tag:blogger.com,1999:blog-53033074821589225652021-02-24T14:01:13.577-08:00Math Mama Writes...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger583125tag:blogger.com,1999:blog-5303307482158922565.post-9449786685116036472021-01-03T18:30:00.006-08:002021-01-03T18:30:53.302-08:00LaTex, a curse and a blessing<p>I've been making teaching materials on computers for over 25 years. Maybe 15 years ago, I was introduced to MathType, and it made my equations so much nicer. Now it doesn't work with Word, and you have to pay a yearly fee. No thanks. It seems crazy to me that MS Word doesn't have a better equation editor. (I don't really remember what I don't like about it, but I think it has annoyed me lots over the years.)<br /></p><p>I got a new computer in the Spring, and since then, whenever I need to make a formula, I've been using my old computer with an old version of Word, and my very old copy of MathType. Today I wondered if it was time to bite the bullet, and make a quiz using LaTex.</p><p>I've tried to learn a bit of Latex a number of times before, and it just felt overwhelmingly weird. I especially hated that I couldn't see what I was doing. This time was better in a number of ways. First, my colleague showed me <a href="https://www.overleaf.com/project" target="_blank">overleaf</a>, where I <i>can</i> see what I'm doing. You can choose split screen, and hit recompile after every little change.</p><p>The next thing that helped was that I got a bunch of materials from the author of the book I'll be using. (<a href="http://discrete.openmathbooks.org/dmoi3.html" target="_blank">Oscar Levin, <i><b>Discrete Mathematics: An Open Introduction</b></i></a>.) I used those as templates for my own work. I deleted what I didn't want, and began to add what I did want. (If you want to learn LaTEx (or TEx), and you don't have a bunch of materials someone else made that you can modify, <a href="https://www.overleaf.com/project/5d5cc0923947e379fd90a6bb" target="_blank">this quiz template</a><a href="https://www.overleaf.com/project/5d5cc0923947e379fd90a6bb" target="_blank"> </a>might be helpful.)</p><p>The reason I was using LaTex was the equations, but that was one of the things I didn't know how to do. This site, <a href="https://latex.codecogs.com/eqneditor/editor.php" target="_blank">codecogs</a>, came to the rescue!</p><p>I also needed to include an image of a Venn diagram. I read up (googled latex image), tried to do what they said, and my image ended up in a weird place, next to the questions. I guessed, and added a line that I saw in other places in my documents from Levin (\vskip 1em). I figure that's a vertical skip. I have no idea what the 1em is. (I tried 5em for more space. Nope.) It worked!</p><p>But the image was still too big. Read up again, use [scale=0.5], put it in the wrong place, so it doesn't work. Figure out the right position, it works! And now the image doesn't look right hanging out on the left. <a href="https://tex.stackexchange.com/questions/53862/how-do-i-align-an-image-to-centre" target="_blank">I read up</a>, use "the centered environment," and it is all just prefect!</p><p>Here's the centering:</p><p style="margin-left: 40px; text-align: left;">\begin{center}<br /> \includegraphics[scale=0.5]{venn10}<br />\end{center} <br /></p><p>That took me over an hour. (Maybe two.) I made a second version of that quiz in ten minutes. </p><p> </p><p>I'm learning...</p><p><br /></p><p><br /></p><p><b>Summary</b></p><p>Does LaTex seem way too complicated, but it still might be the answer to your problems?<br /></p><ul style="text-align: left;"><li>Use a simple environment like <a href="https://www.overleaf.com/project" target="_blank">overleaf</a> where the split screen lets you see what you've done.<br /></li><li>Start with <a href="https://www.overleaf.com/project/5d5cc0923947e379fd90a6bb" target="_blank">a template</a> you can modify.</li><li>Use something simple like <a href="https://latex.codecogs.com/eqneditor/editor.php" target="_blank">codecogs</a> to build your equations.</li><li>google your questions.</li></ul><p>Good luck! <br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com6tag:blogger.com,1999:blog-5303307482158922565.post-68173573210953162352020-12-31T19:10:00.001-08:002020-12-31T19:46:22.916-08:00Square & Triangular Numbers<p> It's my vacation. And here I am, playing with math. Woo hoo.</p><p><br /></p><p>If you've played with this problem before, perhaps this is boring and old hat. But I've seen the question many times, and never before have I followed up on it.</p><p><br /></p><p>I just got a book I ordered. <i><b>A Friendly Introduction to Number Theory</b></i>, by Joseph Silverman. THe very first problem he asks the reader to attempt is:</p><p style="margin-left: 40px; text-align: left;">Exercise 1.1. The first two numbers which are both squares and triangles are 1 and 36. Find the next one, and if possible, the one after that. Can you figure out a way to efficiently find triangle-square numbers? Do you think there are infinitely many?</p><p style="text-align: left;">I found the next one easily, by making lists on paper of the square and triangular numbers. It was about 6 times as big as 35 (which is about 6 times as big as 1). So I figured it would take too long to find another by hand. I wrote <a href="https://sagecell.sagemath.org/?z=eJx1kM0KwyAQhO-C7yA5aSs06c3A9lEKgWoRgrbu5v2r_UkNmLm5s_M5ijBwhs9lShbweuaMyoCSn8J9tkAHScdBnbLhYhJe-CBStqw0xvS9GjkTWd6JD-PyS36NIoKMqI4NdpnbGevUigRoMIseyQeSnYtLuIkY7Nhp1KTzK_SKVtsIAtZVGu3eW9V3bFZ3mjfa71z2J78AKSpZsw==&lang=sage&interacts=eJyLjgUAARUAuQ==">a Sage script</a>. (It took me a few tries. I had lots more print statements until I was sure it was working.) I now have 7 of them. But more importantly, I've found a pattern. If you want to play with this, I would recommend not reading further.</p><p style="text-align: left;"><br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;">.</p><p style="text-align: left;"><br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;">.</p><p style="text-align: left;"><br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;">.</p><p style="text-align: left;"><br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;">.</p><p style="text-align: left;"><br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;">.</p><p style="text-align: left;">The business about each one being about 6 times as big as the one before looked promising. So I checked. Let's call them m (for matching numbers), where the actual number is m<sup>2</sup>.</p><p style="text-align: left;">m0 = 1, </p><p style="text-align: left;">m1 = 6*m0=6,</p><p style="text-align: left;">m2 = 6*m1 - 1 = 35,<br /></p><p style="text-align: left;">m3 = 6*m2 - 6 = 204,</p><p style="text-align: left;">m4 = 6*m3 - 35 = 1189.</p><p style="text-align: left;">At this point, it becomes clear that m(i) = 6*m(i-1) - m(i-2). And that's where I am now. I don't really know that this will continue to work forever. But it does continue for all the numbers I've found using Sage. And I just found one more to see if it continues further. It does.</p><p style="text-align: left;">Next step, proof. I will see if that's something I can do.</p><p style="text-align: left;"><br /></p><p style="text-align: left;">Edited to add:</p><p>I just found a closed form for the formula. It's ugly but it works. (I learned how to do that step from Oscar Levin's <i><b>Discrete Mathematics: An Open Introduction</b></i>, in 2.4, Solving Recurrence Relations. That's the book I'll be using to teach discrete math from this coming semester.) </p><p> </p><p><i><b>Now</b></i> the next step is proof.... <br /></p><p style="text-align: left;"> <br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-2208584645715885982020-12-15T20:39:00.001-08:002020-12-23T07:39:37.674-08:00Getting Better at Canvas<p> I am not a Canvas expert, but I've learned a lot this past semester, and hope to keep learning more.</p><p><br /></p><p>This post is a compilation of some of the things I've learned that make Canvas better for me and my students.</p><p> </p><p><b>Images</b><br /></p><p>I took a course offered by my employer (Contra Costa Community College District) called Becoming an Effective Online Instructor (BEOI). In the course they recommended using lots of pictures in our Canvas pages. I haven't gotten to the point of "lots" yet, but I'm trying to become more aware of what images will help students learn mathematical concepts, and also what mathematical images bring beauty to the screen. </p><p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-NA4YWvFUWMM/X9Ra7NLDSxI/AAAAAAAAF-A/QXrbsmTaTAw7_yv97UJMcgSJ0WZud-QKQCLcBGAsYHQ/s1800/c1%2BTorus%2B4D%2Btb%2Bdc.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1358" data-original-width="1800" height="301" src="https://1.bp.blogspot.com/-NA4YWvFUWMM/X9Ra7NLDSxI/AAAAAAAAF-A/QXrbsmTaTAw7_yv97UJMcgSJ0WZud-QKQCLcBGAsYHQ/w400-h301/c1%2BTorus%2B4D%2Btb%2Bdc.png" width="400" /></a></div><br /><p>I love this image, titled Banded Torus, by <a href="http://www.math.brown.edu/tbanchof/TFBCON2003/art/welcome.html">Thomas Banchoff</a> and Davide Cervone. I recently realized that part of its power for me was its black background. So I changed the cover images for my calculus and precalculus courses, to incorporate a black background. Both of these are done on desmos in reverse contract. The originals, with white background, were nowhere near as lovely.</p><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-RboKUNvjLWU/X9RbSA_lLXI/AAAAAAAAF-M/fuRI-qRpqaUMVC_CzXNA3v-lfnY6tGqAQCLcBGAsYHQ/s850/calculus%2Bimage.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="700" data-original-width="850" height="235" src="https://1.bp.blogspot.com/-RboKUNvjLWU/X9RbSA_lLXI/AAAAAAAAF-M/fuRI-qRpqaUMVC_CzXNA3v-lfnY6tGqAQCLcBGAsYHQ/w285-h235/calculus%2Bimage.png" width="285" /></a></div><p></p><p><br /></p><p>For calculus, I wanted to show both slope and area. <br /></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-4x3G0ClROxc/X9RbPyv0gXI/AAAAAAAAF-I/CWTuf2LDK2UpMyvWu4atlUwiU7GyTEobQCLcBGAsYHQ/s1804/171%2Bhome%2Bimage.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="978" data-original-width="1804" height="187" src="https://1.bp.blogspot.com/-4x3G0ClROxc/X9RbPyv0gXI/AAAAAAAAF-I/CWTuf2LDK2UpMyvWu4atlUwiU7GyTEobQCLcBGAsYHQ/w345-h187/171%2Bhome%2Bimage.png" width="345" /></a></div><p></p><p><br /></p><p>For precalculus, I wanted to show all of the functions we study (along with the circle). I did leave out the rational functions, not wanting the image to look too busy. <br /></p><p><br /></p><p><br /></p><p><b><br /></b></p><p><b>Orientation</b></p><p>That BEOI course offered very specific ideas about how to set up an orientation module. (I had to do one their way for the course, and then I modified it to make it my own for my students.) One of the items in it is a quiz. I loved putting that together. I tell students where the answer to each question is (as part of the question), so they can look it up. Partly, it's a way to emphasize certain things from all of the pages I am hoping they will have read. (Yes, you can call me at home! But not after 8pm.), and it's also a chance to be silly (how many chickens does Sue have?). It also allows students to start out the semester with a perfect quiz score (hopefully!).<br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tOmMaPWDBOk/X9Rn9qtBDtI/AAAAAAAAF_I/hMij-agymMYkzwHEnGQG8xJ9pv9wWlWqwCLcBGAsYHQ/s1434/orientation%2Bquiz%2Bquestion.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="732" data-original-width="1434" height="204" src="https://1.bp.blogspot.com/-tOmMaPWDBOk/X9Rn9qtBDtI/AAAAAAAAF_I/hMij-agymMYkzwHEnGQG8xJ9pv9wWlWqwCLcBGAsYHQ/w400-h204/orientation%2Bquiz%2Bquestion.png" width="400" /></a></div><p><b>Zoom Recordings</b></p><p>I guess Zoom saves these already, but I wanted them listed in my modules. So I had a module with links to each day's recording. In a mid-semester survey, two students requested that the various topics covered be listed with timestamps. I don't have time to do that, but I figured out a way to allow students to do it for each other. I have one page in each unit where I link to each recording by date, and list the topics we covered underneath. I set that page so that students can edit it. (They didn't this semester, but if we start out this way, and they get a bit of extra credit for it, we might be able to jointly build a great resource.)</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vDMxfzE1sKA/X9ReN5rBPxI/AAAAAAAAF-k/WqkJZHp8sZ0vhTYJIR7sU8f4DLYDZ8zHgCLcBGAsYHQ/s1540/unit%2B3%2Bzoom%2B%2Brecordings%2Bimage.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1540" data-original-width="1382" height="400" src="https://1.bp.blogspot.com/-vDMxfzE1sKA/X9ReN5rBPxI/AAAAAAAAF-k/WqkJZHp8sZ0vhTYJIR7sU8f4DLYDZ8zHgCLcBGAsYHQ/w359-h400/unit%2B3%2Bzoom%2B%2Brecordings%2Bimage.png" width="359" /></a></div><br /><p><b>Quiz & Test Retakes</b></p><p>Until this semester, I did not use the Canvas grades function. I do my grading using Excel, and it has lots more flexibility for my crazy formulas that calculate the grade four different ways and take whichever is best for the student. But everything was online this time. So that's where the grades were. I turned off the totals, so students wouldn't see the wrong scores that Canvas figured.</p><p>I allow students to take quizzes multiple times. (New version each time, of course.) And they get two chances on most tests. I started out building a new Canvas assignment for each retake. What a mess to figure grades! I finally realized that Canvas would accept multiple attempts on an assignment, and allow me to look at each one. That feature works great.</p><p>There is a "hide grades" feature that is supposed to hide the grades until I'm ready to post them. But it apparently doesn't hide my comments, which defeats the purpose. (Since I explain my grading in the comments.) Maybe there's a better way to do that, and I'll learn it soon. [Edit: After I wrote this post, I found out that there is indeed a better way. In the gradebook, go to the assignment, at the name of it, click on the three dots, choose 'Grade Posting Policy', and choose manually. Then remember to 'Post Grades' when you're done.]<br /></p><p> </p><p><b>Organizing Content </b><br /></p><p>The Canvas "modules" serve as containers for each of my units. So each one starts with a "unit sheet", giving an introduction to the ideas they'll be learning about, objectives, and a schedule. That schedule is what I want my students to think of as their home base in my class. I add details to it daily, I highlight the current class session, and I link to pages and assignments in it. I add more detail to it when I'm prepping my next class. It works great for me, and I want it to work great for my students. I put a link to it on the Home page, so it's easy to get to.<br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-iK6gHhUyvIg/X9Rh2PfR7YI/AAAAAAAAF-w/WrlkBzdNdIouxpLXYdwbn8G95IFPR0btwCLcBGAsYHQ/s1634/unit%2B1%2Bunit%2Bsheet.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1424" data-original-width="1634" height="349" src="https://1.bp.blogspot.com/-iK6gHhUyvIg/X9Rh2PfR7YI/AAAAAAAAF-w/WrlkBzdNdIouxpLXYdwbn8G95IFPR0btwCLcBGAsYHQ/w400-h349/unit%2B1%2Bunit%2Bsheet.png" width="400" /></a></div><p><b> </b></p><p><b>Community Page-Building</b></p><p>Canvas pages start out as editable only by the teacher. But you can change that to allow students to edit a page. Our fist topic in our second unit (in trigonometry) was radians, and I wanted them to do something after our first test, before that next class session. So I created this page, and I told them to find the best videos online that explain radians. I think comparing video explanations was a great way for them to be thinking about whether they really understood the concept.