tag:blogger.com,1999:blog-28973056780426984872024-02-19T01:47:16.173-05:00Beautiful MathThis blog is devoted to sharing my high school mathematics teaching ideas! I have been using interactive student notebooks (ISNs) for three years now.. I love them and my students love them! Much of what I use originated on the internet on some math teacher blog and I personalized it (except most of the precalc ones, there is little out there for precalc). So I'm sharing so others can do the same! I hope you find some good content in my posts, feel free to comment or ask questions! mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.comBlogger113125tag:blogger.com,1999:blog-2897305678042698487.post-30175231331136532252017-08-05T09:47:00.000-04:002017-08-05T09:47:12.484-04:00Solving Trig Equations Part Two<div class="separator" style="clear: both; text-align: center;">
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<span style="font-family: Arial, Helvetica, sans-serif;">I find that my students have a very difficult time with solving trigonometric equations. Part of this stems from weak algebra skills. This is frustrating because these are students in an honors level Precalculus class who have successfully completed honors level Algebra One and Algebra Two. Perhaps it is because it is "out of context". Which is not a good excuse as it shows me they didn't learn conceptually what it means to solve an equation (and what those algebra skills, such as factoring, are really about).</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">So I start solving trig equations in the introduction unit on trig. The focus of that chapter really is understanding trig functions and the trig ratios that correspond to various angles (having learned right triangle trig in Geometry & Alg 2). </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Then we see solving trig equations again in the Analytic Trig unit - where we also simplify and verifying trig expressions using various identities (and algebra techniques). Later we solve equations in this same unit using more complex sum & difference and multiple angle identities.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">This blog post looks at solving trig equations including basic trig identities. We work on these right after we spend a few days on the basic identities (reciprocal, quotient and Pythagorean) and simplifying trig expressions. The important thing for my students is to practice. So many of my ISN pages are examples. I have students do these together in their small groups and we share work in the larger group setting.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">I usually introduce this section with a few complicated equations that I ask the small groups to solve. I remind them of what we did last unit and that so far in this unit we have some new tools available to use (those previously mentioned identities - it's a good idea to ask students to share what new tools we have and get feedback from class before they solve). After we do this introduction work (solve, share, discuss). I give them the information sheet for their notebook (see above). </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Then we dive into doing examples Some we do together, some we do in small groups. Ten examples in all. </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Now this year a student got confused with "solving trig equations" and "evaluating inverse trig functions". They thought they were the same thing and got stuck in thinking about the limited domain & range for inverse functions. I was like "uh-oh" - we discussed that misconception. Then next class I gave them this foldable to summarize our conversation:</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Since I had a page opposite that little addition in our ISNs, I gave students another solving example for a warm up for the blank page:</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">In retrospect, I will save some paper this coming school year by having students do those 10 original examples directly into their notebooks. A composition notebook does have 200 possible pages and this year in PreCalculus we used only 147 pages. </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">My files<a href="https://app.box.com/s/opo6l3dusbuwj5yhykl5pmkrruamuop0"> on equation solving can be found HERE.</a> These include the two ISN pages and a summary practice sheet of examples. </span></div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com1tag:blogger.com,1999:blog-2897305678042698487.post-20562693364661036112017-07-20T17:26:00.002-04:002017-07-20T17:26:41.042-04:00Solving Trig Equations part one revisited<div class="separator" style="clear: both; text-align: center;">
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I've already blogged about solving trig equations in a <a href="https://beautifulhighschoolmath.blogspot.com/2016/03/solving-trigonometric-equations-in.html">previous post found here.</a> I've updated my lessons a bit by redoing the inserts for my ISN. This blog post shows the updated inserts. See the previous post for more details on the actual lesson.</div>
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inside that foldable:</div>
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Once again a review of some basic algebra skill is necessary and then they are applied to trig equation solving.</div>
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Students then work through a few practice examples:<br />
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All my equation solving foldables <a href="https://app.box.com/s/fcucn9kbzuozifmq3a89dl5c032mwhdn">can be found HERE.</a> You'll also see some worksheets of equation solving problems and answer keys.<br />
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-71086989300632758452017-07-19T17:57:00.000-04:002017-07-19T17:57:14.367-04:00Inverse Trig Functions Revisited<div class="separator" style="clear: both; text-align: center;">
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I'm updating my Inverse Trig Functions post because I updated my ISN pages in the 2016-2017 school year. I still like my introduction information <a href="http://beautifulhighschoolmath.blogspot.com/2016/02/introducing-inverse-trig-functions.html">found at this original blog post.</a></div>
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We still explore the various inverse trig functions using our graphing calculator (love the draw inverse option in the DRAW menu). From there we summarize our findings with three ISN pages for each of the big three trig functions. (I've veered away from doing as many foldables in my precalculus ISN - we have lots of room so don't need to conserve space with folding inserts).</div>
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From there the discussion heads towards how to evaluate inverse trig functions. The key word here is "functions". So we have to look at the domain & range restrictions of the inverse functions when determining what angle we have.</div>
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I have these two pages - the yellow is a flap over the white notebook page (hmmmm what did I just saw about not using foldables - but it makes sense to do it here). We do the white page first to consider and discuss how to do the evaluations in a general way with a few guided examples. Then students do the yellow page for independent/group practice as I circulate around the room.</div>
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From there we go into "double evaluations". I am able to get to all of this because we have an 82 minute block schedule. I usually start off by showing a few on the board so they can see what I mean by "double". I like to show at least one where the function and inverse "cancel" out and one where they do not (see first two examples on white page 95 - something like those two). Because of the restrictions we cannot always assume that a function and its inverse will "cancel".</div>
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So after a bit of discussion we summarize how to "double evaluate" on this page with a few examples to do together.</div>
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At this point I usually do run out of time. So I assign a few problems in our textbook which mostly focus on the single evaluate. Then in our second class I use this insert as our warm up:</div>
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After discussing any questions students have on their homework (and maybe a basic homework quiz) they work in their small groups on a summary worksheet of many different inverse evaluations and double evaluations. You can find my worksheet in the files for this topic.</div>
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All files for inverse trig functions <a href="https://app.box.com/s/44d4s7b6xdszwslht366hsm0gxv0o98f">can be found HERE</a>.</div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-27281716333676016802017-06-20T12:11:00.001-04:002017-06-20T12:11:01.099-04:00Impact of Interactive Notebooks<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFi6_x5azyB8FDwGWmNFTAIjDTdB6XHYCdM35TiWaXqsMH2PK81HlEpOhqn6RCx4a62lS0ay2HBmltN320hhqJwVypFIWPX2XTqMumixjfTmF33lWTZnilCm5ZJkANDgdpigsgDAcju9M/s1600/alg+one+2017.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="536" data-original-width="415" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFi6_x5azyB8FDwGWmNFTAIjDTdB6XHYCdM35TiWaXqsMH2PK81HlEpOhqn6RCx4a62lS0ay2HBmltN320hhqJwVypFIWPX2XTqMumixjfTmF33lWTZnilCm5ZJkANDgdpigsgDAcju9M/s320/alg+one+2017.JPG" width="247" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSFuDLwJL4pZqapN5UN6HSbMbs40rAPH_LUDLkV6G-ZztWXkjSBwB6towRX3bFw70N-tx9WaqlyTRFa7bC07nIZacCn8C1c22YrdCJhVA0i3NgrHIOJIdAeHVohzFHh4xRFxPbdcqSSXc/s1600/precalc+2017.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="538" data-original-width="403" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSFuDLwJL4pZqapN5UN6HSbMbs40rAPH_LUDLkV6G-ZztWXkjSBwB6towRX3bFw70N-tx9WaqlyTRFa7bC07nIZacCn8C1c22YrdCJhVA0i3NgrHIOJIdAeHVohzFHh4xRFxPbdcqSSXc/s320/precalc+2017.JPG" width="239" /></a><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">A reflective post today.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I am just finishing my third year of using interactive student notebooks in my classes. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">How are they working? Do I like using them? Do students like them? Do students find them useful?</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">Yes, I think the interactive notebooks are a success in my classes. Over the three years I've made some adjustments from what I've found on line to match my style and what I think it useful for students. I use them in my two advanced classes - Algebra One and Pre-Calculus. The students mostly love them. There were some groans from some of my students in PreCalculus at the beginning of this year. But even those "groaners" got into doing them and keeping them up to date. Of the 70 students who used these notebooks this year only one student did not keep up with it (a struggling student in my precalculus class).</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">Now that I've done three years of ISNs I have a pretty good file of foldables to use for each class. I've worked hard this year to update them, both before and after using them if necessary. That way any changes I want for 2017-2018 were fresh in my mind for my updating. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I like how the ISNs fit into my instruction. I do have a block schedule (82 minutes) with Algebra One meeting every day and PreCalculus meeting every other day. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">My general approach is to first go through a concept with students the usual way - either with some discovery, whole group examples or guided individual work. Then I use the ISN to summarize the concept with some practice after the summary (usually on the page opposite). </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">What I LOVED seeing this year during our exam review days was students actually accessing the information in their notebooks. They were looking topics up in their table of contents and referring to them as they did problems in the exam study guide. Seeing that happening throughout our three days of exam review in Algebra One was wonderful! I actually had to do very little whole group re-teaching and review. Now, of course, I haven't scored the exams yet. But at least I saw students actively and productively using them.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I've tried a few different approaches to the Table of Contents and for me the best approach is the most basic. At the start of the year students label the first 4-6 pages as TABLE OF CONTENTS (TOC for short). And we fill in it as we go. Every day I post on the board "TOC update" with the latest page numbers and topics for them to fill in.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I stopped being fancy with my inside cover too. There I just put a summary syllabus and the kids create a title page. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">On the inside back cover, I have the students glue in a 6 by 9 manilla envelope for any foldables that we don't quite finish in class. I'm gong to keep that (for my algebra one students). </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I do not use them in my low level Applied Algebra One class. I tried them my first year and it did not work out. That class has quite a bit of absenteeism and that gets very difficult with keeping up in the notebook. However I think I will try again next year but really scale it down. Currently in my Applied Algebra One class I maintain a notebook checklist system with the students (which I will continue as it helps keep the notebooks organized). In that system we would have a page of summary notes at various times in a chapter. What I would do instead now is do that summary in our ISN. The main reason is that I do not see my Applied Algebra students accessing those notes in their notebook when they do problems. So I am going to see how it works to maintain an ISN of the basics in each chapter. I am going to have them keep their ISN in the classroom unless they want to take it home to help with their work. I imagine the conscientious ones will take them home and use them. Those who won't use them will leave them in my room (and then they won't get lost!). Also when a student is absent I'll be able to insert the missing foldables in their notebook as they will have kept the ISN in my classroom. We'll see.... I'll report back on how this goes next year in June 2018. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">My goal on this blog is to finish posting some of my precalculus pages this summer. Then in the 2017-2018 school year post my updated pages in my Algebra One notebook as I teach them. I did do some new things this last year and may tweak them even more this coming year. </span>mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-31831419817157800582017-06-16T14:10:00.001-04:002017-06-16T14:10:36.170-04:00PreCalculus: DeMoivre's Theorem and Calculating Roots of Complex Numbers<div class="separator" style="clear: both; text-align: center;">
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<i><span style="font-size: x-small;">http://mathforum.org/mbower/johnandbetty/p30/p30.htm</span></i></div>
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The next lesson after trigonometric form of complex numbers is working with DeMoivre's Theorem. We calculate powers of complex numbers and then from there look at different roots of complex numbers.</div>
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Most of my classes start with a Q & A on the last lesson and then a POD quiz (problem of the day quiz) based on the last lesson. Something quick to assess understanding.</div>
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I have found that a good way to start this new lesson is to give them a complex number and ask them to square it. (something like -2 - 2i ). Most students can do this pretty well using the distributive property or the box method of multiplying binomials. But then I ask them to raise the same complex number to the 6th power. Since we didn't get to the binomial theorem this year they have no real idea what they could do except keep multiplying. An overwhelming prospect. This sets up some reasonable rationale for today's DeMoivre's theorem. I go through raising it to the 6th power using this new "formula". So much easier!</div>
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I use the following foldable and examples in our composition notebooks (ISNs). </div>
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DeMoivre's Theorem</h3>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgscCeLC6mDdxP3uF-cMqX7Z9TjTb-_XrovojrXzT3JRyMdn3Sn29AgcN2x7P9a6tZoP0JCWdS026oFqBLb_AtYxpCKToc-jLW29DuMy7cvhD56x0cC7fakiXIeNXhlRBn3ojVwzSpmBhI/s1600/demoivres.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="545" data-original-width="366" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgscCeLC6mDdxP3uF-cMqX7Z9TjTb-_XrovojrXzT3JRyMdn3Sn29AgcN2x7P9a6tZoP0JCWdS026oFqBLb_AtYxpCKToc-jLW29DuMy7cvhD56x0cC7fakiXIeNXhlRBn3ojVwzSpmBhI/s640/demoivres.JPG" width="427" /></a></div>
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Then we switch gears to the inverse - what's the inverse of raising a number to a power? Taking roots. I begin this discussion by asking, "What is the square root of 64?". Being an honors class they easily remember there are two square roots, BOTH 8 & -8. Then I asked what is the cube root of 8? Another easy one, 2. But then I tell my students that there are TWO MORE cube roots. And that the number of roots is always equal to the index "n". However some roots might be in the set of complex numbers. </div>
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We look back at example 2 in our last set of examples. The original complex number was raised to the 12th power and the result was 4096. Another 12th root of 4096 is 2. But we started with a complex root in our previous example. We talk about what the other roots look like (there is one more real 12th root in this problem...).</div>
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From there we set up our new formula:</div>
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Nth Roots of Complex Numbers</h3>
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We go through an example very carefully. Notice the foldable tapes on the side of this notebook page and when you lift the flap you see the steps we follow. (a note here on taping in foldables - tape it both on top and then lift it and tape it under to secure it well in the notebook). </div>
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Students start to notice a pattern in the angles of these roots. I use this opportunity to look at the roots on a unit circle and students see how they are evenly spaced. Cool stuff!</div>
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Since this is our last topic for the year we overflowed a bit onto a few more pages.... </div>
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All my <a href="https://app.box.com/s/p58o68zqdooqzlkryykpjkae16pb4gxm">foldable files are found here</a> and a <a href="https://app.box.com/s/0p0bb4x250gx690ajerun8em15v3pnf6">worksheet of this mini unit can be found here</a>.<br />
<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com1tag:blogger.com,1999:blog-2897305678042698487.post-76252474637670078612017-06-14T08:00:00.000-04:002017-06-14T08:00:02.987-04:00Trigonometric Form of Complex Numbers - PreCalculus<div class="separator" style="clear: both; text-align: center;">
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<span style="font-size: x-small;"><i>https://en.wikipedia.org/wiki/Complex_number</i></span></div>
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<span style="font-size: small;">I love starting off my lesson on Trigonometric Form of Complex Numbers. <span style="font-weight: normal;"> </span></span></h3>
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<span style="font-size: small;"><span style="font-weight: normal;">I start off by drawing a real number line and I talk about the set of real numbers. I even do a mini-review of the sets of numbers that make up the set of real numbers.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjASUR9Uo4KfLsxceDssWZtTGsCPIRYOFBTa5q8KzIFgG6UDH52V-UWbVwJG22Oj8Ti18rxXzqawbwl46Xatdg6CuZrUMsjwlld-wDaueDjSZ1EcaYLHbaeKaQy8tH4ZGfQvDkSXYvLdyQ/s1600/real+numbers+venn.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="552" data-original-width="736" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjASUR9Uo4KfLsxceDssWZtTGsCPIRYOFBTa5q8KzIFgG6UDH52V-UWbVwJG22Oj8Ti18rxXzqawbwl46Xatdg6CuZrUMsjwlld-wDaueDjSZ1EcaYLHbaeKaQy8tH4ZGfQvDkSXYvLdyQ/s400/real+numbers+venn.jpg" width="400" /></a></div>
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<i><span style="font-size: x-small;">https://www.pinterest.com/marlab327/real-numbers/?lp=true</span></i></div>
