tag:blogger.com,1999:blog-53033074821589225652023-06-02T03:31:28.350-07:00Math Mama Writes...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger593125tag:blogger.com,1999:blog-5303307482158922565.post-57547173742020057772023-02-24T17:26:00.002-08:002023-02-24T17:28:58.721-08:00Althea's Math Mysteries<p> I've been working for a few years on <i><b>Althea and the Mystery of the Imaginary Numbers</b></i>. It's almost ready for the illustrator. But I wanted to dive deeper into the characters, and started working on the second book in the series, <i><b>Althea and the Mysteries of Pi</b></i>. I'm about 80 pages in on my first (very rough) draft. It has been a blast writing this, because I pretty much know where I'm headed. (Although sometimes I worry that there's too much math, and not enough character development. And then I back up and think about Althea, Kiara, Sofia, and Aiden some more.)</p><p>Today I wanted a good place to put links that the book refers to, so I made a temporary website for all the books. It's a google site (for now). And I made mock-ups for the book covers. It helps me to organize my thoughts, and it is super exciting to see. So even though the books won't be published for another year (or 2?), maybe this will tantalize you. <a href="https://sites.google.com/view/altheas-math-mysteries/home" target="_blank">Here's the site for Althea's Math Mysteries</a>.</p><p>When this book is pretty much done, I'll start working on the third one, <i><b>Althea and the Mysteries of Infinity</b></i>. I have lots of ideas for that one, but they have no structure. I have no idea where I'll start or end. </p><p>When I'm all done, and these 3 books are published, maybe I'll have realized that there are more books in the series. For now, it's looking like just the three.<br /></p><p><br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-64306468634530216122022-09-08T22:10:00.002-07:002022-09-08T22:10:46.465-07:00What does it mean when we feel we "understand" something?<p>On facebook, I'm in a group for people who use <a href="https://beastacademy.com/" target="_blank">Beast Academy</a> (even though I'm not using it), because Beast Academy fascinates me. I love most of what they do.</p><p>A parent today posted that she was confused about the BA way of multiplying 59*59. They have you draw a 60 by 60 box, and then take off one row (of 60) and one column (which is now 59). Your box is now 59 by 59, and its area is 60*60 - 60 - 59. Cool. <br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiT8DIywWj1-tm2XmnEaQi5wuSCf52K17cVL2R3-4NiluU6zVpm-7IoBPLFzZ8wGOxPScmY1U0PRTignwqZ7vOSLUSa1MGf13H5cbOEUP3HqrddklUkHKKedvSPMeexenXGzcFUPeSoNITUvY1EzmNn7XEck36lnU5a-bI8zIbiCLZFVnPDTHSrYNYOBQ/s1567/beast%20Multiplication.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="839" data-original-width="1567" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiT8DIywWj1-tm2XmnEaQi5wuSCf52K17cVL2R3-4NiluU6zVpm-7IoBPLFzZ8wGOxPScmY1U0PRTignwqZ7vOSLUSa1MGf13H5cbOEUP3HqrddklUkHKKedvSPMeexenXGzcFUPeSoNITUvY1EzmNn7XEck36lnU5a-bI8zIbiCLZFVnPDTHSrYNYOBQ/w400-h214/beast%20Multiplication.jpg" width="400" /></a></div><br /><p><br />She wasn't seeing it, so she taught her kid the standard algorithm. Lots of people were giving her flak for that. (We each do our best, so I don't see why folks would jump on her.) She replied to them that she thought learning it multiple ways was a good thing. </p><p>I wrote: "<span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en">Sure
it's great to do things multiple ways, but does he really understand
the algorithm you showed him? (Do you really understand why it works?) I
think that's why you're getting pushback here."<br /><br />She said they both understood it. I replied that I'd have trouble explaining to a young kid why you "put a 0". She wrote: "</span><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en">I
just told him we put it to show that the one number is done. I don’t
know if it’s accurate but he understood it. I don’t really remember it
ever being explained in school."<br /><br />So what she originally meant when she said he understood it, was that he could follow the steps and get it right. Not that he understood <i><b>why</b></i> it worked.<br /><br />I think this is common with math. People think 'understand' means the same as 'can follow the steps'. But I'm afraid that doing math without really seeing why each step makes sense is part of why a lot of people don't like math. It's surely why we easily forget how to do those things. <br /><br /><a href="https://teamone.msuurbanstem.org//wp-content/uploads/2014/07/Skemp-Relational-Instrumental-clean-copy-AT-1978.pdf" target="_blank">Here's an article by Richard Skemp</a>, written back in 1978, about why the deeper understanding, which he calls "relational understanding" is a better way to approach math. (He calls being able to follow the steps "instrumental understanding".) I wrote about <a href="https://mathmamawrites.blogspot.com/2012/06/teaching-for-understanding.html" target="_blank">this topic and this article ten years ago here</a>, but people's ideas about math haven't changed much in that time.<br /></span></p><p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en">Of course, this parent can still explain to her son why the standard algorithm works, so she hasn't somehow wrecked the beauty of Beast Academy, as some people seemed to feel. And that's what got me writing - I want to see how well I can explain the standard algorithm.</span></p><p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en">I figure that the standard algorithm packs in a lot with as little writing as possible. (Maybe when we didn't have calculators, and had to do lots of by-hand multiplication, writing as little as possible was considered an important goal for the way we write things down?) So I figured that it needs to be unpacked a little. That's what I tried to do here. <br /><br /> </span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinF1J0P5pTPS5UBSdHgK33kbkW2htBoVteCqeqVcWeji_m7oZQX-01-l3_syEzqeeuGWr15F9UGM3G7CfwhVPBQr5gBWJ41LvF30vPa5w-szd2xA9ohSYbUPLsYIR9r58D5j4Lgj7bsz3YKzIbsNiYMrE9oT6FCTJPAHfjuslhik-9ZwASVLfTiLgmHw/s1597/Multiplication%20standard%20algorithm%20unpacked.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="827" data-original-width="1597" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinF1J0P5pTPS5UBSdHgK33kbkW2htBoVteCqeqVcWeji_m7oZQX-01-l3_syEzqeeuGWr15F9UGM3G7CfwhVPBQr5gBWJ41LvF30vPa5w-szd2xA9ohSYbUPLsYIR9r58D5j4Lgj7bsz3YKzIbsNiYMrE9oT6FCTJPAHfjuslhik-9ZwASVLfTiLgmHw/w400-h208/Multiplication%20standard%20algorithm%20unpacked.jpg" width="400" /></a></div><br /><p></p><p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en">The first calculation is adding up all 4 areas. The one to the far right is the standard algorithm. The first number in the standard algorithm (531) is the 81 and the first 450 added together (with carrying), and the second number (2950) is the other 450 and the 2500 added together. It's surely as little writing as possible, but it hides so much! Does my unpacking on the left help?