombe.k12.nc.us//cms/lib5/nc01000308/centricity/domain/583/ready%20logo.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://www.buncombe.k12.nc.us//cms/lib5/nc01000308/centricity/domain/583/ready%20logo.jpg" height="200" width="188" /></a>The end of the semester is upon us. Teachers and students are preparing for final exams. In North Carolina there is the End of Course exam for Math 1. This test is used to evaluate schools and teachers. It is one of the measures used by the state to determine the "grade" on the school report card. There are also NC Final Exams for Math 2, Math 3, Discrete, Advanced Functions and Modeling, and Pre-Calculus. These are used to measure teacher effectiveness.<br /><br />Although I am tempted to rant about the fact that a high school is "graded" on a math test that only a portion of 9th graders take. (Advance students take Math 1 in the 8th grade.) I will withhold my opinion on the evaluation system and get to sharing resources.<br /><br />My opinion is that it is best practice to spiral review. This means students are presented with problems throughout the semester that require them to continually access the concepts previously studied. Common practice is to take the last few days before exams and work through problems.<br /><br />Both practices require a set of problems to present students. Today I am sharing review material.<br /><span style="color: red;">UPDATE: Solution sets have been posted on each course page as well as on this <a href="http://www.mcphersonmath.com/2016/05/ncfe-review-solutions.html" target="_blank">post</a></span><br /><ul><li><a href="https://drive.google.com/file/d/0B2jr-aJ3vP5iUk1Jc1gyZ0JFbnc/view?usp=sharing" target="_blank">Math 1 EOC Review</a></li><li><a href="https://drive.google.com/file/d/0B2jr-aJ3vP5iV2M0NDR4SnJGZ00/view?usp=sharing" target="_blank">Math 2 NCFE Review</a></li><li><a href="https://drive.google.com/file/d/0B2jr-aJ3vP5idHNnWFgweHRTcUE/view?usp=sharing" target="_blank">Math 3 NCFE Review</a></li></ul>As with all resources I post, it is possible there are mistakes. If you find any, I would appreciate it if you would let me know. Just leave a quick note here.Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com24tag:blogger.com,1999:blog-8633439154508225255.post-83405764839592605692014-12-17T11:22:00.000-08:002014-12-17T11:22:05.005-08:00Advocating for the Common CoreI support the Common Core. In 2010, when I was first introduced to the standards, the geek in me was excited about diving in, making sense of what they said, and creating learning opportunities for students. I embraced the change because the new standards are better than what we had.<br /><br />Here's a comparison.<br /><div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-dYFkLTRZYG4/VGjdvYm1S9I/AAAAAAAABk0/i2ysoIrgvlc/s1600/Comparison%2Bof%2BStandards.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-dYFkLTRZYG4/VGjdvYm1S9I/AAAAAAAABk0/i2ysoIrgvlc/s1600/Comparison%2Bof%2BStandards.PNG" height="360" width="640" /></a></div><br /></div><div><div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">Previously, students were to <span style="color: red;"><b>use</b></span> linear functions and now they are expected to <span style="color: blue;"><b>create</b></span> linear equations. The change in verbs is a critical transition from the old standards to the new standards. The expectation of what students are to know and be able to do increased. When I noticed this difference in the expectations I began to seek a better understanding of the standards and what I really needed to get my students to do.<br /><br />In 2012, I became a math coach and a large part of my job is supporting teachers to implement the new standards. One standard in particular is<br /><div style="text-align: center;"><span style="color: blue;"><span style="font-family: inherit;">G-GPE.6 <span style="background-color: white; line-height: 19.2019214630127px;">Find the point on a directed line segment between </span></span><span style="font-family: inherit;"><span style="background-color: white; line-height: 19.2019214630127px;">two given points </span></span><span style="background-color: white; font-family: inherit; line-height: 19.2019214630127px;">that partitions the segment in a given ratio.</span></span></div><div style="text-align: left;"><span style="color: blue;"><span style="background-color: white; font-family: inherit; line-height: 19.2019214630127px;"><br /></span></span></div><div style="text-align: left;"><span style="background-color: white; line-height: 19.2019214630127px;">What? I didn't know how to do this much less teach students how to do this. Thus, I began my journey into learning about partitioning. Did you know that students in elementary grade partition? Yep - it's true.</span></div><div style="text-align: left;"></div><ul><li><span style="background-color: white; line-height: 19.2019214630127px;">1st grade: Partition circles and rectangles into equal shares.