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<span style="font-size: small;">This shouldn't take a lot of time, it's just a quick overview on how cool numbers are and how they relate to each other. I summarize by saying the set of real numbers can be shown graphically with a number line. Very cool, very simple.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb8IqEV_KODxGqpdokbncLn-ItIgR8YT0FkWI1IuRY6Lj6qkIIK12RlWyhv7NO8c3aS36ktRETzX6NdoxKZlU1MKmzDXV8xUyUkq28XC1WLgraM4AS3w4NyHrys-kkD8mgPK_MvGAdhRA/s1600/real+number+line.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="144" data-original-width="360" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb8IqEV_KODxGqpdokbncLn-ItIgR8YT0FkWI1IuRY6Lj6qkIIK12RlWyhv7NO8c3aS36ktRETzX6NdoxKZlU1MKmzDXV8xUyUkq28XC1WLgraM4AS3w4NyHrys-kkD8mgPK_MvGAdhRA/s400/real+number+line.gif" width="400" /></a></div>
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<i style="font-size: small;">http://www.onlinemathlearning.com/real-numbers-gre.html</i></div>
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<span style="font-size: small;">And how do imaginary numbers relate to all this? Again, quick reminder of why we have imaginary numbers (ever try to take the square root of -1?). Again this is </span><span style="text-align: center;">very quick and culminates in us having the set of complex numbers! Tada!</span></div>
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<i><span style="font-size: x-small;">https://www.tes.com/lessons/uIuFvVVAvuMA8g/complex-numbers</span></i></div>
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<span style="font-size: small;">So from there I get to show them how the imaginary numbers relate to the real number line with the imaginary axis! I stress that it is not a coordinate plane even though it looks like one. </span></div>
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Now this is something they should have learned in their earlier math courses so it is a quick intro to set the stage for Trig Form of Complex numbers. </div>
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From there we compare the three different ways we can write a complex number (with a greater focus on the first two): Standard Form, Trigonometric Form, and Polar Form. We look at them graphically with my first foldable.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7eD8TY-O_Y0RFN_tqlwmJiGNLJAEo3wz0723ekZo2Bbp9Hc5gadnSNQk0uSbgxyEZnZyzM4nU_QCKOoHDbBqUGefHsCdzg6m5HZEDga5Yp7gLzFR7wkxGBwtlcOxsLM2Y9CWcT2WgM9o/s1600/different+forms.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="545" data-original-width="379" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7eD8TY-O_Y0RFN_tqlwmJiGNLJAEo3wz0723ekZo2Bbp9Hc5gadnSNQk0uSbgxyEZnZyzM4nU_QCKOoHDbBqUGefHsCdzg6m5HZEDga5Yp7gLzFR7wkxGBwtlcOxsLM2Y9CWcT2WgM9o/s400/different+forms.JPG" width="277" /></a></div>
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Inside the foldable we look at converting back & forth between standard form and trig form.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggDdMLyNIECEtWIHVrcDAe6JtSfGjC1EtySRyWlN8alGmClOa0t2Xl7eyUtIEzPyFyBuVjIWykTBgFkMgsjRHNdqC7f61XlHwluHf4dt6kpXV9gohRJEx_XchcKcAlZV_lYzm32_Y4Sfo/s1600/inside+diff+forms.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="460" data-original-width="595" height="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggDdMLyNIECEtWIHVrcDAe6JtSfGjC1EtySRyWlN8alGmClOa0t2Xl7eyUtIEzPyFyBuVjIWykTBgFkMgsjRHNdqC7f61XlHwluHf4dt6kpXV9gohRJEx_XchcKcAlZV_lYzm32_Y4Sfo/s400/inside+diff+forms.JPG" width="400" /></a></div>
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Students then try out some examples on the page opposite.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8xvfBrZ9BJxUVS7zF7WiECPKRBnLLToKN5peDHvd7TzJ0TXUHMMgNA5vP9Rt7QcVlQuBDSLjuPIWouxE9yZhA3T7MXz53_bSA_Hh81V84mUIarLGBlMzrScTZIyqipigqMRcCdX8qiGM/s1600/convert+examples.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="536" data-original-width="399" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8xvfBrZ9BJxUVS7zF7WiECPKRBnLLToKN5peDHvd7TzJ0TXUHMMgNA5vP9Rt7QcVlQuBDSLjuPIWouxE9yZhA3T7MXz53_bSA_Hh81V84mUIarLGBlMzrScTZIyqipigqMRcCdX8qiGM/s400/convert+examples.JPG" width="297" /></a></div>
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Now we have an 82 minute block so I still have time to go over multiplying and dividing complex numbers in trig form. Providing some rationale is a bit challenging since it's pretty straightforward multiplying complex numbers. They know it's like multiplying binomials. Dividing is a little trickier. When the numbers are in trig form there is a nice straightforward method. My foldable shows the official definition/formula on the front.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4wVjfFZOl0VgGuzanG-HDUubcMWd46sV_TjX9mPG878ocRLZOOUkRIp3QxAV78t3hZc-TkWF-QAj0vk5YsaFxfEAqY_LtYCQ_7NDX5B0WgNHvfEy86RIDf3uEbiilLCS3I3jLFZaxd7I/s1600/Multiply+and+Divide.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="550" data-original-width="391" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4wVjfFZOl0VgGuzanG-HDUubcMWd46sV_TjX9mPG878ocRLZOOUkRIp3QxAV78t3hZc-TkWF-QAj0vk5YsaFxfEAqY_LtYCQ_7NDX5B0WgNHvfEy86RIDf3uEbiilLCS3I3jLFZaxd7I/s400/Multiply+and+Divide.JPG" width="283" /></a></div>
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But inside I do give the "quick" way to do it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz6ODmFAQ9KKwdhV5_oZITlDVbNgMWkZbJnQH2qw7UTK1S6e1EAz8iXHa6MXg81vH7PHOqa21hBVlr4-iSHekSEEH3Kyk8zT1xMDklJHLitUZkeCyg-1KWacsHxaqMsi5Y9Hh4YRqVnAk/s1600/INisde+mult+and+div.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="452" data-original-width="592" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz6ODmFAQ9KKwdhV5_oZITlDVbNgMWkZbJnQH2qw7UTK1S6e1EAz8iXHa6MXg81vH7PHOqa21hBVlr4-iSHekSEEH3Kyk8zT1xMDklJHLitUZkeCyg-1KWacsHxaqMsi5Y9Hh4YRqVnAk/s400/INisde+mult+and+div.JPG" width="400" /></a></div>
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I also allow the shortcut way of writing complex numbers in trig form using CIS. </div>
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And on the page opposite we have some examples:<br /><div class="separator" style="clear: both; text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMS2psa3TTGDjFdwZF-wDfD1xCBOkmmmaiy_pnueBi2YB-jSWzr82ZxRSLWK1U3Px5XvLh226k-58bZzfjzgx6zyLehp1RAiuJAJbv975uuEtL6FRdkkevm2WCMG7YpWoOliIcfHM6-2w/s1600/Mult+and+div+examples.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="526" data-original-width="404" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMS2psa3TTGDjFdwZF-wDfD1xCBOkmmmaiy_pnueBi2YB-jSWzr82ZxRSLWK1U3Px5XvLh226k-58bZzfjzgx6zyLehp1RAiuJAJbv975uuEtL6FRdkkevm2WCMG7YpWoOliIcfHM6-2w/s400/Mult+and+div+examples.JPG" width="306" /></a></div>
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This year, I had to be out the day after this lesson. So I gave the students the above examples as a warm up for that class. They had a POD for converting complex numbers and then worked on a practice sheet of the skills we had gone over in this lesson.</div>
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All <a href="https://app.box.com/s/p58o68zqdooqzlkryykpjkae16pb4gxm">foldables can be found here</a> and the <a href="https://app.box.com/s/ibrrxm44c5gazw8sslb34opm51m6wx3t">worksheet can be found here.</a></div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-70861078581541530902017-06-13T09:00:00.000-04:002017-06-13T12:28:42.811-04:00PreCalculus Series 2017<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYi2bcsGuAwInzNxmWcwrKTI6xLWvHebHplpOwyaZit6ZPgIcm55veXL34hTdsmtH10IL8hPNQaeAOXhpOYSfIItWYhR2PPN3bZuMwxse0vxBxLudO9XyziA-Jg6HmIsfDFcmXLYou-kM/s1600/summation+image.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="576" data-original-width="1024" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYi2bcsGuAwInzNxmWcwrKTI6xLWvHebHplpOwyaZit6ZPgIcm55veXL34hTdsmtH10IL8hPNQaeAOXhpOYSfIItWYhR2PPN3bZuMwxse0vxBxLudO9XyziA-Jg6HmIsfDFcmXLYou-kM/s320/summation+image.jpg" width="320" /></a></div>
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<span style="font-size: x-small;">http://study.com/academy/lesson/what-is-summation-notation.html</span></div>
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After we spend a day on reviewing sequences we jump right into series. I'd love to cover it all in one 82 minute block. But we start the class by going over homework problems and then I have a POD (problem of the day homework quiz) so we usually have to finish up infinite series on the next day. (More on my PODs and homework policies on an upcoming blog post).</div>
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Of course, we start Series by defining them. I use a Frayer diagram:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsW9u4PmI__4E-ni3tcIcIhebIDHFL7SRDnUeGiFsMxzedLCswVVz8mGvMhU9ge71tb6e2xPQVa76KWneazMdJXtV8_bHX2vgxc1-crSqBwfXM3oGSjr20joqilX3WgigwnkHkl_ArLlU/s1600/frayer+series.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="542" data-original-width="437" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsW9u4PmI__4E-ni3tcIcIhebIDHFL7SRDnUeGiFsMxzedLCswVVz8mGvMhU9ge71tb6e2xPQVa76KWneazMdJXtV8_bHX2vgxc1-crSqBwfXM3oGSjr20joqilX3WgigwnkHkl_ArLlU/s400/frayer+series.JPG" width="322" /></a></div>
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I have students trim this one a bit so we can glue it into our composition notebook with the orientation shown. That way I have a little room on the bottom for the graphing calculator method of calculating a series. I only let them use this for checking. They have to use other methods to actually calculate the series (as will soon be shown on this blog).</div>
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On the page opposite the definition, we do a page on summation notation.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMV9ZPyB7BRI7llm2wGWFB9wlPSXQG8WnrZgJGXv6NVOAlf-gwtpc96if8zXrV8G8Q1WicOUDyBVVRZd5Lipe6tycC1TAFOKhKi5WmBURgP4myDg-f9pJUT7hQnrnXLzaWlMG2koc-upc/s1600/summation+notation+notes.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="544" data-original-width="404" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMV9ZPyB7BRI7llm2wGWFB9wlPSXQG8WnrZgJGXv6NVOAlf-gwtpc96if8zXrV8G8Q1WicOUDyBVVRZd5Lipe6tycC1TAFOKhKi5WmBURgP4myDg-f9pJUT7hQnrnXLzaWlMG2koc-upc/s400/summation+notation+notes.JPG" width="296" /></a></div>
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Before I start the formulas for series, I tell the famous Gauss story about adding up the numbers one through 100. If you don't know the story, <a href="http://www.coolmath.com/algebra/19-sequences-series/06-gauss-problem-arithmetic-series-01">check out this link</a> (or any other page on the internet that you might when you search this story). I like the link I gave because it references the introduction page for the coolmath lesson on arithmetic series. I LOVE her page and she does a nice job at this link (if you continue to subsequent pages) of developing the formulas. (BTW <a href="http://www.coolmath.com/algebra/19-sequences-series">here is the link</a> for her approach on the whole topic of sequences and series, definitely worth a read through!)</div>
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After we do the "trick" young Gauss used I use that to develop the formula. We fill out a foldable on the Arithmetic Series formula. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6gfyF9gkez0jWuzoNmRkwXQ3EOnKuhEP-1imd6haqocVROBfgsBflrjrKZ7YzSogY6QkHIueuyJm-k7YYmq9fNZJjlxOEZgH8TScWnY2ole6D3r6vtYopwVbRrwzrS3mIKQCHbYl2P4U/s1600/arithm+series.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="470" data-original-width="391" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6gfyF9gkez0jWuzoNmRkwXQ3EOnKuhEP-1imd6haqocVROBfgsBflrjrKZ7YzSogY6QkHIueuyJm-k7YYmq9fNZJjlxOEZgH8TScWnY2ole6D3r6vtYopwVbRrwzrS3mIKQCHbYl2P4U/s400/arithm+series.JPG" width="332" /></a></div>
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Lift the flap up and this is what you will see inside this foldable:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilBhXkTWD5wFVa43A1RRCPZ-Zi7bkhFQugMZArRR_u60WvEYGYkfBkip6WfiqrGQLtP2UpU9Jko85n9C78w5-GCNUyxmiX0m6GxukSsn0Z69OH0powUfdPSYGRKDvpF4Xp2a7cHZjopSE/s1600/inside+arithm+series.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="541" data-original-width="228" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilBhXkTWD5wFVa43A1RRCPZ-Zi7bkhFQugMZArRR_u60WvEYGYkfBkip6WfiqrGQLtP2UpU9Jko85n9C78w5-GCNUyxmiX0m6GxukSsn0Z69OH0powUfdPSYGRKDvpF4Xp2a7cHZjopSE/s640/inside+arithm+series.JPG" width="267" /></a></div>
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On the page opposite, I include some examples for students to work through. This is pretty quick. We don't have to work through them together. They work on them and I walk around and check answers.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE0Hl9moOz66nsOCRb_DfFhnAKa9AajjTgGLazb6UK6PxW9dGXCyOLb6IZcQGApc5xCTPlsvHYcmbBMhyts7nD4imBvC-0KKJ7DOyWOoIfq3DQ_5v8YdpNrW2aGK3ub3kAZ2ClFUBl8VI/s1600/arithm+series+examples.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="524" data-original-width="395" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE0Hl9moOz66nsOCRb_DfFhnAKa9AajjTgGLazb6UK6PxW9dGXCyOLb6IZcQGApc5xCTPlsvHYcmbBMhyts7nD4imBvC-0KKJ7DOyWOoIfq3DQ_5v8YdpNrW2aGK3ub3kAZ2ClFUBl8VI/s400/arithm+series+examples.JPG" width="301" /></a></div>
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There is no cute little story to help my students remember the Geometric Series formula. But fortunately we do have a formula. I introduce that with my foldable.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjH6RzpiC3ke9CQZapZfU4yWBW2JItspfRW8ohqKEoEzm0AY6GmpNi1aOAbekhEWCoBnvsObMpeBT_B5J6Hn1q-EAiXEbQdmBxU0ukCDuqdxTQDdEAeZ0Tcw-5s8FoBLffiFJc8ZOL7RzM/s1600/Geometric+Series.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="534" data-original-width="375" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjH6RzpiC3ke9CQZapZfU4yWBW2JItspfRW8ohqKEoEzm0AY6GmpNi1aOAbekhEWCoBnvsObMpeBT_B5J6Hn1q-EAiXEbQdmBxU0ukCDuqdxTQDdEAeZ0Tcw-5s8FoBLffiFJc8ZOL7RzM/s400/Geometric+Series.JPG" width="280" /></a></div>
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Lift the foldable flap and you see see the formula and what it's all about.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-9_nfBW8D4CQfE8kkbRXgnqN_H7Ghr_YhbgJe12bD4VpmlMYbc0QhUkmP2i-Nj4rHH5ZIcqbWEgR1F4Zzd5-0aWW9GcPbjNxQjaOd-WYKz-7djRExW-p15UufzspMOd_ba2Me8jtYiJA/s1600/inside+geom+series.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="566" data-original-width="250" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-9_nfBW8D4CQfE8kkbRXgnqN_H7Ghr_YhbgJe12bD4VpmlMYbc0QhUkmP2i-Nj4rHH5ZIcqbWEgR1F4Zzd5-0aWW9GcPbjNxQjaOd-WYKz-7djRExW-p15UufzspMOd_ba2Me8jtYiJA/s640/inside+geom+series.JPG" width="282" /></a></div>
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And again, we have a page of examples on the page opposite. The bouncing ball example is a tricky one!</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrcUS_CRdzqVbAZxsQ-IbP4_05qQ-pW-Vb3X5I3zQpuvfBRz7QTNGn3B5_VIKfhPjRWdmGyNySgVKquFAs9R2ATJPxXTYfdykXlAfHjMfPSa0mFufLfd8zlaIzskafWgFdjolZsnijFws/s1600/geom+series+examples.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="534" data-original-width="370" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrcUS_CRdzqVbAZxsQ-IbP4_05qQ-pW-Vb3X5I3zQpuvfBRz7QTNGn3B5_VIKfhPjRWdmGyNySgVKquFAs9R2ATJPxXTYfdykXlAfHjMfPSa0mFufLfd8zlaIzskafWgFdjolZsnijFws/s400/geom+series+examples.JPG" width="276" /></a></div>
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Finally I do some intro-stuff on the sum of an infinite series. I write two infinite series on the board (the first 4 or 5 terms of each, one with |r|>1 and the other with |r|<1). We talk about what it would look like if we added the terms in the sequence - as many as we can anyhow. Students pretty quickly see that the one with |r|>1 just gets larger and larger. But the one with |r|<1 they are not so sure on. So we work with the graphing calculator way to add series. And we find the sum of the first 10 terms. Then the first 20 terms, then 50, then 100, then 150, etc. Usually by the 50th term we get the sum it converges to. So we talk about what this means. I take the finite Geometric Series formula and ask them what happens when n = infinity? What happens to the formula? This helps them to see how the infinite formula is derived from the finite formula. It's pretty cool to talk through it rather than just give them one more formula to memorize.</div>
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Here is my foldable on the formula:</div>
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and on the page opposite we have some examples:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUqKInxAh1xFLQ9sJgGe_OrcDgoie0SSM8dWtOu_QJB2xmPzKl6CHDhzk6skf1uDvUZ8CQK_2Dcohs-DZ6G7QGZ-ggyhLFgV8nSjNvZEPwp1ovo5uOPN3aUdreST-ajU9o4j_l9qPV-iM/s1600/infinite+geom+examples.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="526" data-original-width="372" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUqKInxAh1xFLQ9sJgGe_OrcDgoie0SSM8dWtOu_QJB2xmPzKl6CHDhzk6skf1uDvUZ8CQK_2Dcohs-DZ6G7QGZ-ggyhLFgV8nSjNvZEPwp1ovo5uOPN3aUdreST-ajU9o4j_l9qPV-iM/s400/infinite+geom+examples.JPG" width="282" /></a></div>
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Now this school year (2016-2017) this was all I could do with Discrete Math. So the rest of the "overflow" block that I had to use the first bit for infinite series I gave the students a <a href="https://app.box.com/s/qtg4g5rmak5537sxcrzmlt1lzbbjx9xt">review sheet </a>to practice problems on sequences and series. Then we had a quiz the next day.</div>
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<a href="https://app.box.com/s/yf5euswa7rg6cuja71jfv3zeuq1vfrt6"><br /></a></div>
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<span style="background-color: white; color: black; font-family: "arial" , "helvetica" , sans-serif; font-size: 13.2px;"><b><a href="https://app.box.com/s/yf5euswa7rg6cuja71jfv3zeuq1vfrt6">Here is the folder on box.com that contains the ISN masters for both sequences & series.</a></b></span></div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com1tag:blogger.com,1999:blog-2897305678042698487.post-79825650519017203202017-06-12T15:06:00.002-04:002017-06-12T15:10:26.842-04:00PreCalculus Sequences 2017<div class="separator" style="clear: both; text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3KlysxJdirPTGsxyM1CEtYaRUXFco2TNFOqD8LXQUIRQbpgnlXv70UQyWNDfK34jgmJiv0Zopg2_FCX9G41JT5K3df03_V_9gquawQdIiqz8M3LFdJ6WXAye7plj9t8-IGVQuHHM5gn8/s1600/foxtrot+cartoon.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="201" data-original-width="613" height="130" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3KlysxJdirPTGsxyM1CEtYaRUXFco2TNFOqD8LXQUIRQbpgnlXv70UQyWNDfK34jgmJiv0Zopg2_FCX9G41JT5K3df03_V_9gquawQdIiqz8M3LFdJ6WXAye7plj9t8-IGVQuHHM5gn8/s400/foxtrot+cartoon.JPG" width="400" /></a></div>
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<span style="font-size: x-small;"><i>https://risesa.wordpress.com/tag/foxtrot/</i></span></div>
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We end the year in PreCalculus with two mini-units. This is the first lesson in the first mini-unit "Sequences & Series". is on sequences. Students learned about sequences in Algebra One (many courses ago) and learned the general formulas. We apply those formulas a bit more and then progress to series. When we have time we do a full unit on "Discrete Mathematics" which includes combinatorics (permutations & combinations), binomial theorem and probability. But this year we were running short of time. The Calculus teachers wanted us to be sure to get to series and summation notation - thus the inclusion of this mini unit!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheDemfUeMdMlgMrBQjbCpBEGS5A7H0nMkeehN7uva9cFv_06JhyphenhyphenvBLubp4eZ8VtUXtLgdJx25NT0oNWQyYpyvYXaZw3h0exoLgNIvke09lngo6n0aqR-yH7AhFScsArwD0Z_sZQ-27Crs/s1600/frayer+sequences.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" data-original-height="511" data-original-width="332" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheDemfUeMdMlgMrBQjbCpBEGS5A7H0nMkeehN7uva9cFv_06JhyphenhyphenvBLubp4eZ8VtUXtLgdJx25NT0oNWQyYpyvYXaZw3h0exoLgNIvke09lngo6n0aqR-yH7AhFScsArwD0Z_sZQ-27Crs/s200/frayer+sequences.JPG" width="128" /></a>Back to sequences.<br />
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I start the lesson at the end of the last unit test. As they finish the test, students take a <a href="https://app.box.com/s/isc88oopnc5l7vt4g906knymcnpgpvom">sequences review sheet</a> that reminds them of the preconcepts they had learned in Algebra One.<br />
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We start the actual lesson with a frayer diagram on sequences.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9OdVzgk4ZFrYKGmrN8fbDXpCoIzrEy9skU-XvAJq3IYZpHoUxNk39EE1YKtz92WgfZgdPljVXRFF_tA5uIZNKFljX_RU_7k_DYlDWUnFfaw3MWbbwqZz7jJE1oYwtXKEXk7mlmVqhdfI/s1600/fayer+2+sequences.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="272" data-original-width="410" height="263" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9OdVzgk4ZFrYKGmrN8fbDXpCoIzrEy9skU-XvAJq3IYZpHoUxNk39EE1YKtz92WgfZgdPljVXRFF_tA5uIZNKFljX_RU_7k_DYlDWUnFfaw3MWbbwqZz7jJE1oYwtXKEXk7mlmVqhdfI/s400/fayer+2+sequences.JPG" width="400" /></a></div>
