tag:blogger.com,1999:blog-53033074821589225652024-09-02T02:02:43.239-07:00Math Mama Writes...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger601125tag:blogger.com,1999:blog-5303307482158922565.post-81989238290907628062024-08-23T13:30:00.000-07:002024-08-23T13:30:57.130-07:00Farzanah and the 17 Camels<p>My publisher, Natural Math, uses crowdfunding to support each book they publish. I'm excited about <a href="https://www.kickstarter.com/projects/magicturtle/farzanah-and-the-17-camels" target="_blank">this book</a>.</p><p></p><p></p><p></p><p></p><p>Check it out, and contribute (which is really the same as making an advance order). I think you'll enjoy it.</p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT2H7YS954HExPOj-JQCBXWwIWg7Xid8RRH1kSm1F6iQK059UE9pIOeEhEhppay88r5xzn_SqSEMK4P4jrhVyQUOD3xL27sVtK8XFwoCM0lQ5IsKvcWH9TCY_X3PJhi1wv57Fk7Xdmj-Ff5zZ2OuKtL9mEBuygnjPMT0csghlTHEF1jXdStj6gogWUTPMS/s1600/farzanah.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1002" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT2H7YS954HExPOj-JQCBXWwIWg7Xid8RRH1kSm1F6iQK059UE9pIOeEhEhppay88r5xzn_SqSEMK4P4jrhVyQUOD3xL27sVtK8XFwoCM0lQ5IsKvcWH9TCY_X3PJhi1wv57Fk7Xdmj-Ff5zZ2OuKtL9mEBuygnjPMT0csghlTHEF1jXdStj6gogWUTPMS/w400-h640/farzanah.jpg" width="400" /></a></div><br /><p><br /></p><p><a href="https://www.kickstarter.com/projects/magicturtle/farzanah-and-the-17-camels" target="_blank"><em><b>Farzanah and the 17 Camels</b></em></a> celebrates the excitement and the
rewards of solving a challenging and intriguing math problem. Set
against the backdrop of the ancient Silk Road, with bustling markets,
stunning carpets, fun characters, and camels, the story draws readers
into the magic of Farzanah's surroundings. </p><p>As Farzanah searches
for an unusual approach, a way of solving the problem that no one else
could think of, she follows the wise advice of her mother: </p><blockquote><p><em>"My
dear Farzanah, don't be discouraged,” said Mama. “Sometimes, being
stuck is exactly where you need to be. I find the best thing I can do is
to step away. I free my mind to think about other things. It is in that
space that the magic happens. I am able to look at things from a
different perspective. With wait time and wishful thinking comes the
solution.”</em></p></blockquote><p> </p><p style="text-align: center;"><a href="https://www.kickstarter.com/projects/magicturtle/farzanah-and-the-17-camels" target="_blank"><span style="font-size: medium;"><b>Join the crowdfunding campaign here. </b></span></a><br /></p><p> </p><p> </p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-6117590091289725162024-08-14T18:31:00.000-07:002024-08-23T13:35:44.397-07:00Playful Math Carnival #174 (the June, July, and now August edition): On Fractions & Division<p><span style="font-size: medium;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ8GfuZ99P475eG2MVcgCcFbEjurKO3BnrlmXCDS0H9RxNIdBoGqkPLXQwMh-i2QFni3rHFafFOx6tpSoDzU9ERx88CVfwZBKe9bScNr-II-Wqpyoe7McQ8TVuT74VATd89sXW_LcF2vcaoaKm91d71xJihpViBka588o_s2cnzGRMrurTKlEk0RyC_vKE/s1395/mtap%20midway-1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="832" data-original-width="1395" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ8GfuZ99P475eG2MVcgCcFbEjurKO3BnrlmXCDS0H9RxNIdBoGqkPLXQwMh-i2QFni3rHFafFOx6tpSoDzU9ERx88CVfwZBKe9bScNr-II-Wqpyoe7McQ8TVuT74VATd89sXW_LcF2vcaoaKm91d71xJihpViBka588o_s2cnzGRMrurTKlEk0RyC_vKE/w405-h242/mtap%20midway-1.jpg" width="405" /></a></span></div><span style="font-size: medium;"> </span><p></p><p><span style="font-size: medium;">The <b>Playful Math Carnival</b> is a collection of blog posts and articles from around the internet, putting lots of goodies in one place for your enjoyment. The theme for this issue is <i>fractions and division</i>. Why are division and fractions so much harder than what came before? And how can we explore them in playful, delightful, engaging ways? This carnival includes lots of perspectives, and approaches the topic from many levels, elementary to college.</span></p><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;"> </span></p><p style="text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCibJmLatKgzhyphenhyphenrP5AtOZlKtoApx2fjavPNrsbMoIVumYvYTWoURaU6Z2LkB_bsfYoQY9nB5OYrrRaxMEvzCxC3fgi9GZxQIPRZ8ELhPuBDYkfbpIO7I-HW6lTFX-bkqmxxxHFStAvpEzD90tNLVIUMePph508M2DrxlLNFm6UxwZ7bmtv0vO2NXLYWwIa/s299/174.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="169" data-original-width="299" height="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCibJmLatKgzhyphenhyphenrP5AtOZlKtoApx2fjavPNrsbMoIVumYvYTWoURaU6Z2LkB_bsfYoQY9nB5OYrrRaxMEvzCxC3fgi9GZxQIPRZ8ELhPuBDYkfbpIO7I-HW6lTFX-bkqmxxxHFStAvpEzD90tNLVIUMePph508M2DrxlLNFm6UxwZ7bmtv0vO2NXLYWwIa/s1600/174.png" width="299" /></a></span></p><p><span style="font-size: medium;"><b>A puzzle for 174: </b> <b>What are all the factors of 174?</b> </span><span style="font-size: medium;">Learning how to find factors goes hand in hand with division and fractions. </span><span style="font-size: medium;">It's
easy to see that 2 is a factor of 174. Can you see any others before
you divide by 2? There's a "trick" for 3 (and 9), but everything in math
has a reason. Do you know why that "trick" works? I see that 1 plus 7
plus 4 is 12, and I know that 3 goes into 12. Why would that tell me
something about whether or not 3 is a factor of 174? [Solutions at end.