</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">2nd grade: Partition a rectangle into rows and columns of same size squares.</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">2nd grade: Partition circles and rectangles into two, three, or four equal shares and describe the shares.</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">3rd grade: Interpret whole-number quotients of whole numbers (56/8 is 56 objects partitioned into 8 equal shares)</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">3rd grade: Understand a fraction 1/b as a quantity formed by 1 part when a whole is partitioned into b equal parts</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">3rd grade: Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">3rd grade: Partition shapes into parts with equal areas.</span></li><li><span style="background-color: white; line-height: 19.2019214630127px;">5th grade: Interpret the product (a/b) x q as a parts of partition of q into b equal parts.</span></li></ul><span style="line-height: 19.2019214630127px;">Now, in high school Math 2, students must partition a directed line segment.</span><br /><div style="text-align: left;"><span style="line-height: 19.2019214630127px;"><br /></span></div><div style="text-align: left;"><span style="line-height: 19.2019214630127px;">I looked for resources and found examples of how (procedural understanding). I found a few examples of why (conceptual understanding) but they came from the approach of vectors (not a prior understanding for Math 2 students). I didn't find anything that built upon the previous standards (coherence). So, what did I do? I wrote an investigation which can be found <a href="https://drive.google.com/open?id=0B2jr-aJ3vP5iVUtaT3ZEWUFsWGc&authuser=0" target="_blank">here</a>. </span><br /><span style="line-height: 19.2019214630127px;"><br /></span><span style="line-height: 19.2019214630127px;">This is the stuff I love - learning, creating, making sense of the things. The new standards have challenged me to think and that is a good thing. The new standards have prompted me to learn about how students are thinking about and doing mathematics in grades K-8. The new standards have introduced me to new concepts.</span><br /><span style="line-height: 19.2019214630127px;"><br /></span><span style="line-height: 19.2019214630127px;">I am an advocate for the CCSS because of what it offers teachers and students - a focused, coherent, and rigorous mathematics education.</span></div></div><div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"></div></div>Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-76357234637387661022014-10-29T21:18:00.001-07:002014-10-29T21:31:02.187-07:0044th Annual NCCTM ConferenceI am excited about this year's NCCTM State Math Conference. It is a great time to talk with others who are as interested (and yes, passionate) about teaching and learning mathematics. This year's theme is Big Ideas for Teaching and Learning Mathematics.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-00uHvags5fY/VFG3DD3HlgI/AAAAAAAABkI/VHH_OvYEqZA/s1600/NCCTM%2BCover%2BArt%2B2014.PNG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://1.bp.blogspot.com/-00uHvags5fY/VFG3DD3HlgI/AAAAAAAABkI/VHH_OvYEqZA/s1600/NCCTM%2BCover%2BArt%2B2014.PNG" height="200" width="155" /></a></div><br />Each year I try to share one workshop that examines a specific content area and one session that looks at a teaching practice.<br /><br />This year's content workshop is "<a href="https://drive.google.com/open?id=0B2jr-aJ3vP5iRkZGOTBtUmJvYlk&authuser=0" target="_blank">Transforming the Way We Teach Transformations</a>." I was inspired by an article with a similar title in the August 2010 Mathematics Teacher written by Eileen Fulkenberry and Thomas Fulkenberry. (You can find the article <a href="http://www.nctm.org/publications/article.aspx?id=27797" target="_blank">here</a>.)<br /><br />"<a href="https://drive.google.com/open?id=0B2jr-aJ3vP5iVURtTVQ2SlkyNkE&authuser=0" target="_blank">What Does a Grade Say?</a>" is my session addressing a teaching practice. I will discuss Outcomes Based Grading. This is a practice that I started in my classroom in 2011 and now, as I coach, I support and help teachers implement this practice in their classrooms. It's been an interesting endeavor to put into a presentation all of the complexities regarding something that seems simple.<br /><br />For those at the conference, I hope you have an opportunity to join me. Resources from the presentations can be found on the Presentations page.