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-bk3c07-nQSA/X9Rlq0WHZII/AAAAAAAAF-8/KMxpyLtiq98KrmGUpPlnKvQusGIw0mGmQCLcBGAsYHQ/s1828/radian%2Bvideos%2Bpage.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1454" data-original-width="1828" height="319" src="https://1.bp.blogspot.com/-bk3c07-nQSA/X9Rlq0WHZII/AAAAAAAAF-8/KMxpyLtiq98KrmGUpPlnKvQusGIw0mGmQCLcBGAsYHQ/w400-h319/radian%2Bvideos%2Bpage.png" width="400" /></a></div><br /><p><br /></p><p><br /></p><p><b>Next Semester</b></p><p>I am still thinking about how to get students to participate more, and will be looking for ideas to help with that. I know I should make a few videos where I explain some of the key concepts. But I seem to be resisting doing that.<b> <br /></b></p><p><br /></p><p>What have you learned recently about how to use Canvas well?<br /></p><p></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-62949390883429474402020-11-12T20:42:00.001-08:002020-11-12T20:42:10.021-08:00Note-Taking & Learning Something New at 64<p>I've been teaching for over 30 years, almost all of it at the community college level. So I've gotten pretty used to what I do. (But not bored. I still discover new ideas every semester, and I still love connecting with students.)</p><p>That changed with quarantine. Before 2020, I was pretty sure that I never wanted to teach online. It looked like way more work, and it was clear to me that I wouldn't be able to have the same level of connection with my students in an online class. I was right about both things, but (amazingly, to me) I am enjoying teaching online. </p><p>I meet my students in Zoom two days a week. Most of them won't turn their cameras on, and I want to respect that. (I offered extra credit for cameras on, and I get to see 2 to 5 faces each day. It's better than none.)</p><p>I have a light load this semester. Just two classes. And it still feels like full-time work. Next semester I'll have over twice as many units (in 3 classes). I'm starting to prepare ahead of time, so I don't drown.</p><p>I started taking notes for the Discrete Math book I'll be using, and after I wrote up some notes, I went back and wrote an introduction to note-taking. Tonight I described it to my bother (who's becoming a teacher), and realized that it was a bit of an epiphany for me.</p><p>I have terrible handwriting, and always thought I didn't know how to take good notes. I copy the board in a math class, just like everyone else. That's not really note-taking to my way of thinking. I highlight the good bits when I'm reading, and when I come to an example, I try to do it myself before looking at the author's steps. But notes? Nah, that just never seemed like one of my skills.</p><p>Well, I was a little excited as I finished up my notes for the first section of the textbook. I had set the Canvas page so that students could edit it too, and so I had purposely left some parts of my notes incomplete. As I looked at what I had written and did a bit of rearranging, I saw some patterns.</p><p>So I wrote this introduction:</p><p style="margin-left: 40px; text-align: left;">How do you take notes when you read? My reading notes may surprise you. I see 4 types of things that I'm doing in my notes:</p><ul style="margin-left: 40px; text-align: left;"><li>The first, organizing by making lists, will be familiar to you. </li><li>But I am also trying to connect a new term to other meanings outside of math. </li><li>And I am reacting to what I read (surprise, and noticing how powerful something feels). </li><li>I also made up my own example.</li></ul><p>That seemed kind of cool.</p><p>Then, when I talked to my brother, I realized that I had always thought I was no good at taking notes. (I didn't think I really needed to be any better at it, because I am good at most academics anyway. But...) I never thought I could teach students how to take better notes. And I realized that this one task I gave myself, to make some reading notes for the textbook, suddenly showed me that I know a lot about reading math and taking notes that I can share with students.<br /></p><p>So that's my epiphany. I do know how to take good notes, and <b><i>now</i></b> I know how to describe that process to students.</p><p><br /></p><p>What helps you conquer a text you're reading? Do you take "good notes"? What does that mean to you?<br /></p><p><br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-31555209085099675602020-09-20T10:57:00.002-07:002020-09-20T10:57:18.517-07:00Division by 0<p>[Once again, I have written something for my class that I think will be valuable for others.]<br /></p><p><span style="font-size: 14pt;"><strong>Big question:</strong> What are the values of <img alt="LaTeX: \frac{3}{0}" class="equation_image" data-equation-content="\frac{3}{0}" height="44" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B3%257D%257B0%257D" title="\frac{3}{0}" width="20" />, <img alt="LaTeX: \frac{0}{3}" class="equation_image" data-equation-content="\frac{0}{3}" height="44" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B3%257D" title="\frac{0}{3}" width="20" />, and <img alt="LaTeX: \frac{0}{0}" class="equation_image" data-equation-content="\frac{0}{0}" height="45" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B0%257D" title="\frac{0}{0}" width="20" />?</span></p><p> </p><p><span style="font-size: 14pt;">We want to be able to look at each of these fractions, know what it equals, and understand why. This becomes vital in calculus. [Note: Many students have trouble with this. It may be because elementary teachers are often uncomfortable with division, and teach it by memorization, instead of as something deep to understand. Or it may be that this is deep, and our brains need more time to really make sense of it.]</span></p><p> </p><p><span style="font-size: 14pt;">To help ourselves understand this, we tie it to something simpler that we understand better. Division is the <em>inverse</em> of multiplication (ie they undo each other). So it will help to explore how the two operations are connected.</span></p> <p><span style="font-size: 14pt;">We start with a very concrete and simple problem: <img alt="LaTeX: \frac{6}{3}=2" class="equation_image" data-equation-content="\frac{6}{3}=2" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B6%257D%257B3%257D%253D2" title="\frac{6}{3}=2" /></span></p><p><span style="font-size: 14pt;">[Note: One notational problem with division is that it's written in different ways that place the numbers in opposite orders. <img alt="LaTeX: \frac{6}{3}=6\div3" class="equation_image" data-equation-content="\frac{6}{3}=6\div3" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B6%257D%257B3%257D%253D6%255Cdiv3" title="\frac{6}{3}=6\div3" />, but these are also equal to</span><span style="font-size: 14pt;"><a href="https://1.bp.blogspot.com/-XOn_Gr4n-AA/X2eW877-d1I/AAAAAAAAF7s/MeeFBDGS0U05ErQDoRJY3JYplkEluLggACLcBGAsYHQ/s1024/3%2Bgoes%2Binto%2B6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="656" data-original-width="1024" height="24" src="https://1.bp.blogspot.com/-XOn_Gr4n-AA/X2eW877-d1I/AAAAAAAAF7s/MeeFBDGS0U05ErQDoRJY3JYplkEluLggACLcBGAsYHQ/w38-h24/3%2Bgoes%2Binto%2B6.jpg" width="38" /></a></span><span style="font-size: 14pt;"></span><span style="font-size: 14pt;"></span><span style="font-size: 14pt;">. When I was young, I had trouble keeping track of which was which, so I would write down an easy problem, like this one, to help me remember.]</span></p> <p><span style="font-size: 14pt;">Now we consider the multiplication problem that goes with this division problem: <img alt="LaTeX: \frac{6}{3}=2\Longleftrightarrow3\cdot2=6" class="equation_image" data-equation-content="\frac{6}{3}=2\Longleftrightarrow3\cdot2=6" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B6%257D%257B3%257D%253D2%255CLongleftrightarrow3%255Ccdot2%253D6" title="\frac{6}{3}=2\Longleftrightarrow3\cdot2=6" />, and we can say that 6 divided by 3 is 2 <em><strong>because</strong></em> <img alt="LaTeX: 3\cdot2=6" class="equation_image" data-equation-content="3\cdot2=6" src="https://4cd.instructure.com/equation_images/3%255Ccdot2%253D6" title="3\cdot2=6" />.</span></p><p> </p><p><span style="font-size: 14pt;">Let's use T for top, B for bottom, and A for answer, and rewrite this equivalence of a division problem and its associated multiplication problem, in a way that will always be true: <img alt="LaTeX: \frac{T}{B}=A\Longleftrightarrow B\cdot A=T" class="equation_image" data-equation-content="\frac{T}{B}=A\Longleftrightarrow B\cdot A=T" src="https://4cd.instructure.com/equation_images/%255Cfrac%257BT%257D%257BB%257D%253DA%255CLongleftrightarrow%2520B%255Ccdot%2520A%253DT" title="\frac{T}{B}=A\Longleftrightarrow B\cdot A=T" /></span></p><p><span style="font-size: 14pt;">In the fraction (or division), we have top over bottom gives answer, and that gives us a multiplication problem where the original bottom times the answer from the division gives us the original top. [Note: I am purposely avoiding the proper terms: numerator or dividend, denominator or divisor, and quotient (for the answer). For anyone who gets those terms mixed up, it's easier just to focus on position for the moment.]</span></p> <p><span style="font-size: 14pt;">Now we are ready to consider each of the three original questions, using this correspondence.</span></p><p><span style="font-size: 14pt;">1. Let's think about the multiplication associated with <img alt="LaTeX: \frac{3}{0}" class="equation_image" data-equation-content="\frac{3}{0}" height="44" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B3%257D%257B0%257D" title="\frac{3}{0}" width="20" />:</span></p><p><span style="font-size: 14pt;"><img alt="LaTeX: \frac{3}{0}=A\Longleftrightarrow0\cdot A=3" class="equation_image" data-equation-content="\frac{3}{0}=A\Longleftrightarrow0\cdot A=3" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B3%257D%257B0%257D%253DA%255CLongleftrightarrow0%255Ccdot%2520A%253D3" title="\frac{3}{0}=A\Longleftrightarrow0\cdot A=3" /></span></p><p><span style="font-size: 14pt;">So what do we multiply 0 by to get 3? Hmm. It seems that nothing works. There is no number that can multiply with 0 and give us 3. So the division problem (or fraction) has no solution, and we say that <img alt="LaTeX: \frac{3}{0}" class="equation_image" data-equation-content="\frac{3}{0}" height="36" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B3%257D%257B0%257D" title="\frac{3}{0}" width="16" /> is <em>undefined</em>. This is why we say "division by 0 is undefined".</span></p><p> </p><p><span style="font-size: 14pt;">2. <img alt="LaTeX: \frac{0}{3}=A\Longleftrightarrow3\cdot A=0" class="equation_image" data-equation-content="\frac{0}{3}=A\Longleftrightarrow3\cdot A=0" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B3%257D%253DA%255CLongleftrightarrow3%255Ccdot%2520A%253D0" title="\frac{0}{3}=A\Longleftrightarrow3\cdot A=0" />. Ahh, this one is easier. <img alt="LaTeX: 3\cdot0=0" class="equation_image" data-equation-content="3\cdot0=0" src="https://4cd.instructure.com/equation_images/3%255Ccdot0%253D0" title="3\cdot0=0" /> so the answer is 0.</span></p><p> </p><p><span style="font-size: 14pt;">3. <img alt="LaTeX: \frac{0}{0}=A\Longleftrightarrow0\cdot A=0" class="equation_image" data-equation-content="\frac{0}{0}=A\Longleftrightarrow0\cdot A=0" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B0%257D%253DA%255CLongleftrightarrow0%255Ccdot%2520A%253D0" title="\frac{0}{0}=A\Longleftrightarrow0\cdot A=0" />. Hmm, this time A could be any number, and the multiplication would be correct. This is still division by 0, so it is still undefined, but it is very different from the first case. We call it <em>indeterminate. </em>We can see why by looking at a rational function example.</span></p><p><span style="font-size: 14pt;">Example: <img alt="LaTeX: y=\frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}" class="equation_image" data-equation-content="y=\frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}" src="https://4cd.instructure.com/equation_images/y%253D%255Cfrac%257B%255Cleft(x-1%255Cright)%255Cleft(x-2%255Cright)%257D%257B%255Cleft(x%252B2%255Cright)%255Cleft(x-2%255Cright)%257D" title="y=\frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}" /></span></p><p><span style="font-size: 14pt;">When x= -2 or 2, this function will be undefined (because we have division by 0). But the function's behavior for x values very close to -2 is very different from its behavior for x values very close to 2.</span></p><p><span style="font-size: 14pt;"> <img alt="LaTeX: x=-2" class="equation_image" data-equation-content="x=-2" src="https://4cd.instructure.com/equation_images/x%253D-2" title="x=-2" /> is a vertical asymptote for the graph. This means that as x approaches -2, the y values approach <img alt="LaTeX: \pm\infty" class="equation_image" data-equation-content="\pm\infty" src="https://4cd.instructure.com/equation_images/%255Cpm%255Cinfty" title="\pm\infty" />. (This can be written "as <img alt="LaTeX: x\longrightarrow-2,\:y\longrightarrow\pm\infty" class="equation_image" data-equation-content="x\longrightarrow-2,\:y\longrightarrow\pm\infty" src="https://4cd.instructure.com/equation_images/x%255Clongrightarrow-2%252C%255C%253Ay%255Clongrightarrow%255Cpm%255Cinfty" title="x\longrightarrow-2,\:y\longrightarrow\pm\infty" />".) You can verify this by trying these x values: -2.1, -1.9, -2.01, -1.99,... (You can also use desmos to view the function.)<br /></span></p><p><span style="font-size: 14pt;">What happens near <img alt="LaTeX: x=2" class="equation_image" data-equation-content="x=2" src="https://4cd.instructure.com/equation_images/x%253D2" title="x=2" />? We see that the y value does not depend on the factor <img alt="LaTeX: \left(x-2\right)" class="equation_image" data-equation-content="\left(x-2\right)" src="https://4cd.instructure.com/equation_images/%255Cleft(x-2%255Cright)" title="\left(x-2\right)" />, because it cancels. So, as long as <img alt="LaTeX: x\ne2" class="equation_image" data-equation-content="x\ne2" src="https://4cd.instructure.com/equation_images/x%255Cne2" title="x\ne2" />, <img alt="LaTeX: y=\frac{\left(x-1\right)}{\left(x+2\right)}" class="equation_image" data-equation-content="y=\frac{\left(x-1\right)}{\left(x+2\right)}" src="https://4cd.instructure.com/equation_images/y%253D%255Cfrac%257B%255Cleft(x-1%255Cright)%257D%257B%255Cleft(x%252B2%255Cright)%257D" title="y=\frac{\left(x-1\right)}{\left(x+2\right)}" />. At <img alt="LaTeX: x=2" class="equation_image" data-equation-content="x=2" src="https://4cd.instructure.com/equation_images/x%253D2" title="x=2" />, this <i>would</i> equal 1/4. The function is not defined here, but now we can see that as <img alt="LaTeX: x\longrightarrow2,\:y\longrightarrow\frac{1}{4}" class="equation_image" data-equation-content="x\longrightarrow2,\:y\longrightarrow\frac{1}{4}" src="https://4cd.instructure.com/equation_images/x%255Clongrightarrow2%252C%255C%253Ay%255Clongrightarrow%255Cfrac%257B1%257D%257B4%257D" title="x\longrightarrow2,\:y\longrightarrow\frac{1}{4}" />.</span></p><p><span style="font-size: 14pt;">So why was <img alt="LaTeX: \frac{0}{0}" class="equation_image" data-equation-content="\frac{0}{0}" height="36" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B0%257D" title="\frac{0}{0}" width="16" /> called indeterminate? Because the value associated with it in a particular function is <em>determined</em> by other parts of the function. Although <img alt="LaTeX: \frac{0}{0}" class="equation_image" data-equation-content="\frac{0}{0}" height="36" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B0%257D" title="\frac{0}{0}" width="16" /> is undefined, we saw that, in this particular function the value of the function got close to 1/4 as the x value got close to 2, which is the number that would give us <img alt="LaTeX: \frac{0}{0}" class="equation_image" data-equation-content="\frac{0}{0}" height="36" src="https://4cd.instructure.com/equation_images/%255Cfrac%257B0%257D%257B0%257D" title="\frac{0}{0}" width="16" />. This concept goes with the concept of <em>limits</em>, one of the 3 major topics in calculus.