Hint: It's got to do with 10 being 9 plus 1.]<br /></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;"></span></p><span style="font-size: medium;"><br /><br /></span><p></p><p><span style="font-size: medium;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6WDDQJwpG885qTz9SGgGQbsCtVqL5TBHtTIXAdAlLTne0QF5pDpbJsM9WjPjxFTMGZHL-XXOM72-1nGV1c4fUrVlvapPpt4bbRXExo9wiM1xZjqnsFUeSnqfWpTIq0odtSgbNk0twupeSh-RBBohZoCMNM5QzZkxt5fA5IKn_h3zYuItNLkTatZ95H64Q/s200/fraction%20strips%20%232.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="200" data-original-width="200" height="301" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6WDDQJwpG885qTz9SGgGQbsCtVqL5TBHtTIXAdAlLTne0QF5pDpbJsM9WjPjxFTMGZHL-XXOM72-1nGV1c4fUrVlvapPpt4bbRXExo9wiM1xZjqnsFUeSnqfWpTIq0odtSgbNk0twupeSh-RBBohZoCMNM5QzZkxt5fA5IKn_h3zYuItNLkTatZ95H64Q/w301-h301/fraction%20strips%20%232.png" width="301" /></a></span></div><span style="font-size: medium;"><br /></span><p></p><p><span style="font-size: medium;">Before I started teaching, I had no idea that fractions might be hard. Part of what makes fractions difficult for some students is how many meanings fractions can have: a fraction of one whole, a fraction of some collection, a fraction of a measurement, etc.</span></p><p><span style="font-size: medium;"><br /> </span></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK2Q8Fl0yj5a2LMnadqplRneSX0qkS0w1yTMvkWSyvmNNE6n8LjSxbEXpiBU1YG7VbVeR3G9jdaQvCklENl8F-LOaHs3dby7Ic9KGN6BWoinvqv7WzJGWw66QXSxXMG4QSsHiDbG8pXiEWNZs9EfH5KLAuoUkpnNVBgOpKd9IaNhtLfwQYR0jV1EJdPnOE/s225/division%20symbol.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="225" data-original-width="225" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK2Q8Fl0yj5a2LMnadqplRneSX0qkS0w1yTMvkWSyvmNNE6n8LjSxbEXpiBU1YG7VbVeR3G9jdaQvCklENl8F-LOaHs3dby7Ic9KGN6BWoinvqv7WzJGWw66QXSxXMG4QSsHiDbG8pXiEWNZs9EfH5KLAuoUkpnNVBgOpKd9IaNhtLfwQYR0jV1EJdPnOE/s1600/division%20symbol.png" width="225" /></a></div><span style="font-size: medium;"> </span><p></p><p><span style="font-size: medium;">My own troubles with division came from a slight case of (undiagnosed) dyslexia. Why is it that we write a / b, but then we have b going into a, with the numbers in the opposite order? The way we write it made no sense to me. And I got confused if the numbers were big ones. Because of this challenge for me, I learned one of my first problem-solving lessons: <b><i>Make a simpler problem with the same structure</i></b>. If I saw 158 <span class="kY2IgmnCmOGjharHErah" style="-webkit-line-clamp: 3;"><span>÷ </span></span> 79, I could think to myself, "That's like 6 <span class="kY2IgmnCmOGjharHErah" style="-webkit-line-clamp: 3;"><span>÷ 3." And then I knew what to do - find out how many 79s in 158. Aha, it's 2, just like 6 </span></span><span class="kY2IgmnCmOGjharHErah" style="-webkit-line-clamp: 3;"><span>÷ 3!<br /></span></span></span></p><p><span style="font-size: medium;">I used to hate the words divisor and dividend. I could not keep them straight. And I still don't know which is which (but if I care, the internet is my friend). And I, my friends, am a math professor. I tell my students often that my bad memory has helped me learn math, because I always tried to make sense out of it, instead of memorizing.</span></p><p><span style="font-size: medium;"></span></p><span style="font-size: medium;"><b></b></span><p></p><p><span style="font-size: medium;"><br /> </span></p><p><span style="font-size: medium;">My personal favorite division issue now is why division by 0 is undefined. I wrote about it in my forthcoming book, <a href="https://sites.google.com/view/altheas-math-mysteries/triangles-circles-pi" target="_blank"><b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b>.</a> I'll share that passage at the end of this post. It's written at about high school level. We can go even higher level with the math and explore 0 / 0, an important concept for calculus that took mathematicians 150 years to come to terms with, which I did in <a href="https://mathmamawrites.blogspot.com/2020/09/division-by-0.html" target="_blank">a post a few years back</a>.<br /></span></p><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;">John Golden is the <b>Math Hombre</b>. </span></p><ul style="text-align: left;"><li><span style="font-size: medium;">He wrote <a href="https://mathhombre.blogspot.com/2010/03/conquer-and-divide.html" target="_blank">Divide and Conquer</a> in 2010. It's still golden. </span></li><li><span style="font-size: medium;">John loves games and works with future teachers. This post, <a href="https://mathhombre.blogspot.com/2022/09/fraction-reaction.html" target="_blank">Fraction Reaction</a>, written mostly by one of his students, really gives you a feel for how anyone can create a new game. </span></li><li><span style="font-size: medium;">When we make fractals, we can explore what fraction of the area is shaded. <a href="https://www.geogebra.org/m/b4dhw892" target="_blank">Here's something John made in geogebra</a> to allow students to play with fractals.<br /></span></li></ul><p><br /></p><p><span style="font-size: medium;">Denise Gaskins writes at </span><span style="font-size: medium;"><b>Let's Play Math!</b> </span></p><ul style="text-align: left;"><li><span style="font-size: medium;">John and Denise have something in common ... What is it? Games, of course! Check out a perennial favorite from Denise: <a href="https://denisegaskins.com/2006/12/29/the-game-that-is-worth-1000-worksheets/" target="_blank">The Game That's Worth 1,000 Worksheets</a>. (It's just variations on the card game of war, but 'just' is entirely the wrong word for how much you can do with that! And there is a Fractions War.)</span></li><li><span style="font-size: medium;">Her <a href="https://denisegaskins.com/2024/05/15/if-not-methods-dividing-fractions/" target="_blank">Dividing Fractions article</a> helps you see how to <i>think about </i>dividing fractions so that you're not tempted to fall into using a procedure that might make no sense to you. </span></li><li><span style="font-size: medium;">Besides writing articles, collecting math games, and publishing books with all of that goodness, Denise has also written a lovely series of stories in which </span><span style="font-size: medium;">Alexandria Jones solves some sort of problem her archeologist dad, Dr. Fibonacci Jones, encounters. Here's a small taste, where the two of them are puzzling out<a href="https://denisegaskins.com/wp-content/uploads/2008/02/egyptian-fractions.pdf" target="_blank"> Egyptian Fractions</a>. </span></li><li><span style="font-size: medium;">If that story entices you to want to learn more, David Reimer has written a </span><span style="font-size: medium;">lovely book, <a href="https://press.princeton.edu/books/hardcover/9780691160122/count-like-an-egyptian" target="_blank"><b><i>Count Like An Egyptian</i></b></a>. (<a href="https://www.biblio.com/book/count-like-egyptian-hands-introduction-ancient/d/1608491354" target="_blank">Used copies are available at biblio.com.</a>)</span></li></ul><p><span style="font-size: medium;"><br /></span></p><p><span style="font-size: medium;">Who the heck is Professor Smudge? </span></p><ul style="text-align: left;"><li><span style="font-size: medium;">Here's what they say in their twitter bio: </span><span style="font-size: medium;"><span class="css-1jxf684 r-bcqeeo r-1ttztb7 r-qvutc0 r-poiln3" style="text-overflow: unset;">"Wrote <a href="https://www.mathsmed.co.uk/" target="_blank">Maths Medicine</a>. Rumoured to be called Sigi, after Sigismund (hello surds, hello pi) Arbuthnot, and to have personated Dietmar Küchemann on occasions." Hmm, that's a bit mysterious. Anyway, I found some good looking fraction puzzles on their twitter feed, and you can probably find lots more. </span></span></li><li><span class="css-1jxf684 r-bcqeeo r-1ttztb7 r-qvutc0 r-poiln3" style="text-overflow: unset;"><a href="https://x.com/ProfSmudge/status/1815721769535226135" target="_blank">Here's one</a>: <br /></span><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLdGVgksGdAFTfzYzdqyHg12K3hIRhQiW-cEOob4J3MrSds6WrMRjEUPFPSbSRMkStAhuWCKAyqyL5JebjfG6YiehwdzmVZHqZGwgKonJ1nvMYE78_bpex5Kfc9P08oHS1KAMV36JjWwNiQ3qUzwmnMZIAlppEK1Dng-S7938ppQ3gbR-RSZ1-CQ0oW_Za/s1200/profsmudge%20%231.