<br /></span></p><p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en"><br /></span></p><p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en">It all makes sense to me, but the Beast way feels more fun. (And I don't have to write <i>anything</i> that way. I can hold it all in my head!) What do you think?<br /></span></p><p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en"><br /></span></p><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en"></span><span class="gvxzyvdx aeinzg81 t7p7dqev gh25dzvf tb6i94ri gupuyl1y i2onq4tn b6ax4al1 gem102v4 ncib64c9 mrvwc6qr sx8pxkcf f597kf1v cpcgwwas m2nijcs8 hxfwr5lz k1z55t6l oog5qr5w tes86rjd pbevjfx6" dir="auto" lang="en"></span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-69909993278719747042022-08-16T15:25:00.001-07:002022-08-16T19:51:56.892-07:00Prepping for Fall, Calc II: Lovely Arc Length Example<p>I'll be teaching Calc II for the first time in a few years. This is my first time starting out online with it. So I'm preparing my Canvas shell and thinking about how I want to explain each topic in Canvas. (I know the material well enough that I didn't have to prep this much when we were in person.) The extra prep before we start is so much work, but today it feels totally worthwhile. <br /></p><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;"><div class="separator" style="clear: both; text-align: right;"><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div style="text-align: left;"><div style="text-align: right;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyuRcelRZQKN6-KVp-FF7KyIbNEF_7n-xcc7EJN3cOFM081lww0sY8KK80K7FAcCJTr-B9NA1_ODs6wxP11wnkY7SCU7yvD-dMLkrI6_KeaGc1E3wslY7HXxMOVutVsKU3rSf8GZtzZZbsklhv0I30CNO0-J1yPP6HzclZCfSXtGID813y4r7saQbNXw/s640/wavy%20wall.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="640" data-original-width="517" height="297" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyuRcelRZQKN6-KVp-FF7KyIbNEF_7n-xcc7EJN3cOFM081lww0sY8KK80K7FAcCJTr-B9NA1_ODs6wxP11wnkY7SCU7yvD-dMLkrI6_KeaGc1E3wslY7HXxMOVutVsKU3rSf8GZtzZZbsklhv0I30CNO0-J1yPP6HzclZCfSXtGID813y4r7saQbNXw/w241-h297/wavy%20wall.jpg" width="241" /></a></div> </div><div style="text-align: left;"> </div><div style="text-align: left;">For
arc length I was excited to use "crinkle crankle walls" as an example.
Isn't that a pretty wall? And you can actually use fewer bricks this
way than for a straight wall, because one layer of bricks here is
stronger than it would be straight (so the straight wall would need
extra bricks for support). I'm thinking we'll try to prove that assertion in my Calc II class.<br /></div><div dir="auto" style="text-align: start;"> </div></div></div></div><div dir="auto" style="text-align: start;"><br /></div><div dir="auto" style="text-align: start;"><br /></div></div><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;">It turns out that arc length uses an integral which often has no "elementary solution", meaning there is no anti-derivative using the functions we are familiar with. </div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;">The arc length for y=sin x is...<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicqPQVb2Mc2JO8VLK-bf8WqfA_UVsMvtZjlvdUNyYKYvqzLG4XEw8Bt394osW0n0RUCPkjHe67FzNMZru8FHauOEZrZCTHsP6IWGatnfrC0IEJLP1CdnZHEC0rxONhJp0sj6vCA24OjEmG0_g_Z9wfUGnV1Hym7DyolQusdhiWzA8s8BohwIs3fdrKAw/s211/wall%20length.gif" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="46" data-original-width="211" height="46" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicqPQVb2Mc2JO8VLK-bf8WqfA_UVsMvtZjlvdUNyYKYvqzLG4XEw8Bt394osW0n0RUCPkjHe67FzNMZru8FHauOEZrZCTHsP6IWGatnfrC0IEJLP1CdnZHEC0rxONhJp0sj6vCA24OjEmG0_g_Z9wfUGnV1Hym7DyolQusdhiWzA8s8BohwIs3fdrKAw/s1600/wall%20length.gif" width="211" /></a></div><br /></div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;">And this has no "elementary solution".</div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;"> <br /></div><div dir="auto" style="text-align: start;">I often tell my students that we study infinite series to solve the integrals with no easier solution, but I just realized that that won't work here. (Can't do a square root of an infinite series!) </div><div dir="auto" style="text-align: start;"> </div></div><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;">Ok, no problem. I'm also teaching numerical integration. So I made a <a href="https://docs.google.com/spreadsheets/d/1CK28xLTU1OnW9z-Kk4sBa54-HzAlQwTBbuw-lk4FUJY/edit?usp=sharing" target="_blank">google sheet to do Simpson's method</a>, and it turns out beautifully!! (Beautifully meaning that my answer matched <a href="https://math.stackexchange.com/a/2471308/263729" target="_blank">the answers on Math SE</a> that people explained in ways that were above my head. I don't know a thing about "elliptical integrals".)</div><div dir="auto" style="text-align: start;"> </div></div><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;">I still need to remember how to explain Simpson's rule, but I'll get that back easily enough. </div><div dir="auto" style="text-align: start;"> </div></div><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;">If this wall follows a sine wave, then for 6.28 feet (2π feet) of straight distance covered, it has a length of 7.64 feet. That's just over 20% extra length. (Now to think with my students about whether that's better than the straight wall with supports.)<br /></div></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-17824371149614783762022-08-03T22:50:00.001-07:002022-08-04T10:08:37.973-07:00Technology Woes and Cheers: Venn diagram edition<p>I'm writing questions for my Discrete Math course that will be available to my students (and others) through MyOpenMath, a free online homework system. I'm not very good at programming in their environment, but I'm learning. The cool thing about MyOpenMath is that it uses random numbers in the questions so that each student might get a (slightly) different question.<br /></p><p>I wanted a way to ask, for a random Venn diagram: What is the set notation for this?</p><p>First, I needed a way to make lots of Venn diagrams, all pretty, and all in the same style. I searched the internet for a free online Venn diagram maker. Nothing right showed up. I looked at over a dozen sites. Many wanted me to sign in. That should not be necessary and I skipped those. None of the others were even close to what I wanted, which is pretty simple. Really?! Isn't this something lots of people would want? </p><p>I asked about it on <a href="https://matheducators.stackexchange.com/questions/25455/looking-for-venn-diagram-maker" target="_blank">Math Educators Stack Exchange</a>. Within hours, Cameron Williams posted an answer. He made it on desmos <i><b>for me</b></i>. (How sweet is that?! Amazing.) I know desmos well, so I was able to modify his version to be <a href="https://www.desmos.com/calculator/zfctz77m42" target="_blank">exactly what I wanted</a>, in less time than I had already spent searching. (I suggest you go play with it - it's lovely.) And, if you want orange shading instead of blue, it's very easy to modify this to get exactly what <i><b>you</b></i> want.