</span></p><p> </p><p> </p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-57516849955161872852020-09-16T08:09:00.001-07:002020-09-16T08:09:14.057-07:00What sorts of things are impossible?Here's <a href="https://www.quantamagazine.org/when-math-gets-impossibly-hard-20200914/" target="_blank">an interesting article in QUanta, by David Richeson</a>. I'll be thinking about what else I might add to this post...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-30942048881334026082020-09-11T09:45:00.000-07:002020-09-11T09:45:34.524-07:00Solving Application Problems (in Trigonometry)<p>I started this blog in 2009, was active for about 6 years, and then not so much for the past 5 years. I wrote two posts in the spring, both related to online teaching. We were all trying to learn how to teach well as we scrambled to do it while learning. I was happy to keep seeing my students online, and Zoom was our class. I used Canvas a little but not much.</p><p>Over the summer I learned a lot about effective online teaching. (I'm still not sure it can ever be nearly as effective as in-person, but...) I developed my Canvas shells for each course, and I started the semester readier than I had expected to be. My Canvas shells are not done. I created a "module" that orients students to online learning and my course. And I created a module for our first unit. The rest is still in progress.</p><p>Today I added a page for my trig students, on solving application problems. I want to share it here. (And I may share lots of my Canvas "pages" here, sometimes with modifications.)<br /></p><p>Years ago, I modified George Polya's wonderful outline of problem solving steps. <a href="https://docs.google.com/document/d/1p-oSxNsF0KkH50I9hXuRTfRLZgG4OxXsaxMMeQciExw/edit" target="_blank">We start with that</a>. It's a good idea to print it out, <span style="font-size: 14pt;"><span class="instructure_file_holder link_holder ally-file-link-holder"><span style="font-size: 14pt;"><span class="instructure_file_holder link_holder ally-file-link-holder"><span style="font-size: small;">and turn to it whenever you're stuck.</span> </span></span> </span></span><span style="font-size: 14pt;"><span class="instructure_file_holder link_holder ally-file-link-holder"></span></span></p><p><span style="font-size: 14pt;"><span class="instructure_file_holder link_holder ally-file-link-holder"></span></span></p><div class="inline-block ally-enhancement ally-user-content-dropdown ally-grey-arrow-download-button"><br /></div><span style="font-size: 14pt;"></span><span style="font-size: 14pt;"></span><span style="font-size: 14pt;"></span><span style="font-size: 14pt;"></span><p></p><p><span style="font-size: 14pt;"><img alt="tree with shadow, pretty vs helpful" data-api-endpoint="https://4cd.instructure.com/api/v1/courses/55833/files/6499936" data-api-returntype="File" data-id="6499936" height="313" src="https://4cd.instructure.com/courses/55833/files/6499936/preview" style="float: right; max-width: 150px;" width="150" /></span></p><p><span style="font-size: 14pt;"><strong>Draw a Diagram.</strong></span></p><p><span style="font-size: 14pt;">Always start by drawing a diagram. This step is vital, and is a major part of "Understanding the Problem".<br /></span></p><p><span style="font-size: 14pt;">Your diagram does not need to be artistically good. It does need to show relationships well. An artist might show my shadow going off at an angle. But for a math diagram, it is better to show the right angle involved, <em>as</em> a right angle. </span></p><p><span style="font-size: 14pt;">In the diagrams on the right, the top drawing is prettier, and the shadow is more evocative, but the bottom drawing shows the right angle between a vertical object and its horizontal shadow, which is what will help you do your mathematical analysis.</span></p><p><span style="font-size: 14pt;"><strong>Example</strong> (#22 in 2.4, page 93): <span style="font-family: 'Times';">If the angle of elevation of the sun is 63.4° when a building casts a shadow of 37.5 feet, what is the height of the building? </span></span></p><p><span style="font-size: 14pt;">Draw your diagram now, labeling it with everything given and a variable for the value requested. (My drawing is below.)</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p><span style="font-size: 14pt;"><img alt="building with shadow, labeled" data-api-endpoint="https://4cd.instructure.com/api/v1/courses/55833/files/6500008" data-api-returntype="File" data-id="6500008" height="372" src="https://4cd.instructure.com/courses/55833/files/6500008/preview" style="float: right; max-width: 201px;" width="201" /></span></p><p><span style="font-size: 14pt;">.</span></p><p> </p><p><span style="font-size: 14pt;">I labeled the height of the building h. </span></p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p><span style="font-size: 14pt;"><strong>Write a Trig Equation.</strong></span></p><p><span style="font-size: 14pt;">In a simple problem, with only a few pieces of information this is all you need for the "Devising a Plan" step. We are given the value of the side adjacent (next to) the given angle, and we want to find the value of the side opposite the angle. (The hypotenuse is neither given nor asked for.) Which trig function uses adjacent and opposite? (Two of them do, but the one we use most of the time is...) </span></p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;">... <img alt="LaTeX: \tan\theta=\frac{opp}{adj}" class="equation_image" data-equation-content="\tan\theta=\frac{opp}{adj}" src="https://4cd.instructure.com/equation_images/%255Ctan%255Ctheta%253D%255Cfrac%257Bopp%257D%257Badj%257D" style="max-width: 81px;" title="\tan\theta=\frac{opp}{adj}" /><span class="hidden-readable"><span class="mjx-chtml MathJax_CHTML" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>&#x2061;</mo><mi>&#x3B8;</mi><mo>=</mo><mfrac><mrow><mi>o</mi><mi>p</mi><mi>p</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>j</mi></mrow></mfrac></math>" id="MathJax-Element-1-Frame" role="presentation" style="font-size: 114%; position: relative;" tabindex="0"><span aria-hidden="true" class="mjx-math" id="MJXc-Node-1"><span class="mjx-mrow" id="MJXc-Node-2"><span class="mjx-mi" id="MJXc-Node-3"><span class="mjx-char MJXc-TeX-main-R" style="padding-bottom: 0.331em; padding-top: 0.331em;"></span></span><span class="mjx-mfrac MJXc-space3" id="MJXc-Node-7"><span class="mjx-box MJXc-stacked" style="padding: 0px 0.12em; width: 1.196em;"><span class="mjx-denominator" style="bottom: -0.804em; font-size: 70.7%; width: 1.691em;"><span class="mjx-mrow" id="MJXc-Node-12"><span class="mjx-mi" id="MJXc-Node-15"><span class="mjx-char MJXc-TeX-math-I" style="padding-bottom: 0.472em; padding-top: 0.425em;"></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.28em; width: 1.196em;"></span></span></span></span></span></span></span></span><span style="font-size: 14pt;">, and this gives us <img alt="LaTeX: \tan63.4=\frac{h}{37.5}" class="equation_image" data-equation-content="\tan63.4=\frac{h}{37.5}" src="https://4cd.instructure.com/equation_images/%255Ctan63.4%253D%255Cfrac%257Bh%257D%257B37.5%257D" style="max-width: 107px;" title="\tan63.4=\frac{h}{37.5}" /><span class="hidden-readable"><span class="mjx-chtml MathJax_CHTML" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>&#x2061;</mo><mn>63.4</mn><mo>=</mo><mfrac><mi>h</mi><mn>37.5</mn></mfrac></math>" id="MathJax-Element-2-Frame" role="presentation" style="font-size: 114%; position: relative;" tabindex="0"><span aria-hidden="true" class="mjx-math" id="MJXc-Node-16"><span class="mjx-mrow" id="MJXc-Node-17"><span class="mjx-mi" id="MJXc-Node-18"><span class="mjx-char MJXc-TeX-main-R" style="padding-bottom: 0.331em; padding-top: 0.331em;"></span></span><span class="mjx-mfrac MJXc-space3" id="MJXc-Node-22"><span class="mjx-box MJXc-stacked" style="padding: 0px 0.12em; width: 1.399em;"><span class="mjx-denominator" style="bottom: -0.604em; font-size: 70.7%; width: 1.978em;"><span class="mjx-mn" id="MJXc-Node-24"><span class="mjx-char MJXc-TeX-main-R" style="padding-bottom: 0.378em; padding-top: 0.378em;"></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.28em; width: 1.399em;"></span></span><span class="mjx-vsize" style="height: 1.359em; vertical-align: -0.427em;"></span></span></span></span></span></span></span></p><p><span style="font-size: 14pt;"><span class="hidden-readable"></span><br /></span></p><p> </p><p><span style="font-size: 14pt;"><strong>Do a bit of algebra.</strong></span></p><p><span style="font-size: 14pt;">This is the "Carry out the Plan" step. To solve for h, we multiply both sides of the equation by 37.5:</span></p><p><span style="font-size: 14pt;"><img alt="LaTeX: 37.5\cdot\tan63.4=37.5\cdot\frac{h}{37.5}\:\:\Longrightarrow\:\:h=37.5\cdot\tan63.4=74.8857..." class="equation_image" data-equation-content="37.5\cdot\tan63.4=37.5\cdot\frac{h}{37.5}\:\:\Longrightarrow\:\:h=37.5\cdot\tan63.4=74.8857..." src="https://4cd.instructure.com/equation_images/37.5%255Ccdot%255Ctan63.4%253D37.5%255Ccdot%255Cfrac%257Bh%257D%257B37.5%257D%255C%253A%255C%253A%255CLongrightarrow%255C%253A%255C%253Ah%253D37.5%255Ccdot%255Ctan63.4%253D74.8857..." style="max-width: 466px;" title="37.5\cdot\tan63.4=37.5\cdot\frac{h}{37.5}\:\:\Longrightarrow\:\:h=37.5\cdot\tan63.4=74.8857..." /><span class="hidden-readable"><span class="mjx-chtml MathJax_CHTML" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>37.5</mn><mo>&#x22C5;</mo><mi>tan</mi><mo>&#x2061;</mo><mn>63.4</mn><mo>=</mo><mn>37.5</mn><mo>&#x22C5;</mo><mfrac><mi>h</mi><mn>37.5</mn></mfrac><mspace width="mediummathspace" /><mspace width="mediummathspace" /><mo stretchy="false">&#x27F9;</mo><mspace width="mediummathspace" /><mspace width="mediummathspace" /><mi>h</mi><mo>=</mo><mn>37.5</mn><mo>&#x22C5;</mo><mi>tan</mi><mo>&#x2061;</mo><mn>63.4</mn><mo>=</mo><mn>74.8857...</mn></math>" id="MathJax-Element-3-Frame" role="presentation" style="font-size: 114%; position: relative;" tabindex="0"><span aria-hidden="true" class="mjx-math" id="MJXc-Node-25"><span class="mjx-mrow" id="MJXc-Node-26"><span class="mjx-mn MJXc-space3" id="MJXc-Node-51"><span class="mjx-char MJXc-TeX-main-R" style="padding-bottom: 0.378em; padding-top: 0.378em;"></span></span></span></span></span></span></span></p><p><span style="font-size: 14pt;"><span class="hidden-readable"></span></span></p><p><span style="font-size: 14pt;">I pulled out my calculator for that last step (making sure it was in degree mode). Since our given length was given to tenths of a foot, I round, and give my final answer as <strong>74.9 feet</strong>.</span></p><p> </p><p><span style="font-size: 14pt;"><strong>Check your Solution.</strong></span></p><p><span style="font-size: 14pt;">This is the "looking back" step on the handout. If we look at our diagram, does a height of about 75 feet seem reasonable? Well, the height seems bigger than the shadow, and maybe about twice as big, so yes, it seems reasonable.</span></p><p> </p><p> </p><p><span style="font-size: 14pt;"><strong>Practice</strong>.</span></p><p><span style="font-size: 14pt;">If you get stuck on application problems, a good way to practice is to re-do problems that you've watched someone else do (perhaps on youtube). Try not to look at your notes. If you need to, go ahead and look. Do as much of the problem on your own as you can. If you looked at your notes at all, do it again the next day.</span></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-22631731360228947042020-06-25T23:10:00.000-07:002020-07-13T12:33:42.277-07:00Playful Math Education Carnival #139 (formerly known as Math Teachers at Play or MT@P)<div style="text-align: center;"><i>"It’s like a free online monthly magazine of mathematical adventures." (Denise Gaskins)</i></div><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-Md2kd7DWlfM/XvWN_XKA4QI/AAAAAAAAF4s/hbVZ2-1xB28o06eKKZHh-hKeMck8x_pqACLcBGAsYHQ/s1600/juneteenth%2Bnumbers.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="960" data-original-width="671" height="320" src="https://1.bp.blogspot.com/-Md2kd7DWlfM/XvWN_XKA4QI/AAAAAAAAF4s/hbVZ2-1xB28o06eKKZHh-hKeMck8x_pqACLcBGAsYHQ/s320/juneteenth%2Bnumbers.jpg" width="223" /></a><b> </b><br /><br /><br /><br /><b>Black Lives Matter.</b> How does that idea and movement intersect with math and play? It's hard to imagine play intersecting with the painful history of racism in the U.S. We can <a href="https://apps.urban.org/features/school-funding-do-poor-kids-get-fair-share/" target="_blank">collect data</a> to show how pervasive anti-Blackness has been and is. We can discuss how <a href="https://www.nsf.gov/news/news_summ.jsp?cntn_id=101776" target="_blank">math courses have been used</a> to filter out students from desirable professions (doctors, engineers, lawyers). We can discuss how Black people are more involved in the history of math than you'd guess from the Eurocentric naming. (<a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle#History" target="_blank">Check out who knew Pascal's triangle before Pascal!</a>) None of that is playful. But celebration can be playful. Let's celebrate <a href="https://www.juneteenth.com/history.htm" target="_blank">Juneteenth</a>!<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><h2><b>139</b></h2>Every number is cool.* Here are some ways 139 is cool:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-eCnjz09_BvE/XvWMvB5EuxI/AAAAAAAAF4Y/HcGp4KBGtYgRzQwteH9jhHSFGN8-P2_6ACLcBGAsYHQ/s1600/139.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="200" src="https://1.bp.blogspot.com/-eCnjz09_BvE/XvWMvB5EuxI/AAAAAAAAF4Y/HcGp4KBGtYgRzQwteH9jhHSFGN8-P2_6ACLcBGAsYHQ/s200/139.jpg" width="200" /></a></div><ul><li>139 is the sum of 5 consecutive prime numbers (19 + 23 + 29 + 31 + 37). </li><li>139 is the smallest prime before a prime gap of length 10. </li><li>137 and 139 form the 11th pair of twin primes. </li><li>139 is the 34th prime number. </li></ul><br /><b>Puzzle</b>: The digit sum is the result after adding the digits repeatedly until you get down to one digit. 139’s digit sum is 4. If you write 139 in base two, you get 100 1011, which still has a digit sum of 4. Does this always happen? If not, does it happen in any other bases?<br /><br /><br /><br /><br /><b>New Homeschoolers </b><br />I have a hunch the quarantine has moved lots of families from school to homeschooling. If you’re new to homeschooling, get ready to have fun playing with math. Most mathematicians are in it at least partially for the fun of it. We like to play with numbers, shapes, and logic. The more you play with math with your kids, the more likely they are to enjoy it.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><a href="https://1.bp.blogspot.com/-2nOHeH_Mlpk/XvWOdRfHh3I/AAAAAAAAF40/LDGWjBhZo7A7C8vHWxXow_qQfrmMmRlOQCLcBGAsYHQ/s1600/beast.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="523" data-original-width="540" height="193" src="https://1.bp.blogspot.com/-2nOHeH_Mlpk/XvWOdRfHh3I/AAAAAAAAF40/LDGWjBhZo7A7C8vHWxXow_qQfrmMmRlOQCLcBGAsYHQ/s200/beast.png" width="200" /></a>There are vast resources online to help you. Until 3rd grade, just play games, cook, measure, read mathy stories, and have fun with it all. If your kid wants a curriculum before that because they love math, then check out <a href="https://beastacademy.com/" target="_blank">Beast Academy</a>. It has levels 2 to 5 (topics correspond to grades 2 to 5, difficulty levels are a grade or two higher). Some families never use a curriculum; if you’re interested, you may want to explore unschooling. Math lovers eventually want to take classes, which you can do either through your local community college (I’ll be teaching trigonometry, pre-calculus, and calculus I online this fall) or <a href="https://artofproblemsolving.com/" target="_blank">Art of Problem Solving</a>. There are lots of other great resources; these are just my personal favorites.