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="845" data-original-width="1200" height="281" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLdGVgksGdAFTfzYzdqyHg12K3hIRhQiW-cEOob4J3MrSds6WrMRjEUPFPSbSRMkStAhuWCKAyqyL5JebjfG6YiehwdzmVZHqZGwgKonJ1nvMYE78_bpex5Kfc9P08oHS1KAMV36JjWwNiQ3qUzwmnMZIAlppEK1Dng-S7938ppQ3gbR-RSZ1-CQ0oW_Za/w400-h281/profsmudge%20%231.jpg" width="400" /></a></div></li><li><a href="https://x.com/ProfSmudge/status/1812423219439112236" target="_blank">And here's another</a>: </li></ul><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYrJ1OIwJQYPIrFrutb42_AlcbvQfUDCuS8gBRL2oqK3a-R8RtQuWF04aHOoK3IuCy6krG-0160ZJ646niFvTP-cEuaIrXNF8wp4tYJtNyjiC-r43ddB3FAGanGC_rpjNmTpjJR9tRTgMZ6Pjjfu4Tbu_nTDDFWBMYbTTdjd1nmNqa7EVxxBnv5V1ZxAqG/s1200/profsmudge%20%232.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="846" data-original-width="1200" height="283" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYrJ1OIwJQYPIrFrutb42_AlcbvQfUDCuS8gBRL2oqK3a-R8RtQuWF04aHOoK3IuCy6krG-0160ZJ646niFvTP-cEuaIrXNF8wp4tYJtNyjiC-r43ddB3FAGanGC_rpjNmTpjJR9tRTgMZ6Pjjfu4Tbu_nTDDFWBMYbTTdjd1nmNqa7EVxxBnv5V1ZxAqG/w400-h283/profsmudge%20%232.jpg" width="400" /></a></div><br /><p></p><p> </p><p><span style="font-size: medium;">Shayla Heavner (aka SJ Bennett) created </span><span style="font-size: medium;"><b><a href="https://www.mathbait.com/" target="_blank">MathBait</a></b>, and is the author of <b><i>Marcos the Great and the History of Numberville</i></b>. She brings us two factoring games.<br /></span></p><ul style="text-align: left;"><li><span style="font-size: medium;"><b><a href="https://www.mathbait.com/post/mathbait-mastering-multiplication-part-3" target="_blank">Emoji Mystery</a></b> asks students to use some logic to help them find the factors of two numbers. <a href="https://www.mathbait.com/post/mathbait-mastering-multiplication-part-3" target="_blank">Scroll down this page until you see it.</a> </span></li><li><span style="font-size: medium;">A digital version of the <b><a href="https://www.mathbait.com/post/part-7-primes-and-factoring" target="_blank">Taxman</a></b> game, one of my favorite factoring games (again, scroll down to find it), is also on her wealth of pages.</span></li></ul><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;">Maria Droujkova, founder of <b>Natural Math</b> (my publisher), is conducting a crowdfunding campaign for a lovely book, <b><i><a href="https://www.kickstarter.com/projects/magicturtle/farzanah-and-the-17-camels" target="_blank">Farzanah and the 17 Camels</a></i></b>, by Dr. Sue Looney, which tells the story of an ancient math puzzle. One part of that puzzle asks: How can we possibly give one heir half of the 17 camels? <a href="https://www.kickstarter.com/projects/magicturtle/farzanah-and-the-17-camels" target="_blank">Join that campaign here</a> (your donation is basically an advance order of the book).<br /></span></p><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;">Do you want more?! The <a href="http://ontariomath.blogspot.com/" target="_blank">Ontario Math Links blog</a> is updated weekly. Browse to your heart's content. (That's where I found Professor Smudge.)<br /></span></p><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;"> </span></p><p><span style="font-size: medium;">The <b>Math Teachers at Play Blog Carnival </b>was created in 2009. Its name changed to <b>Playful Math Carnival </b>along the way, and it's been going strong for 15 years! (15 years online feels like a century anywhere else.) Links to all past posts available <a href="https://denisegaskins.com/mtap/" target="_blank">here</a>. I used to include dozens of bloggers in my posts. This one only includes 5 people. (When Google evilly got rid of Google Reader, it really devastated the "math blogosphere".) If you have written something you think we'd like to see, please add a comment.<br /></span></p><p><span style="font-size: medium;"><br /></span></p><p><span style="font-size: medium;"><br /></span></p><p><b><span style="font-size: medium;">Puzzle Solutions:</span></b></p><p><span style="font-size: small;">The factors of 174 are 1, 2, 3, 6, 29, 58, 87, and 174. (There are 8 of them. Do all numbers have an even number of factors, or do some have an odd number of factors? Which are which?)</span></p><p><span style="font-size: small;">Understanding that factoring "trick" for 3 and 9: Add the digits of your number. If 3 or 9 goes into the sum, then it goes into the original. Why? Let's consider 174. The sum of the digits is 12, and 3 goes into 12. Hmm. 174 means 1*100 + 7*10 + 4, and that can be written 1*(99+1) + 7* (9+1) + 4. If I distribute, I get 1*99 + 1 + 7*9 +7 + 4. 99 and 9 are multiples of 3. So we have 1*99 + 7*9 + (1+7+4). Each term is a multiple of 3. The last term is that sum of the digits we looked at. After reading this, could you explain to someone else why the 3 and 9 factoring "tricks" work?</span></p><p><span style="font-size: small;"> </span></p><p><span style="font-size: small;"> </span></p><p><span style="font-size: small;"> </span></p><p><span style="font-size: small;">p.s Here's that ... </span><span style="font-size: small;"></span></p><p><b><span style="font-size: large;">Sneak Preview</span></b><span style="font-size: medium;"> from <b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b>:</span></p><p>
</p><p class="MsoNormal" style="text-align: justify;">Sofia nods. “I messed up. I had 1 over 0, so I wrote 0 for
my final answer. I’m not really sure why it’s supposed to be undefined instead.
Can you explain that? It did feel kind of tricky to me.”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Mom says, “That’s a great question. And to answer it, we
actually need to go back to some basics. The problem is that division doesn’t
always work. It turns out that dividing by 0 doesn’t make sense. But to see
why, we have to go back and look at how we define division." </p>
<p class="MsoNormal" style="text-align: justify;"><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitv2-_R6mwR4Y-4jVn90Ufs6X48l-AJTZV2im5OXGc6x_r7PQt_ejsm7ohF4yGtRfB1TFuXodwy9dsXIIyyA8I46cqzdTCBSnr-rXRuDfzO_7mjSLx1OjkB8SYF2vFBLbKdqcd5S0QO-4LskPNCByY63NeaTMatutu8-rPCAqMT4wZNchCTavdDO08xViB/s581/Division%20By%200-1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="352" data-original-width="581" height="243" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitv2-_R6mwR4Y-4jVn90Ufs6X48l-AJTZV2im5OXGc6x_r7PQt_ejsm7ohF4yGtRfB1TFuXodwy9dsXIIyyA8I46cqzdTCBSnr-rXRuDfzO_7mjSLx1OjkB8SYF2vFBLbKdqcd5S0QO-4LskPNCByY63NeaTMatutu8-rPCAqMT4wZNchCTavdDO08xViB/w400-h243/Division%20By%200-1.jpg" width="400" /></a></div><p class="MsoNormal" style="text-align: justify;"><br /></p><p class="MsoNormal" style="text-align: justify;"><br /></p>
<p class="MsoNormal" style="text-align: justify;">She starts writing on the whiteboard and
explaining at the same time. “We know that 6 over 2 is 3, because 2 times 3 is
6. I want to take that relationship and write it in a more generic way. I’m
going to use T for top, B for bottom, and A for answer. When I was younger, I
think I had trouble remembering numerator and denominator. That might be why I
like saying top and bottom. Or maybe I just like shorter words. Anyway, now I
can look at multiplication to help me think about weird division problems.</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">I look at Sofia. It seems like she’s deep in thought.