<br /></p><p>Then I made 17 screenshots of various combinations of the basic regions, and named them based on the set notation. So "(A un B) int not C.jpg" is the filename for ...</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLvhuUMwG6aLKhi2RvfNBXejDIkcDKm-0Osh8-ZjdVva-EYdIk1oT6Xa8vEZTdLAuas30ni52t5UUn8quN7ol56I-8c5T9upIOYw-7ESD8ssQLAxmIXOaEoaNXBjkWdnDH8CTPwydE0IH_5L27Oi_Rt5ap4AyVpv8aah6shLBPchpCPh4u7t-SH2kziw/s1270/(A%20un%20B)%20int%20not%20C.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1110" data-original-width="1270" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLvhuUMwG6aLKhi2RvfNBXejDIkcDKm-0Osh8-ZjdVva-EYdIk1oT6Xa8vEZTdLAuas30ni52t5UUn8quN7ol56I-8c5T9upIOYw-7ESD8ssQLAxmIXOaEoaNXBjkWdnDH8CTPwydE0IH_5L27Oi_Rt5ap4AyVpv8aah6shLBPchpCPh4u7t-SH2kziw/s320/(A%20un%20B)%20int%20not%20C.png" width="320" /></a></div><p> </p><p>Next I went back to MyOpenMath, and wrote most of my multiple choice problem. I'm still stuck on how to get it to display a randomly chosen image file. I think the folks at the help forum there will help me out on that. Once I finish fixing it, I'll edit this post to show the question. MyOpenMath allows attached videos to explain how to answer the questions. I think I might do a video for this one. </p><p><br /></p><p>So if you want a <span style="font-size: large;"><b><a href="https://www.desmos.com/calculator/zfctz77m42" target="_blank">free online Venn diagram maker, </a></b><b><a href="https://www.desmos.com/calculator/zfctz77m42" target="_blank">it's here</a>.</b></span> I don't know how to help google move this up in the searches so people can find it. Do you?<br /></p><p><br /></p><p><br /></p><p><br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-61098136119722342802022-07-30T11:36:00.004-07:002022-07-30T20:33:35.563-07:00Math Teachers at Play (aka Playful Math Education, Blog Carnival #157)<p> </p><p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGLaHpE70V1ulqCzSD_nVtYtryRgHeLTdxeNQ5t27nXiVBbQ7jPJMqs_-24B5at7FBv3wj7BmvisQrPHCKzzpqIwBDE5n9xZM_dktui-2Pj_vQM7NTr1rxD5MArwJHucTRT7-ibK2zZWwlSy7WNyHSpUVviBADpOO-tuqVEXZXssTj0eG1x3QW6-5ANA/s1598/mtap%20midway-1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="836" data-original-width="1598" height="334" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGLaHpE70V1ulqCzSD_nVtYtryRgHeLTdxeNQ5t27nXiVBbQ7jPJMqs_-24B5at7FBv3wj7BmvisQrPHCKzzpqIwBDE5n9xZM_dktui-2Pj_vQM7NTr1rxD5MArwJHucTRT7-ibK2zZWwlSy7WNyHSpUVviBADpOO-tuqVEXZXssTj0eG1x3QW6-5ANA/w640-h334/mtap%20midway-1.jpg" width="640" /></a></div><br /><p></p><p> </p><p>About once a year, I sign up to host this long-running blog carnival. Ever since Google Reader was snatched away, blogs seem to have fewer readers and less activity. Mine certainly has straggled along in recent years. (I guess I needed a very long rest after finishing my big book.) Today, I'm looking forward to exploring the new ideas I'll find online and gather here.</p><p> </p><p></p><p>We start with cool facts about 157, and a puzzle...</p><p> </p><p><span style="font-family: verdana;"><span style="font-size: large;"><span style="color: #800180;"><b>Cool Little Facts </b></span></span></span><br /></p><p></p><ul style="text-align: left;"><li>157 is the 37th prime number. (37 is prime too.)</li><li>157 is the largest known prime <i>p</i> for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mfrac>
<mrow>
<msup>
<mi>p</mi>
<mrow>
<mi>p</mi>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>p</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {p^{p}+1}{p+1}}}</annotation>
</semantics>
</math></span><img alt="\frac{p^p+1}{p+1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b153bd05cf4c07666a2fb55ba9d6eff243e1a394" style="height: 5.843ex; vertical-align: -2.338ex; width: 7.068ex;" /></span> is also prime (see <span class="nowrap external"><a href="https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a class="extiw" href="https://oeis.org/A056826" title="oeis:A056826">A056826</a></span>).<br />157 is a <a href="https://en.wikipedia.org/wiki/Palindromic_number" title="Palindromic number">palindromic number</a> in bases 7 (313<sub>7</sub>) and 12 (111<sub>12</sub>).</li><li>157 is the largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1. </li><li>
157 is the smallest three-digit prime that produces five other primes by
changing only its first digit: 257, 457, 557, 757, and 857. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Opao">Opao</a>] </li><li>157 is the largest rating on the <a href="http://www.nhc.noaa.gov/aboutsshws.php">Saffir-Simpson Hurricane Wind Scale</a> occurs at sustained winds of 157 mph or higher. </li><li>If we use the English alphabet code a = 1, b = 2, c = 3, … , z = 26, then <a href="https://translate.google.com/?sl=es&tl=en&text=n%C3%BAmero%20primo&op=translate">número primo</a> = 157. </li></ul><p> </p><p><span style="color: #800180;"><span style="font-size: large;"><span style="font-family: verdana;"><b>Puzzle</b></span></span></span></p><p>How many 3-digit numbers can we find where the last digit equals 2 times the first digit plus 1 times the second digit? 157 is one answer. How would you find the others without tediously checking each 3-digit number? (I use a spreadsheet when I want enough data to see patterns, but I worked hard to get the digits apart. Once you find the first few answers by hand, you might see the pattern...)</p><p>[Solution at bottom.] <br /></p><p><span style="font-family: verdana;"><span style="font-size: large;"><br /></span></span></p><p><span style="color: #800180;"><span style="font-family: verdana;"><span><span style="font-size: large;"><b>Not Just Blogs...</b></span></span></span></span></p><p>I'm working on another book, much smaller this time. <i><b>Althea and the Mystery of the Imaginary Numbers </b></i>should be ready sometime next year. Since I'm working on a book, I've been thinking a lot about what makes a fun mathy book. It needs a good storyline. It needs interesting math. And if it's for young kids, it needs lovely illustration. </p><p>There's a prize for good mathy books, called the <a href="https://www.mathicalbooks.org/" target="_blank">Mathical Book Prize</a>. It started in 2015 and doesn't seem to include small publishers like <a href="https://naturalmath.com/goods/" target="_blank">Natural Math</a> (my publisher), so some of my favorites are missing. I think my favorite book on their list might be the picture book <a href="https://bookshop.org/books/which-one-doesn-t-belong-playing-with-shapes/9781580899468" target="_blank">Which One Doesn't Belong</a>, by math blogger Christopher Danielson.</p><p>Here are a few of my favorites that aren't on their list:</p><p><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b><i>Quack and Count</i></b>, by Keith Baker</span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> (for ages 2 to 7),</span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> a board book good for the youngest child who will sit and