<br /><br />You might find ideas that work for you in my book, <a href="https://naturalmath.com/playingwithmath/" target="_blank"><i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i></a>. Or from other books from my publisher, <a href="https://naturalmath.com/goods/" target="_blank">Natural Math</a>. I also highly recommend Denise Gaskins’ <a href="https://denisegaskins.com/blog/" target="_blank">blog</a> (<a href="https://denisegaskins.com/2020/04/09/how-to-homeschool-math/" target="_blank">especially this post on homeschooling math</a>), <a href="https://denisegaskins.com/" target="_blank">website</a>, and <a href="https://tabletopacademy.net/playful-math-books/" target="_blank">books</a>. <a href="https://mathforlove.com/2016/04/how-to-help-your-kids-fall-in-love-with-math-a-guide-for-grown-ups/" target="_blank">Dan</a> and <a href="https://talkingmathwithkids.com/2015/08/31/let-the-children-play/" target="_blank">Christopher</a> have some good ideas about playing mathematically with kids too.<br /><br /><br /><br /><br /><b>Talking Math With Your Kids (#TMWYK)</b><br /><div class="separator" style="clear: both; text-align: center;"><b><a href="https://1.bp.blogspot.com/-QqQ8jlupCPg/XvWOoeQMLbI/AAAAAAAAF44/wRnlYp9TPFw9no1dIZ5BRIJSEStiYR5uwCLcBGAsYHQ/s1600/unschool.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="528" data-original-width="792" height="133" src="https://1.bp.blogspot.com/-QqQ8jlupCPg/XvWOoeQMLbI/AAAAAAAAF44/wRnlYp9TPFw9no1dIZ5BRIJSEStiYR5uwCLcBGAsYHQ/s200/unschool.jpeg" width="200" /></a></b></div><br /><br /><ul><li><a href="http://tjzager.com/2016/04/19/talking-math-in-ghirardelli-square/" target="_blank">Tracy Zager's daughter was thinking hard while mom sat there, exhausted.</a></li><li><a href="https://betweenthenumbers.wordpress.com/2019/03/08/tmwyk-overheard-on-muni-edition/" target="_blank">On the Muni, overheard by Breedeen Pickford-Murray.</a> </li></ul><br /><br /><br /><br /><br /><br /><b>Math & Language Play </b><br />One of my favorite math educators, <a href="http://www.marilynburnsmathblog.com/" target="_blank">Marilyn Burns</a>, invented a game where students look for $1 words. A=1¢, B=2¢, etc. You could combine math and any other subject by making $1 phrases. Sometimes kids like the simplest games. This might be a craze at your house. (My son used to love Shut the Box, a simple dice game that did nothing for me. It sure was good number practice for him.)<br /><br /><a href="https://aperiodical.com/2020/06/poetry-competition-%cf%80-ku/" target="_blank">π-ku, a competition</a>, in which all their favorites will be posted at the Aperiodical blog. I'll try:<br /><div style="text-align: center;">Three One Four.</div><div style="text-align: center;">Hmm.</div><div style="text-align: center;">Not very hard.</div><br /><br /><b>Games</b><br />So much of math is based on logic, any logic games you play will deepen your students' affinity for math. Here are a few others:<br /><ul><li><a href="https://blog.tanyakhovanova.com/2020/06/set-tic-tac-toe/" target="_blank">Set Tic Tac Toe</a>, described by Tanya Khovanova, invented by her students. You may want to play the basic game of Set for a few months before attempting this. But if I could figure out a way to do this at a distance, I'd love to try this out. </li><li><a href="http://planarity.net/" target="_blank">Planarity game</a>. (This is connected to a field of math called graph theory.) </li><li><a href="http://math.hws.edu/eck/js/symmetry/wallpaper.html" target="_blank">Play with wallpaper symmetries.</a></li></ul><br /><br /><br /><b>Math History</b><br />Podcasts aren't my thing. Yet. But if this series is as good as it sounds, I'll just have to figure this newfangled genre out. <a href="http://intellectualmathematics.com/opinionated-history-of-mathematics/" target="_blank">Opinionated History of Mathematics</a>. With an <a href="https://aperiodical.com/2020/06/podcasting-about-opinionated-history-of-mathematics-podcast/" target="_blank">interview and glowing review at Aperiodical</a>. <br /><br /><br /><br /><br /><b>Online Events</b><br />This summer <a href="https://www.artofinquiry.net/" target="_blank">Art of Inquiry</a> is hosting free science webinars on space, astrobiology, and AI for school children and their families. The webinars are led by university professors and industry experts. You can register for the events on <a href="https://www.eventbrite.com/o/art-of-inquiry-5603297429" target="_blank">Eventbrite</a>. Here is their June-July 2020 schedule:<br /><ul><li>Living Through a Revolution: Multi-messenger Astrophysics - Dr. Roopesh Ojha, GSFC NASA, June 26th </li><li>Figuring out the Earth from inside out - Dr. Kanani Lee, Lawrence Livermore National Laboratory, June 30th </li><li>Mars Rovers - Dr. Allan Treiman, Lunar and Planetary Institute, July 3rd </li><li>The search for life on Mars in XXI century - Dr. Alex Pavlov, GSFC NASA, July 10th </li><li>Where in the Universe did we come from? - Dr. Ethan Siegel, science author, "Starts with a Bang" Forbes contributor, July 23rd </li><li>Why we should build a Moon base - Dr. Ian Crawford, University of London, July 31st </li></ul> If you know of other math-related online events, please mention them in the comments. <br /><br /><br /><br /><br />This series of blog carnivals was founded and is kept going by the fabulous Denise Gaskins. <a href="https://denisegaskins.com/mtap/" target="_blank">You can find out more at her blog.</a> <a href="https://mathhombre.blogspot.com/2020/05/playful-mathematics-carnival-138.html" target="_blank">Last month's carnival was hosted by John Golden, the Math Hombre. Check it out!</a> <br /><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-o2T1asrfc4w/XvWMEA0cXmI/AAAAAAAAF4Q/6pEQduaCf4Mn_19xw1mCYs-7C8kNDb7TQCLcBGAsYHQ/s1600/bad%2Bnumbers.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="952" data-original-width="1080" height="176" src="https://1.bp.blogspot.com/-o2T1asrfc4w/XvWMEA0cXmI/AAAAAAAAF4Q/6pEQduaCf4Mn_19xw1mCYs-7C8kNDb7TQCLcBGAsYHQ/s200/bad%2Bnumbers.jpg" width="200" /></a> <br />-----<br /><span style="font-size: x-small;">*Well, sometimes their coolness is in their bad reputation (sounds like a few people I knew in high school) ... </span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com1tag:blogger.com,1999:blog-5303307482158922565.post-77760522413141998652020-06-18T19:44:00.001-07:002020-06-18T19:44:05.912-07:00<div class="_5pbx userContent _3576" data-ft="{"tn":"K"}" data-testid="post_message" id="js_aq4">The Math Teachers at Play Blog Carnival (aka Playful Math Education Carnival) will be a bit late this month.<br /><br /> I am looking for good posts now. If you can send me any links by Saturday, that would be great. I am hoping to put it together on Sunday.<br /><br /> Send your links to mathanthologyeditor@gmail.com, or post them here.<br /><br /> Want to know what a blog carnival is? Check out <a href="https://mathhombre.blogspot.com/2020/05/playful-mathematics-carnival-138.html" target="_blank">last month's, by my pal John Golden</a>.<br /><br /><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-55152477259805827432020-04-19T00:06:00.000-07:002020-04-19T00:06:37.770-07:00Corona Post #2: Teaching Online[#2 because my previous post in March on <a href="http://mathmamawrites.blogspot.com/2020/03/online-math-circle-pythagorean-triples.html" target="_blank">my online math circle</a> was due to people needing to take their math circles online when the shelter-in-place orders were just starting.]<br /><br /><a href="https://1.bp.blogspot.com/-tuXmtCoRr3c/XpvvoUB9jWI/AAAAAAAAF1E/p4Jb5KPO9QIjEwp1bb3tiaGIED4Ul81LQCLcBGAsYHQ/s1600/me%2Bwriting.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1600" data-original-width="1113" height="320" src="https://1.bp.blogspot.com/-tuXmtCoRr3c/XpvvoUB9jWI/AAAAAAAAF1E/p4Jb5KPO9QIjEwp1bb3tiaGIED4Ul81LQCLcBGAsYHQ/s320/me%2Bwriting.jpg" width="221" /></a>I've been teaching online for 4 weeks now, two before our spring break and two after. At first I was just trying to learn how to manage teaching on zoom. I bought a whiteboard that's still sitting on two chairs in my living room, and I sit in a tiny chair while I write on it. Not ideal, but I get to see my students, and I feel like I'm still working with them where they are, not just lecturing.<br /><br />(Some day I'll finally install it on my living room wall. I procrastinate with tasks like that. I'm not sure why it feels intimidating...)<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-Xt5CvgFsJ4I/XpvtyZWGSCI/AAAAAAAAF00/iK-bWlPA4pMjfsaFP4VMoipPNpJRF3d0ACLcBGAsYHQ/s1600/stars%2B%2526%2Bbars.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="448" data-original-width="868" height="165" src="https://1.bp.blogspot.com/-Xt5CvgFsJ4I/XpvtyZWGSCI/AAAAAAAAF00/iK-bWlPA4pMjfsaFP4VMoipPNpJRF3d0ACLcBGAsYHQ/s320/stars%2B%2526%2Bbars.png" width="320" /></a><br /><br /><br />A few weeks ago I made a google slides presentation for my Discrete Math course to explain a way of counting possibilities called <a href="https://docs.google.com/presentation/d/1a7uWiQrjy49chq1RxkvyHy6EB4ujwB4wR27g9rQyUGI/edit" target="_blank">Stars & Bars</a>. I had fun doing it. You're welcome to modify it and use it in your teaching.<br /><br /><br /><br /><br /><br /><br /><a href="https://1.bp.blogspot.com/-zE7Uwgh6zxo/XpvvfniJybI/AAAAAAAAF1A/hpxvm8yId_cmUCFkcaU3Mpw9qclJHpQAgCLcBGAsYHQ/s1600/maclaurin.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="370" data-original-width="664" height="178" src="https://1.bp.blogspot.com/-zE7Uwgh6zxo/XpvvfniJybI/AAAAAAAAF1A/hpxvm8yId_cmUCFkcaU3Mpw9qclJHpQAgCLcBGAsYHQ/s320/maclaurin.png" width="320" /></a><br /><br />Just now I made another. This one is for Calculus II, on <a href="https://docs.google.com/presentation/d/1nCUcTNIbgXQ5c5SJtH352WbYMLEMWQ1F1kRJMClrwOQ/edit#slide=id.g83b4e189f7_0_58" target="_blank">Taylor & Maclaurin Series</a> (really just a Maclaurin series). I was motivated by knowing that there would be too much writing for my little whiteboard. This presentation has a <a href="https://drive.google.com/file/d/1JtrcbZ7hLlxHQj3QIZOwUFiofgbVky6d/view?usp=sharing" target="_blank">handout</a> to go with it.<br /><br /><br /><br /><br /><br /><br /><br /><br />I'm also teaching Calculus I. I haven't made any cool new materials for that course yet. But I will...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-89413990039697147382020-03-23T15:00:00.000-07:002020-03-29T15:05:16.464-07:00Online Math Circle: Pythagorean Triples<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-G4UUOAGkiYw/XnkviZR20gI/AAAAAAAAFzw/68ZRFSJkKPQt3q5QxSZw8cV2KXNgMGMMACEwYBhgL/s1600/3-4-5-triangles.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="332" data-original-width="691" height="191" src="https://1.bp.blogspot.com/-G4UUOAGkiYw/XnkviZR20gI/AAAAAAAAFzw/68ZRFSJkKPQt3q5QxSZw8cV2KXNgMGMMACEwYBhgL/s400/3-4-5-triangles.jpg" width="400" /></a></div><br /><br />The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a<sup>2</sup>+b<sup>2</sup> = c<sup>2</sup>. Usually that makes at least one side something ugly like square root of 2. But a few combinations make all three sides whole numbers. Those are called Pythagorean triples. Here are a few of them: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 20-12-29.<br /><br /><br />Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)<br /><br /><br /><br />Edited to add:<br /><br />This <a href="https://4cd.zoom.us/rec/play/vsUvJeD5_Do3GtDAtwSDB6AqW9XrKq6sgCAar6JbzUe2BnMKMFfyb7YWMbOY7Ulru-Cg7C8-GK2QCS85?continueMode=true&_x_zm_rtaid=Ttrr4k29TpW-Z7JJMVK5Ug.1585518711595.2dc48300251a4ba1f7e3cdf8ef175510&_x_zm_rhtaid=898" target="_blank">online math circle happened on <b>Friday, March 27, at 10am PDT</b> (1pm EDT)</a>. [This link is to the zoom recording, along with its automatically produced (therefore hilariously bad) audio transcript.]<br /><br />I promised to write up some of it here.<br /><br />Way back in 2007, I read Bob and Ellen Kaplan's book, <a href="https://www.betterworldbooks.com/product/detail/Out-of-the-Labyrinth--Setting-Mathematics-Free-9780195147445" target="_blank"><i><b>Out of the Labyrinth: Setting Mathematics Free</b></i></a>, about the math circles they lead. It was such a discovery for me! I went to their first Summer Math Circle Teacher Training Institute, held at Notre Dame, and fell in love with this community. I kept going back for years, craving a discussion of math among equals, figuring out new ways of seeing. One summer we discussed Pythagorean triples, and that December I tried to rebuild what I had learned. I am blessed with a very bad memory, so what I did in December looked very different from what we had done in the summer.<br /><br />I was also exploring online, and ended up putting together a book that collected some of the best resources I had found: <a href="https://naturalmath.com/playingwithmath/" target="_blank"><i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i></a>.<br /><br />Our circle was prompted by Rodi Steinig's request for help learning how to use zoom for online math circles. I offered one of my favorite topics, and off we went. Participants came from as far away as Colombia (and farther?).<br /><br />We proved a few things, and explored a bunch more. I hope some participants went home eager to prove more on their own.<br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-87101425623023152232020-01-04T19:31:00.001-08:002020-01-04T19:31:20.882-08:00Multiplication Chart with PicturesIn the story I'm writing, Althea remembers a multiplication chart that was posted in their bathroom. It had cool pictures around the edges for many of the facts.<br /><br /><ul><li>2x3 was a 6-pack of soda.</li><li>2x6 was a carton of eggs.</li><li>8x8 was a chessboard.</li><li>The fives were sometimes collections of nickels, but 5x12 was the 60 minutes on a clock, and 5x6 was time too.</li></ul><br />I thought I knew of more iconic sets like these, but I can't think of any more as good as these. I'm hoping for help. Do you have images in your head for any of the multiplication facts?<br /><br />Maybe threes will be 3-leaf clovers. 6 of them have 18 petals. That doesn't seem nearly as iconic as the ones above, though.<br /><br />Fours could be legs on dogs. 6 dogs have 24 legs. Twos could be eyes on friends...<br /><br /><br /><b>What are your favorite images for multiplication facts?</b>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-46379242717000628332019-09-21T20:10:00.000-07:002019-09-21T22:15:35.717-07:00The History of Imaginary Numbers is a Soap OperaI first read about this history in <a href="https://www.abebooks.com/servlet/SearchResults?isbn=0471500305&cm_sp=mbc-_-ISBN-_-all" target="_blank"><i><b>Journey Through Genius</b></i></a>, by William Dunham, in the chapter titled Cardano and the solution of the cubic. It reads like a soap opera in which the math is done for glory, not for any possible connection to the real world. The people who came up with imaginary numbers thought they were fictitious elements in the process of solving a cubic, and never expected them to have any real meaning. Turns out they do. Imaginary numbers help scientists describe electrical current and probability distributions, among other things.<br /><br />Years ago, on Living Math Forum, a mom wrote in to ask for help. Her son had asked: “The square root of 1 is 1, so what's the square root of -1 ?” That inspired me to write a math poem, <a href="https://mathmamawrites.blogspot.com/2009/03/math-poems.html" target="_blank">Imaginary Numbers Do the Trick</a>.