Kiara’s taking notes, even though she seemed to know this. I think Mom has
shown me this before. </p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">“So 0 over 5 equals A becomes 5 times A equals 0. So what’s
A?”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Sofia says, “It can only be 0.”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Mom nods. “And there are other good ways to think about this
one. But for the one that tripped you up, this is the only way I know of to
make it really make sense. So now 5 over 0 equals A. What does that become?”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;"><span style="mso-no-proof: yes;"><span class="msoIns"></span></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkQ7KGP54_f_YfVJtckhEUmWIB07oaeKgQ6_GQthG8FhUufmKpGNis4xrrtA90_4P8Cp6XKpoLUtvwoDT5ih6EfshUVFeRVFZSxvelIPsuJ7L0syMDMTQGcBt9OJqJkJugxgRYpcRXEVuOf9Ig2W2WX1-1HSmEYuVpiV1G4ZqHoYcDNrZBHG8MDTILrtum/s820/Division%20By%200%20Part2-1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="492" data-original-width="820" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkQ7KGP54_f_YfVJtckhEUmWIB07oaeKgQ6_GQthG8FhUufmKpGNis4xrrtA90_4P8Cp6XKpoLUtvwoDT5ih6EfshUVFeRVFZSxvelIPsuJ7L0syMDMTQGcBt9OJqJkJugxgRYpcRXEVuOf9Ig2W2WX1-1HSmEYuVpiV1G4ZqHoYcDNrZBHG8MDTILrtum/w400-h240/Division%20By%200%20Part2-1.jpg" width="400" /></a></div><br /><ins cite="mailto:VanHattum,%20Sue" datetime="2023-07-27T17:18"></ins> <br /><p></p>
<p class="MsoNormal" style="text-align: justify;">Aiden starts to talk, but Sofia gives him a look. She says,
“I’m the one who doesn’t get this, so let me try.<span style="mso-spacerun: yes;"> </span>It turns into 0 times A equals 5. But Miss
Annie, you can’t get 5. If you have 0 times anything, you’ll get 0.”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Mom nods and waits.</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Sofia continues, “So there is no A that works in this one,
and that means there’s no A for the first one. So it has no answer, and that’s
why they say undefined?”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Mom nods again.</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Kiara says, “Whoa! I just knew it was supposed to be
undefined, but I definitely did not know why. And until this moment, I would
not have known I was missing something.”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Aiden is nodding too. “I always knew one of those was
undefined, but sometimes I mix up which one is which. I don’t think I’ll have
that problem anymore.”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Sofia looks at him. “So you didn’t get it either?”</p>
<p class="MsoNormal" style="text-align: justify;"> </p>
<p class="MsoNormal" style="text-align: justify;">Aiden says, “I had number 5 right, but I think it was a
lucky guess.”</p>
<p><style>@font-face
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{page:WordSection1;}</style></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-48164140158710671202024-02-18T14:59:00.000-08:002024-02-18T14:59:01.551-08:00Free Online Math Circle Has a Few Spots Open Still<p>
</p><p class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">We picked
a time. We're meeting for nine weeks, each Saturday from March 2 to April 27,
for an hour, at 3pm PT / 6pm ET. We still have a few spots open. We'll be
playing with Triangles, Circles, and Pi, along with the fictional Althea and
her friends. Participants will get an introduction to geometry, proof, and
trigonometry.</span></p><p class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p>
<p class="MsoNormal"><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">I'm writing a new
book series, <b>Althea's Math Mysteries</b>. In four young adult novels, Althea
and her friends explore some</span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> <span style="background: white; color: black;">of the
mysteries of mathematics. The first two books are nearing publication, and the
second book needs folks to test it out.</span> <span style="color: black;">In
<b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b>, </span><span style="color: #212121;">Althea and friends, with the help of Althea’s mom, explore
geometry and proof in order to then learn the basics of trigonometry. </span></span></p>
<p class="MsoNormal"><span style="color: #212121; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p>
<p class="MsoNormal"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">We'd like to find some eager math
students to join us in an online math circle, led by me, to explore along with
Althea and her friends. Students will participate in 9 weeks of lively
small-group sessions: in part a deep and friendly math course, and also a
unique book club, with the author refining the story based on student
reactions.</span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span></p>
<p class="MsoNormal"><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p>
<p class="MsoNormal"><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Do you know any
students who enjoy math, know a bit of algebra, and would enjoy "user
testing” <b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b>?
We're looking for a few more young people to try out the activities in this
book together. </span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span></p>
<p class="MsoNormal"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"><br style="mso-special-character: line-break;" />
<br style="mso-special-character: line-break;" />
</span></p>
<p class="MsoNormal"><b><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Commitments:</span></b><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p><ul style="text-align: left;"><li class="MsoNormal"><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Attend 9 weeks of 60-minute live online sessions from
March 2 to April 27, each Saturday at 3pm PT / 6pm ET.</span></li><li class="MsoNormal"><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Read and comment on 1 to 3 chapters of the book each
week.</span><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></li><li class="MsoNormal"><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Keep an informal math journal during this time.</span><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span></li></ul>
<p class="MsoNormal"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"><br style="mso-special-character: line-break;" />
<br style="mso-special-character: line-break;" />
</span></p>
<p class="MsoNormal"><b><span style="background: white; color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">For all who stay the
course:</span></b><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p><ul style="text-align: left;"><li class="MsoNormal"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">You'll learn the foundations of geometry and trigonometry (and will get a
certificate for completing the course).</span></li><li class="MsoNormal"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Y</span><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">ou'll get a signed copy of the published book.</span></li><li class="MsoNormal"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Your name or alias will appear in the book's acknowledgements, and you
will receive a letter of appreciation for your help with this STEM
project. (If you’d like a letter of recommendation later, we will be happy
to write one for you.)</span></li><li class="MsoNormal"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">You’ll get to build community with math friends and mentors.</span></li></ul>
<p class="MsoNormal" style="background: white; margin-left: 47.0pt; vertical-align: baseline;"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p><p class="MsoNormal" style="background: white; margin-left: 47.0pt; vertical-align: baseline;"><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"> </span></p>
<p class="MsoNormal" style="background: white;"><b><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">Interested</span></b><span style="color: black; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;">? Please email me at mathanthologyeditor@gmail.com for more information, or to sign up.</span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-font-kerning: 0pt; mso-ligatures: none;"></span></p>
<p><style>@font-face
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{margin-bottom:0in;}</style></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-56668388399672343662024-02-07T09:22:00.000-08:002024-02-07T09:33:53.840-08:00Openings Now in Free Online Math Circle<p><span style="font-size: large;"> </span></p><p style="text-align: center;"><span style="font-size: large;">Join an online math circle for students ages 12 to 15 in March and April, exploring geometry, proof, and the basics of
trigonometry.</span></p><p style="text-align: left;"> </p><p style="text-align: left;">As most of you know, I'm writing a new book series. In four young adult novels, Althea and
her friends will be exploring some of the mysteries of mathematics. The first two
books are nearing publication at Natural Math.</p><p>In <i>Althea and the Mysteries of Triangles, Circles, and Pi</i>,
Althea and friends, with the help of Althea’s mom, explore geometry and
proof in order to then learn the basics of trigonometry. My publisher and I would like to
find some eager math students to join me in an online math circle, exploring some math mysteries along with Althea and her friends.