<br /><br />#1325 Transforming the Way We Teach Transformations 10:30 - 12:00 Colony A<br />#1538 What Does A Grade Say 12:30 - 2:00 MeadowbrookKaren McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-83082758906916989752014-10-12T07:46:00.001-07:002014-10-12T07:46:41.054-07:00Creating Curriculum ResourcesI don't remember when I started developing my own resources for teaching and learning. (To be clear, I am referring to engaging learning tasks and not worksheets for practice.) It certainly wasn't when I was a beginning teacher. In my second year of teaching I was responsible for the textbook adoption at my school. What influenced my decision the most? The amount of "extras" that was included. Yes, I was impressed by the boxes of ancillary materials and the promise of consumable workbooks for each year of the adoption.<br /><div><br /></div><div>The year after the adoption, when I was using the new textbooks with all the "extras", I was introduced to a new curriculum resource <a href="http://wmich.edu/cpmp/" target="_blank">Core-Plus Mathematics Project</a>. It didn't have all of the extras. As a matter of fact, the textbook wasn't even in color unless you consider pink accents as color. The use of these materials has shaped my philosophy on teaching and learning. After years of using Core-Plus, whenever I needed a resource that was not included in the textbook, I began creating my own. It was out of necessity because those boxes of "extras" didn't provide learning tasks that assisted me in facilitating students thinking about and constructing their own understanding of the mathematics.<br /><br />Fast forward 14 years. I am now a high school math coach and part of my responsibility is to assist my teachers in curriculum resources. This includes obtaining, evaluating, developing, and using the resources. One of my goals in coaching is to influence teachers to become "standards based" teachers and one strategy is using quality learning tasks. This is a challenge for a myriad of reasons of which I will not expand upon here. Suffice it to say, as a result, I find that I am writing curriculum resources.<br /><br />One of my professional goals is to create curriculum resources that<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://images.clipartpanda.com/writing-clip-art-9T4bq7XTE.jpeg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://images.clipartpanda.com/writing-clip-art-9T4bq7XTE.jpeg" height="136" width="200" /></a></div><ul><li>are good, interesting, and beneficial for teaching and learning math</li><li>explicitly address the instructional shifts (especially for concepts that are lacking in our primary curriculum resources),</li><li>include instructional strategies for use with students</li><li>and inspire others to create learning opportunities specific to their students.</li></ul>You will find these resources (and some created by my colleagues) on this site. I offer them to you to try, share, tweak, critique, and learn from. All I request in return is feedback. What worked? How can it be improved? What did students say and do? I value your input and appreciate you joining me in this important work.</div><div><br /><br /><br /></div>Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-87802503358079171982014-09-17T14:36:00.000-07:002014-09-18T09:35:00.179-07:00Trouble with File Access - RESOLVED<span style="color: blue;">The file links are working once again. If, in the future, they are not working, please let me know.</span><br /><br />It has come to my attention that visitors are not able to access some of the files that I have shared. This is apparently a problem with Google Drive which is where I store all of my files. The Google Forum has posted that they are aware of the problem and are investigating.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-L7z0N9pntcY/VBn7AiSmuUI/AAAAAAAABjI/sx6bfETdOuc/s1600/403%2Berror.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-L7z0N9pntcY/VBn7AiSmuUI/AAAAAAAABjI/sx6bfETdOuc/s1600/403%2Berror.PNG" height="125" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">I will be monitoring the progress. If not solved soon, I will look into other options for file sharing. If there is something you need, please comment below. I will email you the file as soon as possible.</div>Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com1tag:blogger.com,1999:blog-8633439154508225255.post-60976820182363075602014-09-07T20:15:00.000-07:002014-09-07T20:15:56.783-07:00To Reason QuantitativelyThe other day I was listening to a class discussion about independent and dependent variables. The teacher provided students with two related variables and asked students to determine which one was dependent on the other. For example, price of a ticket and number of customers.<br /><br />What I observed is that few students responded. More importantly, after the discussion was over, I still had no idea which students could identify and justify which variable would be dependent and which would be independent.