{"id":529,"date":"2021-12-08T13:50:41","date_gmt":"2021-12-08T18:50:41","guid":{"rendered":"https:\/\/blogs.ams.org\/mathmedia\/?p=529"},"modified":"2022-06-28T11:56:21","modified_gmt":"2022-06-28T15:56:21","slug":"math-digests-november-2021","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-november-2021\/","title":{"rendered":"Math Digests November 2021"},"content":{"rendered":"<h4 style=\"margin-top: 0px;margin-bottom: 0px;font-family: 'trebuchet ms', geneva, sans-serif;font-size: 22px\"><a id=\"1\" href=\"https:\/\/www.quantamagazine.org\/the-secret-math-of-hot-dogs-and-buns-20211118\/\">What Hot Dogs Can Teach Us About Number Theory<\/a><\/h4>\n<p style=\"font-family: 'trebuchet ms', geneva, sans-serif\"><em>Quanta Magazine,<\/em> November 18, 2021<\/p>\n<p>No hot dog is complete without a bun. So it\u2019s maddening that you typically have to buy eight buns at a time, but you can\u2019t buy exactly eight hot dogs. Getting one package of each will leave you with two extra hot dogs and no buns<em>\u2014<\/em>useless leftovers. Patrick Honner explains how the Chinese Remainder Theorem helps you avoid this situation. If you buy the right number of packages, you\u2019ll end up with exactly one bun for each hot dog. Even when there are a few extra hot dogs or buns hanging out in the fridge, the Chinese remainder theorem guarantees you\u2019ll be able to make things work out. If that\u2019s too much math for you, there\u2019s also an easier solution: In my experience, veggie sausages come in packages of four.<\/p>\n<p style=\"margin-bottom: 0px;margin-top: 20px\"><strong>Classroom activities: <\/strong><em>number theory, modular arithmetic, cryptography<\/em><\/p>\n<ul>\n<li>(High school) Have students read the article and complete the exercises at the end.<\/li>\n<li>(High school) Honner mentions how the Chinese remainder theorem comes into cryptography. Have students use the <a href=\"https:\/\/planetcalc.com\/4884\/\">A1Z26 cipher<\/a> to decrypt codes in class:\n<ul>\n<li>Break students into pairs. Have each student think of a secret word and encode it using the A1Z26 cipher. Then, have them encode it even further by finding the numbers mod 4 and mod 9. Thus the word CODE, encoded as (3-15-4-5), becomes (3-3-0-1) mod 4 and (3-6-4-5) mod 9.<\/li>\n<li>Have students trade their codes mod 4 with their partner and try to break them. After 5 minutes, allow them to share the code mod 9. According to the Chinese remainder theorem, they should now have enough information to break the code.<\/li>\n<li>Repeat the activity with new secret words, but this time have students share the code mod 9 first. Have students write about which way was easier and explain the reasoning they used.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Leila Sloman<\/em><\/p>\n<hr style=\"margin-top: 20px;height: 3px;border-width: 0;color: #0057b8;background-color: #0057b8\" \/>\n<h4 style=\"margin-top: 0px;margin-bottom: 0px;font-family: 'trebuchet ms', geneva, sans-serif;font-size: 22px\"><a id=\"2\" href=\"https:\/\/spectrumlocalnews.com\/nc\/charlotte\/politics\/2021\/11\/04\/redistricting-in-n-c---new-maps-approved--favoring-gop\">Redistricting in N.C.: New maps approved, favoring GOP<\/a><\/h4>\n<p style=\"font-family: 'trebuchet ms', geneva, sans-serif\"><em>Spectrum News 1, <\/em>November 4, 2021<\/p>\n<p>Every 10 years, all the states in the US go through the process of <em>redistricting:<\/em> drawing new lines for congressional and state legislative districts based on the latest census data. The current redistricting cycle has been extremely contentious, with partisan map-drawers in many states creating maps heavily biased in favor of their political party. North Carolina is a perfect example. When mathematician Jonathan Mattingly\u2019s team compared the state\u2019s proposed maps to tens of thousands of alternatives, \u201cWe found that the map that has been proposed for the North Carolina House really dramatically under-elects Democrats. We have similar analysis for the Senate.