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Then we defined each of the types of sequences with this foldable. Inside the foldable you will find the details.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHnXBX0finL3-Zgs5O3FqFvFdB4g6WySe6BrJXCya5Ge_IKHRaZ6ddAkB4Go_zGXXM2VpZwyZPFKOmUFlyKkkCHPrkwcja6Y9DUk3XY0NcuR7P8XRiGEurNEi73vo9ZAiDgBzv2jE30yg/s1600/types+of+seq.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="538" data-original-width="394" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHnXBX0finL3-Zgs5O3FqFvFdB4g6WySe6BrJXCya5Ge_IKHRaZ6ddAkB4Go_zGXXM2VpZwyZPFKOmUFlyKkkCHPrkwcja6Y9DUk3XY0NcuR7P8XRiGEurNEi73vo9ZAiDgBzv2jE30yg/s400/types+of+seq.JPG" width="292" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_0xIjVJu34QkmlZB4Ldwg3KCPKnH3DfPMhpNay3YrB3VIcVQnqHobzw_Eq2mrRo7qQRKsK3zcAyrzsHfLKjsA_dvkVWk68utHsXRzYu5vL4yMEE8EfArz_0pHQG6s4hu2-3wz2UsTovw/s1600/types+of+seq+inside.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="463" data-original-width="598" height="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_0xIjVJu34QkmlZB4Ldwg3KCPKnH3DfPMhpNay3YrB3VIcVQnqHobzw_Eq2mrRo7qQRKsK3zcAyrzsHfLKjsA_dvkVWk68utHsXRzYu5vL4yMEE8EfArz_0pHQG6s4hu2-3wz2UsTovw/s400/types+of+seq+inside.JPG" width="400" /></a></div>
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I did include a "neither" category because Fibonacci sequences are very cool and the sequence of perfect squares is pretty common yet is neither arithmetic or geometric.<br />
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Next we reviewed the arithmetic and geometric sequence formulas. I do like to spend a little time showing how each formula relates to example. I like them to see how the arithmetic involves the repeated addition of the common difference (thus appears as multiplication in the formula) and the geometric involves repeated multiplication of the common ratio (thus appears as a power in the formula). I use this foldable to summarize the formulas.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqRF0gCaHLd-yqOAYZ4RmIMEFYIM6CavUpPya30bz7LDXs0tjwnvycdvCODynGaTznwE-PH4hANxKOedVb-M44TzjVMOEg_kg99bwG1a5IjqqAklRbQB7NyrSmiWgzcWotWTWiSLIqoys/s1600/general+formulas.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="534" data-original-width="389" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqRF0gCaHLd-yqOAYZ4RmIMEFYIM6CavUpPya30bz7LDXs0tjwnvycdvCODynGaTznwE-PH4hANxKOedVb-M44TzjVMOEg_kg99bwG1a5IjqqAklRbQB7NyrSmiWgzcWotWTWiSLIqoys/s400/general+formulas.JPG" width="291" /></a></div>
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Inside there are examples.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8qR5Lanw2b2mpR7oI_PdW2JFWLC7ZBiGGKGwTUWTJOPVljEvFYivbcAIPo35d19B-QFIdVTyJEgyQrzYODcqKX1HEZuz0d5L-K-ZvyGmaRCL9yzrquz5a0t1KDN_cr4AcBPeK5n8DvDE/s1600/inside+general+formulas.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="464" data-original-width="603" height="307" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8qR5Lanw2b2mpR7oI_PdW2JFWLC7ZBiGGKGwTUWTJOPVljEvFYivbcAIPo35d19B-QFIdVTyJEgyQrzYODcqKX1HEZuz0d5L-K-ZvyGmaRCL9yzrquz5a0t1KDN_cr4AcBPeK5n8DvDE/s400/inside+general+formulas.JPG" width="400" /></a></div>
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I like my precalculus students to apply these formulas in constructing sequences also. These problems involve creating a system of equations using the general formulas. The geometric one is interesting because you solve that system using division, not a typical operation used in solving systems of equations.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_6gQU9IQqs0sd9Rt1IFOU3Wk0QVil83QEwHvhbIL1NKJZIS9FfB7JqroxAON4lc2mBYDvHdHh3bGOmB1pZtS57md6obxb0w5EMWVtcsdV9ypNmjAQMJrJSR4VhWaLAi1DLVi9Av6DuYI/s1600/constructing+sequences.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="516" data-original-width="376" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_6gQU9IQqs0sd9Rt1IFOU3Wk0QVil83QEwHvhbIL1NKJZIS9FfB7JqroxAON4lc2mBYDvHdHh3bGOmB1pZtS57md6obxb0w5EMWVtcsdV9ypNmjAQMJrJSR4VhWaLAi1DLVi9Av6DuYI/s400/constructing+sequences.JPG" width="291" /></a></div>
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Finally if I have time (we do have an 82 minute block) I like to do a quick comparison of explicit versus recursive rules. The focus in this course is the explicit rules but recursive rules are useful, especially in computer programming. These pages show my overview.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggK4gSFUoPJmFN9JygP1WzUJWidog0tUYn7mvo6XyoE72KY-FSB7WSkFiR8f6ZOc-wMYqYLJ2LFOEVB6CcfvsxpSVOQJKjlOtn3Jt3pzZkZldLHoxVTlMZa9tqxJi_hqLvFhhZYbv4y40/s1600/explicit+vs+recursive+p1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="555" data-original-width="405" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggK4gSFUoPJmFN9JygP1WzUJWidog0tUYn7mvo6XyoE72KY-FSB7WSkFiR8f6ZOc-wMYqYLJ2LFOEVB6CcfvsxpSVOQJKjlOtn3Jt3pzZkZldLHoxVTlMZa9tqxJi_hqLvFhhZYbv4y40/s400/explicit+vs+recursive+p1.JPG" width="291" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimCi_5kgYdSzqw4NFs-2eDfIvVQfjV_uKmsFSqzL6UTh7L6YIzf0Ih1gO1-eg6i3l_rZlnhq7KK2-HQB-wZq6qog03IlfzgbMwiqoEi6sDecfBjSSiLIQVveddMyy_5K8W7oItc-QwQVs/s1600/explicit+vs+recursive+p2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="522" data-original-width="395" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimCi_5kgYdSzqw4NFs-2eDfIvVQfjV_uKmsFSqzL6UTh7L6YIzf0Ih1gO1-eg6i3l_rZlnhq7KK2-HQB-wZq6qog03IlfzgbMwiqoEi6sDecfBjSSiLIQVveddMyy_5K8W7oItc-QwQVs/s400/explicit+vs+recursive+p2.JPG" width="302" /></a></div>
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I like to use our TI 84 graphing calculators to experience the recursive rules too. We go through an arithmetic sequence - first we type in the first term (let's use 8) and press enter. This will give you the first term and establishes the first term in your calculator. Then we type the rule (let's say d = 5, so they would type +5). That establishes the rule in your calculator. And it will give us the second term. With those two steps the recursive formula has been programmed into your calculator. Now you can use this recursive program to generate additional terms. Each time you press enter you will get successive terms starting with the third term.<br />
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For homework, I give my students examples in our textbook to complete.<br />
<a href="https://app.box.com/s/yf5euswa7rg6cuja71jfv3zeuq1vfrt6"><br /></a>
<a href="https://app.box.com/s/yf5euswa7rg6cuja71jfv3zeuq1vfrt6">Here is the folder on box.com that contains the ISN masters for both sequences & series.</a>mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-57054965822436473842016-03-08T08:00:00.000-05:002016-03-08T08:00:18.175-05:00WORD LADDERS - a cool way to shift thinking for verifying identities in precalc<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 11.5pt;">A word ladder is a
sequence of words formed by changing just one letter each time, it’s a fun word
puzzle invented by Lewis Carroll, the author of <u>Alice in Wonderland</u>. Doing
these puzzles in PreCalculus gives students an example of how something must be changed "one step at a time". In our next unit of
“AnalyticTrigonometry” we verify identities, where we change a trig identity one step at a time to
create a new identity. (word ladders we change words, one letter at a time.)<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 11.5pt;">For example: changing
BACK to FIRE could look like this. One letter was changed at a time, each time
a new word was formed.<o:p></o:p></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">I personally find these quite challenging! (surprisingly maybe, I am a big reader). I like the ones that have clues to help me out.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">But even with clues this one is difficult!</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_7NQrOYY2mmHvwpgjtV5mBDBd8fN_G_lRUUlODggbaeDyxpxZiLfwmUECB_zElUJqJB5h2jRdr_BoXSxhfv-S4rXBN1wC42_bzZ8QdrHfk0-fe5tuQp2sKxGk1qojkx1QtGS_DsR1PdY/s1600/WL4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_7NQrOYY2mmHvwpgjtV5mBDBd8fN_G_lRUUlODggbaeDyxpxZiLfwmUECB_zElUJqJB5h2jRdr_BoXSxhfv-S4rXBN1wC42_bzZ8QdrHfk0-fe5tuQp2sKxGk1qojkx1QtGS_DsR1PdY/s640/WL4.jpg" width="363" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">How do I incorporate these into my busy precalculus curriculum? Students get <a href="https://app.box.com/s/iahd2w5imwczcnlgz4a8v3itnnfab65w">this packet </a>after out unit test (today & tomorrow in my classes) and are asked to do at least two pages for homework. Some students get really hooked and complete the whole thing.</span></div>
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mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-79522187867010917362016-03-07T11:08:00.004-05:002016-03-07T11:10:28.664-05:00Solving Trigonometric Equations In PreCalculus part one<span style="font-family: "arial" , "helvetica" , sans-serif;">Well this year we have rearranged our Pre-Calculus curriculum (honors level) a bit since Algebra 2 is now covering a fair amount of our old topics. Our "new" unit 4 is limited to 4 topics: sinusoidal graphs, inverse trig functions, applications of sinusoidal graphs and the basics of solving trigonometric equations. Seems pretty short but we do have an every-other-day block schedule and graphing took a few classes. But I am confident my students are good at graphing all kinds of transformations on sine or cosine graph and comparing their graphs to the original parent functions, woo-hoo!</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">One thing they are not so good at is remembering their trig ratios of basic special and quadrantal angles. Well, they're pretty good at it. They've been under a little pressure this year because every day we've had a 5 question mini-quiz asking to evaluate some trig ratios. Questions read by me aloud to the class with 5-10 seconds for students to write down the response. Do they all get 5s every time (especially after doing this for over a month)? Sadly, no. But I keep telling them - you have to know these, they come up again & again in this course and future math courses. And lo & behold when we were evaluating inverse trig functions - there they were. Now when we solve trig equations, yup here they are again.</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Okay, enough rambling - on to solving trig equations.</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I like for students to have an understanding of what the heck it is they are learning and how it relates to everything else we've been doing. We just recently finished evaluating inverse trig functions but now we are going to solve trig equations that work with the full trig function. I do a lot of repeating myself in making it clear that we are visualizing all the solutions on the trig function graph, NOT only considering the little inverse function graph and its limited range of angles. So we start with some questions - what's the graph of the sine function? How would we find all instances where the sine ratio equals 0.2? </span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Our conversation was supported by this ISN foldable.</span><br />
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<i style="font-family: arial, helvetica, sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd7ga8javsxmNb_a9xHMEowH_svEKN6-PCXo2RFY8Q1RpWfPUNsEiZPeixjDgvvmJYzTYhJItkJadMkPgPlUflorfb72TYZSMvTrRht-1xIEpMVJti5QB5VqxyMSz6ITaT7t-k6MrD4NE/s1600/solve1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd7ga8javsxmNb_a9xHMEowH_svEKN6-PCXo2RFY8Q1RpWfPUNsEiZPeixjDgvvmJYzTYhJItkJadMkPgPlUflorfb72TYZSMvTrRht-1xIEpMVJti5QB5VqxyMSz6ITaT7t-k6MrD4NE/s640/solve1.JPG" width="425" /></a></i></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">We looked at solutions as defined in the standard period interval of [0. 2pi) and the general form that indicates ALL the solutions that infinitely appear on the sine curve as it oscillates off to both negative & positive infinity. That's a little tricky for students, in past years I get a lot of complaining about not being able to remember which way to write the solutions (a clear indication they have no idea what the +2kpi means and how the solution relates to the sine curve, sigh). If I had more time I think I would have them map out a longer version of a coordinate plane that has maybe 6 periods and find all the solutions for say sin(x) = 1/2 and then discuss a shorter way to illustrate those solutions in a general way. But we ran out of time this year (no excuses but I had to be out for my daughter's wisdom teeth removal for a few days). </span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Then we moved on to special angle solving (which means knowing those trig ratios). I used the inside of the ISN to go over some examples.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT4Cldh9lPZGwPt7-F3Nkul1717I0TAkef08fLbuxMuzgUwIy-XoCiMFQyYV70yiZbQ93vLerFLAlnI3E3IbrrN-rLv9WzsBMNoo7cYmPLHy235KyDfy7_S2ptXp-WkJQOS6UpaBwyAVY/s1600/solve+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT4Cldh9lPZGwPt7-F3Nkul1717I0TAkef08fLbuxMuzgUwIy-XoCiMFQyYV70yiZbQ93vLerFLAlnI3E3IbrrN-rLv9WzsBMNoo7cYmPLHy235KyDfy7_S2ptXp-WkJQOS6UpaBwyAVY/s400/solve+2.JPG" style="cursor: move;" width="262" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlWZBsBDpCUJjEECpUnrHc5NThTkAyrHPyvG5R4pGLuf_D9zP74Kbyng1a8VneBRh_O5H85hnKV2cwlUHGXwOZFzYTSq5KwOY7InCybEJItE0VHGeoZgmZ-nhDPwYSg6DAYTJWDhLIubI/s1600/solve3.JPG" imageanchor="1" style="font-family: 'Times New Roman'; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlWZBsBDpCUJjEECpUnrHc5NThTkAyrHPyvG5R4pGLuf_D9zP74Kbyng1a8VneBRh_O5H85hnKV2cwlUHGXwOZFzYTSq5KwOY7InCybEJItE0VHGeoZgmZ-nhDPwYSg6DAYTJWDhLIubI/s400/solve3.JPG" width="271" /></a></span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Then we finally looked at using some algebra skills to solve some trig equations that had a little more to them (no identities yet, that's our next unit). But before doing that I wrote some review equation solving examples on the board. Not to teach how to do these but just to refresh everyone's memory. </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2L9WFzPNHHULWNp2pNgCUK8_NuK93GAoIqrlOm8eiI0r07QHUBgX5YysbQJoMdFc3gugWiqQREN-Ozh5zLVd1Iz0NKCfcQ5cBiHricSYkJcP-3VCj1mbGQzxTK8UQFvqqtJqhkXbRH-w/s1600/solve8.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2L9WFzPNHHULWNp2pNgCUK8_NuK93GAoIqrlOm8eiI0r07QHUBgX5YysbQJoMdFc3gugWiqQREN-Ozh5zLVd1Iz0NKCfcQ5cBiHricSYkJcP-3VCj1mbGQzxTK8UQFvqqtJqhkXbRH-w/s320/solve8.JPG" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">And I always love to tell students how much I did not appreciate the zero product property when I was a student. </span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJXY5Ax_ZG21qJLf2anlKyESY-dhGJAjwj4R9v1d2-0cvyK3d_80AWv8mHQhmc1c4rb73OC3eEGL-5B4T-gPYdEuwHkDCFIy8z8GYWvUPN4IhmXghe_SOrXd58u8XnYnZsG9vwuedBXEk/s1600/zero+product+property.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="95" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJXY5Ax_ZG21qJLf2anlKyESY-dhGJAjwj4R9v1d2-0cvyK3d_80AWv8mHQhmc1c4rb73OC3eEGL-5B4T-gPYdEuwHkDCFIy8z8GYWvUPN4IhmXghe_SOrXd58u8XnYnZsG9vwuedBXEk/s320/zero+product+property.JPG" width="320" /></span></a><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">Always seemed like an obvious little statement but is really so cool and so powerful when it comes to equation solving. Students tend to forget about it when they do their little short cut of solving something like (x + 3)(x - 2) = 0 when they say that x = -3 or 2. They forget (or never really learned?) how they can get those solutions from those two factors. </span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I used this ISN to illustrate examples that use these algebra skills.</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPpBY9vl4ERhVoZy-kcZ8uHo6cC-tMjjmDg5jVF9J6hu8eV-lzafdHwKNzSgiguohv1NUqaQEmcmnfMtcIasgzZgF7pgoNNpiHrMY4efATF7ENvoJ04sKmD-HLrvzS0E6j7Kwyevmn1Zc/s1600/solve4.JPG" imageanchor="1" style="font-family: 'Times New Roman'; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPpBY9vl4ERhVoZy-kcZ8uHo6cC-tMjjmDg5jVF9J6hu8eV-lzafdHwKNzSgiguohv1NUqaQEmcmnfMtcIasgzZgF7pgoNNpiHrMY4efATF7ENvoJ04sKmD-HLrvzS0E6j7Kwyevmn1Zc/s640/solve4.JPG" width="451" /></a></span><br />
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<i>inside left:</i></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_OiYudDEMkfxfo9jW4w335n8HJtfWgHl8dta9cVokaJygwYdtERyCilm1byPKhyphenhyphennl_ZCgE3liXwMlvlUxsrfoyuEGTQ-JeDaenTONyWb2niQM3WkFAVBZ3ZqM_hb7YPCdEt3YE_cQmaU/s1600/solve5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="314" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_OiYudDEMkfxfo9jW4w335n8HJtfWgHl8dta9cVokaJygwYdtERyCilm1byPKhyphenhyphennl_ZCgE3liXwMlvlUxsrfoyuEGTQ-JeDaenTONyWb2niQM3WkFAVBZ3ZqM_hb7YPCdEt3YE_cQmaU/s320/solve5.JPG" style="cursor: move;" width="320" /></a></div>
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<i>inside right:</i></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5RhvTo0PBNsFWu5JuBHPh-cCbsCYVmXJdCJ65A9TvVuP-tk4Ef4EX5RG7ZtP6elajbc6pv5rNtH3k-90yllhT7qUtW70EPlpcmopuWDoo3GBnuEDAteGLCdPad8iYLT-7efT8CfZOrUU/s1600/solve+6.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5RhvTo0PBNsFWu5JuBHPh-cCbsCYVmXJdCJ65A9TvVuP-tk4Ef4EX5RG7ZtP6elajbc6pv5rNtH3k-90yllhT7qUtW70EPlpcmopuWDoo3GBnuEDAteGLCdPad8iYLT-7efT8CfZOrUU/s320/solve+6.JPG" width="318" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Our textbook does have some good homework problems that I assign, but they don't have any of the basic equations to solve. I created a worksheet for that with equation solving using the calculator (graph) instructions on the back.</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">All my files can be found<a href="https://app.box.com/s/fcucn9kbzuozifmq3a89dl5c032mwhdn"> HERE. </a></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6qZQKAbgTLOz5TZ5dY_2z2JIAXT1_izEjyA1TBoc0QYL85bid-hwU0lcgBrt71szDGJIqzQCC3x-cp41tAqIiWljZvwriE2gI250ujrSgWzR2L8w4eE-N8Md3Z1BIS4HOa4tmjMWUf7k/s1600/solve+trig+equations.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6qZQKAbgTLOz5TZ5dY_2z2JIAXT1_izEjyA1TBoc0QYL85bid-hwU0lcgBrt71szDGJIqzQCC3x-cp41tAqIiWljZvwriE2gI250ujrSgWzR2L8w4eE-N8Md3Z1BIS4HOa4tmjMWUf7k/s640/solve+trig+equations.JPG" width="640" /></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><br /></span><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;">see above for a cool basic example, but I also draw a unit circle to show that perspective too. </span><i><span style="font-family: "times" , "times new roman" , serif;"> http://www.slideshare.net/timschmitz/higher-maths-123-trigonometric-functions-358346</span></i></span><br />
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mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-37283125499502737302016-02-25T08:36:00.002-05:002016-02-25T08:36:31.263-05:00Applications of Sinusoidal Graphs<div class="separator" style="clear: both; text-align: center;">
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<span style="font-family: "arial" , "helvetica" , sans-serif;">My pre-calculus students are now very good at graphing sine and cosine with all sorts of transformations. We've completed our work on inverse functions. So now we are ready to dive into applications of sinusoidal functions.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">We start by revisiting the Ferris wheel. This is how I like to introduce sine and cosine graphs this unit (after spending time with the unit circle and rotations it is a great way to see how we get the sinusoidal graph from a circle, see <a href="http://beautifulhighschoolmath.blogspot.com/2016/02/introduction-to-sinusoidal-graphs.html">my blog post here for details</a>).</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">We begin with this scenario:</span></div>
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<!--[if !supportLists]-->1<span style="font-size: 7pt; font-stretch: normal;"> </span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 14.0pt; line-height: 107%;">A Ferris wheel has a diameter of 30 m, with
the center 18 m above the ground. It makes one complete rotation every 60
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<span style="font-family: "arial" , "sans-serif"; font-size: 14.0pt; line-height: 107%;">I </span><span style="font-family: "arial" , "helvetica" , sans-serif;">I have students draw a graph for this, showing one complete rotation. Then I pose the question, what would a sine equation for this scenario look like? How can we go about calculating that? What do we need for our equation?</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> We use the standard form of the sine equation and talk about what we can use for values for a, b, c, & d.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhidm5W09GsGlsBAMqzG61rVn1D85unoA6Hw8WQHRtc74fqSNIJWFpAzuvPhDjXBvgFczWHc7Y0-HwNa1gTvxYMvbzzeCYfrNXtZjJmtObbvtHMbmhDlTU16Or-rXy_DhyKYDKKzKn0HPk/s1600/sine+equation.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="45" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhidm5W09GsGlsBAMqzG61rVn1D85unoA6Hw8WQHRtc74fqSNIJWFpAzuvPhDjXBvgFczWHc7Y0-HwNa1gTvxYMvbzzeCYfrNXtZjJmtObbvtHMbmhDlTU16Or-rXy_DhyKYDKKzKn0HPk/s320/sine+equation.JPG" width="320" /></a></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> from there students identify the amplitude, the period and the midline. We use the lowest point ( ) to calculate our "c" value. And ta-da we have our sine equation.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> then we use this equation and what we know about the scenario to answer questions:</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> Here is the work used to calculate these:</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> Our next example looks at situations that take place over a year's time (thus fixing the period length to 365 days). We start with this group example:</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "calibri" , "sans-serif"; font-size: 14.0pt; line-height: 107%;"> Events
that are cyclic (or periodic), such as seasonal variations in temperature, can be
modeled with trigonometric functions.