<br /><br />Recently I was having a lovely discussion with my editor, <a href="https://naturalmath.com/goods/" target="_blank">Maria Droujkova</a>, and another author, about math storytelling. I realized this topic might possibly make for a good children's story. I'm working on it now. As I think about it, I'm not sure how to find the right age range. The math seems like it requires high school, but the story could interest younger kids, I think.<br /><br /><br /><br /><b>The History</b><br />Here's a short version:<br /><ul><li><b>Scipione del Ferro</b> solves equations of the form <i>x<sup>3</sup> + mx = n</i> (called depressed cubics). On his deathbed, in 1506, he passes his method on to his student, <b>Antonio Fior</b>.</li><li><b>Niccolo Tartaglia</b> boasts that he can solve cubics of the form <i>x<sup>3</sup> + mx<sup>2</sup> = n</i>, so in 1535, Fior challenges him with 30 depressed cubics. (These challenges were a common feature of life as a mathematician in 1500's Italy, and provided a way for mathematicians to get more recognition and paying students.) Tartaglia's return problem list to Fior has a variety of problems. Tartaglia does not yet have a solution for the depressed cubic, and sweats it, working feverishly to try to figure it out. At the last moment, he succeeds, and solves all 30 problems. Fior does not do so well, and is humiliated. </li><li><b>Gerolamo Cardano</b> comes to Tartaglia, asking him to disclose his method. He begs repeatedly, and Tartaglia, now Cardano's guest in Milan, finally concedes. Cardano takes an oath of secrecy. Tartaglia writes his solution in cipher, <a href="https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-poem" target="_blank">as a poem</a> (!). </li><li>Cardano takes on a brilliant student, <b>Ludovico Ferrari</b>, with whom he shares the secret. Together, they solve the general cubic, and then Ferrari goes on to solve the quartic. But all their work depends on reducing these to the depressed cubic, which Cardano has sworn not to tell about.</li><li>Cardano and Ferrari travel to Bologna, and are able to inspect the papers of ... Scipione del Ferro, where they find the solution. Cardano figures that relieves him of his oath and publishes, in his 1545 book, <a href="https://www.maa.org/press/periodicals/convergence/mathematical-treasure-cardanos-ars-magna" target="_blank"><i>Ars Magna</i></a>. He gives both del Ferro and Tartaglia credit, but Tartaglia is furious.</li><li>In the book, Cardano lays out the steps for solving the general cubic. But in doing so, he introduces a mystery. The depressed cubic <i>x<sup>3</sup> - 15x = 4</i> clearly has solutions x = 4 and x = -2+-√3. And yet the formula found by Ferro, Tartaglia, Cardano, and Ferrari includes a √-121 for this equation. Cardano threw up his hands at the mystery. It was explored but not truly understood 30 years later by Rafael Bombelli. It took another almost two centuries for Euler to finally solve the mysteries of complex numbers.</li></ul><br /><ul></ul><a href="https://www.quora.com/What-were-the-renaissance-mathematics-competitions-in-Italy" target="_blank">Here's a nice write-up</a> I found online, but it suggests different facts than the version in Dunham's book. I will keep reading while I write, so I can hopefully get my facts right.<br /><br /><br /><br /><b>My Request</b><br />I'm looking for kids who would like to read my draft versions and tell me what parts they like. If you have kids who understand (at all) the notion of a square root and the idea of what a cubic equation is, would you ask them if they'd like to read my story? (I would, of course, mention them in my book if it gets published.) You, or your kids, can email me at mathanthologyeditor on gmail.<br /><br /><br /><br /><b>Just a Beginning</b><br /><div style="text-align: center;"><b>The Saga of the Imaginary Numbers</b></div><div style="text-align: center;"></div>“Mom, I’ve been thinking… If the square root of 1 is 1, what is the square root of -1?”<br /><br />“What a fun thing to think about, Althea! What have you figured out so far?”<br /><br />“I know that when I square 1 I get 1, and that’s why the square root of 1 is 1. But when I square negative 1, I get 1 too, so shouldn’t the square root of 1 be negative 1 too? But how can it be two things?”<br /><br />“Hmm, that’s a strange one, isn’t it? I think there are too many ones in this for me to keep track of things. Let’s switch to 3.<br /><br />"I’m going to try to say what you said, but with 3 and 9. 3 squared is 9, so the square root of 9 is 3. But negative 3 squared is still 9, So why isn’t the square root of 9 equal to negative 3 also? Is that basically the same question you asked?”<br /><br />“Yes. Except the square root of 9 can’t have two answers. Can it?”<br /><br />“Well, somebody a long time ago decided that there should be just one answer for the square root of a number. And so we say that there is the square root of 9 and also the negative square root of 9.” Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com10tag:blogger.com,1999:blog-5303307482158922565.post-35465726222910762332019-08-25T21:46:00.004-07:002019-08-25T21:48:16.916-07:00Another Semester, Starting with Good IntentionsMy handouts are copied, the piles are organized on my desk. My rosters are printed. (I entered student names into excel, so I can organize things my way.) I've looked over my computer folders and found a few more things to share tomorrow. And I'm getting better at using Canvas' features - I plan to have students evaluate the new activities online, to help me decide whether each activity stays, goes, or gets improved.<br /><br />Will I manage to set up a new student survey in Canvas for each new activity?<br />Will I blog about my classes, like I'd like to?<br />Will I do more activities and less lecture in each class?<br /><br />Once the semester gets rolling, it's hard for me to change things up. It's so much easier to do what I have done before. May my passion keep me improving, all through the term. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-41195648085784163842019-08-17T11:28:00.004-07:002019-08-17T20:01:21.697-07:00First Day, Once AgainI've seen some great advice for the first day of class. (<a href="https://www.chronicle.com/interactives/advice-firstday" target="_blank">Here's the one I've read more than once.</a> I've seen other great ideas, but I don't see them now.) I'd summarize my take on this article as:<br /><ul><li>(<i>Community</i>) Start learning names, and get them learning each other's names. </li><li>(<i>Expectations</i>) Don't spend much time on the syllabus; there are more important things to do. (Have them read it, and then you can quiz them on it the next day, or just ask for their questions.)</li><li>(<i>Learning, Expectations</i>) If you use group activities (which are a very effective way to help groups of students learn), then you want to introduce students to this on day one.</li><li>(<i>Curiosity</i>) An activity that helps them see what's coming in the course would be especially nice.</li></ul><br />Our classes start in <strike>ten</strike> nine days. I am getting ready...<br /><br /><br /><b>First day activity for Calculus I </b><br />I have them graph a parabola (y=x<sup>2</sup>), then draw a tangent to it at x=3. (Some don't know what that means, so I walk around checking.) And then <i><b>estimate</b></i> its slope. After they're done, I get to talk about what makes actually finding the slope hard - such a good intro to the course. And they've had time to review graphing a parabola.<br /><br /><br /><b>First day activity for Linear Algebra </b><br />I have them solve a system of 3 equations in 3 variables. I ask them to:<br /><ul><li>Write down a description of the process, </li><li>Solve the system, </li><li>Now figure how to check whether your answer is correct. (Naysayers, has the group done enough to be sure that the answer is correct?) </li><li>Extra: What does each equation represent geometrically? What does the solution represent geometrically? </li></ul>Once again, a great introduction to the themes of the course.<br /><br /><br /><b>First day activity for Geometry</b><br />I taught geometry earlier this summer for the first time. I had them draw a triangle (and make sure it was different than their neighbor's). Find the midpoints of each side (they could measure or fold). Connect each midpoint to the opposite vertex. I hoped most would do it well enough that the 3 connecting lines would intersect at one point. My goals were to highlight: vocabulary, shapes, construction (which we were not doing with straightedge and compass - yet), conjecture, and the possibility of proof.<br /><br /><br /><br /><b>First day activity for Precalculus</b> <br />I've been thinking for the last few weeks about what I'd like to do for Precalculus. I have found exciting activities in the past that turned out to be way too hard, and intimidated the students. I have a lovely fractions activity, but that doesn't represent what we'll do going forward.<br /><br />I am working hard to create an activity that looks at functions (and circles too) from 4 perspectives: equations, graphs, tables of values, and stories. I have 7 types of relationships (linear, quadratic, polynomial, rational, exponential, periodic, and circles). I don't have stories yet for the polynomial and rational. (My eternal gratitude to anyone who can give me a story I like for either of these.) And I won't show an equation for the periodic. (A trig function wouldn't make sense yet. But we'll get to discuss that.) So that makes 25 "clue sheets".<br /><br />I have 40 students, who I'll put in 10 groups of 4. Each group will start with two clue sheets. [So 5 of the sheets will not be handed out at first. I can label those as graphs on the back, keep them at the front, and let student turn them over once they're pretty sure they didn't find a graph match to their set.] Each will describe a different type of function/relation from one perspective and ask them to do a few things. Then they pair up with a clue sheet for each pair, and go looking for the matching clue sheets (same function/relation, different perspective). They go back to their group and explain to each other what they found. (I'll have extras up front, so anyone done early can work on a 3<sup>rd</sup> function/relation.) When we're done, we'll have a summary of the function types we'll be studying all semester.<br /><br />I dreamed some of this up late last night. When I started working today, I worried that it would be too hard. (I make up some crazy stuff sometimes when I'm falling asleep.) So my goal as I put this together has been to scaffold it enough. I am assuming some comfort with linear functions, and some familiarity (but not comfort) with quadratics and exponentials. They may not have encountered the others. (And most will not know any trig.)<br /><br />I put my <strike>first</strike> second draft into <a href="https://drive.google.com/file/d/1u93faKFuoc7UbTJCSK7QETdUBBX-H844/view?usp=sharing" target="_blank">a google doc here</a>. Your suggestions may help me improve it. (I decided to leave out the rational function. 6 functions with 4 clues each would be 24. One story and one equation are left out. That's 22. The last two clues will sit up front.)<br /><br />Edited (8/17): This is a great activity, but too complex for day one. I will do it on day two. On day one, we will review linear functions in a similar, but much simpler way. <a href="https://drive.google.com/file/d/119M2fMb-7qfptog4d2ldfguGHbeqswoa/view?usp=sharing" target="_blank">Here is my handout</a>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-VFwpR4i0q0w/XVi8jvTc8_I/AAAAAAAAFbc/x0-yMuiaE0k_1zMmFd_zjfmHyEdZJCn2gCLcBGAs/s1600/day%2Bone%2Bhandout.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="647" data-original-width="499" height="640" src="https://1.bp.blogspot.com/-VFwpR4i0q0w/XVi8jvTc8_I/AAAAAAAAFbc/x0-yMuiaE0k_1zMmFd_zjfmHyEdZJCn2gCLcBGAs/s640/day%2Bone%2Bhandout.png" width="492" /></a></div><br /><br /><br /><b>My Goals</b>:<br /><ul><li>Review plugging values in for x to find y. (Do they remember that b<sup>0</sup>=1? What else might trip them up?)</li><li>Review graphing.</li><li>See functions/relations in the context of modeling a situation.</li><li>Identify functions/relations by type.</li><li>See precalculus as a place to strengthen their understanding of all of this. </li></ul><br />If you use this activity, please let me know what changes you decide to make and how it goes.<br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-34334853681258422332019-06-17T20:02:00.001-07:002019-06-17T20:02:29.449-07:00Teaching Geometry - Journal Post #1I usually try to write blog posts with a point. I don't often have the energy for that lately. But I do want to keep track of how this class goes... <br /><br />Today was the first day of my geometry class. (It is the first time ever for me to teach geometry. I have done lots and lots of prep.) It goes for 6 weeks, and meets 3 1/2 hours a day. (Yikes!) My students are mostly high school students, though this is a college class. The college screwed up registration. I had 28 students on my list, but a bunch thought they were on a waitlist (only happens when a class is full with 40 students), and came in not yet registered. I now have 35 or 36 students. Of course I ran out of handouts.<br /><br />I gave what I thought were good directions for our first activity, and students managed to do strange other things.<br /><br /><br />My directions:<br /> <br /><div align="center" class="MsoNormal" style="text-align: center;"><b style="mso-bidi-font-weight: normal;">Welcome and Triangle Activity</b></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This activity is an introduction to the themes of the course. The more thought and care you put into it (the activity and the course), the more learning and joy you will get out of it.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">With a straightedge, draw a big triangle on this page. Make it as weird as you want as long as it is a triangle. Make sure it looks different than the triangle your neighbor draws.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Now you are going to draw three lines inside it. Each line will go from a corner (aka <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">vertex</i></b>) to the middle of the opposite side (aka <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">midpoint</i></b>). These lines are called the <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">medians</i></b> of the triangle. How will you find the midpoints? Do this as accurately as you can. Now look at what others have drawn. What do you notice?</div><div class="MsoNormal"><br /></div><div class="MsoNormal">[Can you guess how this activity connects with the themes of the course?]</div><div class="MsoNormal"><br /></div><div class="MsoNormal">***</div><div class="MsoNormal"><br /></div><div class="MsoNormal">I should have had them find the midpoints first. And <b><i>then</i></b> talk about connecting those to opposite vertices. (Fixing the handout now.) Some people connected the midpoints to each other. Some drew a new triangle inside the first triangle. (Pretty but nothing like what I asked for.)<br /><br />So the activity took lots longer than I expected. Sadly, I didn't note the time. (I have a tutor working with me in class. I will ask her to keep track of how long things take.)<br /><br />I thought I could manage to lecture on 4 sections of our textbook in about an hour and a half. No way. I enjoyed what I did with the students, but we only got through 2 sections.