\u201d The mathematicians\u2019 methods, which have already featured prominently in court cases in recent years, provide rigorous quantitative evidence that the maps in North Carolina and elsewhere intentionally give one party an unfair edge. (<a href=\"https:\/\/www.newsobserver.com\/news\/politics-government\/article255511481.html\">A <em>Raleigh News &amp; Observer <\/em>article<\/a> also quotes Mattingly.)<\/p>\n<p style=\"margin-bottom: 0px;margin-top: 20px\"><strong>Classroom activities: <\/strong><em>gerrymandering, voting, geometry, Markov chain Monte Carlo<\/em><\/p>\n<ul>\n<li>(Middle school) Teach students about gerrymandering with <a href=\"https:\/\/curriculum.illustrativemathematics.org\/MS\/teachers\/1\/9\/6\/preparation.html\">this lesson plan from Illustrative Mathematics<\/a>. The lesson begins with simple examples of elections for a school mascot and school board, then gives students the opportunity to draw district maps that favor one candidate over another.<\/li>\n<li>(Middle school \/ high school) Play <a href=\"https:\/\/www.jrmf.org\/activities\/gerrymandering\">this gerrymandering game<\/a> by the Julia Robinson Mathematics Festival with 28 levels that increase in difficulty. Have students work in groups, then share their solutions with the class.<\/li>\n<li>(High school) Teach students about gerrymandering with <a href=\"https:\/\/resources.corwin.com\/tmsj-highschool\/student-resources\/chapter-8\/lesson-82-gerrymandering\">this lesson plan from Corwin<\/a>. The worksheets include problems related to contiguity, the efficiency gap, and two mathematical measures of compactness.\n<ul>\n<li>(Advanced) Introduce the concept of <a href=\"https:\/\/towardsdatascience.com\/mcmc-intuition-for-everyone-5ae79fff22b1\">Markov chain Monte Carlo<\/a> methods, which underlie the mathematical analysis of the North Carolina maps mentioned in the article.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong><em>Related Mathematical Moments poster and interview:\u00a0<\/em><a href=\"https:\/\/www.ams.org\/publicoutreach\/mathmoments\/mm132-gerrymandering-podcast\">Countermanding Gerrymandering.<\/a><\/strong><\/p>\n<p style=\"text-align: right\"><em>\u2014Scott Hershberger<\/em><\/p>\n<hr style=\"margin-top: 20px;height: 3px;border-width: 0;color: #0057b8;background-color: #0057b8\" \/>\n<h4 style=\"margin-top: 0px;margin-bottom: 0px;font-family: 'trebuchet ms', geneva, sans-serif;font-size: 22px\"><a id=\"3\" href=\"https:\/\/www.sciencenews.org\/article\/pearl-symmetry-round-formation-oysters-nacre-math\">How an oyster builds a perfectly round pearl<\/a><\/h4>\n<p style=\"font-family: 'trebuchet ms', geneva, sans-serif\"><em>Science News<\/em>, November 10, 2021<\/p>\n<p>At the center of an oyster is something beautiful: a pearl. At the center of a pearl is something less so: a misshapen lump of debris. For years, scientists have wanted to figure out how oysters grow perfectly round pearls over irregular bits of junk. Now, the mystery is solved, Rachel Crowell writes. The answer is a mathematical pattern found commonly in nature, called 1\/f noise or <em>pink noise<\/em>. Oysters build tiny layers of mineral and protein, called nacre, around the debris or grains of sand, gradually smoothing out the odd shape. Each layer of nacre has a different thickness that depends mathematically on the one below it. This <em>inversely proportional<\/em> relationship also shows up in seismic activity, classical music, and heartbeats.<\/p>\n<p style=\"margin-bottom: 0px;margin-top: 20px\"><strong>Classroom activities: <\/strong><em>pink noise, inverse proportionality, waves<\/em><\/p>\n<ul>\n<li>(All levels) Hear what pink noise sounds like <a href=\"https:\/\/en.wikipedia.org\/wiki\/Pink_noise\">on Wikipedia<\/a>, and watch videos about pink noise in <a href=\"https:\/\/www.youtube.com\/watch?v=9T978ES0LdQ\">human biology<\/a> and <a href=\"https:\/\/www.youtube.com\/watch?v=Qkb2yRKJxSc\">music production<\/a>.