Regina is the capital of Saskatchewan, a province in Canada. The average
temperature for Regina is hottest at 27<sup>o</sup> C on July 28, and coolest
at -16<sup>o</sup>C on January 10.</span></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">We jump right in to finding the equation for this situation. Again using the basic sine equation and deciding what we already know (period and using the dates and temperatures we can calculate the amplitude and midline). Again we use the minimum temperature to solve for "c". The result is not as perfect as our Ferris Wheel sine equation as the real life data is not as perfect as a perfectly circular Ferris wheel rotating at a constant speed. But it's very close. (you can see the problem with it by calculating the hottest temperature using the equation, it doesn't give you exactly 27 degrees). </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> I like to draw the graph for these yearly events after I have the equation. It's easier to do that than just working with two pieces of data. I can get some other points on the graph with my equation.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Then I have students answer a few questions using the equation.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">From here I now have students work on a packet of problems related to those two examples. They work together in small groups and do some problems for homework as well. This packet is used over a two day period. You will see those two class examples embedded in the worksheet (I put them there so there is a hard copy record of them, also my scanned answer key has all the details of "how to" for all the problems. This is helpful for absent students). </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">On the second day (or third depending on scheduling etc( we do a little bit of summarizing with some ISN inserts for our notebooks. I have a two page spread where we model two examples. One in which the given information is the description of a scenario:</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtZ1s4sq-kEJeVb1k-5Z5EIZ4ztOsEfQnWoH7-04y_pa-2kw4CdcRjMSwavxbKgT4jZRWIJiPGMhYeZ268JZdeny-Q9NIwdC3USTDzMgZ0wxXBhIQ5e43JQUyiJF67oBCo8BfCS3gul8E/s1600/apps+p1.JPG" imageanchor="1" style="font-family: 'Times New Roman'; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtZ1s4sq-kEJeVb1k-5Z5EIZ4ztOsEfQnWoH7-04y_pa-2kw4CdcRjMSwavxbKgT4jZRWIJiPGMhYeZ268JZdeny-Q9NIwdC3USTDzMgZ0wxXBhIQ5e43JQUyiJF67oBCo8BfCS3gul8E/s400/apps+p1.JPG" width="301" /></a></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">The second example uses a set of data. We "hand-calculate" the sine equation but then we compare it to the regression equation from the graphing calculator.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuYdQhnQgFTyHQ1MxPIWTb99o_r23nrxHsxWmaF7WR9UZcy3uYzo65qmMQ0W_rp0nwIDdn76zCgp6mbJFMlBX5Emg7sjtpWxLQ1RH0akxoUMgxgHWbp4qY57JfFvGb6bVgWBYHgmrbG1A/s1600/apps+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuYdQhnQgFTyHQ1MxPIWTb99o_r23nrxHsxWmaF7WR9UZcy3uYzo65qmMQ0W_rp0nwIDdn76zCgp6mbJFMlBX5Emg7sjtpWxLQ1RH0akxoUMgxgHWbp4qY57JfFvGb6bVgWBYHgmrbG1A/s400/apps+2.JPG" width="275" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">I have students attach the problem only on the left margin with tape (both front & back to keep it secure) so they can show the graph on the reverse side. (we don't tape it in until after they draw the graph). We show both our "hand-calculated" graph and the calculator regression graph. Very similar.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw6zbDgEvpAUOkkD7JSQLOTpfrioWd9LHWYZ0LBC2s0EPG6z0YFrb2FDa_8QLehIzbuvI8REuLIsd1Z6X-6KK_wsvBPKz_44wc531p2RImMlkFgXvHX2dXCIglyEvRDdbU20PATxTuzNM/s1600/apps+2B.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw6zbDgEvpAUOkkD7JSQLOTpfrioWd9LHWYZ0LBC2s0EPG6z0YFrb2FDa_8QLehIzbuvI8REuLIsd1Z6X-6KK_wsvBPKz_44wc531p2RImMlkFgXvHX2dXCIglyEvRDdbU20PATxTuzNM/s320/apps+2B.JPG" width="320" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">At some point I do discuss the difference between the form of the equation we use to create our equation</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhidm5W09GsGlsBAMqzG61rVn1D85unoA6Hw8WQHRtc74fqSNIJWFpAzuvPhDjXBvgFczWHc7Y0-HwNa1gTvxYMvbzzeCYfrNXtZjJmtObbvtHMbmhDlTU16Or-rXy_DhyKYDKKzKn0HPk/s1600/sine+equation.JPG" imageanchor="1" style="font-family: arial, helvetica, sans-serif; margin-left: 1em; margin-right: 1em;"><img border="0" height="45" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhidm5W09GsGlsBAMqzG61rVn1D85unoA6Hw8WQHRtc74fqSNIJWFpAzuvPhDjXBvgFczWHc7Y0-HwNa1gTvxYMvbzzeCYfrNXtZjJmtObbvtHMbmhDlTU16Or-rXy_DhyKYDKKzKn0HPk/s320/sine+equation.JPG" width="320" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Versus the equation the TI graphing calculator uses to create the regression equation: </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">y = a sin (bx+c)</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Two big differences. First the "b" value is not factored out. We always factor out our "b" value to isolate the true "c" value which illustrates the right/left shift in our graph. In the graphing calculator model this shift is obscured by the horizontal stretch/compress action. Also the calculator uses +c and we use -c to indicate the counter-intuitive nature of the right/left shift. </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">So what can happen (as seen in the work in the above example) the regression equation can be very similar to our equation in the amplitude and period and midline but be quite different in the "c" value. Even if you factor out the value of b. This comes from the fact that the calculator zeroed in on a different "standard period" than we did in our calculations. Basically my understanding is that a sine equation transformations indicate what transformations take place on the parent function of sine with the basic period of 0 to 2pi. It show where the "new standard period" shifts to. Our lines are very close, we just have different "new standard periods". Hope this makes sense!</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">And finally we wrap things up with a performance task that has students collect data on number of daylight hours at various latitudes. The project introduction reads:</span></div>
<blockquote class="tr_bq" style="line-height: 115%; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "arial" , sans-serif; font-size: 12pt; line-height: 115%;">In this project you will
develop equations and apply sinusoidal functions that model the number of hours
of daylight for locations in the world at different latitudes. To study the
relationship between latitude and the number of hours of daylight, data will be
collected and analyzed. Using the example of the number of daylight hours, you
will then investigate other sinusoidal phenomena.</span></blockquote>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">This will be the second year I am using this project. It is a bit time consuming so I do give them some time in class to get started. Then students need to manage their time well to complete the project. I warn them it's a bit involved so not something they should leave until the night before it's due to work on (as so many seem to do!).</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKPNWW0OGf9Tvgwg24mepRHo-8QAdAnlTFLuG2qlEPXQ_kgdv1cqahRLlgdFIDB2YCMKTOzyn6lVXoB26Py4FWuireGTqjgSpl9YmQUSkLGTRIwZ0HxGUdmumbSgzc7-XVGjlvgM3UneM/s1600/daylight+graph.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="223" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKPNWW0OGf9Tvgwg24mepRHo-8QAdAnlTFLuG2qlEPXQ_kgdv1cqahRLlgdFIDB2YCMKTOzyn6lVXoB26Py4FWuireGTqjgSpl9YmQUSkLGTRIwZ0HxGUdmumbSgzc7-XVGjlvgM3UneM/s400/daylight+graph.JPG" width="400" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">This <a href="https://app.box.com/s/4fwn5wx2r016a68ctotptp12p6endmu9">link</a> will bring you to all my graphing sinusoidal function files including applications. </span></div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-57803572884720114462016-02-17T15:01:00.001-05:002016-02-17T15:03:12.102-05:00Inverse Trig ISN and Evaluate<span style="font-family: "arial" , "helvetica" , sans-serif;">We set the stage with our <a href="http://beautifulhighschoolmath.blogspot.com/2016/02/introducing-inverse-trig-functions.html">"exploring inverse trig functions" </a>and now we move into how we use them in this course. We take a few minutes at the start of class to summarize what we learned with the explore by filling in a little ISN booklet of three pages (one each for sine, cosine & tangent):</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4nV_nO89cjv4JSJBjHiBoKCwxRSRfnSt6LCjrkQYcb_MsRDIvu-ePc7IijUpk9UfVCM4qK0IM9Fe9SrEU4GHupJ0Ss6Lm5hJENq29w7FOC23xH-f5y_ntQ_l_E9dWtYf9dFV4u-BKoSo/s1600/inverse+summary+sine.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4nV_nO89cjv4JSJBjHiBoKCwxRSRfnSt6LCjrkQYcb_MsRDIvu-ePc7IijUpk9UfVCM4qK0IM9Fe9SrEU4GHupJ0Ss6Lm5hJENq29w7FOC23xH-f5y_ntQ_l_E9dWtYf9dFV4u-BKoSo/s400/inverse+summary+sine.JPG" width="262" /></a> </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_Ow7swDBJ_GVzJbe4nxFOMd6zmKLR_okj00TCn2ng979FW6gMKEjAFI1HvFdZn-cIyKNdt48ouEGhlaiXLH686o7D3VPODpsf00-SY4vgbqqtBYTFF49vYSDGpxGeXhOnS8EQg5t9DDM/s1600/inverse+summary+cos.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_Ow7swDBJ_GVzJbe4nxFOMd6zmKLR_okj00TCn2ng979FW6gMKEjAFI1HvFdZn-cIyKNdt48ouEGhlaiXLH686o7D3VPODpsf00-SY4vgbqqtBYTFF49vYSDGpxGeXhOnS8EQg5t9DDM/s400/inverse+summary+cos.JPG" width="278" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOwZlFYnW7NTyaTPvhynJSJF7d2-iq_mqCulW34WOsl_XH4QmiB7bVazfsozcXqWmwOB8senpxuC9eVRgi1Ymq16UjUyctRm2gR2SBLJQX27FrPqobHq5d98E2aWOablFh7VltDwsg2Cg/s1600/inverse+summary+tan.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOwZlFYnW7NTyaTPvhynJSJF7d2-iq_mqCulW34WOsl_XH4QmiB7bVazfsozcXqWmwOB8senpxuC9eVRgi1Ymq16UjUyctRm2gR2SBLJQX27FrPqobHq5d98E2aWOablFh7VltDwsg2Cg/s400/inverse+summary+tan.JPG" width="255" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Recall that you can "undo" any trig operation with its inverse to find the angle (such as in solving right triangles or other triangle application problems). What I am asking students to do now is to find the range value for a given domain value of a trig inverse function. </span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Very subtle concept here. Students have to be very solid with their trig ratios to be successful here. Basically what we are doing is <span style="font-size: large;">finding the angle </span>when we are <span style="font-size: large;">given the ratio.</span> This can be very difficult for some students.</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">What I do is go over a few problems with them such as</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWRNbEz1ZMVYoDo9gW7DiXLoGqZQRk6Qq60i77xdWV8vXq6vjqHinuH8DOQ6-vcM4p8NHyVPJ4OwGqhr-o5tbB0Ra7eyaUBcTwTDvsLiultOe5QynmpmbQ2UxFSeYWrd8cGzqQ1avfBek/s1600/inverse+sine.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="96" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWRNbEz1ZMVYoDo9gW7DiXLoGqZQRk6Qq60i77xdWV8vXq6vjqHinuH8DOQ6-vcM4p8NHyVPJ4OwGqhr-o5tbB0Ra7eyaUBcTwTDvsLiultOe5QynmpmbQ2UxFSeYWrd8cGzqQ1avfBek/s200/inverse+sine.JPG" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIU8lET7cfccyzMFXlpnjWXGOODNP28kRRf1M0JGy6LuwpB2Sy6fAlt-SwYjeMYM8ZYyY88jHbLWXAi0Zht9Tkq5w92VJFdRzRUcumRa3fouJ1zcsuANxFuxvA4QN7YcoWYZqSKU5PAqo/s1600/inverse+cos.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="93" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIU8lET7cfccyzMFXlpnjWXGOODNP28kRRf1M0JGy6LuwpB2Sy6fAlt-SwYjeMYM8ZYyY88jHbLWXAi0Zht9Tkq5w92VJFdRzRUcumRa3fouJ1zcsuANxFuxvA4QN7YcoWYZqSKU5PAqo/s200/inverse+cos.JPG" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjv-XOmdMjWUCrdIVYNGI9uKZk7sjXzUmcUIouV9NohvMzO3ZDc1yldBoxkZjvWxswusZBydZZmTMyRY9v3j3pFCXnuuxTlpz0MIA0gqydqFU58UUMzQPkmmDloFzES3-Kay1Q45QmWKfk/s1600/inverse+tan.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="65" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjv-XOmdMjWUCrdIVYNGI9uKZk7sjXzUmcUIouV9NohvMzO3ZDc1yldBoxkZjvWxswusZBydZZmTMyRY9v3j3pFCXnuuxTlpz0MIA0gqydqFU58UUMzQPkmmDloFzES3-Kay1Q45QmWKfk/s200/inverse+tan.JPG" width="200" /></a></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">and refer to the unit circle visuals for the inverse trig function</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjj2Sv-DaHbHwAcdbxm2Rd-nYMsmBoE6EII_QX-Zb8Dk3A0OXbyxAlnLCo7jD9NO78hcaCs4Bt193YNPNUmNXlt6BsBw3D9_pQ3nOs89bJOee0b1QZduY7fbGzG06Tb0S9bi_ueeM6xKcc/s1600/inverse+visual2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjj2Sv-DaHbHwAcdbxm2Rd-nYMsmBoE6EII_QX-Zb8Dk3A0OXbyxAlnLCo7jD9NO78hcaCs4Bt193YNPNUmNXlt6BsBw3D9_pQ3nOs89bJOee0b1QZduY7fbGzG06Tb0S9bi_ueeM6xKcc/s320/inverse+visual2.JPG" width="244" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB1tA011sFf5AhvoicjcoxzANmZgGmpRjw0qotK4mdUO6M3aU16EDLvYuECszICs7ODsooS6510WNyppHh5V5i77h2vWoXjaN4acTQrFfWKQQZID24xhoh-9mJIhWgPCBjuZc9gOvZmCE/s1600/inverse+visual.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/AVvXsEjB1tA011sFf5AhvoicjcoxzANmZgGmpRjw0qotK4mdUO6M3aU16EDLvYuECszICs7ODsooS6510WNyppHh5V5i77h2vWoXjaN4acTQrFfWKQQZID24xhoh-9mJIhWgPCBjuZc9gOvZmCE/s320/inverse+visual.JPG" width="256" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUP0H1TunOUuQ6QsF-FC1msdjmN3CH0iZyRd_VPerI9AfvVlDJmE7C_AuB96_FA7iWhYsB9-zNafchskM6fJJajogW4npYJDFfPMXpGg_GnopS5VVL5mG35Q2uOwWN3aITOJ5ESP3XrnU/s1600/inverse+visual3.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/AVvXsEgUP0H1TunOUuQ6QsF-FC1msdjmN3CH0iZyRd_VPerI9AfvVlDJmE7C_AuB96_FA7iWhYsB9-zNafchskM6fJJajogW4npYJDFfPMXpGg_GnopS5VVL5mG35Q2uOwWN3aITOJ5ESP3XrnU/s320/inverse+visual3.JPG" width="227" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I remind them that the domain of the inverse (for sine & cosine) is very limited. We can only evaluate values between [-1, 1]. When the ratio is negative I ask them questions to help them remember that the original sine ratio is the y-coordinate in the function and where is the y-coordinate negative in the restricted unit circle visual?</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Basically students came up with these "thinking steps" for evaluating:</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">1) look at the ratio - what special or quadrantal angle results in this ratio?</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">2) now be careful - what is the sign of the ratio? If it's negative think about which inverse trig ratio we are using and where the negative ratios reside in the "unit circle connection" we drew above. Use that to determine the angle.</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I also ask,</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"> "which inverse trig ratios can have negative angles in their domain? which can only have positive angles in their domain?"</span></blockquote>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I like to keep them thinking with this one</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; margin-left: 1em; margin-right: 1em; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj111YTA7PhpVF56lr4uDnb1WbSs9vM6RCsEhKmx1uAubYdRdaLg4hDHfi0nLniYGCL0YqiH1Pydracs_EAnSTkeaXmmN7sPq2nuVFqr3vmZD7761udqW5uJIY4w7MaaMbMjYNIa-bEWqs/s1600/inverse+sine2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj111YTA7PhpVF56lr4uDnb1WbSs9vM6RCsEhKmx1uAubYdRdaLg4hDHfi0nLniYGCL0YqiH1Pydracs_EAnSTkeaXmmN7sPq2nuVFqr3vmZD7761udqW5uJIY4w7MaaMbMjYNIa-bEWqs/s1600/inverse+sine2.JPG" /></a></span><br />
(I ask them "what is the domain of arcsine?".....and what is the numerical value for pi?)<br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">After a few examples students practice "how to evaluate" inverse trig ratios with this foldable:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj41AxqF0-4S1VIyXTJ9rxr1AoGqExEva5zghIes5pvnrpXXkx7eMZXRRo5KoDRC8sa6z9zdX7WPxFFsh-rBcZyLB_n-lRC8rD2U6pCfGsOEMyV1U7dl_p2Ii3KsCdITcm5j-dYOFup4Sc/s1600/inverse+evaluations.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj41AxqF0-4S1VIyXTJ9rxr1AoGqExEva5zghIes5pvnrpXXkx7eMZXRRo5KoDRC8sa6z9zdX7WPxFFsh-rBcZyLB_n-lRC8rD2U6pCfGsOEMyV1U7dl_p2Ii3KsCdITcm5j-dYOFup4Sc/s640/inverse+evaluations.JPG" width="488" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">And then I work through a few "double evaluations" such as:</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; margin-left: 1em; margin-right: 1em; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1c52hDkm-TvsyRH3CLtulY1Ok_F89Rjl8BT9WrVjrNamRfklkThp2BjyUOhTqKGzhdSwMcfIOSNqPffR8sLuLtxMjk2V3BJj6gYnqP98v1GTjF11VJONBsvTK6gJGwJKvv3-s4DDFrQs/s1600/inverse+double1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="87" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1c52hDkm-TvsyRH3CLtulY1Ok_F89Rjl8BT9WrVjrNamRfklkThp2BjyUOhTqKGzhdSwMcfIOSNqPffR8sLuLtxMjk2V3BJj6gYnqP98v1GTjF11VJONBsvTK6gJGwJKvv3-s4DDFrQs/s200/inverse+double1.JPG" width="200" /></a> </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxn6u7eCO87BKmVB28TtuhKgO1Uh9-2ujL15IBtLdzdSQIyZsQh3vcu0MYQ02wXk6fIlZrgaDzfE5Cx8cbEzg3dSokO3R1mmeSjcJEULsVeXo_3nnnWTwtmddHgjaExmQTAGB8E6HwZMc/s1600/inverse+double2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="90" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxn6u7eCO87BKmVB28TtuhKgO1Uh9-2ujL15IBtLdzdSQIyZsQh3vcu0MYQ02wXk6fIlZrgaDzfE5Cx8cbEzg3dSokO3R1mmeSjcJEULsVeXo_3nnnWTwtmddHgjaExmQTAGB8E6HwZMc/s200/inverse+double2.JPG" width="200" /></span></a><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">You've got to be careful with the second one as the inverse doesn't always just undo the operation with the result being the given angle. </span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">And then to keep them on their toes I give them one of these. All kinds of confusion zings around the room as students try to remember what special angle has that trig ratio. When I confirm that there isn't one, I tell them that we really don't need to know the angle. We are ultimately just finding the trig ratio for cosine of some mystery angle. </span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; margin-left: 1em; margin-right: 1em; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn2Mpbq2MGJ4HhtaR-pHEil8QcuCXst8ElBI9jxrJZ5ZamBfGfjupOx1tWGuzIt2CW8XJkBFegT6JkmwluuR3QQwCRVm2sLjhOQ2ZEvEIBTkSNaWJ3e7dhKaW95k8mXMWEGimW9MJQqj0/s1600/inverse+double+weird.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn2Mpbq2MGJ4HhtaR-pHEil8QcuCXst8ElBI9jxrJZ5ZamBfGfjupOx1tWGuzIt2CW8XJkBFegT6JkmwluuR3QQwCRVm2sLjhOQ2ZEvEIBTkSNaWJ3e7dhKaW95k8mXMWEGimW9MJQqj0/s1600/inverse+double+weird.JPG" /></a></span><br />
with some prompting the hope is that someone will remember that the tangent ratio is y/x and the cosine ratio is x/r so all they need to do is figure out r with the Pythagorean relationship for x, y & r.<br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Finally students use this foldable on the page opposite to practice these evaluations.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX9Kxts3B3ejrJl6V5RLuc5I7nOvyDs_lPphpLApKwx6PBEsLMf4Vz2mG_ECgdyPHsRiwqoMPxxK_rq_3xOd1Bm1zfgAIN04w-gzCUM21nBXvxvGDudJlaZIp893VjOYNQBNrz5tDOtlY/s1600/inverse+double+evaluations.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX9Kxts3B3ejrJl6V5RLuc5I7nOvyDs_lPphpLApKwx6PBEsLMf4Vz2mG_ECgdyPHsRiwqoMPxxK_rq_3xOd1Bm1zfgAIN04w-gzCUM21nBXvxvGDudJlaZIp893VjOYNQBNrz5tDOtlY/s640/inverse+double+evaluations.JPG" width="427" /></a><br />
<i>this double evaluation goes on top of the evaluation page - I glue the evaluation page onto the notebook page and this page gets taped across the top creating a little flap over the evaluation page. </i><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I also have a worksheet of practice for students. Preview of first few problems:</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">All my files on Inverse Trig Functions can be found<a href="https://app.box.com/s/44d4s7b6xdszwslht366hsm0gxv0o98f"> HERE</a> (and this is everything - introduction sheets, evaluate sheets, ISN materials, answer keys)</span>mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-84873715202456630352016-02-11T06:00:00.000-05:002016-02-11T06:00:23.064-05:00Introducing Inverse Trig Functions<div class="separator" style="clear: both; text-align: center;">
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So I've been reading a bit about inverse trig functions on line and how people approach teaching them. It got me thinking about what my focus is with inverse trig functions and inverse functions in general in PreCalculus. My thinking is that students in Algebra 2 learn about inverse functions and the mechanics of inverse functions (how to create graphs, how to create inverse function equations, what they mean in terms of real life situations). PreCalculus is a time to dissect the idea of inverse functions more deeply in a bit of a more theoretical way. Are there limitations to the inverses of functions? Is every inverse of a function a function itself? We explored these questions back in the fall and it was hard for students. I have a good set of classes, honors level but they have been used to learning processes and procedures and not really developing a deeper understanding.</div>
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I like to start out these lessons on inverse trig functions by having a discussion about the difference between inverse operations and inverse functions. This sort of maps out our discussion (I lead the discussion with questions - what is an operation? what is the inverse of that operation? Let's consider an operation we all know pretty well - "squaring" - what does that look like as an operation? as a function? as an inverse operation? as an inverse operation?).</div>