<br /><br />The teacher and I met afterwards and planned an activity that we thought would generate more student engagement and provide information about what students really understood. The plan was to ask students to generate variables they thought were related. Next, each student would share the variables and the rest of the students would move to one side of the room or the other as to which variable they thought was the dependent variable. Students would provide their reasoning for their choice. Repeat...<br /><br />So what happened? Here are some samples of what students wrote:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-vt0edquUHHo/VA0eUIIV4OI/AAAAAAAABi4/rV6euPbOCr8/s1600/variables.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-vt0edquUHHo/VA0eUIIV4OI/AAAAAAAABi4/rV6euPbOCr8/s1600/variables.png" height="103" width="320" /></a></div><br />Hmmmm... Not what we anticipated. It appears that students identified "things" that are related but without thought about the measurement or quantities. How can we discuss independent and dependent variables when students don't recognize the quantities that are represented by the variables?<br /><br />The second Standard for Mathematical Practice is to "Reason abstractly and quantitatively." What does this mean, especially to reason quantitatively? What does this mean students can do? How do we, as teachers, design experiences that increase students' ability to reason quantitatively?<br /><br /><a href="http://www.nctm.org/publications/article.aspx?id=42697" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img alt=" 6 Principles for Quantitative Reasoning and Modeling" border="0" src="http://4.bp.blogspot.com/-weX8U_AQ9do/VAKlKLbS8yI/AAAAAAAABio/OJJAtHoxXa4/s1600/Quantitative_Reasoning_Article.png" height="131" width="200" /></a><br />I recently read the article "6 Principles for Quantitative Reasoning and Modeling" by Eric Weber, Amy Ellis, Torrey Kulow, and Zekiye Ozgur (<i>Mathematics Teacher </i>August 2014 Vol. 108 pp. 24-30.)<br /><i><br /></i>The authors describe quantitative reasoning as a "specific way of thinking about mathematics" and focuses on its role in the modeling process. The first of six principles for integrating quantitative reasoning in instruction is "Rewrite a problem situation or prompt so that students must identify the quantities that they believe are relevant to solving the problem." The key idea that really stands out to me is the <span style="color: blue;"><b>students</b></span> identifying the quantities.<br /><br />I am coaching the teacher to infuse this principle through every activity she can throughout the next module and then repeat the activity again. We agree that this must be an ongoing theme and not something a single lesson will address.<br /><br />Through this experience I have become "hypersensitive" to whether students are reasoning quantitatively and if teachers are providing opportunities for students to develop this habit of thinking about mathematics. So, when a teacher asks me for advice on a lesson plan or an activity this is one of lenses I am using.<br /><br />How do you address this mathematical practice? Share your experiences and strategies.Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-41427597000206540472014-08-16T19:44:00.003-07:002014-08-16T19:44:59.277-07:00The 1st Day of School<a href="http://3.bp.blogspot.com/-JN-GZYz_Who/U_ASOzebUwI/AAAAAAAABiA/tyb5zuSaCF0/s1600/welcomeback.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://3.bp.blogspot.com/-JN-GZYz_Who/U_ASOzebUwI/AAAAAAAABiA/tyb5zuSaCF0/s1600/welcomeback.jpg" height="200" width="153" /></a><span style="font-family: Arial, Helvetica, sans-serif;">What should students do on the first day of math class? It seems that the standard introduction includes a syllabus, grading policy, rules, and procedures. What seems to be debatable is whether or not students should do math on the first day. Huh?</span><br /><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><span style="font-family: Arial, Helvetica, sans-serif;">When I think about the first day, I want to clearly communicate expectations. One of the expectations is that everyone will engage in the teaching and learning of math on a daily basis. This means every day - including the first day.</span><br /><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span><span style="font-family: Arial, Helvetica, sans-serif;">The first day would include making sure the students are in the right class, briefly introducing myself, go over three rules (Be Ready, Be Responsible, Be Respectful), and then do a math task.