Participants will join lively small-group sessions: in part a deep and
friendly math course, and also a unique book club, allowing me to refine the story based on student reactions.</p><p><i>Althea and the Mysteries of Triangles, Circles, and Pi</i>
is a fictional story set in the present, in which the characters
discuss math, with Mom throwing in a few true stories from the past.
Like <i>The Number Devil</i> and <i>Math Girls</i>, this book gives you more the more you put into it by doing the math yourself.</p><p>Do you know any students who enjoy math, know a bit of algebra, and would enjoy user testing <i>Althea and the Mysteries of Triangles, Circles, and Pi</i>? We’re looking for 5 to 8 young people to try out the activities in this book together. <a href="https://docs.google.com/forms/d/e/1FAIpQLSc-2juefWtXQA_Ct_f8SPbeRQvemuipqbV8rAmivO-YR6cs0A/viewform" target="_blank"><b>If interested, please add your name and information here.</b></a></p><p><b> </b></p><p><b>Commitments:</b></p><ul><li>Attend
9 weeks of 60-minute live online sessions in March and April. Times to
be determined, most likely 4 p.m. EST / 1 p.m. PST, on Saturday or a
weekday (whichever works for more students).</li><li>Read and comment on 1 to 3 chapters of the book each week.</li><li>Keep an informal math journal during this time.</li></ul><p><b> </b></p><p><b>For all who stay the course:</b></p><ul><li>You’ll learn the foundations of geometry and trigonometry (and will get a certificate for completing the course).</li><li>You’ll get a signed copy of the published book.</li><li>Your
name or alias will appear in the book’s acknowledgements, and you will
receive a letter of appreciation for your help with this STEM project.
(If you’d like a letter of recommendation later, we will be happy to
write one for you.)</li><li>You’ll get to build community with math friends and mentors.</li></ul><p style="text-align: left;"> </p><p style="text-align: center;"><a href="https://docs.google.com/forms/d/e/1FAIpQLSc-2juefWtXQA_Ct_f8SPbeRQvemuipqbV8rAmivO-YR6cs0A/viewform" target="_blank"><img alt="Sign Up Button" class="alignnone size-full wp-image-25665" src="https://naturalmath.com/s/wp-content/uploads/2024/02/Sign-Up-Button.png" /></a></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-69773647370564375812024-02-05T13:00:00.000-08:002024-02-05T18:58:24.795-08:00The Storytellers of Math<p>Anyone here reading knows that I'm working on my series of young adult novels with math at the center - Althea's Math Mysteries.</p><p> </p><p>But did you know that this drive to tell math stories is growing among budding storytellers across the lands? </p><p style="text-align: center;"><b>Sue</b> in California (me!) is writing Althea's Math Mysteries. Four of them!<br /></p><p style="text-align: center;"><b>Shayla</b> (aka SK Bennett) in New Mexico is writing the next book after <a href="https://www.mathbait.com/marco-the-great" target="_blank">Marco the Great and the History of Numberville</a>. (I'm loving this one. I'm so glad there will be another.)<br /></p><p style="text-align: center;"><b>Sarah</b> in Washington has written some wonderful fairy tales about physics and math. I'm reading <a href="https://www.mathwithsarah.com/books" target="_blank">Newton's Laws: A Fairy Tale</a> right now. (Currently free.)</p><p style="text-align: center;">And of course there are about a dozen lovely stories from the authors who work with <b><a href="https://naturalmath.com/goods/" target="_blank">Natural Math</a></b>. <br /></p><p><br /></p><p>Who else is out there, writing tales of mathjoy that I haven't discovered yet?!<br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-88688317145652047652023-12-08T18:08:00.000-08:002023-12-08T18:10:03.220-08:00Illustrating Althea<p>I'm not much good at drawing, but most of the illustrations in <b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b> are math work. I can do those. So I've put my own illustrations into the manuscript as placeholders. There will be a professional illustrator, later.<br /></p><p>This past week, I was looking for where more illustrations are needed. I decided Althea would draw a map of California while thinking about their summer trips. So I drew it. Their home is in Berkeley, they go to camp in Quincy, and they're planning a trip to San Diego to visit Legoland (because her younger brother Rudy would love that, and their moms met in San Diego).</p><p>I had fun drawing the map. First I downloaded a map of California into goodnotes, as the template for my document. Then I outlined it, and then changed the template to make the original fancy map go away. Finally, I got to add the places of interest to Althea.</p><p>As one friend on facebook pointed out, it would help to make the line weights different for the outline versus the routes she's imagining. The professional illustrator can either take care of that, or show me how it will work best for the published book.<br /></p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-RnDI6esXysUYEkCRfUaJAVLqwx9NAQoh0j7P-7OPCL2XmEiD2r7GQlK4wRpbh6KxKAaVryscJBT9m9ZVRsWr_7C6itZRa7Ucs3q7xoohGXDSfvbHJhFSh6eQWTvjT9wZkKy3f92WrJDXSxLwZlilj995mt8ygaxCwjame57ehl-9GiDGDE7FppM-pYBM/s1690/map%20of%20california%20(1)-1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1690" data-original-width="1492" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-RnDI6esXysUYEkCRfUaJAVLqwx9NAQoh0j7P-7OPCL2XmEiD2r7GQlK4wRpbh6KxKAaVryscJBT9m9ZVRsWr_7C6itZRa7Ucs3q7xoohGXDSfvbHJhFSh6eQWTvjT9wZkKy3f92WrJDXSxLwZlilj995mt8ygaxCwjame57ehl-9GiDGDE7FppM-pYBM/w354-h400/map%20of%20california%20(1)-1.jpg" width="354" /></a></div><br /><p><br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-82697156230303455952023-11-26T20:27:00.000-08:002023-11-26T20:27:50.498-08:00Althea's Math Mysteries<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh72lo5zcZbUSzqIxSi7VAwuh1WEKviOX7Fk4_0lWM68wvnjXNteeXuRkrbOolIcv_0rXDxoJeVw5eFrq51GFoxCHNvwlMMDJ-qz5f4Xm1tU3P68zrxHioNjVjN1RSo4fsIDhRuWCPoLr_lsN1uXwlRYNm4uTgEkVQ1g9ix2IJkdsxxZGFq3dSp638ijDP9/s2323/me%20with%20book.