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Then I steer the discussion to trig ratios as operations, as functions, etc. I try to do this by asking as many questions as possible.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipmSRxNg6rRQR9XKQ4MXq0dxtotP4_HbrRpvAuTyFeFmLmXf_xPXNx0gCmNCqVj4l4rqASqJs0SKSJ_nDuECD9IrJz8eOkX2v0R5dosARCwrnD3ttFGl9z9JxPzsww0RSGy_hasxXUQpM/s1600/sin+inverse.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="352" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipmSRxNg6rRQR9XKQ4MXq0dxtotP4_HbrRpvAuTyFeFmLmXf_xPXNx0gCmNCqVj4l4rqASqJs0SKSJ_nDuECD9IrJz8eOkX2v0R5dosARCwrnD3ttFGl9z9JxPzsww0RSGy_hasxXUQpM/s400/sin+inverse.JPG" width="400" /></a></div>
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Then after the discussion I distribute an "explore" packet to students. We do the first page together (it's just a review of inverses with quadratics and the domain & range restrictions). Then students work together in their small groups and using their graphing calculators work on visually representing and defining domains and ranges of inverse trig functions. I put a big focus on the visual (it's how I learn best and I find students often learn better that way too),</div>
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I thought I'd visually go through what we do & students do with the explore packet. You can open a blank electronic version of this worksheet at the end of the post.</div>
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We use TI-84 or TI-83 graphing calculators. We graph a basic quadratic and state the domain & range:</div>
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Then using the DRAW menu we can actually see the true inverse without restrictions of this function. We do that and draw it. Discussion ensues with "what's wrong with this picture?"</div>
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From there we remember that the inverse of the quadratic is the square root function. We draw that. We carefully examine the graph and the function equation to be clear about the domain and range. </div>
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A little "mantra' we chant is the the domain of the inverse comes from the range of the original. This is verified and we see how it restricts the range. All this is review of something we did back in the fall.<br />
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Now students get to work on their own doing a similar process to sine, cosine and tangent. These bits below show ideally what they should be coming up with. If there is time I have three small groups present what they come up with for each and explain. Then I lead the discussion into how these relate to the unit circle. That will continue in my next post (we pull it all together in some ISN inserts).<br />
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<b>SINE:</b></h2>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpiJfVMwZc2jazgAw6NWRPE4DN5htCeHmkbRKjCBUOTRHF8ZCjiPjxfVYaIdLCwacbnqtPu5rGGGzVGyrHzidQpCq6NtPSmoG6ds8ty71z3OMxfeBOPFoatOPeV2RjnPnzOMz3cnp6EeM/s1600/inversetrig4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpiJfVMwZc2jazgAw6NWRPE4DN5htCeHmkbRKjCBUOTRHF8ZCjiPjxfVYaIdLCwacbnqtPu5rGGGzVGyrHzidQpCq6NtPSmoG6ds8ty71z3OMxfeBOPFoatOPeV2RjnPnzOMz3cnp6EeM/s400/inversetrig4.JPG" width="343" /></a><br />
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<h2>
<b>COSINE:</b></h2>
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<h2>
<b>TANGENT:</b></h2>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiot3_S-kjQtX3wmjuQYHGlmpH2nVZcINXDmkcznZ7Osg0tyKncsQCUkHt9LTNCRGAdcvAaVs_hCbkEKZl9-Ixtcgy2CjAAFB6MWphyphenhyphenMARZ_NEa2feKMZU1elNg_YKA6yMGVYqzh1wjJuI/s1600/inversetrig9.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiot3_S-kjQtX3wmjuQYHGlmpH2nVZcINXDmkcznZ7Osg0tyKncsQCUkHt9LTNCRGAdcvAaVs_hCbkEKZl9-Ixtcgy2CjAAFB6MWphyphenhyphenMARZ_NEa2feKMZU1elNg_YKA6yMGVYqzh1wjJuI/s400/inversetrig9.JPG" width="322" /></a></div>
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Is this good math teaching? I hope so. I just remember when I first started teaching PreCalculus 10+ years ago. I was equipped with my previous learning as a precalculus student many many years previous and a bunch of textbooks. No real curriculum. When I got to inverse trig functions I had to ask myself - what are these really? What do I want to know about them? How can I make sense of them in relationship to everything else I know about mathematics? I have a hard time using problem sets in textbooks to drive my teaching. I like having a deeper understanding of how things work rather than just know how to solve given problems.</div>
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We do then go onto apply these ideas to problems. But I'd like to think of another way to assess understanding than just evaluate inverse trig functions. Maybe some writing prompt for the students to respond to.</div>
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All my files on Inverse Trig Functions can be found<a href="https://app.box.com/s/44d4s7b6xdszwslht366hsm0gxv0o98f"> HERE</a> (and this is everything - introduction sheets, evaluate sheets, ISN materials)</div>
<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com1tag:blogger.com,1999:blog-2897305678042698487.post-45572968390278726672016-02-08T12:25:00.003-05:002016-02-08T12:25:17.632-05:00Graphing Trig Functions part 2<div class="separator" style="clear: both; text-align: center;">
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Next we spend time playing around with graphs of sine and cosine. I work on developing some "number sense" and use a somewhat intuitive approach to graphing by combining what we know about the characteristics of the graphs.</div>
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We do a bunch of examples together - do these in our ISN:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCqsBcaUtJnl0W3XdI3Hmu8w-9-IfLvbHWk-Wl13_Q7Mn2E9Rp5w4U-vBWWbmsc13Z603TtS_WaUo-lFXgZsuZXX9zn5QYCF9FeG_vapRyQcvXDAAiXHVdHeKBDYoEyDD7hyphenhyphenWukYTQpdA/s1600/graph+examples.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCqsBcaUtJnl0W3XdI3Hmu8w-9-IfLvbHWk-Wl13_Q7Mn2E9Rp5w4U-vBWWbmsc13Z603TtS_WaUo-lFXgZsuZXX9zn5QYCF9FeG_vapRyQcvXDAAiXHVdHeKBDYoEyDD7hyphenhyphenWukYTQpdA/s640/graph+examples.png" width="440" /></a></div>
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We do a lot of graphing with all sorts of variations on transformations. Before graphing we list the transformations and characteristics and use them to plan our graph space. I use the midline, amplitude and the period start & end to determine the boundaries of the "single period" of the new transformed function. I like to show the parent function graph on the same graph too so we can see how it changed. We show one period but I continually remind them that this is only one of an infinite number of periods and we are just showing where the initial period from the parent [0, 2pi) ends up after the transformations.</div>
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Years ago I learned a cool technique of finding the start & end of a period when there is both a right/left shift AND some type of horizontal stretch/compression. We set up a compound inequality with the standard boundaries of 0 & 2pi. And we put the "argument" in the compound inequality. Then we solve the inequality. Which applies the two transformations to the boundaries of the standard period showing us where the new transformed period starts & ends. See my examples below. </div>
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Here are two examples we put in our ISN. My files below have many other examples.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_RyOwiNpeOpXHrlb7uhp1Q_-zTv4q9cUPUdTZ9OydKNAGuR3pfckbhcaICaM3EtmpLNUuR3GHSs5Rvka6SSPhBUbglUg0h9beZMiodDn-iEXM3cOdzzv2JXEpWsEeYlTwA0LGrdRUNFo/s1600/graph+ex2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="492" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_RyOwiNpeOpXHrlb7uhp1Q_-zTv4q9cUPUdTZ9OydKNAGuR3pfckbhcaICaM3EtmpLNUuR3GHSs5Rvka6SSPhBUbglUg0h9beZMiodDn-iEXM3cOdzzv2JXEpWsEeYlTwA0LGrdRUNFo/s640/graph+ex2.JPG" width="640" /></a></div>
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<span style="text-align: left;">All my graphing trig functions files are </span><a href="https://app.box.com/s/4fwn5wx2r016a68ctotptp12p6endmu9" style="text-align: left;">HERE</a><span style="text-align: left;"> (everything - a bunch of lesson materials, ISN materials, etc). </span></div>
<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-6403620125036195302016-02-07T11:11:00.002-05:002016-02-07T11:11:28.093-05:00Graphing Trig Functions part 1<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhORZa5kzfvVcrE46351BiO6NLzfz1gImEE3sIp3Ff_BLNCo5QS7Fy6NTFWAiaIrpUCldC4Gl0H8A4e1miehyIDu06oQRetnrj4kWUgEtsPpRj-Lx5pOcuo2ojYyXhGjs3vQaCwmLHi0s4/s1600/sine+and+cosine.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhORZa5kzfvVcrE46351BiO6NLzfz1gImEE3sIp3Ff_BLNCo5QS7Fy6NTFWAiaIrpUCldC4Gl0H8A4e1miehyIDu06oQRetnrj4kWUgEtsPpRj-Lx5pOcuo2ojYyXhGjs3vQaCwmLHi0s4/s640/sine+and+cosine.JPG" width="640" /></a></div>
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The stage has been set! We've learned about angles as rotations, we've visualized trig ratios of those angles, we are thinking in radians and we've made a connection between the circular (Ferris wheels) and the sinusoidal shaped graphs. Time to get serious and do Sine, Cosine, and (for some variety) Tangent.<br />
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I start by drawing a circle on the board with students fill in for me the angles and the ordered pairs along each axes.<br />
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Then I set up a set of axes to go with this unit circle. The x-axis is the angle and the y-axis will be our trig values. I'm sure to use the language of "independent variable" and "dependent variable" to help them make the connection with function work they did back in Algebra One. I also ask them what the maximum sine value is and the minimum sine value (and thus the range).</div>
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From here I give each student group a whiteboard marker and one student from each group goes up one at a time and takes a point from the unit circle and marks its location on the coordinate plane. Starting with (0, 0) and rotating around to (2pi, 0). Then someone uses a marker to connect all the data points to make a nice smooth sinusoidal curve.</div>
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We do this again for cosine. Then we do something similar for tangent with very different results. With tangent we get to talk about what "undefined" looks like on a graph and how can we get more points (pi/4 is a good reference angle to use with tangent). I wish I had taken pictures of the students' work on my whiteboard. </div>
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It was great to get them all out of their seats and being involved in the construction of the sinusoidal curves. They knew sine and cosine had these curved graphs but with this they could see how they were constructed. <u>*side note</u> I have in the past tried more creative and involved ways to make the connection between the unit circle and the sinusoidal curve from use uncooked spaghetti to a simulation on the graphing calculator. But this simple "student out of their seats, everyone contribute" basic drawing approach is so nice and clear and concise. Not fancy but really does the trick!</div>
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After all this drawing and discussing students fill in a foldable for their ISN.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjf49dtsLQvo3xViJJxLCIeDKN7lcbNPZhXLB64VMj-I9LXvG0b3dzqeqtAfFZCMtTb2Juy0krqP-d2piE5i-lCD8NEDh3Otw9drahmRK_hTvdXLdQDPmg5HUxObTHo-5r5MDi-EAqOwsc/s1600/graphing+sine+cosin+etc.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjf49dtsLQvo3xViJJxLCIeDKN7lcbNPZhXLB64VMj-I9LXvG0b3dzqeqtAfFZCMtTb2Juy0krqP-d2piE5i-lCD8NEDh3Otw9drahmRK_hTvdXLdQDPmg5HUxObTHo-5r5MDi-EAqOwsc/s640/graphing+sine+cosin+etc.JPG" width="494" /></a></div>
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This is my fancy space saving four square fold. You can see above there are two on this page. The graphing one opens up showing two squares to this:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbTlV3kBK00sOrpPk-zbGyCBtqN98hNPqfcdmXxgXeRdc_KoeTBNpRaZ4KpqbulAy28bL-jTzSHoyPb6tVYiQXKXJhKMoxGKYIfoupcHlJoTHxhQXks6ihAJNeuN2wssS8mqueSGwQUXg/s1600/first+fold+graph.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="268" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbTlV3kBK00sOrpPk-zbGyCBtqN98hNPqfcdmXxgXeRdc_KoeTBNpRaZ4KpqbulAy28bL-jTzSHoyPb6tVYiQXKXJhKMoxGKYIfoupcHlJoTHxhQXks6ihAJNeuN2wssS8mqueSGwQUXg/s640/first+fold+graph.JPG" width="640" /></a></div>
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and then opens up to four squares to show this:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiKB4njj1UDsz2LDzcIy_pcKJrKoKF7QD1BU2ghY_ytFvbEa5MwKVECP-PCBzefCGyblcV7VrSgXmc4Nv9Sxov1JGdz7XBu3CbMIRT21Q-wiZoPSbQJfzr0urmW60G34XMBf0OCQT76H8/s1600/full+inside+graphing.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="504" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiKB4njj1UDsz2LDzcIy_pcKJrKoKF7QD1BU2ghY_ytFvbEa5MwKVECP-PCBzefCGyblcV7VrSgXmc4Nv9Sxov1JGdz7XBu3CbMIRT21Q-wiZoPSbQJfzr0urmW60G34XMBf0OCQT76H8/s640/full+inside+graphing.JPG" width="640" /></a></div>
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Students use our classroom experience to fill in these foldables in their small groups as I walk around and prod them along but mostly listen to really cool conversations about trig and graphing. </div>
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Now since I have an 82 minute block I use about 15 minutes at the end to quickly review transformations and relate them a bit to these functions. I do this with a foldable that we also put on the graphing page. Another "modified" 4 square fold. Transformations should be review but they have some trouble with the horizontal stretch and compress action especially as it interferes with seeing the true nature of the right/left shifts if they occur in the same function.</div>
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the transformation unfolds to this "two square":</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihiS44_W2rxsfg6gpg1lDNKRSww1mFSpTuXijKP0VgJ8AitnGyMhfmk5E_KHvZwYsp0u3sDyfwCqZXZZe4Ean3TZ_JJrBKrcm8VqJgJsTYP9aElhSC27jbtEBHfGU20oJKJrfXRJ-ax7w/s1600/inside+transform.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihiS44_W2rxsfg6gpg1lDNKRSww1mFSpTuXijKP0VgJ8AitnGyMhfmk5E_KHvZwYsp0u3sDyfwCqZXZZe4Ean3TZ_JJrBKrcm8VqJgJsTYP9aElhSC27jbtEBHfGU20oJKJrfXRJ-ax7w/s640/inside+transform.JPG" width="640" /></a></div>
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Which actually have four little vertical doors that we cut. And under each door there is information on what a, b, c & d represent. We don't have much time to talk about these and mostly students copy the information on the foldable as I display it with the document camera.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe8j5CA0Y6estrr19aGpovCc_tmqE9rsTLt7evqnFFqjud-Qu3Lpra_UN2962Jj3sggblNud7Y-PdTwGvrtFNciWaOfM-v0olkaMzurIhEd_xVPNLlpV3B6v3lc2oEXpJHjUskOXzxFGM/s1600/inside+tranform+flaps.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe8j5CA0Y6estrr19aGpovCc_tmqE9rsTLt7evqnFFqjud-Qu3Lpra_UN2962Jj3sggblNud7Y-PdTwGvrtFNciWaOfM-v0olkaMzurIhEd_xVPNLlpV3B6v3lc2oEXpJHjUskOXzxFGM/s640/inside+tranform+flaps.JPG" width="640" /></a></div>
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My textbook has some good basic problems that dabble with characteristics and transformations so I give them a bunch of those to work through.</div>
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<span style="text-align: left;">All my graphing trig functions files are </span><a href="https://app.box.com/s/4fwn5wx2r016a68ctotptp12p6endmu9" style="text-align: left;">HERE</a><span style="text-align: left;"> (everything - a bunch of lesson materials, ISN materials, etc). </span></div>
<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-74486598915971205602016-02-06T15:18:00.001-05:002016-02-06T15:18:14.176-05:00Introduction to Sinusoidal Graphs<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg62punKqJG8mZlkDiimFJxgw_UVwhj0jFSlhXTY6MkWCXJB1q6idHTSkF-ieQ6xVsCLyvoRGw-kEWhnDR8j48VT2rl8c1J9ezrarlTTiIE31BVQRoWcswV_2dZwOUtefFxmL23unhABgU/s1600/intro.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="428" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg62punKqJG8mZlkDiimFJxgw_UVwhj0jFSlhXTY6MkWCXJB1q6idHTSkF-ieQ6xVsCLyvoRGw-kEWhnDR8j48VT2rl8c1J9ezrarlTTiIE31BVQRoWcswV_2dZwOUtefFxmL23unhABgU/s640/intro.JPG" width="640" /></a></div>
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Ever since I started teaching PreCalculus 10+ years ago I have used Ferris wheels to introduce sinusoidal functions. Well not real Ferris wheels, although that would be really cool. But the idea of them. And we do have a little discussion about Ferris wheels in general and it's interesting to see that there are a handful of students who just don't like them. huh.</div>
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Anyhow - way back when I started teaching PreCalculus I found in this book <u><a href="http://www.amazon.com/Functions-Modeling-Change-Preparation-Calculus/dp/0470039191">Functions Modeling Change</a></u> (but an earlier edition) the Ferris Wheel approach. Very cool. </div>
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So I start off the discussion with a few PowerPoint slides. The one you see above is projected as students come in. Then we use ...</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIFiv6GUdNCO2P-D83evkli9zw4BSB4A-ZI5haR_ynIjhpl8FUNGmVpkjEdRQA7Fx4-_ilIxqxFGYVA0-zx0ZJdAlnrER_fkXvG_SP_qTKHifio2RhPVL1a6O8c2i_VG4_EF4jmaMEa5k/s1600/london+eye.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIFiv6GUdNCO2P-D83evkli9zw4BSB4A-ZI5haR_ynIjhpl8FUNGmVpkjEdRQA7Fx4-_ilIxqxFGYVA0-zx0ZJdAlnrER_fkXvG_SP_qTKHifio2RhPVL1a6O8c2i_VG4_EF4jmaMEa5k/s400/london+eye.JPG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnA0j32bFAYbUYIFpwtMLGr2RgUH3c-Z16fUSpq-ECJapozjlHXzNTzdkrew5oH2GOTX6lYEY06FsJrUd2t5QxC0xo6eqmQieV-tFEZ4sfnK2ln2ZcrNJOJqOKzNawkbuu5T02kJAf4Tk/s1600/more+london+eye.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="301" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnA0j32bFAYbUYIFpwtMLGr2RgUH3c-Z16fUSpq-ECJapozjlHXzNTzdkrew5oH2GOTX6lYEY06FsJrUd2t5QxC0xo6eqmQieV-tFEZ4sfnK2ln2ZcrNJOJqOKzNawkbuu5T02kJAf4Tk/s400/more+london+eye.JPG" width="400" /></a></div>
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I actually went on the London Eye about 5 years ago. Very cool. And sometimes I have a student in my class who has been too.</div>
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Then I tell the class - okay make a graph of the height of a person on the Ferris wheel with respect to the time (in this case minutes). They work in their small groups and come up with nice sinusoidal graph. Wish I had taken pictures from my whiteboards of what they came up with. </div>
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From here we have a discussion of periodic and sinusoidal graphs. We define those terms. we talk about characteristics and students work with more Ferris wheel scenarios. We look at some variations in the location and the height of the loading platform. Of course times and diameters change too. It's a great way to get students to think sinusoidally, relating to something they have experience with.</div>
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We use this foldable in our ISNs for periodic functions:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHGGqQcZubj8zem_oCVjonmZqlERTBtjDFwp4F5tMqAnImqQ_ciFDiABdIgYTisq_6SK3gVIhvmCL_38Fefg_r_XsLJxIViTjti_hEvAWgOyThgyRdbXrJNxZSIwE7eT5Pa9Kh6IT0jDA/s1600/periodic+functions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHGGqQcZubj8zem_oCVjonmZqlERTBtjDFwp4F5tMqAnImqQ_ciFDiABdIgYTisq_6SK3gVIhvmCL_38Fefg_r_XsLJxIViTjti_hEvAWgOyThgyRdbXrJNxZSIwE7eT5Pa9Kh6IT0jDA/s640/periodic+functions.png" width="465" /></a></div>
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The four doors on the bottom two-thirds of the foldable open up:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpO4I3Av9ghh9u1whEm-1LO3yw9n4vKCEpjBAK98uZwZYyyzzavtl731wWBfhfR-jzjUKRe5hhz4qL-QsDYIXlqwcSAxiwZpupM9V43_tvfbkbI7yVyUT-LybDzf1Bd3d7MKhioOBcGC8/s1600/inside+periodic.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="348" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpO4I3Av9ghh9u1whEm-1LO3yw9n4vKCEpjBAK98uZwZYyyzzavtl731wWBfhfR-jzjUKRe5hhz4qL-QsDYIXlqwcSAxiwZpupM9V43_tvfbkbI7yVyUT-LybDzf1Bd3d7MKhioOBcGC8/s640/inside+periodic.JPG" width="640" /></a></div>
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After students do a bunch together (we have 82 minute blocks) we summarize with this example on an ISN foldable. I have students do the front and the second inside page.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRx2MXsTZ_9j91xdz4geT-XqVrK2tUKWYVVjs_two_xr6M-mnsAMx3qBISlFUlaF40vu4WIVHapK_e3O0rcQbgy2hL6YDpIMFzy94w2xAwxQr03ZbaOj9ev14NFoKWVt9s-ZaKFQBy5ME/s1600/ferris+wheel.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRx2MXsTZ_9j91xdz4geT-XqVrK2tUKWYVVjs_two_xr6M-mnsAMx3qBISlFUlaF40vu4WIVHapK_e3O0rcQbgy2hL6YDpIMFzy94w2xAwxQr03ZbaOj9ev14NFoKWVt9s-ZaKFQBy5ME/s640/ferris+wheel.png" width="422" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMr6kRercWxIjg9zx4ffRYe5MfokzwqjGmaltxKHcJcHmhfx9Pzubx-rDkF9iBCYtzmwuu1g5sde4QRE-wROwURC7Lt-652Up_QjLkZYThfBCLUZ3DHOz1faHTU4dK-wIlR0HZwggFOHA/s1600/inside+ferris2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMr6kRercWxIjg9zx4ffRYe5MfokzwqjGmaltxKHcJcHmhfx9Pzubx-rDkF9iBCYtzmwuu1g5sde4QRE-wROwURC7Lt-652Up_QjLkZYThfBCLUZ3DHOz1faHTU4dK-wIlR0HZwggFOHA/s640/inside+ferris2.JPG" width="532" /></a></div>