listen to a story. And it stays good because it's so luscious. Great
illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7
ducklings are enjoying themselves in every combination. “Slipping, sliding,
having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, 4 plus 3, 3 plus 4, and
so on.) It would be great to have a book like this for each number, showing all
the number pairs that make it.</span></p><p><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuzf-8ZxStiaGDqR3R5A61fjhFQL7nzAu9xTaC08jyzaHlDoENIUR6lnG28Rd7lqsxiSxYeWHpG7P1oBIPyLK494LTdkTWWeV0I6_brGnxmUHqtLEd0ZI8tzB9G9T8HWTk_aOtq552sMj4I5mzAjup1teQgDzDIyp0_tWkESW7hWrVBvlOnxTmKuUV7A/s1280/quack%20and%20count.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1280" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuzf-8ZxStiaGDqR3R5A61fjhFQL7nzAu9xTaC08jyzaHlDoENIUR6lnG28Rd7lqsxiSxYeWHpG7P1oBIPyLK494LTdkTWWeV0I6_brGnxmUHqtLEd0ZI8tzB9G9T8HWTk_aOtq552sMj4I5mzAjup1teQgDzDIyp0_tWkESW7hWrVBvlOnxTmKuUV7A/w385-h217/quack%20and%20count.jpg" width="385" /></a><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> <br /></span></span></div><p></p><p><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b><i>How Hungry Are You?</i></b> by Donna Jo Napoli and Richard Tchen</span></span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> (for ages 3 to 12), on equal sharing.</span> <span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">The picnic starts with just two friends, rabbit is bringing 12
sandwiches and frog is bringing the bug juice. Monkey wants to come, "My
mom just made cookies. I could take a dozen." They figure out how much of
each goody each friend will get. In the end, there are 13 of them, and the
sharing becomes more complicated. One of the delights of this book is the
little icons showing who’s talking. It would make a good impromptu play. [</span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">There are lots of good books on equal sharing. Another lovely one is <b><i>The Doorbell Rang</i></b>, by Pat Hutchins.]</span></p><div class="MsoNormal">
<span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b><i>The Cat in Numberland, </i></b>by Ivar Ekeland</span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> (for ages 5 to adult)</span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">, starts when Zero knocks on the door of the Hotel Infinity. He’d like
a room, but they’re all full (with the number One in Room One, and so on).
Turns out that’s no problem. The cat who lives in the lobby gets confused - if
the hotel is full, how can the numbers make room for zero just by all moving up
one room? Things get worse when the fractions come to visit. This story is
charming enough to entertain young children, and deep enough to intrigue
anyone. Are you ready to learn about infinity with your 5 year-old?</span></div><div class="MsoNormal"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> </span></div><div class="MsoNormal"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b><i>The Man Who Counted</i></b>, by <span class="ptbrand">Malba Tahan </span></span>
<span class="ptbrand"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">(for ages 6 to adult), was</span></span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span class="ptbrand"> written in Brazil, and set in the Middle East. We
follow the adventures of Beremiz, an accomplished mathematical problem-solver.
He uses math to settle disputes, solve riddles and mysteries, and entertain his
hosts. The series of 34 adventures, each with a math puzzle, is reminiscent of
the Arabian Nights. If you read one chapter a night, your audience will be
begging for more – and isn’t that the way it should be?</span></span> <br /></span></div><div class="MsoNormal">
<b><i> </i></b></div><div class="MsoNormal"><b><i>Carry On, Mr. Bowditch</i></b><span style="font-style: normal; font-weight: normal;">, by Jean Lee Latham (for ages 7 to adult)</span>, <span style="font-style: normal; font-weight: normal;">is a
slightly fictionalized account of the life of Nathaniel Bowditch, who loved
math, but had to leave school when his family needed his help. He was
indentured to a ship chandlery for 9 years. Although that dashed his
hopes of someday going to Harvard to study math, it was the right place
to learn the mathematics behind navigation. When he finally went to
sea, he invented a new way to ‘do a lunar’, and spent endless hours
correcting errors in the tables used for navigation. Bowditch’s book,
the American Practical Navigator, first published in 1902, is still
regularly updated, and is carried on U.S. naval vessels to this day. </span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"> </span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;">And coming very soon ... Denise Gaskins' 2nd edition of <a href="https://denisegaskins.com/2022/07/14/sample-my-new-playful-word-problems-book/" target="_blank"><i><b>Word Problems from Literature</b></i></a>. (She'll be using kickstarter to raise some funds to get this out the door. Crowdfunding is how tiny publishers make it work!)</span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-family: verdana;"><span style="font-size: large;"><span style="color: #800180;"><span style="font-style: normal; font-weight: normal;"><br /></span></span></span></span></div><div class="MsoNormal"><b><span style="font-family: verdana;"><span style="font-size: large;"><span style="color: #800180;"><span style="font-style: normal;">...And Now the Blogs (mostly geometry)<br /></span></span></span></span></b></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><a href="https://denisegaskins.com/2022/06/23/how-to-make-time-for-exploration/" target="_blank">Denise Gaskins' How to Make Time for Exploration</a>, in which Denise considers the benefits of Michelle's "Minimalist Math" curriculum, used along with games and books. <br /></span></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiELT78Wxlx_tm57aVGQa-v3cWA-zWYCazR_DjssU2nW9Aw5FeUhnbLyGBhpFF-ZCbbeMxcdloIzZL_QwaPIDAASQBi7L028O4y7OZqREsNPApUTqxbbs6suk1ICScQixJxgLt8UHz9rVdYCQoMx774K-7HLg8aPUYZM84Eyht0QhfQTso3anFbojLnGw/s1074/kandinsky%20box.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="904" data-original-width="1074" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiELT78Wxlx_tm57aVGQa-v3cWA-zWYCazR_DjssU2nW9Aw5FeUhnbLyGBhpFF-ZCbbeMxcdloIzZL_QwaPIDAASQBi7L028O4y7OZqREsNPApUTqxbbs6suk1ICScQixJxgLt8UHz9rVdYCQoMx774K-7HLg8aPUYZM84Eyht0QhfQTso3anFbojLnGw/w197-h166/kandinsky%20box.jpg" width="197" /></a></div><span style="font-style: normal; font-weight: normal;"> </span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><a href="https://www.tumblr.com/blog/view/mathhombre/688093776418324480?source=share&fbclid=IwAR3zvsPP9ypRFMJ_YXBcJSQjHgIsY03K_JOgkCRRsM0yLlNIGyqK2BDop8k" target="_blank">John Golden's Art, Math, and Geogebra Project</a>, in which John has created a way for you to change a Kandinsky box to be new combinations of colors. Fun.