<\/li>\n<li>(All levels) Discuss <a href=\"https:\/\/www.theatlantic.com\/science\/archive\/2016\/02\/white-noise-sound-colors\/462972\/\">why this sound is called pink noise<\/a>. (Hint: sound and light are both waves. What\u2019s the difference between white light and pink light?)<\/li>\n<li>(Middle level) When teaching students to graph equations like $y=1\/x$, use pink noise as an example application. (Note that most graphs of pink noise appear linear because the axes are on a logarithmic scale.)<\/li>\n<li>(High level) What color might you associate with a sound that has a high intensity at middle frequencies but zero for high and low frequencies? What about a sound that has a low intensity at low frequencies but a high intensity at high frequencies? (Answers in the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Colors_of_noise\">\u201cColors of Noise\u201d<\/a> Wikipedia page)<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr style=\"margin-top: 20px;height: 3px;border-width: 0;color: #0057b8;background-color: #0057b8\" \/>\n<h4 style=\"margin-top: 0px;margin-bottom: 0px;font-family: 'trebuchet ms', geneva, sans-serif;font-size: 22px\"><a id=\"4\" href=\"https:\/\/theconversation.com\/pythagoras-revenge-humans-didnt-invent-mathematics-its-what-the-world-is-made-of-172034\">Pythagoras&#8217; revenge: humans didn&#8217;t invent mathematics, it&#8217;s what the world is made of<\/a><\/h4>\n<p style=\"font-family: 'trebuchet ms', geneva, sans-serif\"><em>The Conversation<\/em>, November 21, 2021<\/p>\n<p>Is math real, or is it just a way of describing the world? According to Sam Baron, an Australian professor, math is as real as the paper, the pen, and the brain with which you do it. Baron recently wrote an article for <em>The Conversation<\/em> in which he argues that math is an essential part of nature, not just a way of describing it. He draws upon examples of math in nature to prove his point. Bees make six-sided cells to store honey since hexagons form the most efficient tiles. Periodic cicadas evolved to emerge every 13 and 17 years because prime numbers help them avoid the more regular feasting patterns of predators. \u201cThe world has two parts, mathematics and matter,\u201d Baron writes. \u201cMathematics gives matter its form, and matter gives mathematics its substance.\u201d<\/p>\n<p style=\"margin-bottom: 0px;margin-top: 20px\"><strong>Classroom activities:<\/strong> <em>tiling, prime numbers, nature<\/em><\/p>\n<ul>\n<li>(All levels) Hexagons can tile a flat surface (cover it entirely with no gaps) to infinity. According to the <em>honeycomb conjecture<\/em> in math<em>,<\/em> the hexagon is the best shape for tiling a plane, in the sense that hexagons require the shortest total perimeter to cover a given area. Bees use this fact to build efficient honey storage.\n<ul>\n<li>Watch this It\u2019s Okay to Be Smart video called <a href=\"https:\/\/www.youtube.com\/watch?v=Pypd_yKGYpA\">\u201cWhy Nature Loves Hexagons.\u201d<\/a><\/li>\n<li>What other regular polygons can tile the plane? Using sturdy paper like construction paper, cut out <a href=\"http:\/\/kristinzweigle.blogspot.com\/2015\/06\/shapes.html\">an equilateral triangle, a square, a pentagon, a hexagon, and an octagon<\/a>. Use each shape to sketch a tiling of a plane (if you can!).<\/li>\n<li>(Upper level) Once you\u2019ve finished your tilings, measure the total perimeter and area of each design. Can you see why bees benefit from using hexagons?<\/li>\n<\/ul>\n<\/li>\n<li>(All levels) Suppose that a group of cicadas emerges every 13 years, and their two predators have lifecycles of 3 and 4 years.\n<ul>\n<li>How often will the cicadas face the threat of the 3-year predator?<\/li>\n<li>What about the 4-year predator?<\/li>\n<li>How often will the cicadas face <em>both <\/em>predators in the same year?<\/li>\n<li>How would the answers change if the cicadas emerged every 10 years instead?<\/li>\n<\/ul>\n<\/li>\n<li>(High school) Using Baron\u2019s article and <a href=\"https:\/\/www.youtube.com\/watch?v=X_xR5Kes4Rs\">this TED-Ed video<\/a> to start the conversation, discuss whether you think math is invented or discovered.