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The next class we focus on finding the height at certain times. This can be very confusing and difficult for students but I like how it pulls in the circle and some trig (we stay in radians). We usually end up doing a whole block of problems I make up for various Ferris wheels.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhc9DmsjY9LifNtKqtLN5Vy2IMIc_dnmzGsgk9iM1frPU7lobDkXOhAQRN_ff7HANB9bbVEFd6L_vWWcMUowySeIZLmGd9uzaIHRsp9mznxWfOJKBySjhhpU2owcQjM4iS2TrRpxVHFTxg/s1600/inside+ferris1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhc9DmsjY9LifNtKqtLN5Vy2IMIc_dnmzGsgk9iM1frPU7lobDkXOhAQRN_ff7HANB9bbVEFd6L_vWWcMUowySeIZLmGd9uzaIHRsp9mznxWfOJKBySjhhpU2owcQjM4iS2TrRpxVHFTxg/s640/inside+ferris1.JPG" width="446" /></a></div>
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All this preliminary work gets students ready to work with the more abstract and general sine and cosine graphs.</div>
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All my graphing trig functions files are <a href="https://app.box.com/s/4fwn5wx2r016a68ctotptp12p6endmu9">HERE</a> (everything - a bunch of lesson materials, ISN materials, etc). </div>
<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-27781083663256667712016-02-03T11:02:00.002-05:002016-02-03T11:02:18.052-05:00Mixture Problems in Algebra One<div class="separator" style="clear: both; text-align: center;">
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This intiative by #MTBoS has been great - I've been better about blogging and visually documenting what I do in class. And it's been great fun to read other blog posts on the weekly initiatives. This latest directive is to blog about a lesson. This is mostly what my blog is about - blogging on lessons I do and the interactive student notebook materials I use/create. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd7x1j_R0754MZpPDRSXsJfP_8Yq9sM5h0eAxjuYTkkelrDRI1ph1oHjza3BzFRL9k_icDNIstfJZMMuFw3Yp9GutTJkycpE82dDFIv_krAw-hDf0Xfa75T3Sa2-MsF1IfAu6tttWilgQ/s1600/mylesson.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="221" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd7x1j_R0754MZpPDRSXsJfP_8Yq9sM5h0eAxjuYTkkelrDRI1ph1oHjza3BzFRL9k_icDNIstfJZMMuFw3Yp9GutTJkycpE82dDFIv_krAw-hDf0Xfa75T3Sa2-MsF1IfAu6tttWilgQ/s400/mylesson.jpg" width="400" /></a></div>
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Teaching mixture problems using systems of equations can be challenging because it can be difficulty to visualize what they are about. I'm all about visualizing and understanding concepts (for myself and for my students). This lesson sets the stage for building understanding about the concepts underlying mixture problems.</div>
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The class is Algebra One, honors level, ninth grade. We've been working with systems of equations for about a week and a half now. Students learned about systems last year in 8th grade, mostly focusing on solving systems using graphs. I started this unit by focusing on creating models for situations with two conditions and two unknowns (<a href="http://beautifulhighschoolmath.blogspot.com/2016/01/start-lesson-with-question.html">see this blog post</a>). We then learned how to solve the algebraic models without graphing and I incorporated applications problems as I went. Mixture problems are their own little category of applications though, so I introduce those separately.</div>
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This year I decided to introduce mixtures visually with something all students love - cereal! Last year my student teacher and I talked about this idea and she implemented it in her classes (thank you Tess!) so I took her good work and tweaked it a bit for my class this year. </div>
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I set up a little "Research & Development Lab" in the back of my classroom:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBj53Icejm640UGZrrv7fMh32eF1Gl9vLUJ0nnVNnAi424ngZ1XN_81Zh2kcQTGMtgrpuMWUqFKAh6p_or9FOqLqqxyUYRd6KeoPl-boZ9KtjMy9H1HO8JKXwRlj2YGmaikXtdP_6I00s/s1600/pic6.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBj53Icejm640UGZrrv7fMh32eF1Gl9vLUJ0nnVNnAi424ngZ1XN_81Zh2kcQTGMtgrpuMWUqFKAh6p_or9FOqLqqxyUYRd6KeoPl-boZ9KtjMy9H1HO8JKXwRlj2YGmaikXtdP_6I00s/s640/pic6.JPG" width="408" /></a></div>
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Those ziploc bags each hold an ounce of the labeled cereal (about a heaping cup of each).</div>
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And provided some facts that our research team is using:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhHygo9eyqQbqPoh_CTYXCsVD3qWHilDkIuaXK9toRcitQXpN_ZKjghyphenhyphenI80pQ0a5BRM7i1Z83-OSSrEnHBQCd5Y-VtPxZr2JnklVnnbhfArHUV-hucOg0px1wXAe8RVoB39OtSAjrScts/s1600/pic5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="336" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhHygo9eyqQbqPoh_CTYXCsVD3qWHilDkIuaXK9toRcitQXpN_ZKjghyphenhyphenI80pQ0a5BRM7i1Z83-OSSrEnHBQCd5Y-VtPxZr2JnklVnnbhfArHUV-hucOg0px1wXAe8RVoB39OtSAjrScts/s640/pic5.JPG" width="640" /></a></div>
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The question was how many ounces of each type (Kix & Cocoa Puffs) would have to be in this blend to achieve the size and cost we are aiming for? Students saw that this was a situation with two unknowns and two conditions. So it was time to organize this information into a system of equations (all this comes from student responses):<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6X2qGWPSta2TD8641SGmhS_9UZ3iDJEbe09VsKehpoSZ3JlHZR5NCI13j_O7qiuxjWs_UFLLhEZFyXSLnTyAGhlBwNMQzFcrq0yINyT-KkySJ0T7a_04ob2ihAadWFofouR_Gj2MnMhs/s1600/pic2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="173" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6X2qGWPSta2TD8641SGmhS_9UZ3iDJEbe09VsKehpoSZ3JlHZR5NCI13j_O7qiuxjWs_UFLLhEZFyXSLnTyAGhlBwNMQzFcrq0yINyT-KkySJ0T7a_04ob2ihAadWFofouR_Gj2MnMhs/s400/pic2.JPG" width="400" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhE_8tb23gBJhjVFEY7Jwr9RVVSrRcPTIE7CjmUklRaEr_Vxt_9keun6pKYtDxrfkSiupc-iGfkuVJKwvETlpaKucR2-bdv2T0SASPT7QJMsCucvFkPGg_Jw-6mOnJUf1xnoQlR69OALlg/s1600/pic4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="193" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhE_8tb23gBJhjVFEY7Jwr9RVVSrRcPTIE7CjmUklRaEr_Vxt_9keun6pKYtDxrfkSiupc-iGfkuVJKwvETlpaKucR2-bdv2T0SASPT7QJMsCucvFkPGg_Jw-6mOnJUf1xnoQlR69OALlg/s400/pic4.JPG" width="400" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZ4Zrx752aM-MB72OSPLPbbpmQGOMbuQNs7ZFzAhbUU4lF-mVjakhkZVrN_ziO-qwY5TORy3OsAWZWQDkR-Q35oSSc7lWB-5lNZchtI496gNyMwE00jlNYrFXxOwKvJms_RcDf-wZUuf0/s1600/pic3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="316" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZ4Zrx752aM-MB72OSPLPbbpmQGOMbuQNs7ZFzAhbUU4lF-mVjakhkZVrN_ziO-qwY5TORy3OsAWZWQDkR-Q35oSSc7lWB-5lNZchtI496gNyMwE00jlNYrFXxOwKvJms_RcDf-wZUuf0/s400/pic3.JPG" width="400" /></a><br />
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The tricky bit of course is in setting up the equation that represents the cost. Students have to remember that the sum represents the cost of each individual type of cereal (cost per ounce times number of ounces) so that must be equal to the cost of the new cocoa kix cereal box (cost per ounce times the total number of ounces in our prototype box).<br />
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Once we got this all set up students solve the system. Now the fun part - we actually measured out a "box" of this special blend together. And then students got to sample the cereal (yum). I gave everyone a small cup full of the cereal to munch on.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghUj_7dGPVeFLApbaPMHeFacYdceADBcLYAxIXBx3faN2SkAowPhEZujDkFajz4tzZpSURnfCqj_hkvlo_M3UKWxtA5p_RwH6UQlOdwxCumClCMHXCxqhtd3eUHjJvjwne9VnGb4iTBu0/s1600/pic1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghUj_7dGPVeFLApbaPMHeFacYdceADBcLYAxIXBx3faN2SkAowPhEZujDkFajz4tzZpSURnfCqj_hkvlo_M3UKWxtA5p_RwH6UQlOdwxCumClCMHXCxqhtd3eUHjJvjwne9VnGb4iTBu0/s400/pic1.JPG" width="400" /></a><br />
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So how did this work? Well it was certainly fun and I have a bright bunch of kids this year so they were able to create the system pretty easily. I handed out the cereal samples for students to munch on while we did some problems together but there were only 10 minutes left to class and I think everyone was too excited snacking to really think about any more problems. I started to work on an ISN foldable looking at different types of mixture problems but gave that up pretty quickly. Instead I gave them two problems to try for homework and figured the next day we would pull things together.<br />
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DAY #2 on Mixtures<br />
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Before I went over homework and got any feedback from students I had them work in small groups at the board on mixture problems similar in nature to what we did in class with the cereal. After the group solved theirs they had to switch and analyze the solution and work of the problem next to theirs.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipF7rE8A1Hf-qs9zadKBZjjwBbyFVCOX_Xi3wRVKcQQClrK2vtcNDaZ0rC7wURlp7Mg-y114VRhU8bjWTpksaOXYc5A4DEEi7RoXVmgcpQZLB9TfKI5sqjDihAWlA7EgrSKv-OLFH8uMg/s1600/board1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipF7rE8A1Hf-qs9zadKBZjjwBbyFVCOX_Xi3wRVKcQQClrK2vtcNDaZ0rC7wURlp7Mg-y114VRhU8bjWTpksaOXYc5A4DEEi7RoXVmgcpQZLB9TfKI5sqjDihAWlA7EgrSKv-OLFH8uMg/s400/board1.JPG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdYiMgf8clz_zTUNiS-CI8cAo2OpE1HRqOnrmXNMlq61qCWg1ExyXLe-Al7uYARtZGdNT5NiVsbDC9vPEQ19MKVNvSHDqgmXhploMOgOYXy4o7tZ6Jp_YDUijsmPOXj0AnvvOlpxbfrm0/s1600/board2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdYiMgf8clz_zTUNiS-CI8cAo2OpE1HRqOnrmXNMlq61qCWg1ExyXLe-Al7uYARtZGdNT5NiVsbDC9vPEQ19MKVNvSHDqgmXhploMOgOYXy4o7tZ6Jp_YDUijsmPOXj0AnvvOlpxbfrm0/s400/board2.JPG" width="392" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPXrV3cARAibxlEUVJzJkpzJIqym8Ul2i3XNLEfUHuXS1K_wO3F18dwpahZfXNxYe1UxJizAo3-nogEWTKjrlb1U5VbOgVJS-cIowSxC5zXlnQaOAyrb9D81MbT2ploC-8Hg2bwCgAqSE/s1600/board3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPXrV3cARAibxlEUVJzJkpzJIqym8Ul2i3XNLEfUHuXS1K_wO3F18dwpahZfXNxYe1UxJizAo3-nogEWTKjrlb1U5VbOgVJS-cIowSxC5zXlnQaOAyrb9D81MbT2ploC-8Hg2bwCgAqSE/s400/board3.JPG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsDDgj_Nol2itOeX6kjOXADsHSgLybozoqoNuwWlMDvwe9oL7WqJdj-AmMn_TB1oFKP4BxYHfYn7GcLwjiMdvhjBIMsKqx-esO-0OMlGMSDgFRKFHu9mG8LOyqfH5lEOqWays6NfJoKXQ/s1600/board4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="268" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsDDgj_Nol2itOeX6kjOXADsHSgLybozoqoNuwWlMDvwe9oL7WqJdj-AmMn_TB1oFKP4BxYHfYn7GcLwjiMdvhjBIMsKqx-esO-0OMlGMSDgFRKFHu9mG8LOyqfH5lEOqWays6NfJoKXQ/s400/board4.JPG" width="400" /></a></div>
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This worked really well and students said they were understanding mixture problems better.<br />
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Then we focused on mixture problems that contain percents (including interest problems).<br />
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After that we used our ISN to summarize our learning with the left page containing a foldable showing 4 examples that the students set up systems for...<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlt8vpd3xzkJgx-1pq2yhDfi8_1T0xrMf3M5MSh07R9gA562eQAJ8IYKBv3n-bxW0nSrdgpb5m_CfrQHJILy3ZRsiN7OIj1rBAg1nlIGlBtvMsvcPd50gZJL3Pq9h0owbRTUboJ7syAyM/s1600/ISN1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlt8vpd3xzkJgx-1pq2yhDfi8_1T0xrMf3M5MSh07R9gA562eQAJ8IYKBv3n-bxW0nSrdgpb5m_CfrQHJILy3ZRsiN7OIj1rBAg1nlIGlBtvMsvcPd50gZJL3Pq9h0owbRTUboJ7syAyM/s400/ISN1.JPG" width="316" /></a><br />
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(inside bits here:)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMxSHqstY_Oc14z16LKLUkhfnRzGhrzd_X19Z5Yne98pjMTzeeufh1VEZdcVINo_vBUWF8wcC7J6XLMpo-u0Y157c4bvUCEjIuOogstVkdRXCfuLJLZz_GaAZ9JjgqSOoOFmD8hvVJjWA/s1600/ISN3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMxSHqstY_Oc14z16LKLUkhfnRzGhrzd_X19Z5Yne98pjMTzeeufh1VEZdcVINo_vBUWF8wcC7J6XLMpo-u0Y157c4bvUCEjIuOogstVkdRXCfuLJLZz_GaAZ9JjgqSOoOFmD8hvVJjWA/s640/ISN3.JPG" width="640" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbc-R7LKr6saTGYprwz0y-KSNr-Nx_rynk5IXd4mlDNUxoHdv1Si9Ah5UZ97BqODHf5qv9IhSHrceEv9BDfDznAJTl_EvPDYCd3mKHzScwRG6TbC6VPJp91VJXOR3AMWrSjoE1PcbUD30/s1600/ISN4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="204" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbc-R7LKr6saTGYprwz0y-KSNr-Nx_rynk5IXd4mlDNUxoHdv1Si9Ah5UZ97BqODHf5qv9IhSHrceEv9BDfDznAJTl_EvPDYCd3mKHzScwRG6TbC6VPJp91VJXOR3AMWrSjoE1PcbUD30/s640/ISN4.JPG" width="640" /></a><br />
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and the right page solving those systems.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCVkrHmBX_3iX9Y_pVRhU6pYnqbDHvSuU7lJViPp5dxeQ9xDsruEa-ggC-LviDONvQ1Se04WyFqrEonVwjZOJfQiaJzV1gp2RpyRRrK7IrzuwALhM-SHAykbvmfCatchU0shOzZo6X_tE/s1600/ISN2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCVkrHmBX_3iX9Y_pVRhU6pYnqbDHvSuU7lJViPp5dxeQ9xDsruEa-ggC-LviDONvQ1Se04WyFqrEonVwjZOJfQiaJzV1gp2RpyRRrK7IrzuwALhM-SHAykbvmfCatchU0shOzZo6X_tE/s640/ISN2.JPG" width="498" /></a></div>
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All files used in this lesson (worksheets, board problems & ISN foldables)<a href="https://app.box.com/s/89b08beh4wgrc6w3mwbmardpr79g01b5"> can be found HERE.</a><br />
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-34311199410261497212016-01-31T07:56:00.001-05:002016-02-03T07:45:57.092-05:00Start a Lesson With a QuestionWhenever possible I try to start any lesson with a question to lead into what we are learning that day. This is a real shift from when I started teaching in the 80s - teaching training then focused on "tell tell tell" while students listened. Being the kind of learner that needs to understand I was never a big fan of that and figured out how to make questioning a bigger focus of my teaching. <a href="http://mathsolutions.com/about-us/marilyn-burns/">Marilyn Burns</a> was a great mentor for this and I used her book<br />
<h1 class="a-spacing-none" id="title" style="background-color: white; box-sizing: border-box; color: #111111; line-height: 1.2; margin-bottom: 0px !important; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding: 0px; text-rendering: optimizeLegibility;">
<span style="font-family: "arial" , sans-serif; font-size: small;"><a href="http://www.amazon.com/Collection-Math-Lessons-Grades-6-8/dp/0941355039/ref=la_B000APO7K4_1_9?s=books&ie=UTF8&qid=1454243457&sr=1-9"><span class="a-size-extra-large" id="productTitle" style="box-sizing: border-box; line-height: 1.2 !important; text-rendering: optimizeLegibility;">Collection of Math Lessons, A: Grades 6-8</span> </a> </span><span style="background-color: transparent; font-weight: normal;"><span style="font-family: "times" , "times new roman" , serif; font-size: small;">a great deal when I taught middle school.</span></span></h1>
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Anyhow back to this blog post...Here I am focusing on the 3rd week of the MTSoB blogging initiative<br />
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So to repeat myself - Whenever possible I try to start any lesson with a question.</div>
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Algebra One last week we were starting systems of equations. I knew that students had dabbled in this in 8th grade - they had solved systems of equations using graphing and perhaps had been exposed to some algebraic methods. But did they really know what a system was? I love really knowing what certain math concepts ARE and what they are good for (not always real life applications, just how they interconnect with other things I know about math), It's really the only way I can learn math. </div>
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So I started class with this question..</div>
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<span style="font-size: large;"><b>What are some mathematical models that we can use to solve problems?</b></span></blockquote>
Students came up with great responses (almost like there were cue cards or something) - Graphs, Tables, Equations. Yay! But this wasn't really the question to lead the lesson, this was the warm up. Next I asked them<br />
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<span style="font-size: large;"><span style="color: #e48312; font-family: "calibri";"> </span><span style="color: #404040; font-family: "calibri"; font-weight: bold;">You have a bunch of quarters and dimes. </span></span></div>
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<span style="font-size: large;"><span style="color: #404040; font-family: "calibri"; font-weight: bold;">The value of
your coins is $3.00. </span></span></div>
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<span style="font-size: large;"><span style="color: #404040; font-family: "calibri"; font-weight: bold;">How many of each coin do you have?</span></span></div>
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<span style="line-height: normal; text-indent: 0px;">My students sit in small groups so I let them work on that. They gathered up some graph paper and made beautiful linear graphs of this situation with a lovely equation to accompany it. We talked about whether this graph was continuous or discrete and whether we would consider values in all four quadrants. Also with some prodding they came up with a table showing the possible solutions. Great work!</span></div>
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<span style="line-height: normal; text-indent: 0px;">Then I told them that this question presented them with a situation that had two unknowns (number of quarters & number of dimes) and one condition that established a relationship between the unknowns. It resulted in a number of possible solutions. What if I gave them another condition to partner with this information?</span></div>
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<span style="font-size: 12pt; line-height: 14.4px; text-indent: -9.6px;"><span style="color: #e48312; font-family: "calibri";"> </span></span><span style="color: #404040; font-family: "calibri"; font-weight: bold; line-height: 14.4px; text-indent: -9.6px;"><span style="font-size: large;">What if the number of dimes is two more than the number of quarters?</span></span></div>
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<span style="line-height: normal; text-indent: 0px;">Then students worked on narrowing things down with this condition. They added on to the models they already created (another line on the graph, another equation for that condition). They found that this second condition resulted in a unique solution. Cool.</span></div>
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<span style="line-height: normal; text-indent: 0px;">These initial questions allowed us to explore situations with two unknowns and two conditions that would have a unique solution. From here we could define systems of equations (after we did a few more examples) and keep in mind that is what drives a system of equations in two variables. <b>Two unknowns and Two conditions. </b></span></div>
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<span style="line-height: normal; text-indent: 0px;">I put all our questions on a powerpoint. I also have a homework sheet to accompany this. We created a cool foldable for our notebook of an example (this was used the next class block as our warm up). All supporting materials can be found at this <a href="http://link./">link.</a></span></div>
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Here's the foldable I used with my students to illustrate these models. It's a cool "4 square fold" that my student teacher found last year (thanks Tess!). So it allows you to get a lot of information down in once place.<br />
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Here is the front all folded up into 4s (see how it folds up to use only half the page?):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUL8I4lmWtxO4yrpa_vy0jgc8zFxaE6SxudYxwCU1vl7M3-CIlhrh0M9fElZ6zblqGUosuofWLLLFpbZRWhoqEaez5lYqK9b2LhvTr8BjtW0Z4oUywJoupmRSbbveieZuI72RM1VeEAG0/s1600/ISN1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUL8I4lmWtxO4yrpa_vy0jgc8zFxaE6SxudYxwCU1vl7M3-CIlhrh0M9fElZ6zblqGUosuofWLLLFpbZRWhoqEaez5lYqK9b2LhvTr8BjtW0Z4oUywJoupmRSbbveieZuI72RM1VeEAG0/s400/ISN1.JPG" width="328" /></a></div>
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Open it once and you get two places to have information:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE0XKjekI1aB_blfJpjxgYPeaYzSPr9tmiIAkH5F4vj277Gt5EiF16MWzQ8Y_YCcK5isQwv3VncH2AFfNzzgrnHFnO79FoCjU810Z05c2bXIwCDEYxQl7vW8LOaxGBlyXrhJzdWgSkThE/s1600/ISN2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE0XKjekI1aB_blfJpjxgYPeaYzSPr9tmiIAkH5F4vj277Gt5EiF16MWzQ8Y_YCcK5isQwv3VncH2AFfNzzgrnHFnO79FoCjU810Z05c2bXIwCDEYxQl7vW8LOaxGBlyXrhJzdWgSkThE/s400/ISN2.JPG" width="400" /></a></div>