</span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /> </span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><a href="http://www.sineofthetimes.org/euclid-walks-the-plank/" target="_blank"><span style="font-style: normal; font-weight: normal;">Daniel Scher's Euclid Walks the Plank on Geometric Construction</span></a><span style="font-style: normal; font-weight: normal;">, in which Daniel explores helping students to see the power of circles in building equal length line segments, using Geometer's Sketchpad for his online experiments. Once again, you get to play with the geometry.<br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><a href="https://samjshah.com/2022/07/20/pcmi-2022-post-2-3d-printing/" target="_blank">Sam Shah on 3D Printing</a>, in which Sam shares lots of cool 3D printing projects but decides they aren't really helping his students learn math. Do you have any 3D printing projects that help your students learn math?</span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><a href="https://jsandfordmath.ca/2021/04/18/play-persist-prove/" target="_blank">Joann Sandford's Play, Persist, Prove</a> on thinking about the angles in polygons. Can you use pattern blocks to <i>prove</i> what the angles are?</span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;">I adore Catriona Schearer's geometry puzzles, which she posts on <a href="https://twitter.com/Cshearer41" target="_blank">twitter</a> and elsewhere. <a href="https://www.youtube.com/watch?v=8YOaaRm6-qs&t=4s" target="_blank">Here's a video of her talking about them</a>. (I recommend starting at about 8:30. They wait for participants and talk about Mathigon first.) Here's a lovely puzzle of hers. The big triangle that holds all the others is also isosceles. Find it on her twitter feed, and you'll see lots and lots of thoughts about it.</span></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8vzyMJ8fBoHBfx_Lmogx7U0EUZHIvy8EGlKsackHamqnuXZRBZWnERpVa0mjFgb0aw-GNRB3VlYMGUIKDXI1zP3ofVP5Y6pfYMl6QyULQKmv6_n3ezQFSO1TRedansOiu3oVLpD3nQaqSsf3grQfJujQ0iTvF5QQygyvSY91bRUtEyAj74uC6hABhbg/s943/4%20triangles%20in%201%20all%20isos.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="530" data-original-width="943" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8vzyMJ8fBoHBfx_Lmogx7U0EUZHIvy8EGlKsackHamqnuXZRBZWnERpVa0mjFgb0aw-GNRB3VlYMGUIKDXI1zP3ofVP5Y6pfYMl6QyULQKmv6_n3ezQFSO1TRedansOiu3oVLpD3nQaqSsf3grQfJujQ0iTvF5QQygyvSY91bRUtEyAj74uC6hABhbg/w640-h360/4%20triangles%20in%201%20all%20isos.jpg" width="640" /></a></div>One more way to play with geometry ... <a href="https://sciencevsmagic.net/geo/" target="_blank">this site gamifies geometric construction.</a> I love it. </div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"> </div><div class="MsoNormal">Do you want more info on this blog carnival, or would you like to read old carnival posts? <a href="https://denisegaskins.com/mtap/" target="_blank">Denise Gaskins has got you covered.</a><br /><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"> <br /></span></div><div class="MsoNormal"><span style="font-style: normal; font-weight: normal;"><br /></span></div><p><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span></p>
<p><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> </span></p><p><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> </span><br style="mso-special-character: line-break;" /><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span></p><p> <br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;"><br /></p><p style="text-align: left;"><b>Puzzle solution</b>: There are 8 of these starting with 1: 113, 124, ..., 179, then 6 starting with 2, up to 2 starting with 4, for a total of 20.<br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-78550081512302353582022-03-04T23:21:00.000-08:002022-03-04T23:21:46.891-08:00Logic Puzzle, Supposedly from Einstein...<p> ... but there's no evidence for that. The puzzle originally had folks smoking cigarettes. Yuck. I've changed that to eating candy.</p><br /><div><div class="" dir="auto"><div class="ecm0bbzt hv4rvrfc dati1w0a e5nlhep0" data-ad-comet-preview="message" data-ad-preview="message" id="jsc_c_5w"><div class="j83agx80 cbu4d94t ew0dbk1b irj2b8pg"><div class="qzhwtbm6 knvmm38d"><span class="d2edcug0 hpfvmrgz qv66sw1b c1et5uql oi732d6d ik7dh3pa ht8s03o8 a8c37x1j fe6kdd0r mau55g9w c8b282yb keod5gw0 nxhoafnm aigsh9s9 d3f4x2em iv3no6db jq4qci2q a3bd9o3v b1v8xokw oo9gr5id hzawbc8m" dir="auto"><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;">The situation:</div><ul style="text-align: left;"><li>There are 5 houses in five different colors.</li><li>In each house lives a person with a different nationality.</li><li>These five people drink a certain beverage, eat a certain candy, and keep a certain pet.</li><li>No one has the same pet, eats the same kind of candy, or drinks the same beverage. </li></ul><div style="text-align: left;"> </div></div></span><span class="d2edcug0 hpfvmrgz qv66sw1b c1et5uql oi732d6d ik7dh3pa ht8s03o8 a8c37x1j fe6kdd0r mau55g9w c8b282yb keod5gw0 nxhoafnm aigsh9s9 d3f4x2em iv3no6db jq4qci2q a3bd9o3v b1v8xokw oo9gr5id hzawbc8m" dir="auto"><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div style="text-align: left;"><br /></div><div dir="auto" style="text-align: start;">The question is: Who owns the fish?</div></div><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;">Hints:</div><ul style="text-align: left;"><li>the Brit lives in the red house</li><li>the Swede has a dog</li><li>the Dane drinks tea</li><li>the green house is on the left of the white house</li><li>the green house's owner drinks coffee</li><li>the person who snarfs M&Ms has birds</li><li>the owner of the yellow house loves peanut butter cups</li><li>the person living in the center house drinks milk</li><li>the Norwegian lives in the first house</li><li>the person who adores Heath bars lives next to the one who keeps cats</li><li>the person who has a horse lives next to the peanut butter cup lover</li><li>the person who eats Snickers bars drinks beer</li><li>the German eats Almond Joys</li><li>the Norwegian lives next to the blue house</li><li>the person who eats Heath bars has a neighbor who drinks water</li></ul></div></span><span class="d2edcug0 hpfvmrgz qv66sw1b c1et5uql oi732d6d ik7dh3pa ht8s03o8 a8c37x1j fe6kdd0r mau55g9w c8b282yb keod5gw0 nxhoafnm aigsh9s9 d3f4x2em iv3no6db jq4qci2q a3bd9o3v b1v8xokw oo9gr5id hzawbc8m" dir="auto"><div class="cxmmr5t8 oygrvhab hcukyx3x c1et5uql o9v6fnle ii04i59q"><div style="text-align: left;"> </div><div dir="auto" style="text-align: start;"><p><span class="d2edcug0 hpfvmrgz qv66sw1b c1et5uql oi732d6d ik7dh3pa ht8s03o8 a8c37x1j fe6kdd0r mau55g9w c8b282yb keod5gw0 nxhoafnm aigsh9s9 d3f4x2em iv3no6db jq4qci2q a3bd9o3v b1v8xokw oo9gr5id hzawbc8m" dir="auto"></span></p><div class="kvgmc6g5 cxmmr5t8 oygrvhab hcukyx3x c1et5uql ii04i59q"><div dir="auto" style="text-align: start;">[There is one thing that's unclear: Is "the first house" the one on the left of the bunch? I assumed that. Apparently, you can assume that it's on the right end, and according to Wikipedia, you'll get the same answer. I haven't explored that.]<br /></div></div><p></p> <br /></div></div></span></div></div></div></div></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-61038402823539326702022-02-12T17:29:00.000-08:002022-02-12T17:29:16.475-08:00Still learning, after all these years...<script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); </script> <script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"> </script>