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Some of what I did was impromptu. I asked them to each write down their own definition of angle. Then I asked for volunteers to read their definitions. I talked about courage. I waited. Finally I got a volunteer. And then another. I did not criticize their definitions, even though I didn't like them much. The third one was closer. He was struggling for a word for how far the one side is from the other. I repeated his definition, with a big pause where he had been stuck. And then I said rotation. We talked about things that rotate.<br /><br />One of the book problems we did just asked them to measure a line segment against a ruler printed in the book, and angles on a protractor. It was hard for them. And that was good for me to see. It tells me where we are.</div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"><br /></div><div class="MsoNormal">I hated our textbook when I first started reading it, but eventually it seemed fine to me. I think the first chapter has more problems than the rest of the book.<br /><br />1.1 Reasoning. They talked about types of reasoning and included 'intuition' (along with inductive and deductive reasoning). No. Intuition helps us guess what we might want to try, and helps us see things that might be true. But I don't like them calling it one of 3 types of reasoning.</div><div class="MsoNormal"></div><div class="MsoNormal"></div><div class="MsoNormal"><br /></div><div class="MsoNormal">1.2 on measurement turns out to be more useful than I expected. Looking it over now, there's a lot more there that I haven't gotten to. Good thing I've asked them to turn in their notes on their reading. I hope that helps. (They are supposed ot read the first 4 sections.)</div><div class="MsoNormal"></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The last hour and a half is devoted to labtime. We will be using lots of labs from Henri Picciotto's <a href="https://www.mathed.page/geometry-labs/index.html" target="_blank">Geometry Labs</a> (free). I had hoped to do 3 of them today. We didn't even finish one. (Lab 1.1 involves putting different shapes of blocks around a point.) I had also hoped to introduce them to <a href="http://www.euclidthegame.com/" target="_blank">euclidthegame</a>. We'll do that tomorrow.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Quiz first thing tomorrow. I am hoping to keep them on top of things by quizzing daily.<br /><br />We are part of a program called High School STEM Connection. 20 of my 35 students are part of that program, and are required to come to tutoring after class. Most of them did. They got most of their homework done. Yay them. I love spending time with them where I'm not in charge. I'm just there to help out when they need me.</div><div class="MsoNormal"></div><div class="MsoNormal"><br />I am hoping to complete 5 sections of the book tomorrow. I'm tempted to throw out the cool group activities, but I know I shouldn't. I think I'll just see how far we get, like today. And then tomorrow evening I'll have to think again about how to adjust.<br /></div><style><!-- /* Font Definitions */ @font-face {font-family:"ＭＳ 明朝"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:1 134676480 16 0 131072 0;} @font-face {font-family:"ＭＳ 明朝"; mso-font-charset:78; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:1 134676480 16 0 131072 0;} @font-face {font-family:Cambria; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-1610611985 1073741899 0 0 159 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --></style>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-85576729872421086122019-05-19T20:49:00.000-07:002019-05-19T20:49:39.165-07:00Teaching Basic Geometry<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-yfiK_AiPLwE/XOIiYs3u_CI/AAAAAAAAFOM/T8MRUuXu7tQcTYheZ0_3LKWAsibGK93ZQCLcBGAs/s1600/cat%2Bgeometry.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="514" data-original-width="600" height="274" src="https://4.bp.blogspot.com/-yfiK_AiPLwE/XOIiYs3u_CI/AAAAAAAAFOM/T8MRUuXu7tQcTYheZ0_3LKWAsibGK93ZQCLcBGAs/s320/cat%2Bgeometry.jpg" width="320" /></a></div><br /><br /><br />Our college offers a course that is pretty much the same as high school geometry. I have never before taught it because we only offer it in the summer, and summer semester is squished into 6 weeks. I haven't wanted to teach math on that sort of schedule because it seems like it would be hard for students to learn well that quickly.<br /><br />But I'm under load and I love geometry, so I will be teaching this course in a month. I've been preparing ever since January - reading the text, making plans and handouts, doing constructions. Geometry doesn't have the same flow for me that algebra does. I thought that might be because I've been teaching algebra for 30 years. But other math educators have also noticed how much harder geometry can be. Brain teasers everywhere. I like that, but I also need to figure out how to make this doable for the average student. I have gone through the book twice now, thinking about what they'll be learning, and hoping to notice the places they'll get stuck.<br /><br />We will be doing lots of compass and straightedge constructions. Even though I loved geometry in high school, I hated that compass. I'd press too hard and open it further in the middle of an attempted circle. Or not hard enough, and then the pencil wouldn't draw. Arghh. Computer tools make that a problem of the past. I've mentioned two sites in previous posts, <a href="http://sciencevsmagic.net/">sciencevsmagic.net</a> and <a href="http://euclidea.xyz/">euclidea.xyz</a>. But euclidea won't let you unlock a higher level until you succeed at the lower level, and I found that frustrating. <a href="https://www.mathed.page/" target="_blank">Henri Picciotto</a> pointed out that <a href="http://euclidthegame.com/">euclidthegame.com</a> is much like euclidea, but doesn't require you to solve one level before doing the next. After checking it out, I'm not sure I have any reason to go back to euclidea; euclidthegame is much more satisfying.<br /><br /><br /><b>Translating a Line Segment </b><br />So one of the tools I'll be using with my class is euclidthegame. I had a blast working through the levels back in January, and wanted to redo them now to see which sorts of things might be hard for the students. I got stuck on level 7, translating a line segment. I replicated two solutions given in the comments, but I didn't understand why the steps worked.<br /><br />When I looked up 'translate a line segment', all of the sites I found said to hold your compass at the right size and move it over. That doesn't work on euclidthegame, and wasn't part of the original Greek compass and straightedge protocol. I almost gave up on understanding this construction, and thought about telling students they could skip this one, since our book does it the easy way too.<br /><br />People say believing you can do something helps you do it, but I often succeed right after giving up. Ornery, I guess. I doodled a bit, and realized that thinking about a parallelogram was the key. After that, I was able to solve it quickly. I want to share my solution here. (Turns out, this idea was in the comments at euclidthegame, I just hadn't seen it.)<br /><br />The idea is to make a line segment congruent and parallel to AB, with C as one endpoint of the new segment. If you imagine making a parallelogram with the old segment and the new one, and if you know that diagonals of a parallelogram intersect at their midpoints, it becomes relatively straightforward.<br /><ol><li>Sketching parallelogram ABDC, </li><li>We know that we can construct the diagonal BC, </li><li>And find its midpoint (E). </li><li>Then make a ray from A through E, </li><li>And a circle centered at E through A. </li><li>Where that circle and ray meet will be D. (Because E is now the midpoint of AD.)</li><li>CD is the translated line segment we wanted. </li></ol> Lovely.<br /><br />The students won't have enough information when they get to this construction in the book (chapter 1) to do it the right way. We'll have to wait until chapter 4, where they will find out why those diagonals bisect each other. I sure am glad to know that ahead of time.<br /><br /><b><br /></b><b>Planning the Course</b><br />We will be meeting from 8 to 11:35 am four days a week. Who is going to get there on time at 8am if they don't have to? So I will start with a quiz every day. If they don't do well, they can retake it (new questions, same topics) outside of class time. [So a real sleepyhead could still ace the quizzes, even if they came late. But that will take some serious dedication. That's cool.]<br /><br />Then I'll do my usual combination of lecture, pairwork, groupwork, and guided discussion from 8:15 to 9:45. After they take a break they will have lab time. Partly because I don't believe anyone can concentrate for 3 1/2 hours in a regular classroom, and partly because I think they really need to get their hands on some of these tools. I'll have geoboards and dot paper to record their geoboard results. I'll be using most of Henri Picciotto's lovely <a href="https://www.mathed.page/geometry-labs/index.html" target="_blank"><i>Geometry Labs</i></a> book. (Free. Thank you, Henri!) And they'll be using both euclidthegame and <a href="https://www.geogebra.org/" target="_blank">geogebra</a> to do their constructions. Lab projects are 20% of their grade. They'll have to do constructions, geoboard activities, proofs, and one activity of their choice. <br /><br />Anyone teaching geometry who'd like more details, please ask. And anyone who would like to share tips, I'm all ears.<br /><br /><br /><br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-48025788058788050992019-04-01T14:36:00.000-07:002019-04-01T14:36:30.225-07:00Playful Math Ed Blog Carnival (aka Math Teachers at Play) #126<a href="https://4.bp.blogspot.com/-wUcPqYc5Jso/XKJDh1DwWgI/AAAAAAAAFGw/NRyX9aBF_Vg4RxSRpR1P0pmErDEbOwxXgCEwYBhgL/s1600/baby-chicks-3-300x198.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="171" data-original-width="187" src="https://4.bp.blogspot.com/-wUcPqYc5Jso/XKJDh1DwWgI/AAAAAAAAFGw/NRyX9aBF_Vg4RxSRpR1P0pmErDEbOwxXgCEwYBhgL/s1600/baby-chicks-3-300x198.jpg" /></a>126 = 6*21 = <span style="color: red;">2</span>*<span style="color: blue;">3</span>*<span style="color: blue;">3</span>*<span style="color: lime;">7</span><br /><br /><br />If you want to choose 4 chicks randomly from 9 total chicks, there are 126 ways to do it.<br /><br /><br />Students learn more if they make up the stories for story problems themselves. Can your students make up stories for these ways of making 126?<br /><br />126 = 2<sup>7</sup> - 2<sup>1</sup> (difference of powers of 2)<br />126 = 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> + 7<sup>2</sup> (sum of consecutive squares) <br />126 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers) <br /><div style="text-align: right;"> (from <a href="http://www.archimedes-lab.org/numbers/Num70_200.html">http://www.archimedes-lab.org/numbers/Num70_200.html</a>)</div><br /><div style="text-align: right;"><br /></div><div style="text-align: left;">This blog carnival has evolved from being mainly contributions to being mainly items the blog host has discovered. Since my passion lately is geometry, this issue is dedicated to <span style="color: purple;"><b><span style="font-size: large;">geometry</span></b></span>. (Which of the 3 ways of making 126 above has a geometric interpretation? Hint: There's a picture of it here... somewhere...)<br /><br /><br /><br /><b>Constructions</b><br />I have been intrigued for the past few years with Archimedes' method of finding pi. He didn't have the square root symbol, so he approximated using fractions, getting pi between 3 10/71 and 3 1/7. But if we follow his steps, and keep the square roots, we get a lovely pattern for our answer.You can try it. Construct a hexagon in a circle. If the radius of the circle is 1, then the hexagon's perimeter is 6. Perimeter over diameter = 6/2 = 3. Now create a dodecagon (12-sided polygon) from the hexagon. You can find the side lengths from repeated use of the Pythagorean theorem, and then find perimeter over diameter. Your result will be closer than for the hexagon. You can repeat this process until a pattern emerges. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-3jDmRswYY4g/XKJQ6o6nOMI/AAAAAAAAFHE/uxQd1CiSEuAO3OqGwiUCCWktUuDQ8XEkwCLcBGAs/s1600/construction2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="368" data-original-width="516" height="228" src="https://4.bp.blogspot.com/-3jDmRswYY4g/XKJQ6o6nOMI/AAAAAAAAFHE/uxQd1CiSEuAO3OqGwiUCCWktUuDQ8XEkwCLcBGAs/s320/construction2.png" width="320" /></a></div><br /><br />If you want to get better at geometric construction (straightedge and compass style), play with it at <a href="https://sciencevsmagic.net/geo/" target="_blank">sciencevsmagic</a> or <a href="http://www.euclidthegame.com/" target="_blank">euclidthegame.</a><br /><br />You can improve your geometric reasoning skills with the puzzles in <a href="https://solvemymaths.com/2017/12/03/geometry-snacks/" target="_blank"><i><b>Geometry Snacks</b></i></a> (and <i><b>More Geometry Snacks</b></i>), by Ed Southall and Vincent Pantaloni. There are more puzzles at his blog. If you like them, the book is a treasure trove.<br /><br />Because I've fallen in love with geometry, I decided to teach it this summer, for the first time ever. So I'm doing a lot to prepare. Henri Picciotto is an expert geometry teacher who graciously offered me his time over breakfast. He advised me to download his <a href="https://www.mathed.page/geometry-labs/index.html" target="_blank">Geometry Labs</a> book (free) from his Math Ed Page site. There is so much more there than this. But this alone was a huge gift. I think it may transform my course.<br /><br /><br /><a href="https://4.bp.blogspot.com/-nA-3Mqx03V0/XKJ9O4CY8II/AAAAAAAAFHQ/3IiOaHeetE0D4rHw_ic0JBS9_jDJCis2gCLcBGAs/s1600/pentagram.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="559" data-original-width="591" height="188" src="https://4.bp.blogspot.com/-nA-3Mqx03V0/XKJ9O4CY8II/AAAAAAAAFHQ/3IiOaHeetE0D4rHw_ic0JBS9_jDJCis2gCLcBGAs/s200/pentagram.png" width="200" /></a>I've been collecting geometry mysteries. Medians are the lines from midpoints of the sides of a triangle to the opposite vertices. The 3 medians seem to always cross at one point. Why is that? I tried for weeks to prove it, and just couldn't. I finally gave up and looked at the proof. (And told my students how much fun I had failing!) I then found another proof that followed a very different path. Can you prove it?<br /><br />Here's a simpler mystery: If you make a 5-pointed star (perfectly even, I can't do that without digital help...), what is the angle at each point?<br /><br /><br />One of my favorites for seeing the geometry in math topics you didn't know were geometric is <a href="https://www.facebook.com/magicpi2/videos/1286450834841623/?__xts__[0]=68.ARDmt_2oyE4H0IuYYZQFo4t4sQmCtQBFtLesqoGzsheGcF7sbtzMLAUlzbF1S_E1CedPAr5tiu6pTAprqj4U3AssRvyHfneOkXFr-2ETSBv1GAjUX0NCOQaCwyT4oIbtP2-T2wbzSPCosCtuxXAI5uzk55POVqqvGdEbjZEv0c4TFRXMbQ496JKm6h0eIqii-10c-s5FLNm_9FnMkuS9xyuGwFgYgNnp7827AmcIvw_jxH1T08_fz7xVnZ7zPG2XkyErx9dRXcLyZ-Hc85_Pm0wdN07l5VlbTni2XPVIu_OtLmG0zdbIXIjnf96lmQNQbsrQOTM3ND2iekLJZdi5H7ctbcMTpW24A-rHwA&__tn__=-R" target="_blank">Magic Pi - math animations</a>. I hate that they're only on Facebook because I am not comfortable linking to facebook in class. But they are amazing. (I linked to one that's pure geometry. So cool.) They apparently do most of their animations in <a href="http://geogebra.org/" target="_blank">geogebra</a>. I am a complete novice next to them. Here's a geogebra sketch I made today. It might be my first in their 3D mode.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MF9H7HV0css/XKJOHU-CUKI/AAAAAAAAFG4/-fvEBbMQj_wttCk2fINZYO9hAtCxZ5xuACLcBGAs/s1600/stack%2Bof%2Bsquares.