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2323" data-original-width="2109" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh72lo5zcZbUSzqIxSi7VAwuh1WEKviOX7Fk4_0lWM68wvnjXNteeXuRkrbOolIcv_0rXDxoJeVw5eFrq51GFoxCHNvwlMMDJ-qz5f4Xm1tU3P68zrxHioNjVjN1RSo4fsIDhRuWCPoLr_lsN1uXwlRYNm4uTgEkVQ1g9ix2IJkdsxxZGFq3dSp638ijDP9/s320/me%20with%20book.jpeg" width="291" /></a></div><br /> <p></p><p>This blog may not be as active as it used to be, but it's a good way for me to remember some things. My first post about my Althea stories was in September of 2019, so I've been working on the first two books in this series for four years now. I'm hoping we'll be able to publish them in about a year. </p><p>I have a very hopeful timeline that puts publication in October. But we all know that things never go as well as we hope. (And I'm <i>wishing</i> we could do it just a bit faster than that, so they'd come out in time for Math Storytelling Day, September 25, Maria's and my birthday.)</p><p>I have pretty complete drafts done of <b><i>Althea and the Mystery of the Imaginary Numbers</i></b> and <b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b>. Soon I'll be asking for folks to read the manuscripts and comment on them. (Email me at mathanthologyeditor@gmail.com if you'd like to be one of our readers.) After that's done, we'll do our usual (<a href="https://naturalmath.com/goods/" target="_blank">Natural Math publishing</a>'s usual) crowdfunding campaign. And then there will be illustration, copy editing, proofreading, page layout, and books!<br /><br /><a href="https://sites.google.com/view/altheas-math-mysteries/home" target="_blank">Here's my mock-ups of the covers</a>, and lots of information that goes with the books.</p><p>At a few points, I've really wanted to see what these would look like as actual paperback books. lulu.com made that easy. Two books cost me under $25. I've done that 3 times, while I've polished up the books. What you see in the photo above is me holding the 3rd printed draft copy of <b><i>Althea and the Mysteries of Triangles, Circles, and Pi</i></b>.</p><p><br /></p><p>In other news, I retired on May 20 from my full-time job teaching math at a community college. Teaching online was way too much work, and less satisfaction than teaching in person. I'm still covid-cautious, so I also didn't want to go back to teaching in person. Retirement has been wonderful. I'm working hard on the books, visiting Michigan where I help my dad (who's 90), cleaning up my house a bit, and working on my yard. </p><p>All that was plenty for about the first five months. When I noticed that I sometimes felt like I had nothing to do, I posted in a Beast Academy group on facebook that I was thinking of offering a class. Someone suggested that I apply for a position with AOPS. (<a href="https://artofproblemsolving.com/" target="_blank">Art of Problem Solving</a> is an amazing online resource, providing great math textbooks and classes, and they wrote the fabulous Beast Academy curriculum.) I did that, and I'll start teaching for them soon!<br /><br />I will definitely be blogging more over the next year, to let anyone still reading my blog know what's up with these books. (If you're out there reading this, I'd love to hear from you.)<br /></p><p><br /></p>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-42928635669831736982023-07-31T18:26:00.003-07:002023-07-31T18:32:20.866-07:00Playful Math Blog Carnival #166<p style="text-align: left;"></p><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ1UOAcHba4jF54PsS6RJb7W0VecvdKJHHW1175mW2ULKnrzr2XjzK8m9WyN0-OYfl_MCdd5nBXXHvRYAW209o_KnTyiWUindAckxZGACMDFDgTiaRYS1cDtw-a_O8uBK3eOkax0bGhj3ENVYgJFLjIAzFJFAk2E1Mn3rGd5yFvz9jl7Hb_4ilYs4HvYRP/s1395/mtap%20midway-1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="832" data-original-width="1395" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ1UOAcHba4jF54PsS6RJb7W0VecvdKJHHW1175mW2ULKnrzr2XjzK8m9WyN0-OYfl_MCdd5nBXXHvRYAW209o_KnTyiWUindAckxZGACMDFDgTiaRYS1cDtw-a_O8uBK3eOkax0bGhj3ENVYgJFLjIAzFJFAk2E1Mn3rGd5yFvz9jl7Hb_4ilYs4HvYRP/w400-h239/mtap%20midway-1.jpg" width="400" /></a></div><div style="text-align: left;">This blog carnival has been around for 14 years. Almost every month for 14 years, someone has added a post to this collection. That's quite a long life for an internet phenomenon. (Congratulations, Denise, for keeping this going!) If you'd like to see any of the previous posts in this series, <a href="https://denisegaskins.com/tag/mtap-playful-math-carnival/" target="_blank">check them out here</a>. For many years, blogs were a big part of my time online. But not so much lately. </div></div><p></p><p style="text-align: left;">When our number (166 now) was in the 20s, 30s, or 40s, I'd make sure to have that many links. Nope, I don't have time to find 166 great links (and you'd get tired just looking through them). But they're out there. I have learned so much from bunny hopping around the web of math bloggers over the years. And even though blogs aren't the popular thing now, most of the old ones are still out there, waiting for you to find them and get excited.</p><p style="text-align: left;"> </p><p style="text-align: left;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuygFvPyINA5XYSKPSXNUCBnjri77J7WqPfBthmvNjIXzjJcCcgVNZM4eRAWUoCFTDFKeZm13AEqE3-hOnvvEwSA5CTAa52mE-x5C2Kx_OWw3xXmktSWh2jhrNO8a3CTi5m58ppLPQFojTiF_DlWtzY7j17k2CwOTNtDnTY4bo0yXesiEed6RKAvtiLRB6/s808/centered%20triangular%20numbers.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="202" data-original-width="808" height="93" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuygFvPyINA5XYSKPSXNUCBnjri77J7WqPfBthmvNjIXzjJcCcgVNZM4eRAWUoCFTDFKeZm13AEqE3-hOnvvEwSA5CTAa52mE-x5C2Kx_OWw3xXmktSWh2jhrNO8a3CTi5m58ppLPQFojTiF_DlWtzY7j17k2CwOTNtDnTY4bo0yXesiEed6RKAvtiLRB6/w370-h93/centered%20triangular%20numbers.png" width="370" /></a></div><b>The 166 puzzle: </b>It turns out
that 166 is a 'centered triangular number'. If you start with a dot, and
then you put a triangle around that, and a bigger one around that, etc, you