<p>This semester I'm teaching Calculus I and Linear Algebra. In each class, I've had a moment of discovery in the past week or so.</p><p> </p><p><b>Calculus: Derivatives from Graphs </b><br /></p><p>In calculus, I work with them on what the derivative graph of a function would look like, given just the graph of the function. So if the graph of f is this ...</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhWb_ZxyGsL-8O_GgAyZUVQbgN6n4g1kVM-4oaTd-nA95-dl4ULjKh8jbFtWG7QItCd55Mk6FX_8vrmGqqqPmj_01regeVXFwjdsoGlwPVjw-tiTvSn4Uhzuno2bVHOX3OaN4rJgMJNO3GyZ96wqsR14aOBH73xBRv_iSrbU0FkOvAK-_Dq63yYTmQUGQ=s670" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="382" data-original-width="670" height="182" src="https://blogger.googleusercontent.com/img/a/AVvXsEhWb_ZxyGsL-8O_GgAyZUVQbgN6n4g1kVM-4oaTd-nA95-dl4ULjKh8jbFtWG7QItCd55Mk6FX_8vrmGqqqPmj_01regeVXFwjdsoGlwPVjw-tiTvSn4Uhzuno2bVHOX3OaN4rJgMJNO3GyZ96wqsR14aOBH73xBRv_iSrbU0FkOvAK-_Dq63yYTmQUGQ=s320" width="320" /></a></div><br /><p>... then what would f' look like? The activity (with 8 different graphs) went as it usually does. </p><ul style="text-align: left;"><li>Step 1: Find where the slope is 0, and give f' a value of 0 at that x.</li><li>Step 2: Where the slopes of f are positive, highlight positive values for f' (and similarly for negative slopes). (Actually, the highlighting was new. I usually just draw dotted lines.)<br /></li><li>Step 3: Draw a curve that connects it all.</li></ul><p>We had an absolute value curve and discussed where the derivative is undefined. (Which I marked with vertical red lines.) </p><p> </p><p>And then we got to this one ... <br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiO0xGRZJKh8Gpcavd3FbGjcdXfeehv73fuuszj3Z-gNGuvSvGxXBlV1Kg5C-N8ry1xHNw-9dJ_HCKl-y6Zx0S15gGPyEmiWHP6Ee5GRdQnRUBAcIVuzleP77loDHOKPIkm1FVR89-ZWg7-pe5exf5DtlBfTGB-mDq0waxQSVA3gVjgAskYmQMTMQZS6w=s678" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="678" height="142" src="https://blogger.googleusercontent.com/img/a/AVvXsEiO0xGRZJKh8Gpcavd3FbGjcdXfeehv73fuuszj3Z-gNGuvSvGxXBlV1Kg5C-N8ry1xHNw-9dJ_HCKl-y6Zx0S15gGPyEmiWHP6Ee5GRdQnRUBAcIVuzleP77loDHOKPIkm1FVR89-ZWg7-pe5exf5DtlBfTGB-mDq0waxQSVA3gVjgAskYmQMTMQZS6w=s320" width="320" /></a></div><p></p><p>I said that <i>w'</i> looked like this ...</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgU1IucpXOwHMe6WT12Tn5P5tDUXrLFf5xjVW9hHs7k0Edcg8I8-hOLSi_7kyLM5955GTaPN-HC9P2I8VRKoWcUEWY9XWLPrXnth64ql844zBG65owcJytS-trJ4ZnJokh2yOmDox5i-r3DSsJlsx8FKF9nY_Add4s0jRPr_4BT4pphArY6cgF5hM5ePw=s688" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="328" data-original-width="688" height="153" src="https://blogger.googleusercontent.com/img/a/AVvXsEgU1IucpXOwHMe6WT12Tn5P5tDUXrLFf5xjVW9hHs7k0Edcg8I8-hOLSi_7kyLM5955GTaPN-HC9P2I8VRKoWcUEWY9XWLPrXnth64ql844zBG65owcJytS-trJ4ZnJokh2yOmDox5i-r3DSsJlsx8FKF9nY_Add4s0jRPr_4BT4pphArY6cgF5hM5ePw=s320" width="320" /></a></div><br /><p>And <b>a student asked</b> how I knew the lines were straight. Hmm, <i>do</i> I know that? "I'm not sure. Let's see..."<br /></p><p> </p><p>I thought about the curve given for <i>w</i> and said it looked like a bunch of parabola shapes (which I know have straight line derivatives), ... or like the absolute value of sine. I decided this was a fascinating question, and put both on desmos.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjZXvJrR8qpmTQ6d1m_tOOLBdoNAkbY2cx7fA_Am2-ikWPAzAzVyjxw7pNMJPhcesrrRl-OtJ7xzNc-YPbCrTHXu799RYZ9XphQcLEt91Q_hl4gh0Yx2uzEVHznlG5KHKATwv5Qe7NY8bQfzOr2qUwQeSUt3-efh6snmNO-43_NQ5hvUzbyKfT-gMch-w=s962" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="292" data-original-width="962" height="97" src="https://blogger.googleusercontent.com/img/a/AVvXsEjZXvJrR8qpmTQ6d1m_tOOLBdoNAkbY2cx7fA_Am2-ikWPAzAzVyjxw7pNMJPhcesrrRl-OtJ7xzNc-YPbCrTHXu799RYZ9XphQcLEt91Q_hl4gh0Yx2uzEVHznlG5KHKATwv5Qe7NY8bQfzOr2qUwQeSUt3-efh6snmNO-43_NQ5hvUzbyKfT-gMch-w=s320" width="320" /></a></div><p></p><p>The red is y = |sin(π/2*x)|, and the blue is y = -(x+1) <sup> 2 </sup> + 1 and y = -(x-1) <sup> 2 </sup> + 1. To me, it looks like the original graph of <i>w</i> could be either one. But the derivative is the straight line segments only if <i>w</i> came from parabolas. If it came from a sine wave, then the derivative is curved (coming as it does from cosine). Using orange for the derivative of the sine graph and purple for the derivative of the parabolas graph, I got this in desmos...</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhR2DgL5mKqLt_9PoO43FfaxBHXUrW7Hs4AHdSrTTOHP44IGFH6ZcnOb2GPEd_rfaZ5duj3gsrDs6jDLiodX3pVXRlZu3SW_Yl3-hhzC2XJImORbdPCcmmVMQC09ee7CoaF1PfHC2qQW2veY281Y4HrRphQOfpf_7O4Kc__SML5mBM-SmcIjEp1AcGD_g=s954" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="938" data-original-width="954" height="315" src="https://blogger.googleusercontent.com/img/a/AVvXsEhR2DgL5mKqLt_9PoO43FfaxBHXUrW7Hs4AHdSrTTOHP44IGFH6ZcnOb2GPEd_rfaZ5duj3gsrDs6jDLiodX3pVXRlZu3SW_Yl3-hhzC2XJImORbdPCcmmVMQC09ee7CoaF1PfHC2qQW2veY281Y4HrRphQOfpf_7O4Kc__SML5mBM-SmcIjEp1AcGD_g=s320" width="320" /></a></div><br /><p>Very different look to the derivatives, even though the original <i>w</i> could have been either of the original functions I put onto desmos. Fascinating!</p><p> </p><p> </p><p><b>Linear Algebra: Pivots vs Free Variables</b></p><p>We are using some fabulous activities from the <a href="https://iola.math.vt.edu/" target="_blank">Inquiry-Oriented Linear Algebra project</a>, along with our textbook, <b><i>Linear Algebra and Its Applications</i></b>, by David Lay (we're using the 4th edition). We had just done part 3 of the Magic Carpet project the day before, and I was summarizing. We were talking about the span of a set of 3 vectors in <span> ℝ</span><sup>3</sup>, and saw that the span made a plane through the origin. This was because there were 2 pivot columns. And then <b>a student asked</b>, "But don't we use the number of free variables to decide whether we have a line or a plane?" </p><p> </p><p>To me that felt like a very deep question for a student to be asking this early in the semester. I said I'd answer the next day, since we were almost out of time. The next day I said, "We looked at the pivots because we were asking about span, which is all the linear combinations of the column vectors. Until we started considering span, we more typically asked about all the solutions to a set of equations, which is a different sort of question. For that, we look at how many free variables to determine if all our solutions create a line or a plane (or something more)."