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="479" data-original-width="709" height="216" src="https://1.bp.blogspot.com/-MF9H7HV0css/XKJOHU-CUKI/AAAAAAAAFG4/-fvEBbMQj_wttCk2fINZYO9hAtCxZ5xuACLcBGAs/s320/stack%2Bof%2Bsquares.png" width="320" /></a></div><br /><br /><br /><br /><b>Making Your Own Math</b><br />At the beginning, I mentioned having students make up their own story problems. <a href="https://www.arithmophobianomore.com/starting-points-building-story-problems/" target="_blank">Here's a lovely post from <i>Arithmophobia No More</i></a> about just that. <a href="http://mathstateofmind.blogspot.com/2019/03/numberless-word-problems-and-bar-models.html" target="_blank">Here's another angle on teaching story problems</a>, from Jen at Math State of mind. Leaving out the numbers helps students to slow down. <br /><br /><a href="https://whenlifegaveuslemons.wordpress.com/2019/02/21/create-your-own-math-game/" target="_blank">This blog post, by Amy at <i>When Life Gave Us Lemons</i></a>, is about her son making up his own math games. And <a href="http://mathhombre.blogspot.com/2012/04/multiplying-game-possbilities.html" target="_blank">John Golden has a whole class make up variations</a> on a game he shared with them.<br /><br /><br />Denise Gaskins, founder of this carnival, pulls together so many books and ideas I love in <a href="https://denisegaskins.com/2014/08/13/fractions-15-110-180-1/" target="_blank">this post</a>. I don't know how she does it! The (surface) topic is fractions, but more than that, it made me think about how we can help students learn by saying less. The video she includes, with a teacher asking the two boys questions, and never telling them they're wrong, is fabulous. One of the commenters at Denise's post linked to <a href="https://established1962.wordpress.com/2014/08/14/a-productive-conversation-with-susie/" target="_blank">a discussion of his own with a student</a>. And that made me think about <a href="https://drive.google.com/file/d/0B4Lou9CsLnQxaTdjODd6cFkxMjQ/view" target="_blank">Bob Kaplan's guide to 'becoming invisible'</a> (or not giving away the math). (What math delights have you found lately by following your nose? Bunny hops rock!)</div><div style="text-align: left;"><br /><br /><br />You can check out the <a href="https://aperiodical.com/2019/03/carnival-of-mathematics-167/" target="_blank">Carnival of Mathematics here</a>. And if you'd like to host <i>this</i> carnival (we need help next month!), you can <a href="https://denisegaskins.com/mtap/" target="_blank">learn more and sign up here</a>.<br /><br /><br /><br /> </div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-13491867549079422772019-03-22T11:43:00.000-07:002019-03-22T11:43:05.595-07:00Coming soon: Math Teachers at Play (aka Playful Math Ed) blog carnivalI'll be posting the blog carnival here sometime late next week. Right now I'm beginning to gather links to lovely, playful math ed posts (and sites and videos and ...). If you know of something I should include, please email me at mathanthologyeditor on gmail.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-dkKfh1Uj1M8/XJUsgIJwf5I/AAAAAAAAFFI/bNsUIA1Y4pkmNZK4QzY8yuYzEZh4YWrrACLcBGAs/s1600/fireworks.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="183" data-original-width="276" height="424" src="https://4.bp.blogspot.com/-dkKfh1Uj1M8/XJUsgIJwf5I/AAAAAAAAFFI/bNsUIA1Y4pkmNZK4QzY8yuYzEZh4YWrrACLcBGAs/s640/fireworks.jpg" width="640" /></a></div><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-88159732612540827282019-02-12T12:07:00.000-08:002019-02-12T12:10:05.160-08:00When Math Tells a Story<div class="separator" style="clear: both; text-align: center;"><a href="https://media1.tenor.com/images/3a620033bd9d5725d53b542b00accea4/tenor.gif?itemid=4852871" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="498" data-original-width="498" height="320" src="https://media1.tenor.com/images/3a620033bd9d5725d53b542b00accea4/tenor.gif?itemid=4852871" width="320" /></a></div><br /><br /><br />On the <a href="http://groups.yahoo.com/group/LivingMathForum/" target="_blank">Living Math Forum</a> group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):<br /><br /><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">The equation for a circle is (x-h)<sup>2</sup> + (y-k)<sup>2</sup> = r<sup>2</sup> . Many students see each equation like this as separate from any other equations/formulas they know. I try to get them to look at this deeply. From the structure of it (square plus square equals square), I see that it's really the <a href="https://mathmamawrites.blogspot.com/2012/10/proving-pythagorean-theorem.html" target="_blank">Pythagorean Theorem</a>. Why would something for right triangles show up in the equation of a circle?! (That blew me away a few years back. I've been teaching math for 30 years, but that question seemed deep.) </div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">It's because our coordinate system has the two axes perpendicular to each other. So the distance from the x-coordinate of a point on the circle to the x-coordinate of the center is measured horizontally and the similar y distance is measure vertically. You can build a right triangle from the center to (almost) any point on the circle. The constant radius is the hypotenuse of that right triangle. <br /><br />So this equation tells a little story.</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">How does a whole course tell a story?</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">Algebra is about solving equations and about graphing. We want to see how real life situations (anything with data that has two components, like time and height) can be represented with equations and with graphs. In Algebra 1 students learn to solve simple equations and to graph. And hopefully they learn how the two skills are connected. There's a bit more. Systems of equations allow us to model slightly more complex situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after factoring) is graphing and recognizing equations of parabolas. <br /><br />Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.</div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;"><br /></div><div style="color: black; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 12pt;">The course can tell the tree's story, or any story that can be told through data, graphs, and/or equations. It tells the story of using math to help us think quantitatively about problems we care about. (And of course there are plenty of things we care about that cannot be quantified.)</div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-57200343759117636702019-02-10T21:58:00.001-08:002019-02-10T21:58:13.032-08:00Links to Good Math PostsI am closing tabs, so that <i>maybe</i> my browser will move faster. Here are all the links I couldn't stand losing:<br /><ul><li><a href="https://www.youtube.com/user/misterwootube" target="_blank">Mr. Woo's Math Channel on youtube, aka WooTube</a>. This came recommended by the amazing Julie Brennan (of <a href="https://www.livingmath.net/" target="_blank">Living Math</a>).</li><li>From Math with Bad Drawings, <a href="https://mathwithbaddrawings.com/2018/08/01/the-teacher-who-only-says-gimme/" target="_blank">The Teacher Who Only Says Gimme</a> (part of the <a href="https://samjshah.com/mathematical-flavors-convention-center/" target="_blank"><i>Convention of Mathematical Flavors</i></a> blogfest), on getting students to think creatively in math.</li><li><a href="https://www.mathteacherscircle.org/news/mtc-magazine/f2018/we-the-people/" target="_blank">We the People</a>, the math behind various preferential voting schemes. (And why there is not a best way.) From the Math Teacher's Math Circle Network.</li><li><a href="https://www.nytimes.com/interactive/2018/06/13/upshot/boys-girls-math-reading-tests.html" target="_blank">Sometimes boys and girls do equally well, other times not. What is involved? New York Times article.</a></li><li><a href="http://busynessgirl.com/favorites/contemporary-algebra-collection/" target="_blank">Busyness Girl's Contemporary Algebra Collection</a></li></ul>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-18783376296012761832019-01-10T19:38:00.000-08:002019-01-10T19:40:36.019-08:00Geometry and Visual Thinking Puzzle Collection - Help Needed<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-7ra_rFJvtJg/XDgN28BCMkI/AAAAAAAAE9U/hTP68t-gLdIzrDccvuS9EBa0BI2u9ZYxgCLcBGAs/s1600/4%2Brectangles.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="290" data-original-width="311" src="https://4.bp.blogspot.com/-7ra_rFJvtJg/XDgN28BCMkI/AAAAAAAAE9U/hTP68t-gLdIzrDccvuS9EBa0BI2u9ZYxgCLcBGAs/s1600/4%2Brectangles.png" /></a></div><br /><br /><br /><br />Math lovers, I need your help. (Math-likers welcome too. Math-haters who are into torturing themselves are also quite welcome, if you know any.) If you love puzzles, that might be enough to suck you in. I'm hoping...<br /><br /><br />I am pulling together a collection of problems meant to entice students who aren't all that into math. (So, relatively easy puzzles.) I think we focus too much on algebra and not enough on the visual skills that come along with geometry. So each problem in this collection include<span class="text_exposed_show">s visualizing. Most of them involve geometry.</span><br /><br /><div class="text_exposed_show">I'm calling them PPODs - Puzzle Problem Of the Day. I plan to post them in my classrooms and in the math department of the community college I work at.<br /><br />I have found quite a few good problems in <i>Geometry Snacks</i>, by Ed Southall and Vincent Pantaloni. (I bought the e-version of the book yesterday.) I just now discovered Ed's blog (<a data-ft="{"tn":"-U"}" data-lynx-mode="asynclazy" href="https://l.facebook.com/l.php?u=http%3A%2F%2Fsolvemymaths.com%2F%3Ffbclid%3DIwAR1vkFgv7KbIBWKvUCmNmx9feUiBTQwd8sTFLA5c46CSJtcex5q2gEgFpKs&h=AT2L_1Xq97XC_3LyTdBFc4hqB0BVSjKgm5Xw9rxjungeeAaK6m8Xu-gs300OxOtULWWm8L6Y2kkhQp1oIAFfPBXkLzXNxXYtsB80Y479iSgFCOiFo6VQYCFzXeF6D80C6b-q9gHcRu9waeDRTyGpOOSuoVKnPAaH0RyF" rel="noopener nofollow" target="_blank">solvemymaths.com</a>), and found more goodies. My other favorite source is the beautiful by-hand drawings done by Catriona Shearer, @cshearer41. I couldn't find who to credit for about a quarter of the problems I've collected.<br /><br />I'm aiming for 60 problems, one for (almost) each day (M-Th) of the 16-week semester. I have about 50 already. What I need is help determining the difficulty level of each one. I need volunteers to help me figure that out. I'm using a scale of 1 to 5 myself, and anything I'd call a 4 or 5, I'm throwing out. (I've gotten stuck on a few problems. Those are not in the collection, of course.)<br /><br />If you're interested, I'll send you my collection. And then you post your ratings <a href="https://docs.google.com/spreadsheets/d/1MttNarw13gRA9UMkBVRGbpVKyOXJ3kK5yyiZ1vuQZV4/edit#gid=0" target="_blank">here</a>.<br /> Just let me know in the comments if you'd like to check these out. (Or email me at mathanthologyeditor@gmail.com.) Also, I'm interested in your ratings and thoughts even if you just end up doing a few of the puzzles.</div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-39529956116951815772018-11-28T16:32:00.001-08:002018-11-28T16:32:24.174-08:00Logarithms: How Math Helps Science<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-RV7f8jCIUIQ/W_8vtfc__HI/AAAAAAAAE0k/2Hefz9_QXigjSn5pWHtIWPzW8yJE2aHcACLcBGAs/s1600/kepler2law.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="249" data-original-width="398" height="199" src="https://2.bp.blogspot.com/-RV7f8jCIUIQ/W_8vtfc__HI/AAAAAAAAE0k/2Hefz9_QXigjSn5pWHtIWPzW8yJE2aHcACLcBGAs/s320/kepler2law.JPG" width="320" /></a></div>I'm teaching my algebra students about logarithms today. It is likely the hardest algebra topic there is. When I started at Contra Costa College, after having taught full-time at Muskegon Community College for 6 years, I think I still was not 100% solid on logs. (I could do the problems, but proving those properties was a tangled mess.)<br /><br /> I had developed <a href="https://mathmamawrites.blogspot.com/2010/04/murder-mystery-project-for-logarithms.html" target="_blank">my murder mystery project</a> because my officemate at MCC was a chemistry teacher, and she had complained that teachers weren't teaching logs well enough in Intermediate A<span class="text_exposed_show">lgebra.I hoped that project would help. I think it does, but I don't have anything solid to base that on.</span><br /><span class="text_exposed_show"><br /></span><div class="text_exposed_show"> For many years now, I've told students a bit of history in my introduction to logarithms. John Napier invented them in 1614 as a way of making hard calculations easier. (Can you imagine finding the square root of 192.7 without a calculator? You'd have to guess, check, and modify repeatedly...)<br /><br />Around that time Kepler came up with his three laws of planetary motion. Over the years, I have repeatedly wondered out loud whether Kepler was using logarithms while he figured these things out. <br /> Today when this came up, I was able to turn to our tutor and ask him to look it up. He found a <a href="https://www.mathpages.com/rr/s8-01/8-01.htm?fbclid=IwAR3bUzHl523g6gm-U97PVLLZfXuE5Un2Z2OcxerfjjfU6NsOOJ-Hxf6rwdE" target="_blank">fascinating article</a> on it.<br /><br /> Turns out Kepler came up with the first two "laws" before playing with Napier's new idea. But his third came after.<br /><br /> I. Planets move in ellipses with the Sun at one focus.<br /> II. The radius vector (line from sun to planet) describes equal areas in equal times.<br /> III. "The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances."<br /><br /> This is an improper use of the word proportion. He was making a proportion from the logs of the ratios, and saw that log(T1/T2) = 1.5*log(r1/r2). In modern terms, we say something like: The square of the ratio of the times (that it takes two planets to go around the sun) equals the cube of the ratio of their mean distances from the sun. The fact that he was thinking in terms of logs shows how helpful the new (in 1614) concept was for helping scientists see patterns. (<a href="https://www.mathpages.com/rr/s8-01/8-01.htm?fbclid=IwAR3bUzHl523g6gm-U97PVLLZfXuE5Un2Z2OcxerfjjfU6NsOOJ-Hxf6rwdE" target="_blank">Read that article</a> for more on Kepler and Napier.)<br /><br />I love that math can help us see new things. </div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com1tag:blogger.com,1999:blog-5303307482158922565.post-31119513984073600172018-09-08T16:17:00.002-07:002018-10-04T19:46:13.164-07:00Geometric Construction of the Regular Pentagon<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-oYXNhVqh9us/W4ywV5RDpEI/AAAAAAAAEnc/FMxwu0EqB-k8HFhzUjIQkZE0Fy27evQXgCLcBGAs/s1600/1st%2Bpentagon.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="721" data-original-width="985" height="292" src="https://2.bp.blogspot.com/-oYXNhVqh9us/W4ywV5RDpEI/AAAAAAAAEnc/FMxwu0EqB-k8HFhzUjIQkZE0Fy27evQXgCLcBGAs/s400/1st%2Bpentagon.png" width="400" /></a></div><br />I never particularly enjoyed geometric constructions when I was younger. It may be because I had a tendency to press down too hard, and mess up the width my compass was set to. Like so many things, technology has a fix for that.<br /><br />You can do geometric constructions on a number of sites, and in <a href="https://www.geogebra.org/" target="_blank">geogebra</a>. My favorite site, which <a href="https://mathmamawrites.blogspot.com/2013/07/euclidean-construction-science-vs-magic.html" target="_blank">I discovered about 5 years ago</a>, is <a href="http://sciencevsmagic.net/">sciencevsmagic.net</a>. I having been telling students about it for years, and decided this summer to play with it again, to see how much I still like it. (A lot, it turns out.)