<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr style=\"margin-top: 20px;height: 3px;border-width: 0;color: #0057b8;background-color: #0057b8\" \/>\n<h4 style=\"margin-top: 0px;margin-bottom: 0px;font-family: 'trebuchet ms', geneva, sans-serif;font-size: 22px\"><a id=\"5\" href=\"https:\/\/www.npr.org\/2021\/11\/02\/1051476829\/a-14-year-old-won-a-prestigious-award-for-his-discoveries-on-antiprime-numbers\">A 14-year-old won a prestigious award for his discoveries on &#8216;antiprime&#8217; numbers<\/a><\/h4>\n<p style=\"font-family: 'trebuchet ms', geneva, sans-serif\"><em>NPR,<\/em> November 2, 2021<\/p>\n<p>Since 2011, middle school students have entered their science projects in the Broadcom MASTERS competition in hopes of a $25,000 prize. This year for the first time, a young mathematician won the contest. Nell Clark covered Akilan Sankaran\u2019s accomplishment for <em>NPR<\/em>. Akilan, who is 14, came up with an algorithm to quickly find \u201canti-prime\u201d numbers, numbers with many prime factors. To do this, he used a function whose output depended on an integer\u2019s factorization properties. In the end, the program he wrote delivered mind-bogglingly large anti-prime numbers in a short time. This isn\u2019t just a mathematical curiosity: \u201chighly divisible numbers are useful in computing because they can be used to divide data among computer processors, Akilan explains.\u201d<\/p>\n<p style=\"margin-bottom: 0px;margin-top: 20px\"><strong>Classroom activities: <\/strong><em>number theory, prime factorization<\/em><\/p>\n<ul>\n<li>(Middle school) The <a href=\"https:\/\/en.wikipedia.org\/wiki\/Highly_composite_number\">mathematical definition<\/a> of an anti-prime (or highly composite) number is one that has more divisors than any smaller whole number has. Ask students to find all seven anti-prime numbers from 1 to 40 based on this definition (without looking at the preceding link!). (Hint: 1, 2, and 4 are the first three)<\/li>\n<li>(Middle school) Assign this <a href=\"https:\/\/www.khanacademy.org\/math\/pre-algebra\/pre-algebra-factors-multiples\/pre-algebra-prime-factorization-prealg\/e\/prime_factorization\">prime factorization practice<\/a> from Khan Academy.<\/li>\n<li>(High school) For his project, Akilan came up with <a href=\"https:\/\/projectboard.world\/societyforsciencebcm\/finalist-booth\/akilan-sankaran---exploring-and-analyzing-highly-divisible-numbers\">two functions<\/a><u>,<\/u> $f_s(n)$ and $S_k(n)$ (slide 3), that he used to find anti-prime numbers. He also compared them to the divisor function $d(n)$. On slide 5, he shares some properties of these functions.\n<ul>\n<li>Have students prove the divisor function properties in the first and second rows of the table on slide 5: $d(p) = 2$ for any prime $p$, and $d(mn) = d(m)d(n)$ when $m$ and $n$ don\u2019t share any prime factors.<\/li>\n<li>Have students give an example to show that if $m$ and $n$ share a prime factor, then the multiplicative property of $d$ fails (that is, $d(mn)\\neq d(m)d(n)$).<\/li>\n<li>(Very hard) Have students try to prove the smooth function properties in the first and second rows: $f_s(p) = 1+ 1\/p$ when $p$ is a prime, and $f_s(mn) = f_s(m)f_s(n)$ when $m$ and $n$ don\u2019t share any prime factors.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Leila Sloman<\/em><\/p>\n<hr style=\"margin-top: 20px;height: 3px;border-width: 0;color: #0057b8;background-color: #0057b8\" \/>\n<p>Some more of this month&#8217;s math headlines:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.newsweek.com\/google-doodle-lotfi-zadeh-scientist-fuzzy-logic-mathematical-concept-1654317\">Who Was Lotfi Zadeh? Google Doodle Honors the Azerbaijani American Scientist<\/a><br \/>\n<span style=\"font-style: italic\">Newsweek<\/span>, November 30, 2021<\/li>\n<li><a href=\"https:\/\/www.vaildaily.com\/news\/eagle-valley\/curious-nature-nature-is-a-mathematician\/\">Curious Nature: Nature is a mathematician<\/a><br \/>\n<span style=\"font-style: italic\">Vail