</span></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><span style="font-family: Arial, Helvetica, sans-serif;">What kind of tasks are good for the first day? I look for a task that:</span></div><div><ul><li><span style="font-family: Arial, Helvetica, sans-serif;">is engaging / interesting</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">has students collaborating</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">has multiple entry points for students</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">has multiple solution methods</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">includes math ideas that launches the first unit of study (if possible)</span></li></ul><span style="font-family: Arial, Helvetica, sans-serif;">One of my favorite tasks for Math 1 is <i>Crossing the River</i>. The premise is that there are 8 adults and 2 children that need to cross a river. The boat can hold either one adult OR one child OR two children.</span></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><b><span style="color: blue;">The Launch:</span> </b>(I like to tell stories to engage students in the situation. So I tell them...)</span></div><div><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-16thuMlAz5s/U_AVKb7imrI/AAAAAAAABiM/CJATocJfNAc/s1600/DSC_0014.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-16thuMlAz5s/U_AVKb7imrI/AAAAAAAABiM/CJATocJfNAc/s1600/DSC_0014.JPG" height="133" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Calderwood Lake - The Men's Camping Trip</td></tr></tbody></table><i><span style="font-family: Arial, Helvetica, sans-serif;">My husband and son go on a Men's Camping trip every fall. On one of these trips they were going to hike around the lake. Eight of the men, my son, and another young boy took off early in the morning. When they were about halfway around the lake, the sky began to get cloudy and it appeared that a storm was arriving. One of the men noticed a boat and suggested they take the boat across the lake to quickly get back to the campsite. Testing the capacity of the boat, they determined that only one adult OR one of the boys OR both of the boys could be in the boat at one time. Now what?</span></i></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><span style="font-family: Arial, Helvetica, sans-serif;">Students are provided the opportunity to ask questions. These include:</span></div><div><ul><li><span style="font-family: Arial, Helvetica, sans-serif;">Are the boys capable of rowing the boat on their own? Is it safe?</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">Should they be taking a boat that doesn't belong to them?</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">Is there rope so they can pull it back across?</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">How much time would it take for them to walk back or continue on?</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">How much time does it take to get across the lake?</span></li><li><span style="font-family: Arial, Helvetica, sans-serif;">How many trips would it take to get everyone across the lake?</span></li></ul><span style="font-family: Arial, Helvetica, sans-serif;">The types of questions students ask indicate if students are thinking quantitatively. The first few questions are about the situation but not ones we can explore. The last few questions are ones that involve some quantitative reasoning. After the questions are posed, we discuss which ones can be explored with the information we have at hand. The last question is the one we focus on.</span></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><b><span style="color: blue; font-family: Arial, Helvetica, sans-serif;">The Exploration:</span></b></div><div><span style="font-family: Arial, Helvetica, sans-serif;">I inform students that there are materials available for their use located in the resource center. These include paper, colored paper, graph paper, rulers, chips of different colors, scissors, etc. (Doing this on the first day introduces students to the resource center and sets the procedure that they can access these tools whenever they need something.)</span></div><div><a href="http://4.bp.blogspot.com/-AXg_UZoNO7U/U_AWP0RPRsI/AAAAAAAABiY/4WaA1NO1xR8/s1600/CR%2BStudent%2BB%2Bp2.tif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-AXg_UZoNO7U/U_AWP0RPRsI/AAAAAAAABiY/4WaA1NO1xR8/s1600/CR%2BStudent%2BB%2Bp2.tif" height="150" width="200" /></a><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><span style="font-family: Arial, Helvetica, sans-serif;">Students work in groups to determine an answer to the question. (I prefer groups of 3 and no more than 4.) Some draw pictures while others get the color chips and begin manipulating them back and forth across an imaginary or sometimes drawn lake. Eventually, they determine the number of trips. I ask extension questions to groups. <i>What if there were 15 adults? 30 adults? What if there were 5 kids? </i></span></div><div><i><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></i></div><div><b><span style="color: blue; font-family: Arial, Helvetica, sans-serif;">The Share Out:</span></b></div><div><span style="font-family: Arial, Helvetica, sans-serif;">For this activity I tend to focus on the different ways groups approached the problem. I select and sequence how groups share with the intent of creating opportunities for students to compare their methods; understanding the differences and the similarities.