</p><p>I have never had a student ask a question like this, and was quite intrigued. I told them we'd explore somewhat similar questions in our 3rd unit (chapter 4 of Lay), when we will explore column space and null space. Once again, I was fascinated. </p><p>I've been teaching for over 30 years. I know calculus I inside and out. I've taught linear algebra often enough to feel like I'm a pretty solid expert on the basics. (I'd love to have more expertise on where this class might lead them.) Even so, I learn new things each semester. Even teaching beginning algebra, I have repeatedly seen it from a new perspective when prodded by some unique question a student was asking.<br /></p><p>Yay for student questions.<br /></p><p><br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-82867647582608127712021-12-16T18:45:00.000-08:002021-12-16T18:45:04.518-08:00Geometry Course for Homeschoolers, Spring 2022<p> </p><div id="x_appendonsend">I love geometry! (Well, I love a lot of math topics, but geometry feels especially like playing around.)</div><div><br aria-hidden="true" /></div><div>And I will be teaching a small course online for homeschoolers, starting in January. Here are the details:<br aria-hidden="true" /><br aria-hidden="true" /><div><b><br aria-hidden="true" /></b></div><div><b>Geometry Course</b></div><div><ul><li>Monday, January 10 to Thursday, May 26 (no class on Feb. 14)<br aria-hidden="true" /></li><li>Mondays and Thursdays, 4:30 to 6pm CA time / 7:30 to 9pm East Coast time, on Zoom<br aria-hidden="true" /></li><li>$800 for the course. (Please pay in advance. If you need sliding scale, please contact me to discuss.)</li><li>6 to 10 students</li><li>Using Michael Serra's<i><b> Discovering Geometry</b></i>
(4th edition, which is pretty reasonable used), along with (free)
materials from Henri Picciotto. We will also use geogebra extensively
(also free).<br aria-hidden="true" /></li><li><a data-auth="NotApplicable" data-linkindex="0" href="https://sites.google.com/view/geometry-with-professor-sue/home" rel="noopener noreferrer" target="_blank" title="https://sites.google.com/view/geometry-with-professor-sue/home">Check out my site for more about the course and me</a>. <br aria-hidden="true" /></li></ul></div><span></span><br aria-hidden="true" /></div>Please
contact me soon if interested. (Email suevanhattum@hotmail.com or
mathanthologyeditor@gmail.com.) I'm happy to chat on the phone too, if
you have any questions. You can text me at 510-367-8085, and we can talk
at a time that works for us both.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-58823982252725460372021-07-16T11:29:00.004-07:002021-07-16T18:10:45.276-07:00Sizes of Infinity<p> </p><div dir="auto"><div class="ecm0bbzt hv4rvrfc ihqw7lf3 dati1w0a" data-ad-comet-preview="message" data-ad-preview="message" id="jsc_c_lz"><div class="j83agx80 cbu4d94t ew0dbk1b irj2b8pg"><div class="qzhwtbm6 knvmm38d"><span class="d2edcug0 hpfvmrgz qv66sw1b c1et5uql oi732d6d ik7dh3pa ht8s03o8 a8c37x1j keod5gw0 nxhoafnm aigsh9s9 d3f4x2em fe6kdd0r mau55g9w c8b282yb iv3no6db jq4qci2q a3bd9o3v knj5qynh oo9gr5id hzawbc8m" dir="auto"><div class="kvgmc6g5 cxmmr5t8 oygrvhab hcukyx3x c1et5uql ii04i59q"><div dir="auto" style="text-align: start;">I am floored. Here is a new mathematical result that sounds pretty important. I'm surprised I hadn't heard of it sooner. It was published online in April.</div><div dir="auto" style="text-align: start;"> </div></div><div class="o9v6fnle cxmmr5t8 oygrvhab hcukyx3x c1et5uql ii04i59q"><div dir="auto" style="text-align: start;"><a href="https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/" target="_blank">This Quanta article</a> explains it pretty well. But if the article doesn't make sense to you, I can explain more. This is the field I had planned to go into when I was thinking I'd get a PhD. I loved my two logic courses at Eastern Michigan University. But the one I took at UCSD was not fun. I think because it was too far above me, and I couldn't stay grounded.</div><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;">The one problem with the article is that it made it sound like the big question was resolved. But it's not. I thought it was saying that the continuum hypothesis is false. The continuum hypothesis is about sizes of infinity. The smallest infinity is what you get when you count out all the infinite whole numbers (or all the fractions), and it is called the countable infinity. The continuum hypothesis says that the next size up is what you'd get "counting" the real numbers (like the number line). But there may be a size in between. </div></div><div class="o9v6fnle cxmmr5t8 oygrvhab hcukyx3x c1et5uql ii04i59q"><div dir="auto" style="text-align: start;"> </div></div><div class="o9v6fnle cxmmr5t8 oygrvhab hcukyx3x c1et5uql ii04i59q"><div dir="auto" style="text-align: start;">I hope there is a way to get a meaningful example of that in-between size of infinity. (The are bigger and bigger infinities, but the two things grounded in numbers we know well, integers and real numbers, are the most interesting to me.)</div></div><div class="o9v6fnle cxmmr5t8 oygrvhab hcukyx3x c1et5uql ii04i59q"><div dir="auto" style="text-align: start;"> </div><div dir="auto" style="text-align: start;">A fun way to start thinking about infinity is a book that's accessible even to young kids. It's a five chapter picture book titled <i><b>The Cat in Numberland</b></i>. Sadly, it doesn't seem to be available (unless you want to pay ridiculous prices). <a href="https://naturalmath.com/goods/" target="_blank">My publisher, Natural Math,</a> tried to help the author get it reprinted, but Cricket books (Carus publishing) wouldn't give up their rights, and won't republish. (Maybe we should look into that again...)</div><div dir="auto" style="text-align: start;"><br /></div><div dir="auto" style="text-align: start;"><div dir="auto" style="text-align: start;">[The
Quanta article links to the proof that was published online in April. I
don't expect to understand that, but I'll try reading it. I might quit