<br /><br />There are 40 challenges, in sets of 4. This time around I got every shape but the pentagon. I asked for hints (from math friends on facebook) which I then avoided reading because I wanted to do it on my own. I got nowhere that way, and finally followed instructions for how to construct it. Every other shape I have constructed makes sense on its own, but the construction of a regular pentagon must be proved with algebra. Ok, I don't feel so bad about 'cheating'.<br /><br />I kept constructing it over and over, as I tried to understand it better. The first time (which you see above) I used <a href="http://www.opentutorial.com/How-to-Construct-a-Pentagon/construction" target="_blank">these instructions</a> (which I found by following links from wikipedia) and constructed it at sciencevsmagic, achieving my final shape challenge. Then I did it again using geogebra. I needed to figure out how it worked, and was at a loss. <a href="http://www.cut-the-knot.org/pythagoras/pentagon.shtml" target="_blank">Cut-the-knot</a>, a site I've gone to many times with my math questions, had a different construction - which I followed on both sciencevsmagic and geogebra - and an explanation (by Scott Brodie) that I kept working through as I wrote this.<br /><br />[Sadly, Alexander Bogomolny, the creator of the cut-the-knot site, has died. I hope the math blogging community can find a way to maintain his amazing site. We miss you, Alexander.]<br /><br />I will explain this as I understand it. My explanation will mostly just restate what I learned from these sites. But it may help some people, since some of the reasoning steps in Scott's explanation were hard for me to follow.<br /><br />The constructions are worth doing first, so you can get a feel for what's happening as you do it. But when you're done, the question is still open - is this <i>really</i> a regular pentagon? The following explanation proceeds in two major steps. First it looks at relationships in the regular pentagon and the pentagram (which is a regular pentagon with a five-sided star inside it). Then it looks at the relationships built by the construction method, and we finally see that the side length given in the two ways is the same. Bingo!<br /><br /><br /><br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-leVPM5Oo39Y/W49YYua6XFI/AAAAAAAAEoM/iAakg5reoMYIvOFTrUdzP7a_KvtsFc5kACLcBGAs/s1600/pentagon%2Bv2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="696" data-original-width="709" height="314" src="https://3.bp.blogspot.com/-leVPM5Oo39Y/W49YYua6XFI/AAAAAAAAEoM/iAakg5reoMYIvOFTrUdzP7a_KvtsFc5kACLcBGAs/s320/pentagon%2Bv2.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><b>The Construction</b><br />If you're ready to follow this, I trust that you already know how to construct the perpendicular bisector of a line segment. I will leave out the construction details for each of those.<br /><ul><li>Starting with a circle with center O, construct a horizontal diameter, AB.</li><li>Now construct its perpendicular bisector, CD.</li><li>Construct the perpendicular bisector of AO, intersecting it at E. </li><li>Construct the circle with center at E, through C, labeling its intersection with AB as F.</li><li>Construct the circle with center at C, through F, labeling its intersection with the original circle (above B) as G.</li><li>Go around the original circle, constructing this same size circle from each new point. (Center at G, through C, gives H at the other intersection with the original circle. Etc.)</li></ul>The question remains: Does this really construct a regular pentagon, or is it maybe just pretty close?<br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Blx1DUz_W-Q/W49ZLf3UsPI/AAAAAAAAEoU/wSKrp1CNzqocUpUTfxEqgN2ZX1oXaeMHwCLcBGAs/s1600/pentagram%2Bv2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="700" data-original-width="692" height="400" src="https://4.bp.blogspot.com/-Blx1DUz_W-Q/W49ZLf3UsPI/AAAAAAAAEoU/wSKrp1CNzqocUpUTfxEqgN2ZX1oXaeMHwCLcBGAs/s400/pentagram%2Bv2.png" width="395" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br /><b>Relationships in a regular pentagram</b><br /><ul><li>The angles in a regular pentagon are each 108<span class="ILfuVd yZ8quc">°, so each external angle, like </span>∠HMN, is 72<span class="ILfuVd yZ8quc">°. That makes each angle at a point, like </span><span class="ILfuVd yZ8quc">∠NHM, 36</span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°. Since </span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠HMG is 108</span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°, </span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠MHG and </span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠MGH must each be 36</span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°. </span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"> Since angles </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠GLH and </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠LGH are each 72</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°</span></span>, </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Δ</span></span></span></span></span></span></span></span></span></span></span></span></span>HGL has two equal sides, HL and HG (which also equal CM, HI, etc).</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">ΔCHI is similar to </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Δ</span></span></span></span></span></span></span></span></span></span></span></span></span>IHM. So corresponding sides are in proportion. We get CH/HI=HI/HM, giving us HI<sup>2</sup> = CH*HM = CH(CH-CM) = CH(CH-HI).</span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Let CH = x*HI. Then the equation above gives us </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">HI<sup>2</sup> =x*HI(x*HI-HI), which gives us a (perhaps) familiar equation: 1 = x</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>-x, whose solution (in this case, where x is clearly greater than 1) is x = (1+</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2 aka φ. (At this point, I get excited. φ, aka the golden ratio, shows up in such different contexts!)</li><li>So CH = φ*HI, and (similar triangles) HI = φ*HM and HM = φ*MN.</li><li>We now have relationships between all the sides of the pentagon, star, and line segments on the star, but these are not yet connected to the radius of the circle.</li><li>Now consider ΔODH. Two sides are radii, and the angle at O is 36°. Another triangle similar to all the others we've found. So OD = φ*DH.</li><li>Also note that ∠DHC (inscribed angle on the diameter) is a right angle.</li><li>Let's switch to simple variables names now. Let r=OD, s=CG, t=DH, and d=CI=CH.</li><li>Then d= φ*s, r= φ*t, and t<sup>2</sup>+d<sup>2</sup> = (2r)<sup>2</sup>, giving (r/φ)<sup>2</sup> +(φ*s)<sup>2</sup> = 4r<sup>2</sup>.</li><li>This gives us s<sup>2</sup> = 1/φ<sup>2</sup>*(4r<sup>2</sup> - (r/φ)<sup>2</sup>) = (4/φ<sup>2</sup> - 1/φ<sup>4</sup>)*r<sup>2</sup>. </li></ul><br /><br /><br /><b>Analyzing our construction</b><br /><ul><li>We will find the length of s = CG = CF.</li><li>OE=1/2*r. And EC = <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√(OE</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>+OC</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>) = </code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√(5/4)*r = </code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2*r = EF. OF = (<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2- 1/2)*r.</li><li>So <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">s<sup>2</sup> = </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>CG<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = CF<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = OC<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> + OF<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = (1+(<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2- 1/2)<code><sup>2</sup>)*r<sup>2</sup>.</code><code></code></li><li>We have two very messy expressions for s<sup>2</sup>, one from the regular pentagon/pentagram and one from our construction. Are they equal? It may help to write out positive and negative powers of <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">φ </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>to help simplify the first expression. Yes! They are both equal to (5-<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2*r<sup>2</sup>, making the side length of the regular pentagon inscribed in a circle of radius 1 equal the square root of (5-<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2.</li><li>And that proves that our construction created a regular pentagon. (Whew!) </li></ul><br /><br />I've completed 32 of the 40 challenges at <a href="https://sciencevsmagic.net/geo/" target="_blank">sciencevsmagic</a>. I still haven't figured out how to make some of the shapes within the original circle, and still haven't found the least moves for some of the shapes. I'm so glad there's still something to work toward the next time I revisit this.<br /><br /><br /><br />[Blogger doesn't deal well with superscripts and square roots. Once again, I am noticing that I ought to learn some LaTex. Sigh.] <br /><ul></ul><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com5tag:blogger.com,1999:blog-5303307482158922565.post-29660198845747253032018-07-01T18:20:00.000-07:002018-07-01T18:20:23.674-07:00Math Teachers at Play #118<br /><a href="https://2.bp.blogspot.com/-teS_QhiW1AI/Wzl15QPAiEI/AAAAAAAAEWw/8Iq6MsiunOQyBUY6MtTgB1orslYj3CtegCLcBGAs/s1600/118%2Bv4.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="225" data-original-width="225" height="320" src="https://2.bp.blogspot.com/-teS_QhiW1AI/Wzl15QPAiEI/AAAAAAAAEWw/8Iq6MsiunOQyBUY6MtTgB1orslYj3CtegCLcBGAs/s320/118%2Bv4.jpg" width="320" /></a>In two more months, we'll hit MTAP #120, which <i>should</i> be the ten-year anniversary mark. Is it?<br /><br /><br /><br />The number 118 ...<br />... factors to 59*2.<br />... is 1110110 in base two, 11101 in base three, 1312 in base four, 433 in base five, and 226 in base 7.<br />Not particularly exciting...<br /><br /><br /><br />Aha! Can you make 118 using four 4's? (I did.) I wonder if you can make it using five 5s, or ...? <br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /><b>Summertime Learning</b><br /><a href="https://www.mathmammoth.com/lessons/summer_math.php" target="_blank">Summer Math Resources</a> from Math Mammoth's Maria Miller. <br /><br />Denise Gaskins offers us one of her FAQs, on <a href="https://denisegaskins.com/2018/06/15/faq-forgetting-what-they-learned/" target="_blank">forgetting what they learned</a>. My favorite of all her suggestions? Play games! <br /><br />On <a href="https://emergentmath.com/2018/06/25/routines-the-driving-beat-of-your-class/" target="_blank">Routines</a> and <a href="https://emergentmath.com/2018/06/28/lessons-the-stuff-were-used-to-but-better/" target="_blank">Lessons</a> <br />I like Geoff's perspective: "What’s the shortest amount of time you could possibly do the talking? Go with that. And maybe subtract a few more minutes." After 30 years of teaching, I am still learning, and I still get excited when I see something that might make a difference for my students. This looks helpful, and perfect for my summer meditations on teaching.<br /><br /><br /><br /><b>Young Math </b><br />Can you imagine doing a college math lesson with 2nd graders, and having it work out well? Manan Shah did it! <a href="http://mathmisery.com/wp/2018/06/14/set-theory-for-second-grade/" target="_blank">His lesson was on set theory.</a><br /><a href="https://www.mathmammoth.com/lessons/summer_math.php" target="_blank"><br /></a>Ahh... A whole blog on magical math books. And she wrote <a href="https://mathbookmagic.com/2018/06/18/uncovering-the-complexities-of-counting-with-a-magical-counting-book/" target="_blank">about Christopher Danielson's new book, How Many?</a> If you like this, she also has <a href="https://mathbookmagic.com/blog-book-list/" target="_blank">a list of every book she has posted about</a>. Thank you, Kelly Darke!<br /><br />Remember playing the card game War? If you're a parent, those memories may not be so fond. Kids love it and parents can get sooo bored. Kent wrote <a href="https://www.gamesforyoungminds.com/blog/war-card-game" target="_blank">a great post about his interactions with his son around Addition, Subtraction, and Multiplication War</a>. (If you want a good summary of even more mathy variations on war, <a href="https://denisegaskins.com/2006/12/29/the-game-that-is-worth-1000-worksheets/" target="_blank">check out Denise Gaskins' classic post.</a>)<br /><br /><a href="https://medium.com/@jamestanton/the-global-math-project-uplifting-mathematics-for-all-9253d383e5e6" target="_blank">The first ever Global Math Week</a> went well, with its exploding dots exploding around the world. Thank you, James Tanton, for getting people all over the globe into math.<br /><br />How do you talk about numbers with young children? There are so many ways! Here's another:<a href="http://realteachingmeansreallearning.blogspot.com/2018/06/dice-chats.html" target="_blank"> Counting with Dice</a>, from Dave Martin. <br /><br /><br /><br /><br /><a href="https://3.bp.blogspot.com/-cZfAlzi1CUs/Wzl15qHyafI/AAAAAAAAEXI/lObvsKEgyEUFA1zUBDilQt9NOkawImIeQCEwYBhgL/s1600/118%2Bv3.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="768" height="180" src="https://3.bp.blogspot.com/-cZfAlzi1CUs/Wzl15qHyafI/AAAAAAAAEXI/lObvsKEgyEUFA1zUBDilQt9NOkawImIeQCEwYBhgL/s320/118%2Bv3.jpg" width="320" /></a><b>Calculus</b><br />Which is bigger, e<sup><span class="st">π</span></sup> or <span class="st">π<sup>e</sup>?</span> Sure, you can check it on a calculator. Or <a href="http://blog.mrwaddell.net/archives/1835" target="_blank">you can use areas to see why it's true</a>. Lovely! (from Glenn Waddell)<br /><br />The Fundamental Theorem of calculus says that you can figure out areas by using anti-derivatives. I do a project to help students understand it. Sam Shah does even more. <a href="https://samjshah.com/2018/06/21/part-iii-the-final-turn/" target="_blank">This is Part III of his lessons for this topic.</a> I recommend clicking on the links to Parts I and II first.<br /><span style="background-color: #ead1dc;"><br /></span><br /><br /><b><br /></b><b><br /></b><b>A Few More Goodies...</b><br />Dan McKinnon shares his notes from his <a href="http://www.mathrecreation.com/2018/06/origami-workshop-again.html" target="_blank">Origami Workshop</a>. For those who enjoy doing origami, can you find something new here? <br /><br />Logic and Math go together so well. Check out this <a href="https://mathsvenns.com/nearest-integer-one-decimal-place-two-decimal-places/" target="_blank">blog full of Venn Diagrams to fill in</a>.<br /><br />In his post <a href="https://pballew.blogspot.com/2018/06/from-surds-to-ab-surds.html" target="_blank">From Surds to Ab-surds</a>, Pat Bellew looks at an interesting relationship, which looks like bad simplifying but is still correct. He suggests it as a challenge in an algebra course to produce more correct 'bad simplifying' equations. Hmm, I want to think about how to do that.<br /><br /><br /><br /><br /><b>Until Next Month...</b><br />Our sister blog carnival, <a href="http://aperiodical.com/2018/06/carnival-of-mathematics-158/" target="_blank">The Carnival of Mathematics</a>, often includes posts that are above my head. This month, our big sister is quite approachable. Enjoy!<br /><br />If you have suggestions for next month's MTAP, share via the form at the <a href="https://denisegaskins.com/mtap/" target="_blank">Carnival home page</a>. Sharing in the carnival, or hosting, is a great way to increase connections in the #mtbos/#iteachmath community.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0