Daily<\/span>, November 20, 2021<\/li>\n<li><a href=\"https:\/\/www.quantamagazine.org\/ana-caraiani-delights-in-building-mathematical-bridges-20211117\/\">The Mathematician Who Delights in Building Bridges<\/a><br \/>\n<span style=\"font-style: italic\">Quanta Magazine, <\/span>November 17, 2021<\/li>\n<li><a href=\"https:\/\/theconversation.com\/cancers-are-in-an-evolutionary-battle-with-treatments-evolutionary-game-theory-could-tip-the-advantage-to-medicine-170175\">Cancers are in an evolutionary battle with treatments \u2013 evolutionary game theory could tip the advantage to medicine<\/a><br \/>\n<span style=\"font-style: italic\">The Conversation, <\/span>November 16, 2021<\/li>\n<li><a href=\"https:\/\/www.newyorker.com\/culture\/persons-of-interest\/richard-rusczyks-worldwide-math-camp\">Richard Rusczyk\u2019s Worldwide Math Camp<\/a><br \/>\n<span style=\"font-style: italic\">The New Yorker, <\/span>November 12, 2021<\/li>\n<li><a href=\"https:\/\/www.telegraphindia.com\/my-kolkata\/people\/mathematician-with-roots-in-kolkata-solves-a-century-old-maths-problem\/cid\/1838403\">Mathematician with Kolkata roots solves a century-old maths problem<\/a><br \/>\n<span style=\"font-style: italic\">The Telegraph India<\/span>, November 11, 2021<\/li>\n<li><a href=\"https:\/\/asknature.org\/collection\/banneker\/\">Benjamin Banneker: A Life Told in Cicada Years<\/a><br \/>\n<span style=\"font-style: italic\">Ask Nature, <\/span>November 9, 2021 (reprinted by <a href=\"https:\/\/www.scientificamerican.com\/article\/long-overlooked-benjamin-banneker-is-recognized-for-work-on-cicadas-and-against-slavery\/\"><span style=\"font-style: italic\">Scientific American<\/span><\/a>)<\/li>\n<li><a href=\"https:\/\/www.quantamagazine.org\/mathematicians-find-structure-in-biased-polynomials-20211109\/\">Mathematicians Find Structure in Biased Polynomials<\/a><br \/>\n<span style=\"font-style: italic\">Quanta Magazine, <\/span>November 9, 2021<\/li>\n<li><a href=\"https:\/\/www.nytimes.com\/2021\/11\/04\/us\/california-math-curriculum-guidelines.html\">California Tries to Close the Gap in Math, but Sets Off a Backlash<\/a><br \/>\n<span style=\"font-style: italic\">The New York Times<\/span>, November 4, 2021<\/li>\n<li><a href=\"https:\/\/www.quantamagazine.org\/surprising-limits-discovered-in-quest-for-optimal-solutions-20211101\/\">Surprising Limits Discovered in Quest for Optimal Solutions<\/a><br \/>\n<span style=\"font-style: italic\">Quanta Magazine, <\/span>November 1, 2021<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>What Hot Dogs Can Teach Us About Number Theory Quanta Magazine, November 18, 2021 No hot dog is complete without a bun. So it\u2019s maddening that you typically have to buy eight buns at a time, but you can\u2019t buy exactly eight hot dogs. Getting one package of each will<span class=\"more-link\"><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-november-2021\/\">Read More &rarr;<\/a><\/span><\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2],"tags":[18,35,36,42,50,57,58,61,66,71,72,88,93],"class_list":["entry","author-shershberger","post-529","post","type-post","status-publish","format-standard","category-math-in-the-media-digests","tag-cryptography","tag-geometry","tag-gerrymandering","tag-inverse-proportionality","tag-markov-chain-monte-carlo","tag-modular-arithmetic","tag-nature","tag-number-theory","tag-pink-noise","tag-prime-factorization","tag-prime-numbers","tag-tiling","tag-waves"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/529","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/comments?post=529"}],"version-history":[{"count":1,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/529\/revisions"}],"predecessor-version":[{"id":732,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/529\/revisions\/732"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=529"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=529"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=529"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}