</span></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><b><span style="color: blue; font-family: Arial, Helvetica, sans-serif;">Reflection:</span></b></div><div><span style="font-family: Arial, Helvetica, sans-serif;">Now that the students have a shared common experience I use it to discuss expectations such as making sense of problems, collaboration and student discourse.</span><br /><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span><span style="font-family: Arial, Helvetica, sans-serif;">For the next two weeks students will learn the different procedures for the classroom when they need to know them. For example, where do they put papers that need to be turned in? We will go over this the first time they have something to turn in. Another example, what is the procedure for leaving the classroom? I go over this the first time a student asks. I have determined that students often don't really learn the procedures for the classroom until it becomes something they recognize they need to know.</span><br /><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span><span style="font-family: Arial, Helvetica, sans-serif;">The first day focuses on the expectation that math class will be about developing and exploring ideas using math.</span></div>Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com0tag:blogger.com,1999:blog-8633439154508225255.post-57021463195899234282014-08-09T18:31:00.000-07:002014-08-09T18:31:29.686-07:005 Reasons I Started McPherson Math<span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;"><i><span style="font-size: large;"><b>#1 To become a more reflective educator.</b></span></i></span><br /><span style="font-family: Arial, Helvetica, sans-serif;"> The school year is very busy with each day bringing its own set of priorities. I am constantly asked for help on a variety of issues that teachers deal with on a daily basis. As a reflective teacher, I must be purposeful in thinking about what I have planned, observed, and experienced. How did my actions effect teaching and learning?</span><br /><br /><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif; font-size: large;"><i><b>#2 To document my own professional growth.</b></i></span><br /><div><span style="font-family: Arial, Helvetica, sans-serif;"> My job as a high school math coach involves focusing on the professional development of teachers. I am constantly researching and developing plans specific to my teachers' needs. </span><span style="font-family: Arial, Helvetica, sans-serif;">Whether it is new content, new technology, or a new teaching strategy, </span><span style="font-family: Arial, Helvetica, sans-serif;">I am always learning and growing as a professional.</span></div><div><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;"><i><br /></i></span></div><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif; font-size: large;"><i><b>#3 To engage in conversations that challenge me to think about educational issues.</b></i></span><br /><span style="font-family: Arial, Helvetica, sans-serif;"> I like to debate the issues because it helps me develop and/or clarify my viewpoint. It also helps increase my capacity to communicate with others. I'm invested in teaching and learning and want to be knowledgeable of the issues so I am able to advocate effectively.</span><br /><span style="font-family: Arial, Helvetica, sans-serif;"> </span><br /><div><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif; font-size: large;"><i><b>#4 To impact others by sharing ideas, struggles, and triumphs.</b></i></span></div><div> <span style="font-family: Arial, Helvetica, sans-serif;">For years I have referred to myself as a BASE teacher. (That stands for Borrow And Share Everything.) I enjoy collaborating with others, sharing whatever resources I have, seeking solutions, and celebrating our successes. I've tried other digital platforms to share resources and I hope this one is easier to maintain and for others to use.</span></div><div><br /></div><div><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif; font-size: large;"><i><b>#5 To have fun with math.</b></i></span></div><div><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;"> </span><span style="font-family: Arial, Helvetica, sans-serif;">I like math and I really like a good math problem. It's the geek in me and I own it.</span></div><div><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div><div><span style="font-family: Arial, Helvetica, sans-serif;">So, there you have it. Five reasons why I started <b><i><span style="color: #0b5394;">McPherson Math</span></i></b>. I hope you join me for this journey.</span></div><br />Karen McPhersonhttp://www.blogger.com/profile/08027462766555745915noreply@blogger.com3