get up to 166. How many triangles did you use?<p></p><p style="text-align: left;"><br /></p><p style="text-align: left;"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjx0XMfIwYYAE_x_uX02LifaUOvv4zXfrzhSQe4qDyb13h9fLdBBXz_pJinwSxPUU4-NSCQCgChY7zvcsvvpYMpqbXbycnae-Fb9g2lScTdRFCRtndkhJ-G5T_8rC2D-40_ZNVRbs-_Zm-4yGB47CL_w7VPiEwvvDduHKt_sDI_iC7I2ELcx5Mh4j5Pbnp/s776/envelope.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="776" height="79" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjx0XMfIwYYAE_x_uX02LifaUOvv4zXfrzhSQe4qDyb13h9fLdBBXz_pJinwSxPUU4-NSCQCgChY7zvcsvvpYMpqbXbycnae-Fb9g2lScTdRFCRtndkhJ-G5T_8rC2D-40_ZNVRbs-_Zm-4yGB47CL_w7VPiEwvvDduHKt_sDI_iC7I2ELcx5Mh4j5Pbnp/w142-h79/envelope.png" width="142" /></a></div><br /><p></p><p style="text-align: left;">I have just run out of envelopes. How should I make myself one? (a <a href="https://www.fawnnguyen.com/teach/bu6cpdip0zgt2u99n6f2ygxp771b2l" target="_blank">puzzle</a> from Fawn Nguyen) What shape of paper will you use?<br /></p><p style="text-align: left;"> </p><p style="text-align: left;"><b> </b></p><p style="text-align: left;"><b>Online Mathy games</b></p><ul style="text-align: left;"><li style="text-align: left;"><b>The Dailies:</b> When wordle came out, there were a number of mathy imitators, including <a href="https://nerdlegame.com/" target="_blank">nerdle</a>, <a href="https://numberle.org/" target="_blank">numberle</a> (like nerdle, but you can do longer or shorter equations), <a href="https://www.mathler.com/" target="_blank">mathler</a>, <a href="https://www.thenumble.app/" target="_blank">numble</a>, <a href="https://summle.net/" target="_blank">summle</a>, and lots more. Most of them have vanished. More recently, <a href="https://beastacademy.com/all-ten" target="_blank">Beast Academy added their All Ten game</a>, which I'm hooked on.What's your favorite?<br /></li><li style="text-align: left;"><a href="http://mathhombre.blogspot.com/2023/07/games-before-class.html" target="_blank">Games Before Class</a> (from Math Hombre, not all are math, hence "before class")</li><li style="text-align: left;"><a href="https://denisegaskins.com/2023/07/31/target-ten/#more-50282" target="_blank">Target Ten</a> (from Denise Gaskins)</li><li style="text-align: left;"><span class="r-18u37iz"><a href="https://nrich.maths.org/5468" target="_blank">Factors and Multiples Game</a> (from nrich, I got 48 as my longest possible chain of factors and multiples. Can you beat me?)</span></li><li style="text-align: left;"><span class="r-18u37iz">Do you know of other goodies? Let us know in the comments. <br /></span></li></ul><p style="text-align: left;"><br /></p><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz"><b>Geometry Puzzles</b></span></div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz"> <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmDvsewLU8CZXeK5lv8iBuSa9yKHxPNXa72rNkS-X9p2rCsyZapoqn5Pj70D1Ee6krUbgY5O1Px2McwgEUvuOh2nw9i1W-hBwyw9gk3Bmh-QxEVFvjqKdOQpD-1nRREXsLAtZ9mOKEtoAs14mnA5MP_SmvbombxDeSZ4tYNkNmaIJcsN_1IL7xaWFvARfu/s990/geometry%20challenge.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="924" data-original-width="990" height="207" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmDvsewLU8CZXeK5lv8iBuSa9yKHxPNXa72rNkS-X9p2rCsyZapoqn5Pj70D1Ee6krUbgY5O1Px2McwgEUvuOh2nw9i1W-hBwyw9gk3Bmh-QxEVFvjqKdOQpD-1nRREXsLAtZ9mOKEtoAs14mnA5MP_SmvbombxDeSZ4tYNkNmaIJcsN_1IL7xaWFvARfu/w221-h207/geometry%20challenge.png" width="221" /></a></div><br /></span></div><p><br /></p><p><span class="r-18u37iz"> </span></p><p><span class="r-18u37iz"><a href="https://twitter.com/dment37/status/1685920229199605760" target="_blank"><b>Find the blue area</b>. </a></span></p><p><span class="r-18u37iz">This one stumped me (no trig required).</span><span class="r-18u37iz"> <br /></span></p><p><span class="r-18u37iz"> </span></p><p><span class="r-18u37iz"><a href="https://twitter.com/dment37/status/1685920229199605760" target="_blank"></a></span><br /></p><p><br /></p><p><span class="r-18u37iz"><a href="https://twitter.com/sonukg4india/status/1685932632453025792" target="_blank"></a></span></p><div class="separator" style="clear: both; text-align: center;"><span class="r-18u37iz"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjczIkuGwy_OMLfpBptsrGSoQkXUJ1l8j1qrPQZMLvuXaRYCPpgwGFIzXZADtVv9hV5cZo5Wyc-oEwGU8hy3VKj453emSJHQwmxS1zNBIURe_7RIcToGVim6jpaAFDNIKMBR1vx0V3Eq_QAXeSoQQksP0bOt2Hqr61oyS_nWzCbWXtlSrfYrNOrjI6eMYUQ/s1016/star%20angle%20puzzle.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="950" data-original-width="1016" height="212" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjczIkuGwy_OMLfpBptsrGSoQkXUJ1l8j1qrPQZMLvuXaRYCPpgwGFIzXZADtVv9hV5cZo5Wyc-oEwGU8hy3VKj453emSJHQwmxS1zNBIURe_7RIcToGVim6jpaAFDNIKMBR1vx0V3Eq_QAXeSoQQksP0bOt2Hqr61oyS_nWzCbWXtlSrfYrNOrjI6eMYUQ/w227-h212/star%20angle%20puzzle.png" width="227" /></a></span></div><p></p><br /><p><span class="r-18u37iz"> </span></p><p style="text-align: left;"></p><p style="text-align: right;"> <span class="r-18u37iz"><a href="https://twitter.com/sonukg4india/status/1685932632453025792" target="_blank">This one's a lot easier. </a></span></p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p></p><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><img border="0" data-original-height="487" data-original-width="1522" height="135" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBBCfC53Ot0-xNcuW8C8K7SIHZ5CESZGxtTWKyy8jNYKGtAPtG1ufPUV20SsXgJod-D9WNMEJK9Rr9BtlhxmZ09qQi6q7H9clPSxh3yfrg8Uvxfpq3nt9We2XKWr9dHxk9VpaAqXlpiR8E3gJaHIXDEhAcVP481RBZbQFSnIzDul0yvVL8u_1DtMM_Q9jb/w424-h135/cshearer%20boxes%20rainbox%20shaded.jpg" width="424" /></div>What fraction is shaded? Catriona Shearer (@Cshearer41) made this, along with gobs more, mostly pretty challenging, which she posts on twitter. And <a href="https://drive.google.com/file/d/1hVP8tLURVDphmHsphz5BQLVzHCeTts29/view" target="_blank">here's a collection of over 300 of them.</a><p></p><p> </p><br /><span class="r-18u37iz"><b>Beyond the Games & Puzzles <br /></b></span><div class="css-1dbjc4n"><div class="css-1dbjc4n r-1s2bzr4"><ul style="text-align: left;"><li><a href="https://mathforlove.com/lesson/balance-fractions/" target="_blank">Balance Fractions</a> (from Math for Love) looks fun.<br /></li><li class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz">I must be on a fractions kick. This stood out for me, too.</span> <a href="https://www.mathed.page/early-math/fractions.html" target="_blank">Fractions on Grids</a> (from Henri Picciotto).<br /></li><li class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" id="id__e5chcsijy99" lang="en">In the <a href="https://sites.google.com/view/altheas-math-mysteries/home" target="_blank">series of young adult math novels I'm writing</a>, circles come up in the 2nd book. An early reader pointed out that I might be mixing up circles and disks, so I re-wrote a few phrases. <a href="https://www.saravanderwerf.com/defining-circles/" target="_blank">Here are some lovely circle lessons</a> (from Sara VanDerWerf), where she helps students distinguish between those two ideas.<br /></li></ul><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"> </div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"> </div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"> </div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en">Searching for more? Some good hashtags are: <span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0">