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Completely open it up and there are 4 places to have information:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgleuhOc2hGMPSC4EffQtPSBdhGDTH3DWk-D1XvxgvxPSENFky5xlVa_lTtYxc2lqZod0M7WKdT0yqILpYAN1fYpmhQh-FK2urqxNNMmSY3cBWLHruguf72TAsHXpe09eSRonTL_1-meJ8/s1600/isn3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgleuhOc2hGMPSC4EffQtPSBdhGDTH3DWk-D1XvxgvxPSENFky5xlVa_lTtYxc2lqZod0M7WKdT0yqILpYAN1fYpmhQh-FK2urqxNNMmSY3cBWLHruguf72TAsHXpe09eSRonTL_1-meJ8/s400/isn3.JPG" width="400" /></a></div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-57609784082595753972016-01-27T15:19:00.004-05:002016-01-31T08:01:41.613-05:00Graphing Calculator Pictures - PreCalculus Project<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimd7ZaJFuTxqx-sjP_gfVLdB3Pcl4okOTyO0fgy40ecELkpTCqeA0Kk-3vTWRqrSacOV2llwKwpwbU5JxLNFzo3lrtxfLLHDglNilPuV9Ao7eRnoxxW1C1SCI08f-psKLrB9kme94KQKQ/s1600/amer+gothic.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimd7ZaJFuTxqx-sjP_gfVLdB3Pcl4okOTyO0fgy40ecELkpTCqeA0Kk-3vTWRqrSacOV2llwKwpwbU5JxLNFzo3lrtxfLLHDglNilPuV9Ao7eRnoxxW1C1SCI08f-psKLrB9kme94KQKQ/s1600/amer+gothic.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBu8_9az9OpWPCEHkHQKLBnSc2AMyzAy_z_YNQfpz-_SZd3AQsfamhIkKLTZeNp3riKWvqlsg1BedPoIk4ptzUW7cqk9fY6cgO-TaIYyt4c0ImSj2okKjvJsVKMqIecuuE_U1nkjr1Ip8/s1600/crab.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBu8_9az9OpWPCEHkHQKLBnSc2AMyzAy_z_YNQfpz-_SZd3AQsfamhIkKLTZeNp3riKWvqlsg1BedPoIk4ptzUW7cqk9fY6cgO-TaIYyt4c0ImSj2okKjvJsVKMqIecuuE_U1nkjr1Ip8/s1600/crab.png" /></a></div>
Many teachers in our department had projects in their math classes long before the common core curriculum had us having students do performance tasks. I hate grading projects but love having students do them because they show me that they can really apply what they have learned to something interesting and different from what we are able to do in class. I've taught PreCalculus (honors level) for about 10 years and this particular project is sort of like the pinnacle of all the projects I have students do. It's absolutely amazing what students are able to create on their graphing calculator.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL3kZNrRRednLgpvgk1lN8I4e-gknfTsE3ErHP6VLJW9cLi9PX73TI8qKuavWNHLBxYq1ZhxFH91qHQOhZqRTdUkkzOSL8nXBbCmI-ZBEYz8ms6IqrzDbtX_ZSa4QWHT3xYur_v26-G6I/s1600/clifford.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL3kZNrRRednLgpvgk1lN8I4e-gknfTsE3ErHP6VLJW9cLi9PX73TI8qKuavWNHLBxYq1ZhxFH91qHQOhZqRTdUkkzOSL8nXBbCmI-ZBEYz8ms6IqrzDbtX_ZSa4QWHT3xYur_v26-G6I/s1600/clifford.png" /></a></div>
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So what's this incredible project? Students draw a picture on their graphing calculator using functions they've learned and restricting the domains (essentially creating piecewise functions on their calculators). By the time students are in PreCalculus they have learned all the basic parent functions. By December we've reviewed all those types (& have just started graphing sine, cosine & tangent) and reviewed transformations. The <u>bare minimum</u> requirement of this project is to use 10 equations and three function types. As you'll see in my rubric the minimum only results in a maximum grade of a C. I tell students they should aim for about 40 equations. And they should realize and plan for the fact that this project can take anywhere from 8 to 20 hours. And that the "write up" can take 2 hours on its own. My students are terrible at time management, waiting until the last minute to do most things. That will not work with this project. It takes a lot of time and most students have to have me transfer their images from their calculator to the computer with my trusty old graph link (although they can do it themselves with the cord that comes with the new calculators and some software from TI).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwTPcJin4tfQdI-8NEJt8sA8w2lcRtq2lNlMhA5GiVQ58aY8EUeS3SZyaFvlyeNzuaSXoI_r1wqpGuV1VEcy8M2lgnm2LcIK5AzMI8TVOtNrHTY23VXr_tztaLtidGTQfZdTBdUI_AqJc/s1600/graph+link.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwTPcJin4tfQdI-8NEJt8sA8w2lcRtq2lNlMhA5GiVQ58aY8EUeS3SZyaFvlyeNzuaSXoI_r1wqpGuV1VEcy8M2lgnm2LcIK5AzMI8TVOtNrHTY23VXr_tztaLtidGTQfZdTBdUI_AqJc/s1600/graph+link.jpg" /></a></div>
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So I had the idea to have students do this while sitting in a really boring professional development worksheet many years ago (like 10 years ago) so started playing around on my graphing calculator and came up with Bob the Dog:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_4kmGx3so992S5ZFu1-pjYwEIdiTrY5RZ6zpGK7RKHZ3dWzO2heEF1O2pZDEh-dWVlIGN-qz1Zcm_3je9GdOpQRQ-xWbOuO5AOkB8ko1UH6bhRP8Wjip0Rg8izhOropX2MWwM7i77tX0/s1600/bob+the+dog.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_4kmGx3so992S5ZFu1-pjYwEIdiTrY5RZ6zpGK7RKHZ3dWzO2heEF1O2pZDEh-dWVlIGN-qz1Zcm_3je9GdOpQRQ-xWbOuO5AOkB8ko1UH6bhRP8Wjip0Rg8izhOropX2MWwM7i77tX0/s1600/bob+the+dog.png" /></a></div>
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I was pretty proud of myself, but of course after 10 years of seeing fantastic pictures done by students this is not a really a great picture. Oh well. I am so impressed by what students have done and it's time to share in a blog post.</div>
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At this point, everyone knows this project is coming up in PreCalculus. So in early December I get them started. I tell them to find a horizontal image they would like to draw. They need to print out this image in the size of 7.5 inches by 5 inches. And I show them this fish as an example:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQzjVnFmewPVlrBFT0lPjGjwgTAMyBitYYocTv01tSUcECdW7DcQfU0RwqVUts4sdGvqVXzptIC9dGg5I3njovhsYllMZ0GsguSlDTC4CX_BXJOk3iV9JUsIxgBNpgc8Ayc0SnFc8NpO0/s1600/fish+from+internet.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="228" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQzjVnFmewPVlrBFT0lPjGjwgTAMyBitYYocTv01tSUcECdW7DcQfU0RwqVUts4sdGvqVXzptIC9dGg5I3njovhsYllMZ0GsguSlDTC4CX_BXJOk3iV9JUsIxgBNpgc8Ayc0SnFc8NpO0/s320/fish+from+internet.JPG" width="320" /></a></div>
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I give students a few days to come in with a picture and then we spend about 30 minutes going over how to start the project. I did my "Bob The Dog" freehand without this step but students prefer to have a starting point and this is a great way to get started visualizing how to transfer an image to a coordinate plane and then their calculator. They trace their image onto graph paper that has a [-15,15] by [-10.10] "window". This will match the square window they will use on their calculator [ZOOM] 5:ZSquare.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuLhEtDuixb15aGnsFQvubOM1UYxjJFUnr-EDH75Q1KrH_vz3uOgVm_Sq-PnCv7SSmx2qTyLwchDGZci3Q6LzTNtWXT6MzSb_btfQqn5TSUFVZowvhGLfOzl5Yub0eEmrSfODfh8woeJE/s1600/fish+on+graph+paper.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="231" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuLhEtDuixb15aGnsFQvubOM1UYxjJFUnr-EDH75Q1KrH_vz3uOgVm_Sq-PnCv7SSmx2qTyLwchDGZci3Q6LzTNtWXT6MzSb_btfQqn5TSUFVZowvhGLfOzl5Yub0eEmrSfODfh8woeJE/s320/fish+on+graph+paper.JPG" width="320" /></a></div>
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From there they look at the image and work with a major shape part of it (like the top of the fish) and consider what functions they would use to create those lines. I remind them of the general transformation form of any function y = a (x - h) + k and of the many parent functions they've studied in Algebra 2 and PreCalculus. </div>
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We don't devote any full class blocks to working on this and we still have regular lessons going on. I do give them the day before Christmas break to work on it in class. And I am available to help after school. Some students do this entirely on their own, some need 10 minutes of help getting started, some need an hour of one-to-one help to get started. But everyone comes up with a cool picture!</div>
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Students also have to do a write up of their work. This is time consuming too! But they do a great job. I've included files below that have the rubric and project info, tips for the project, a sample student write up and a powerpoint of images submitted by students over the years (oh about 50 or so slides).</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6YSp2yl6jfoq59o9UfIYypgzG_Jz_8nG-XPQtMI1rvo85h6EB9vDtUuYqmyThuKvRLw9-gQqWZjE9Dglcx-MnEjOSKaRHKlLqy3IbBLwGZiTglpW5qpcBaEZ61IGyf0MORTWDq7ug_TE/s1600/owl.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6YSp2yl6jfoq59o9UfIYypgzG_Jz_8nG-XPQtMI1rvo85h6EB9vDtUuYqmyThuKvRLw9-gQqWZjE9Dglcx-MnEjOSKaRHKlLqy3IbBLwGZiTglpW5qpcBaEZ61IGyf0MORTWDq7ug_TE/s1600/owl.png" /></a> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpleweq-BaDVSumYt_2Fx3kMw3KbIejXkwSV6LKQaBu61kj_0asZpM-bL5kXY5A_H6XR7o9EwhG8PRhR6vbl-svyqR2_TWqKehApMBHv4TtmiL1jZy6g60YBiwvF1Rim6pKjDQiz_W2k8/s1600/pig.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpleweq-BaDVSumYt_2Fx3kMw3KbIejXkwSV6LKQaBu61kj_0asZpM-bL5kXY5A_H6XR7o9EwhG8PRhR6vbl-svyqR2_TWqKehApMBHv4TtmiL1jZy6g60YBiwvF1Rim6pKjDQiz_W2k8/s1600/pig.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHddMrt_6jEB9OWD2tLub3-an9xhVXE2Fc_RJOzknA4PaTAGnfbFE4ScLV0qVJX6qpy9QChxUg0DvIule5yPEaYCIFVABHO5gg7N_I7qaGNMJx-l8U8bxRr0q4VqhGdH3wFyIUNGbuH_Y/s1600/jerry.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHddMrt_6jEB9OWD2tLub3-an9xhVXE2Fc_RJOzknA4PaTAGnfbFE4ScLV0qVJX6qpy9QChxUg0DvIule5yPEaYCIFVABHO5gg7N_I7qaGNMJx-l8U8bxRr0q4VqhGdH3wFyIUNGbuH_Y/s1600/jerry.png" /></a> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5G-HsO-ogKKHXLc32pgvAC_iSAAQfDlqmXqUtzLzon3VppidPPs6l9aONl6MvxgGf9gLITJmmNcFTUrpx0d5gSoKoSWTX4P2jh4xN7CZt5PO85kB3P4v71FQrtNqVgziZHP7WbnA0Hsk/s1600/rabbit2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5G-HsO-ogKKHXLc32pgvAC_iSAAQfDlqmXqUtzLzon3VppidPPs6l9aONl6MvxgGf9gLITJmmNcFTUrpx0d5gSoKoSWTX4P2jh4xN7CZt5PO85kB3P4v71FQrtNqVgziZHP7WbnA0Hsk/s1600/rabbit2.png" /></a></div>
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They are not that difficult to grade. I've gotten the rubric down to a very quantitatively calculated grade with only one piece that can vary according to "creativity".</div>
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After they are all collected and graded I print out everyone's final pictures and create a graphing calculator gallery in the hallway for the school to enjoy. I invite school officials, the superintendent and the members of the board of education to stop by and see the gallery. It really is impressive.</div>
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Probably the best part of this project is how much students like it. They all say it takes a lot of time but they learned a lot and really enjoyed working on it! Yay, success!</div>
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Go to all documents related to this project<a href="https://app.box.com/s/urtarm7se55ny7e0es28s026ebyz0n4q"> HERE. </a><br />
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mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com2tag:blogger.com,1999:blog-2897305678042698487.post-89895837890963033212016-01-21T17:07:00.001-05:002016-01-31T08:01:02.038-05:00My Favorite<div class="separator" style="clear: both; text-align: center;">
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<span style="color: #666a76; font-family: "open sans" , "helvetica neue" , "helvetica" , "arial" , "lucida grande" , sans-serif;"><span style="line-height: 28.432px;">Welcome to week 2 of the blogging initiative! The </span></span><span style="color: #666a76; font-family: "open sans" , "helvetica neue" , "helvetica" , "arial" , "lucida grande" , sans-serif; font-size: 16px; line-height: 28.432px;">week two blogging challenge is to simply blog about one of our favorite things. Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well. It can be anything in teaching that you love! It can also be something that you have blogged or tweeted about before.</span><span style="color: #666a76; font-family: "open sans" , "helvetica neue" , "helvetica" , "arial" , "lucida grande" , sans-serif;"><span style="line-height: 28.432px;"> OK sounds great!</span></span></div>
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I've been teaching for many years and over the years there have been many favorite things. I especially love working with teenagers (most of the time) and love teaching math (especially when we delve into the "why" of things). So I'll try to keep my favorite list short but I can't just write one thing....</div>
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One of the best parts of teaching is seeing how students think and how they break down a problem/concept. I've always worked hard at asking open ended questions a lot and asking why a lot in class. This is just an every day favorite thing for me about teaching math. It can be hard work holding back and not explaining things to students, certainly if they get stuck i give hints or suggestions on some path they can try. It's something I've always done and in the early years got some resistance from other teachers. So I love that all the latest common core math practice standards naturally promote this type of teaching. Yay!</div>
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And my new favorite thing has a few sub-topics (will try to keep this short). I LOVE using <b>interactive student notebooks</b> (ISN) in my classes! I don't use them to "drive" my lessons. Instead they are used to summarize topics after we've explored and figured things out. We no longer have a textbook in my Algebra classes (waiting to see what becomes available for the new common core curriculum) so these are invaluable reference tools for students. My PreCalculus students do have a textbook but the ISN is still a great supplementary resource for them. It also appeals to my creative side. And to my student's creative sides. I don't get too fancy with foldables, just creative enough.</div>
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Managing the ISN in my lessons is made so much easier by a few things that are also FAVORITES:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4OrDMQ_cOAumtKx8aTQgMJ6P0w9ZmBt3cSxNSlvZwa7y91rmFOiuuHCy_-n1s33QP5tZzLOqrghyzsVUCSB7xIwjmTnHi7OUh_H7iRQiONJEdSJV5Za6ZL5j15ZD1tu4egSFLmVYYJbw/s1600/document+camera.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4OrDMQ_cOAumtKx8aTQgMJ6P0w9ZmBt3cSxNSlvZwa7y91rmFOiuuHCy_-n1s33QP5tZzLOqrghyzsVUCSB7xIwjmTnHi7OUh_H7iRQiONJEdSJV5Za6ZL5j15ZD1tu4egSFLmVYYJbw/s320/document+camera.JPG" width="320" /></a><b>Document Camera</b> - great for displaying what we are filling in. Most of our teachers do not have one so I had to beg to get this and it's only on loan from our school library. They are so wonderful about doing this for me.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbr_hRH6k8zm9-Kq8MkTSxb79TyxZ2kkYhnS3sZMQXWdxHol3XaOs10mBZSgw09YSjJTuVT6rhRrg13dhFjZr1yUef1jhtqc5dOW66H0zO5Mg3hUfSRSpYY6A-2y0__hPG7MwhOWu2jdE/s1600/baskets.JPG" imageanchor="1" style="background-color: transparent; clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbr_hRH6k8zm9-Kq8MkTSxb79TyxZ2kkYhnS3sZMQXWdxHol3XaOs10mBZSgw09YSjJTuVT6rhRrg13dhFjZr1yUef1jhtqc5dOW66H0zO5Mg3hUfSRSpYY6A-2y0__hPG7MwhOWu2jdE/s320/baskets.JPG" width="320" /></a><b>Supply Baskets</b> - any supplies we might need (scissors, colored pens, highlighters, tape) I have at the ready for students in little baskets.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-u3Vdl6xHQQUnYiMpWRcNYskXrm7l8NVaPKQ7EG_4WIAfe6LpWqpRtXSakG_-zBBK8zQl3Nsr3A2Tb7BYVKM8EXFnTtkbKsSF8TruoCan8qTOMxCUB9HD0pArDGafgGGAM2mnaPvfcL8/s1600/seats.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-u3Vdl6xHQQUnYiMpWRcNYskXrm7l8NVaPKQ7EG_4WIAfe6LpWqpRtXSakG_-zBBK8zQl3Nsr3A2Tb7BYVKM8EXFnTtkbKsSF8TruoCan8qTOMxCUB9HD0pArDGafgGGAM2mnaPvfcL8/s320/seats.JPG" width="244" /></a><b></b></div>
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<b>Group seating in my classroom</b> - so each group gets a basket of supplies with one student in each group grabbing what they need for the day. This group seating is also great in encouraging students discourse in exploring new topics and problem solving.<br />
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<b>GLUE SPONGES</b> - a neat, permanent way to glue in any foldables we use in our ISN. I blogged about them here in <a href="http://beautifulhighschoolmath.blogspot.com/2015/07/tidying-up-isn.html">this post.</a><br />
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I've blogged a bunch about using the ISN in my classroom and a lot of my posts on certain topics show how the ISN supplements the lesson. I just started blogging in June 2015, here is my <a href="http://beautifulhighschoolmath.blogspot.com/2015/06/starting-out.html">first post</a> that highlights my use of interactive student notebooks.</div>
mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-70126913859970276792016-01-12T12:35:00.002-05:002016-01-31T08:01:54.642-05:00A Day in My Life as a Teacher<h1 class="post-title" style="background-color: #f8f8f9; border: 0px; clear: both; color: #444854; font-family: 'Open Sans', 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 2.665em; letter-spacing: -1px; line-height: 1.333; margin: 0px 0px 1rem; outline: 0px; padding: 0px; position: relative; vertical-align: baseline;">
<a href="https://exploremtbos.wordpress.com/2016/01/10/week-1-of-the-2016-blogging-initiative/" rel="bookmark" style="border: 0px; color: #53a1b8; font-family: inherit; font-size: 42.64px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;">Week 1 of the 2016 Blogging Initiative!</a></h1>
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We often get caught up in our day teaching, and often don’t take the time to be in the present. So we have <em style="border: 0px; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">two options </em>that with that goal in mind. Choose whichever floats your boat and blog about it!</div>
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<span style="border: 0px; font-family: inherit; font-style: inherit; font-weight: 600; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Option 1:</span> We rarely take the time to stop and smell the roses. Even on the most disastrous of days, good things happen. And these good things, when you’re on the lookout for them, pop up. All. The. Time. So for one day (heck, do it for many days), keep a lookout for the small good moments during your day and blog about them. We bet that by keeping an eye out for the good, your whole day will be even better!</div>