very quickly.]</div></div></div></span></div></div></div></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-79960853282653087572021-06-18T18:49:00.001-07:002021-06-18T18:49:16.880-07:00More Tech: Sue Finally Learns How to do Screencasts<p>I broke my ankle a few months ago, and could no longer use my whiteboard. I asked my college for an iPad and got it within a week. I asked in the Math Mamas group on Facebook for software recommendations - goodnotes and one other both got high recommendations. I went with goodnotes and fell in love.</p><p>Teaching online is significantly more work than teaching in person, and this just added to my workload. But I love that students can easily get my notes on Canvas. And this week I made my first screencast. And then my second. It took me a few hours to get the hang of it for the first one. I may have done the second one in under 20 minutes. Both of them are for a basic geometry course I'm teaching at my college, in which most of the students are high school students.<br /></p><p><br /></p><p><b>Indirect Proof (aka Proof by Contradiction)<br /></b></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwWK6Tjht-74V1DTP-cuxMqiqZfxkKYFnJsUTddW6OrWyfCWqzsisRTKidrDaSPzfztoWS9I1ZV01acSk_4vQ' class='b-hbp-video b-uploaded' frameborder='0'></iframe></div><br /><p><br /></p><p><br /></p><p><b>A Direct Proof</b></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dw0IkbGL7kehAj-p_b2CDuHQZvTMqS_1xpUdFxqiCbu0NOmwHo8K3_bFfz2tCUrenwLLQeVR7dB_0znXiWj6A' class='b-hbp-video b-uploaded' frameborder='0'></iframe></div><p> </p><p>I think I could do a few of these a week. Before posting on Youtube, I'd like to find a way to have my face in the corner if possible... Once I feel like I know what I'm doing, the Math Mama's channel gets underway!<br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag: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.com0tag: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.com0tag: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>
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<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 />
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<b>139</b></h2>
Every number is cool.* Here are some ways 139 is cool:<br />
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<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>
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<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 />
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<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 />
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<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 />
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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 />
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<b>Talking Math With Your Kids (#TMWYK)</b><br />
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<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>
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<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 />
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<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 />
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Three One Four.</div>
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Hmm.</div>
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Not very hard.</div>
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<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 />
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<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>
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<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 />
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<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 />
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<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>
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If you know of other math-related online events, please mention them in the comments. <br />
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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 />
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<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.com0tag: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 />
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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 />
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Send your links to mathanthologyeditor@gmail.com, or post them here.<br />
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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 />
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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 />
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<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 />
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(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 />
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<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 />
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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 />
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<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 />
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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 />
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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;">
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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 />
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Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)<br />
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Edited to add:<br />
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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 />
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I promised to write up some of it here.<br />
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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 />
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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 />
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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 />
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<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>
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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 />
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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 />
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Fours could be legs on dogs. 6 dogs have 24 legs. Twos could be eyes on friends...<br />
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<b>What are your favorite images for multiplication facts?</b>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag: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 />
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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 />
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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 />
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<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.com0tag: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.com0tag: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 />
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<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.com0