</span><span class="r-18u37iz"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/MTBoS?src=hashtag_click" role="link">#MTBoS</a></span><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0"> </span><span class="r-18u37iz"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/ITeachMath?src=hashtag_click" role="link">#ITeachMath</a></span><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0"> </span><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0"> </span><span class="r-18u37iz"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/Elemmathchat?src=hashtag_click" role="link">#Elemmathchat</a></span><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0"> </span><span class="r-18u37iz"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/MSmathchat?src=hashtag_click" role="link">#MSmathchat</a></span></div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz"> </span></div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz"> </span></div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz"> </span></div><div class="css-901oao r-18jsvk2 r-37j5jr r-1inkyih r-16dba41 r-135wba7 r-bcqeeo r-bnwqim r-qvutc0" data-testid="tweetText" dir="auto" id="id__e5chcsijy99" lang="en"><span class="r-18u37iz">If you've seen some good math pedagogy out in the wilds of the internet, add a comment. </span> </div></div></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag: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";">
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<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=AD6v5dwcbasZ6wtYjjnysfTvpkTGONOAM73UylGn-ZINPjDFq-pBo8XYgiS2hcODhnK8ay2pmAwXngWqxu46UpY9zg' 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=AD6v5dxY8V55BvcW1dPT_LSkKLbj3IvJEY6E3ELf0bQucD8C8MLxdkBhyorMxY_vYoKbhT39lf7bKie1yeUg7_wOrA' 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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpqs5or20iO_DFLD2tmCYJf4KizpyTqs5gZNtijMSBISluEcXuUf5mNt7-IlRPtza5-7UKYHgYissT31b2UP6c7H0FnHpFlSOavsWGW5PSM9nrc_B_PnVHIbK05hPj1Ny1nAbNcAwPKBPk/s1800/c1+Torus+4D+tb+dc.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1358" data-original-width="1800" height="301" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpqs5or20iO_DFLD2tmCYJf4KizpyTqs5gZNtijMSBISluEcXuUf5mNt7-IlRPtza5-7UKYHgYissT31b2UP6c7H0FnHpFlSOavsWGW5PSM9nrc_B_PnVHIbK05hPj1Ny1nAbNcAwPKBPk/w400-h301/c1+Torus+4D+tb+dc.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAi5-yJy0unRBOpoTucJSHy8WRg4ypvj0gtWdfX7eRK-xbZIqVwwzhKu7ipKg-R1qjWPtFo9MjpqKVPheqUnhqMZgB1k0z4vMaN1uSBr-bQ3BWP7I0epVNv0C5nHEQIq37gvFUrrmqOG0s/s850/calculus+image.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAi5-yJy0unRBOpoTucJSHy8WRg4ypvj0gtWdfX7eRK-xbZIqVwwzhKu7ipKg-R1qjWPtFo9MjpqKVPheqUnhqMZgB1k0z4vMaN1uSBr-bQ3BWP7I0epVNv0C5nHEQIq37gvFUrrmqOG0s/w285-h235/calculus+image.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwlqm4xqtyFGtOabQnmWpOJfSg0vQ4Z9EDpC2Ka_2Hg1v9df_p5dYHsbbxZrqrLGeV2oxDPFcZUKVRUcGXAhCIiaO13ZUNx-y7gFIkTAf0iKIS-uCUiBlTPK1Q7Rkh-F9yfUiboH8Ws1Rh/s1804/171+home+image.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwlqm4xqtyFGtOabQnmWpOJfSg0vQ4Z9EDpC2Ka_2Hg1v9df_p5dYHsbbxZrqrLGeV2oxDPFcZUKVRUcGXAhCIiaO13ZUNx-y7gFIkTAf0iKIS-uCUiBlTPK1Q7Rkh-F9yfUiboH8Ws1Rh/w345-h187/171+home+image.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDYqd7QziBTepM9dHRHZcYBKXo478309sfCiHI9gNpIf3jz8KHfOJNREGwKkh-gTwLaIdYxSdnJ4vZFlnLxWosGPwfICk-SZGO4d1BFgjnJmATR40U_nuv0vvy1-LIxmw7fXJDZI8pc0Gu/s1434/orientation+quiz+question.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="732" data-original-width="1434" height="204" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDYqd7QziBTepM9dHRHZcYBKXo478309sfCiHI9gNpIf3jz8KHfOJNREGwKkh-gTwLaIdYxSdnJ4vZFlnLxWosGPwfICk-SZGO4d1BFgjnJmATR40U_nuv0vvy1-LIxmw7fXJDZI8pc0Gu/w400-h204/orientation+quiz+question.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgATYVMrGANKz_tyH_V53UDiKZC0tOH6iet57rqoUTT-PQ5hM9o3LkQvx6OaZZLz8q29vy4NID7NHmYHnsE_LW7oPO4Mns1JPvIK5MXGaPimEUsO05_CrZlfjdKpB5YKGnjXosnbF4dOkSn/s1540/unit+3+zoom++recordings+image.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1540" data-original-width="1382" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgATYVMrGANKz_tyH_V53UDiKZC0tOH6iet57rqoUTT-PQ5hM9o3LkQvx6OaZZLz8q29vy4NID7NHmYHnsE_LW7oPO4Mns1JPvIK5MXGaPimEUsO05_CrZlfjdKpB5YKGnjXosnbF4dOkSn/w359-h400/unit+3+zoom++recordings+image.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdVWxzN4ibDzhyv9qaGfq2BuJrChZh9lIZ5HWUKwUAsMBD4fa4ITH5R6QtFZbzASyAjchOf5xGMoumhbqAkkMxwbnInui1KE1lTlB4HMtqwmMtZF2vJMKkSIY4O-jxUwYXLF0v2dOKF37W/s1634/unit+1+unit+sheet.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1424" data-original-width="1634" height="349" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdVWxzN4ibDzhyv9qaGfq2BuJrChZh9lIZ5HWUKwUAsMBD4fa4ITH5R6QtFZbzASyAjchOf5xGMoumhbqAkkMxwbnInui1KE1lTlB4HMtqwmMtZF2vJMKkSIY4O-jxUwYXLF0v2dOKF37W/w400-h349/unit+1+unit+sheet.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgb3lemURGfwyEsJh3TNDjm0ZOYGOBQkRNwl0LB6PWPTJS8sMKQQvlGXY-RNIgrNjHlcgbDv3sQAUnlZ5ih48fJTX4qCjUvklk9HlNXGwEeFqSFmiqu9kGpNkCMuq2_8ISYvs5ykGBnaZ1W/s1828/radian+videos+page.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1454" data-original-width="1828" height="319" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgb3lemURGfwyEsJh3TNDjm0ZOYGOBQkRNwl0LB6PWPTJS8sMKQQvlGXY-RNIgrNjHlcgbDv3sQAUnlZ5ih48fJTX4qCjUvklk9HlNXGwEeFqSFmiqu9kGpNkCMuq2_8ISYvs5ykGBnaZ1W/w400-h319/radian+videos+page.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh0dMQPchxjfz0zN-oNdq0QnIF3J3RemfnHSOAweXjkrN-FI6zj92ZIiBbCWoLdOffE3O-pi_onJPNMOXjk7dU2UTS3I01Jvfsnf4lQmH6FjWAhd_5Hmyn9czTe5vKi7zvuFTx8kqKuFyD/s1024/3+goes+into+6.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://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh0dMQPchxjfz0zN-oNdq0QnIF3J3RemfnHSOAweXjkrN-FI6zj92ZIiBbCWoLdOffE3O-pi_onJPNMOXjk7dU2UTS3I01Jvfsnf4lQmH6FjWAhd_5Hmyn9czTe5vKi7zvuFTx8kqKuFyD/w38-h24/3+goes+into+6.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.com0