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<span style="border: 0px; font-family: inherit; font-style: inherit; font-weight: 600; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Option 2: </span>A few years ago, some people in the #MTBoS wanted to share what their teaching lives were like. Partly because we all work in different schools, and so we wanted to get a glimpse of our friends-in-action. At the same time, we also wanted to battle against the idea that teaching is easy. We wanted to share what it is like to be a teacher with non-teachers! So we all blogged about a single day of teaching — from start to finish. So for the first week of the blogging initiative, we thought you blogging about a day in your lives would be a great way to start getting to know each other!</div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Okay a blog post all about a day in the life of a math teacher - how exciting! (LOL). Certainly easy to write as I've lived it. So yesterday January 11, 2016. </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">5:20 AM Phone alarm goes off (I've got to find new wake up music - this is sooo annoying). No snooze for me, this is the latest I can sleep....</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">5:25 - 6:40 Shower, get dressed, feed the cat, make my<a href="http://www.popsugar.com/fitness/Healthy-Caramel-Apple-Smoothie-Recipe-39702446"> morning smoothie (today it's a healthy caramel apple smoothie) </a>& assemble my lunch and then head to school.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">6:50 Arrive at school - check my mailbox and head up to the third floor to my classroom.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">7:00 Students start arriving, one in particular from my Applied Algebra class for extra help on what he missed last class (others are there for my first block class with quick questions about the test they'll be taking once class starts). </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">7:33 to 8:55 - First block - PreCalculus. We're having a test on unit 3 "Circular Functions". After the test students are working on mid-term exam review (they have an outline of topics with suggested review problems in their textbook) and their graphing calculator design project (which I promise I'll be blogging about soon - such a GREAT project). While students take their tests I correct their <a href="https://app.box.com/s/7rop65lvgwgbrethwwj5ht5ezc5j3tq9">study guides</a> (mostly just for completion but some perusal for correctness). After the test I walk around and give assistance on projects. One student is making the slinky dog from toy story and needed help with the functions for the feet. </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">She decided to use log functions and I was helping her with the transformations to get them in the right position in her image (she had already established the stretches and reflections needed but couldn't get the shifts exactly as she wanted). I will include her final result in my pending blog post about this project.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">9:05 to 10:29 Second block class - Algebra One. This class also has an assessment. Unit 5 end of unit quiz. It's a short unit, only one week. Focus is on creating scatter plots and trend lines (by hand & using graphing calculator) and analyzing the results (linear regression, interpolation, extrapolation, correlation, causation?, outliers & influential observations). We go over the <a href="https://app.box.com/s/le2en7kl3fvu0n2q0rbdqg0kbcy442pm">homework </a>first (basically I check to see if there are questions - I post the homework answers on my website so I don't have to read all the answers. Students have to check their answers but only about 1/3 of the class did). After the assessment students work on their midterm exam packet and I start to answer questions. We got through a little review of unit 1 (sequences) and students told me things they had questions on so I could prepare some problems for next class. Exams are coming up this Thursday.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">10:35 to 11:03 I actually went to lunch with the math department today! I usually only make it about once a week but things are a little lighter this week with exams coming up (no lessons to really prepare until classes start again on January 21). The big talk in lunch was about powerball lotto - the jackpot is huge this Wednesday, over a billion $! So of course everyone is buying tickets and/or contributing to a pool. I wish I had time to go to lunch more often but often use that half hour to get a few things done (use bathroom, run copies, clean up classroom and set up for next class on A days). So today was a treat to hang out and talk with grown-ups.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj0Drua8-Je20Kkj8MP4uq4M-eq0nkSsW-qWwQJD6UoNeZONlLYd9vvMTBKqeueFJ2QW7c_cP6DpWFI8ylxJ0O0dr58_W7PTmsejbaFsSuAi2kZFTfTvpzYW7oqsJUza059g_Mc-DxY3k/s1600/Lemon-Meringue-Cookie-Cups1.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj0Drua8-Je20Kkj8MP4uq4M-eq0nkSsW-qWwQJD6UoNeZONlLYd9vvMTBKqeueFJ2QW7c_cP6DpWFI8ylxJ0O0dr58_W7PTmsejbaFsSuAi2kZFTfTvpzYW7oqsJUza059g_Mc-DxY3k/s320/Lemon-Meringue-Cookie-Cups1.jpg" width="213" /></a><span style="font-family: "arial" , "helvetica" , sans-serif;">11:08 to 12:20 This block is my prep block for the day. A nice long prep I know. I usually devote my prep to parent communication & planning. We don't have textbooks for Algebra & Geometry at this points so we are creating materials for our classes. Since I've been teaching my Algebra Ones for a few years I have a lot of materials created but I'm always tweaking them. I save all my correcting for outside of school (at home in evenings or weekends). Today I had to do some exam modification work for my Applied Algebra class. It's a small class but 12 out of the 14 students have an IEP (individualized education plan) that calls for a variety of modifications. In general I create all the materials for the class so that the modifications are covered and everyone gets them. But then I have to create packets for the "exam proctors" - most of the students start/take the exam in an alternative setting. So I organized that. And finished sending out exam materials to my math department coordinator. And gathered data on my latest assessments in Algebra One (unit 4 test and performance task) for my coordinator. And took care of some parent contact tasks. Checked email. And double checked what cupcake recipe I'm going to use for the cupcake baking contest our faculty does at a faculty luncheon the first day of exams. This year I'm going with <a href="http://www.momontimeout.com/2014/01/lemon-meringue-cookie-cups/">Lemon Meringue Cookie Cups.</a></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">12:20 to 12:32 set up my classroom for my next class. This is my Applied Algebra One class and I have a special seating set up for them. I put 15 desks around in a squared off horseshoe facing the main whiteboard with three smaller groups of desks behind them. We start in the main horseshoe to go over homework and have general whole class discussion. then the smaller groupings are used for independent or group work. Generally my classroom is set up with 6 groups of 5 desks each. But I learned this class needed the horseshoe set up to help them with focus and attention. It works really well with them and once they can work independently on a concept they can move to a "table" grouping. Some stay in the horseshoe grouping the whole time because they need constant monitoring. I have a teacher's aid in that class. He's great but doesn't know the math so I give him answer keys with my lesson plan. I work more closely with the kids who need that constant one-on-one while he just monitors that the rest of the students are staying on task and checking answers.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">12:38 to 2:00 Last block of the day - my Applied Algebra One class. Today we wrapped up the "rate of change" concept with a quick review of the homework and a quiz. Then we came back as a full group to review more for the exam. Today we focused on unit 2 expressions & equations.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">2:00 to 2:15 Run down to guidance to check in with a counselor about a students, stop by the mailboxes to distributed the Applied Algebra One exam packets to the special ed teachers.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">2:15 to 3:15 Math Department Meeting - today we learned more about the whole school SATs that we are just starting in CT as our high school "standardized test". All juniors will take the SATs on March 2. So we learned about the logistics of that schedule. And talked about the PSATs (all 10th & 11th graders took that in October and scores are out) and how we can help students use information about their scores to guide their SAT preparation. I'll be spending more time on learning more about the new SATs & using collegeboard.com and Khan academy in my classroom on our next PD day February 12th.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">3:15 to 3:30 head home</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">3:30 to 5:00 some home "stuff" including a 30 minute fitness walk around the neighborhood. Brrr it was cold - in the mid-30s with a lot of wind. Gotta do my exercise to keep up on <a href="https://docs.google.com/spreadsheets/d/1XjACy06i9WUvqpJm6Oa9KFKZ64W8hOIDQfF8n4c1TMk/edit#gid=0">#FitBoS! </a></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">5:00 to 6:00 some correcting of quizzes. Not much time to do this tonight as I have book group later and I'm hosting. Most of the food is all partially prepped but once book group starts I won't get any more work done. So I didn't get much done.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">6:00 - 10:00 set up, book group 7 to 9:30 and clean up. Our book was<a href="https://www.goodreads.com/book/show/21853621-the-nightingale"> The Nightingale.</a> </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">10:00 to 11:00 read to unwind, then finally to bed (to start all over again the next day....)</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Pretty busy, but probably pretty boring to read about. This was a lighter day really because of midterm exams coming up and not having to do much prep right now. Fortunately with exams we get the afternoons "free" (no students) so we can correct exams and I get a lot of prep done then for the first weeks of the new term.</span></div>
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mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-6309232008328357902016-01-07T16:21:00.002-05:002016-01-31T08:02:09.808-05:00Blogging InitiativeTime to dust off the blog....I find it pretty difficult to keep up during the school year. The primary purpose of my blog has been to share the materials I've created. Especially the ISN (interactive student notebooks) materials. Most of my materials have been adapted from things I've found from the internet - so in the spirit of passing on and sharing the "wealth" I started my blog last June (2015).<br />
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There's a pretty amazing group of math education bloggers out there - many of them stay connected through the Math Twitter Blogosphere ( #MTBoS). And now there is a blogging initiative for 2016 led by #MTBoS. You can learn more <a href="https://exploremtbos.wordpress.com/2016/01/03/kicking-off-the-2016-blogging-initiative/">here. </a><br />
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So here is my official statement for this initiative:<br />
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-82388769644579185252015-12-14T05:00:00.000-05:002015-12-14T05:00:05.612-05:00Winter Estimation FunLast summer I found a cool website called Estimation 180 and used an estimation activity with a quarter cup of almonds with my classes. A neat little activity to get them thinking.<br />
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<a href="http://www.estimation180.com/blog/winter-estimates-with-a-crisper-look">Here's another one that would be fun to do on that half day before our Christmas break</a>. Ultimately everyone gets a mug of hot chocolate at the end.<br />
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How Much Hot Chocolate is in the Glass? </h3>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSSCfST1HDXKa_kcn6wk00g7HEOfG-70YsIvbw_0n8dsl0-Y-f1ZQzpvuL5goKdrUHaqyc2q-pyf5Cwg7kBbBKxdMIx6uUPvgyWPprfulu0hMznb5NnupcD8M0ja4dkw7s13SOvWkJELM/s1600/hot-chocolate.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSSCfST1HDXKa_kcn6wk00g7HEOfG-70YsIvbw_0n8dsl0-Y-f1ZQzpvuL5goKdrUHaqyc2q-pyf5Cwg7kBbBKxdMIx6uUPvgyWPprfulu0hMznb5NnupcD8M0ja4dkw7s13SOvWkJELM/s400/hot-chocolate.jpg" width="394" /></a></div>
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I could do this with heat proof paper cups and give everyone a cup and pose that question. </div>
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Then we extend the estimate to marshmallows...</div>
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How many small marshmallows will fit?</h3>
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Of course we'll need to discuss what that means. How many fit on the surface? How many fit to fill the mug completely? </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxqV5uzOnwBQVtUXaryHivHLBSG9HOEUEYeud3yBcWozo8iFadC5A8vYPiPiUOYXjrOChouK1xeysclTz2sSrtO_i4vcSjRZGRg8DAc_p4hIAM31xBlR2SDP9-PeBp5AYXr-xeTaUJkBg/s1600/hot+choc.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxqV5uzOnwBQVtUXaryHivHLBSG9HOEUEYeud3yBcWozo8iFadC5A8vYPiPiUOYXjrOChouK1xeysclTz2sSrtO_i4vcSjRZGRg8DAc_p4hIAM31xBlR2SDP9-PeBp5AYXr-xeTaUJkBg/s400/hot+choc.jpg" width="400" /></a><br />
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and finally</div>
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How many large marshmallows will fit?</h3>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyl77sNbTCwj7PZ1OEA3Cgjjz8ADNywMW4JqHpW2VkfaCX8p84iv7oixDBYAnWvsylmYv0-i55yYNK8Qvv7lTsw51DymQispch7vMW4dbqCkhUP-bgvCX_YWtXtIAjvF2eiCBwabwuXjQ/s1600/hot+choc+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyl77sNbTCwj7PZ1OEA3Cgjjz8ADNywMW4JqHpW2VkfaCX8p84iv7oixDBYAnWvsylmYv0-i55yYNK8Qvv7lTsw51DymQispch7vMW4dbqCkhUP-bgvCX_YWtXtIAjvF2eiCBwabwuXjQ/s400/hot+choc+2.jpg" width="400" /></a></div>
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some discussion about what this means.</div>
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Kids record estimates - can use his site or? create a spreadsheet or survey for my classes?</div>
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Finish up with drinking hot chocolate!</h3>
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<i>NOTE to self - bring in hot water kettle, canister of hot chocolate powder, cups, small marshmallows, large marshmallows. Yum!</i></div>
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<br />mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-89225814943566537752015-11-09T13:31:00.001-05:002015-11-30T13:00:52.813-05:00Domain & Range ToolI blogged about this domain & range tool a bit in a <a href="http://beautifulhighschoolmath.blogspot.com/2015/07/starting-functions-independent.html">post</a> this summer but today I used it in the classroom and it was really great and I wanted to share this again with updated pictures to show how it worked out.<br />
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Domain and range from a graph can be quite tricky. I do this with my advanced algebra one group. This domain & range finder tool is a great way to visualize the domain & range.<br />
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They did have some homework last night that involved identifying which graphs are functions and trying to determine the domain & range of each (we had already defined domain & range and even filled in the DIX ROY bit the last class). Most kids came in to class today saying they didn't understand the homework. After going over homework we created our domain & range finder tool.<br />
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This tool can be a little "fussy" and I warned the students that today is a bit "artsy-craftsy". I really avoid doing anything too complicated with my foldables - there is the danger of it becoming a folding-cutting class rather than an algebra class. So I prepare quite a bit ahead of time. And I time the cutting and folding tasks to be done while I check homework and help students out in their small groups so it doesn't take away as much instruction time.<br />
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I also prepped by photocopying and doing some preliminary cutting with my paper cutter - so worth the time investment. <br />
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First everyone gets a pre-cut domain & range finder tool (we only have to fold it):<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwlWoPDb_04wY6Mmewxgx9jOPR7DsDghl72t9-8xi8u_RERW5Lh1VHO6ql1YSf-ZtqUWYqITcBnI8tF9FkjUiGzRmei2FNiHhgJ4Bv4iODDmu9PNVb816pTiHa4VawsrV3NnpgZOaPfQo/s1600/domain+and+range+tool.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwlWoPDb_04wY6Mmewxgx9jOPR7DsDghl72t9-8xi8u_RERW5Lh1VHO6ql1YSf-ZtqUWYqITcBnI8tF9FkjUiGzRmei2FNiHhgJ4Bv4iODDmu9PNVb816pTiHa4VawsrV3NnpgZOaPfQo/s400/domain+and+range+tool.JPG" width="400" /></a><br />
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And an envelope foldable (partially pre-cut) and two strips of domain & range cards (8 cards all together).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD3BU7AeL7B_zjxSQCMkNT95kB9Sum-z_tMa_m0jypg_-CuTiLk2ZYngnj1fa3Gypx0b725npX3C2gippr0B4THtQHEKeQIQW8a120GdyHwcJIFVfvTrWDChBDW26g7B-s2v84t7rVuAU/s1600/graph+strips2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD3BU7AeL7B_zjxSQCMkNT95kB9Sum-z_tMa_m0jypg_-CuTiLk2ZYngnj1fa3Gypx0b725npX3C2gippr0B4THtQHEKeQIQW8a120GdyHwcJIFVfvTrWDChBDW26g7B-s2v84t7rVuAU/s400/graph+strips2.JPG" width="280" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfP8znyLhF_6lr_QTe3ms1rjp3FVIbxSZ2lz_XP5a_5LgmeBON5ar5av5QJRz87PHRMz4CFu7senAk583vmLwqbhtP-MQzkEbdqVapYxE49dzfpzdUtdbdbcIcJKUikJfg1BZ-rfNjwSQ/s1600/envelope.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="270" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfP8znyLhF_6lr_QTe3ms1rjp3FVIbxSZ2lz_XP5a_5LgmeBON5ar5av5QJRz87PHRMz4CFu7senAk583vmLwqbhtP-MQzkEbdqVapYxE49dzfpzdUtdbdbcIcJKUikJfg1BZ-rfNjwSQ/s400/envelope.JPG" width="400" /></a><br />
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We folded the tool and the envelope and glued everything in our notebooks before proceeding.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_v2jXisf8HtHlROSN4CgGpgEQB4ILkK6-eIpgUZwKKi6bWIWb-i3Yh6qz-C7WTP9uYQlDTBPxzjsRy8uyazBIMvVVIJG6Pv8k5zfIXcRHHjO9w6zm52gJYgsckfa0gBA20hn5NvnFlOw/s1600/first+page.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_v2jXisf8HtHlROSN4CgGpgEQB4ILkK6-eIpgUZwKKi6bWIWb-i3Yh6qz-C7WTP9uYQlDTBPxzjsRy8uyazBIMvVVIJG6Pv8k5zfIXcRHHjO9w6zm52gJYgsckfa0gBA20hn5NvnFlOw/s640/first+page.JPG" width="436" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiab-mZpF9xsDBLjVpOfXEOs_UFeu7Ez3wXJ-yq-PQUgpzkFtuM593SG1D-RWXmrs3bcvhI2MIUBupEAUQEEdyR9sWdGdP8OMjH0ca1ut11IpD5yxRwJtzE5mWq6BjxbCur_LeCNt9tAP4/s1600/second+page.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiab-mZpF9xsDBLjVpOfXEOs_UFeu7Ez3wXJ-yq-PQUgpzkFtuM593SG1D-RWXmrs3bcvhI2MIUBupEAUQEEdyR9sWdGdP8OMjH0ca1ut11IpD5yxRwJtzE5mWq6BjxbCur_LeCNt9tAP4/s640/second+page.JPG" width="427" /></a><br />
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I walked through three cards with them, showing how to use the tool and how it helps us find the domain & range. So cool. So visual.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjThF6AzJ2HJVJF2yJoqkFbvxYqfebLVHJwesHDWHGETZMG_IRnUod6UhlaUDV9OybSZ5rZOmZ-6zF2Acb8BHQ4BZjVzQhvv93D2KRMBd_waAPK94fDYHNu7AA_sdHwC-fUOvB6VHczekk/s1600/cut+up+graph+cards.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjThF6AzJ2HJVJF2yJoqkFbvxYqfebLVHJwesHDWHGETZMG_IRnUod6UhlaUDV9OybSZ5rZOmZ-6zF2Acb8BHQ4BZjVzQhvv93D2KRMBd_waAPK94fDYHNu7AA_sdHwC-fUOvB6VHczekk/s400/cut+up+graph+cards.JPG" width="243" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsVqHbpmDYgMXkDQQXgSINbbN2UJjyL5HEciIcGwBz1iLtyPtULcRyQmuDt2tMica9oxnapVCHXtYz10499u_3BErKStf9T1ntRoxDPiSBOOndyuZuiRx_TO8L_IjLAywjqLN2Lpfyui8/s1600/find+domain.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsVqHbpmDYgMXkDQQXgSINbbN2UJjyL5HEciIcGwBz1iLtyPtULcRyQmuDt2tMica9oxnapVCHXtYz10499u_3BErKStf9T1ntRoxDPiSBOOndyuZuiRx_TO8L_IjLAywjqLN2Lpfyui8/s400/find+domain.JPG" width="338" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1HUx-6I9xKeRWot9kJ-7rnSyZqaN-4EdPFP3SrqvgZUGVoLsYm9bjsNYDTekLXScmNM4M62Y8uaBy70W1Ih1ngyQSWB2uWuZjfUbtidJDeGSA7FhGAChP5MYAY5dKjvAljhx6sv70nak/s1600/finding+range.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1HUx-6I9xKeRWot9kJ-7rnSyZqaN-4EdPFP3SrqvgZUGVoLsYm9bjsNYDTekLXScmNM4M62Y8uaBy70W1Ih1ngyQSWB2uWuZjfUbtidJDeGSA7FhGAChP5MYAY5dKjvAljhx6sv70nak/s400/finding+range.JPG" width="320" /></a><br />
They had to finish the other five cards.<br />
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Then do the purple graphs on their notebook pages.<br />
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Then go back and do the back of the homework (some really challenging ones on that!).<br />
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As students worked I displayed answers via my document camera. More practice on a HW worksheet. Quiz next class!<br />
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All my documents can be found <a href="https://app.box.com/s/fs01s6ih5id51w8x68x3mh0j1a3qot19">HERE. </a><br />
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The original idea comes from math=love http://mathequalslove.blogspot.com/2013/08/algebra-2-interactive-notebook-pages.html - thank you again! So many wonderful ideas in her blog. And she inspired me to do this blog to share my materials.mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0tag:blogger.com,1999:blog-2897305678042698487.post-43117255527112134602015-11-07T05:30:00.000-05:002015-11-07T05:30:01.920-05:00Polynomial Functions part 1We focus on Polynomial Functions now. This takes a few blocks. First we look at the general shape - identify end behavior & middle behavior (zeros & multiplicity) initially without using a grapher of any type and then verifying with their graphing calculator.<br />
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Students start by exploring the first three problems on the intro exploration sheet. Then we discuss anything they noticed (such as how they found the degree, the y-intercept and zeros - all the functions are in factored form at this point). Then we fill in our ISN foldable about polynomial functions.<br />
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inside:</div>
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At this point, we've run of time and I have them finish the exploration sheet for homework.<br />
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We start the next class with a few warm up problems but then get into the main objective of this lesson which is to be able to describe characteristics of a polynomial function and draw the graph without the use of the graphing calculator. (all the examples we do at this point are factored or easily factorable).<br />
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We go through some examples with the right hand reflect page in our ISN.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm9a33IfZMMW70Qviazahm15zMXc0Zkh_0iqY2Yrw3uQr2KhiR22sQhW2wsQ_PEoR0ZfGQc-a17t3QC0a8adKS60cCHt-oejoDZNMdQJJ7VAgtGg5M6VzQKKje10DA06WDXqBka52XbYM/s1600/poly+example.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm9a33IfZMMW70Qviazahm15zMXc0Zkh_0iqY2Yrw3uQr2KhiR22sQhW2wsQ_PEoR0ZfGQc-a17t3QC0a8adKS60cCHt-oejoDZNMdQJJ7VAgtGg5M6VzQKKje10DA06WDXqBka52XbYM/s640/poly+example.png" width="508" /></a><br />
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Then students have a set of problems to work through for homework. I'll also do a formative assessment on this before moving onto Polynomial Functions part 2.<br />
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<a href="https://app.box.com/s/7ejrxzm1wlay59dqw64up88n6vkdah38">All materials used can be found here. </a>mahikerhttp://www.blogger.com/profile/10168411241547834194noreply@blogger.com0