This is the 141st Playful Math Education Blog Carnival. And I will be taking this lower-key than I have past times I was able to host the carnival. I do not have higher keys available this year.
I will start by borrowing a page from Iva Sallay, kind creator and host of FindTheFactors.com, and say some things about 141. I owe Iva Sallay many things, including this comfortable lead-in to the post, and my participation in the Playful Math Education Blog Carnival. She was also kind enough to send me many interesting blogs and pages and I am grateful.
141 is a centered pentagonal number. It’s like 1 or 6 or 16 that way. That is, if I give you six pennies and ask you to do something with it, a natural thing is one coin in the center and a pentagon around that. With 16 coins, you can add a nice regular pentagon around that, one that reaches three coins from vertex to vertex. 31, 51, 76, and 106 are the next couple centered pentagonal numbers. 181 and 226 are the next centered pentagonal numbers. The units number in these follow a pattern, too, in base ten. The last digits go 1-6-6-1, 1-6-6-1, 1-6-6-1, and so on.
141’s also a hendecagonal number. That is, arrange your coins to make a regular 11-sided polygon. 1 and then 11 are hendecagonal numbers. Then 30, 58, 95, and 141. 196 and 260 are the next couple. There are many of these sorts of polygonal numbers, for any regular polygon you like.
141 is also a Hilbert Prime, a class of number I hadn’t heard of before. It’s still named for the Hilbert of Hilbert’s problems. 141 is not a prime number, which you notice from adding up the digits. But a Hilbert Prime is a different kind of beast. These come from looking at counting numbers that are one more than a whole multiple of four. So, numbers like 1, 5, 9, 13, and so on. This sequence describes a lot of classes of numbers. A Hilbert Prime, at least as some number theorists use it, is a Hilbert Number that can’t be divided by any other Hilbert Number (other than 1). So these include 5, 9, 13, 17, and 21, and some of those are already not traditional primes. There are Hilbert Numbers that are the products of different sets of Hilbert Primes, such as 441 or 693. (441 is both 21 times 21 and also 9 times 49. 693 is 9 times 77 and also 21 times 33) So I don’t know what use Hilbert Primes are specifically. If someone knows, I’d love to hear.
Also, at the risk of causing trouble, The Aperiodical also hosts a monthly Carnival of Mathematics. It’s a similar gathering of interesting mathematics content. It doesn’t look necessarily for educational or playful pieces.
The Reflective Educator posted Precision In Language. This is about one of the hardest bits of teaching. That is to say things which are true and which can’t be mis-remembered as something false. Author David Wees points out an example of this hazard, as kids apply rules outside their context.
Simon Gregg’s essay The Gardener and the Carpenter follows a connected theme. The experience students have with a thing can be different depending on how the teacher presents it. The lead example of Gregg’s essay is about the different ways students played with a toy depending on how the teacher prompted them to explore it.
Now I can come to more bundles of things to teach. Colleen Young gathered Maths at school … and at home, bundles of exercises and practice sheets. One of the geometry puzzles, about the missing lengths in the perimeter of a hexagon, brings me a smile as this is a sort of work I’ve been doing for my day job.
Starting Points Maths has a page of Radian Measure — Intro. The goal here is building comfort in the use of radians as angle measure. Mathematicians tend to think in radians. The trigonometric functions for radian measure behave well. Derivatives and integrals are easy, for example. We do a lot of derivatives and integrals. The measures look stranger, is all, especially as they almost always involve fractions times π.
Lowry also has Helping Your Child Learn Time, using both analog and digital clocks. That lets me mention a recent discussion with my love, who teaches. My love’s students were not getting the argument that analog clocks can offer a better sense of how time is elapsing. I had what I think a compelling argument: an analog clock is like a health bar, a digital clock like the count of hit points. Logic tells me this will communicate well.
YummyMath’s Fall Equinox 2020 describes some of the geometry of the equinoxes. It also offers questions about how to calculate the time of daylight given one’s position on the Earth. This is one of the great historic and practical uses for trigonometry.
To some play! Miguel Barral wrote Much More Than a Diversion: The Mathematics of Solitaire. There are many kinds of solitaire, which is ultimately just a game that can be played alone. They’re all subject to study through game theory. And to questions like “what is the chance of winning”? That’s often a question best answered by computer simulation. Working out that challenge helped create Monte Carlo methods. These can find approximate solutions to problems too difficult to find perfect solutions for.
Conditional probability is fun. It’s full of questions easy to present and contradicting intuition to solve. Wayne Chadburn’s Big Question explores one of them. It’s based on a problem which went viral a couple years ago, called “Hannah’s Sweet”. I missed the problem when it was getting people mad. But Chadburn explores how to think through the problem.
Now to some deeper personal interests. I am an amusement park enthusiast: I’ve ridden at least 250 different roller coasters at least once each. This includes all the wooden Möbius-strip roller coasters out there. Also all three racing merry-go-rounds. The oldest roller coaster still standing. And I had hoped, this year, to get to the centennial years for the Jackrabbit roller coaster at Kennywood Amusement Park (Pittsburgh) and Jack Rabbit roller coaster at Seabreeze Park (Rochester, New York). Jackrabbit (with spelling variants) used to be a quite popular roller coaster name.
So plans went awry and it seems unlikely we’ll get to any amusement parks this year. No county fairs or carnivals. We can still go to virtual ones, though. Amusement parks and midway games inspire many mathematical questions. So let’s take some in.
Michigan State University’s Connected Mathematics Program set up set up a string of carnival-style games. The event’s planners figured on then turning the play money into prize raffles but you can also play games. Some are legitimate midway games, such as plinko, spinner wheels, or racing games, too.
Hooda Math’s Carnival Fun offers a series of games, many of them Flash, a fair number HTML5, and mostly for kindergraden through 8th grade. There are a lot of mathematics games here, along with some physics and word games.
Specific rides, though, are always beautiful and worth looking at. Ann-Marie Pendrill’s Rotating swings—a theme with variations looks at rotating swing rides. These have many kinds of motion and many can be turned into educational problems. Pendrill looks at some of them. There are other articles recommended by this, which seem relevant, but this was the only article I found which I had permission to read in full. Your institution might have better access.
Lin McMullin’s The Scrambler, or A Family of Vectors at the Amusement Park looks at the motion of the most popular thrill ride out there. (There are more intense rides. But they’re also ones many people feel are too much for them. Few people in a population think the Scrambler is too much for them.) McMullin uses the language of vectors to examine what path the rider traces out during a ride, and what they say about velocity and acceleration. These are all some wonderful shapes.
And Amusement Parks
Many amusement parks host science and mathematics education days. In fact I’ve never gone to the opening day of my home park, Michigan’s Adventure, as that’s a short four-hour day filled with area kids. Many of the parks do have activity pages, though, suggesting the kinds of things to think about at a park. Some of the mathematics is things one can use; some is toying with curiosity.
Here’s The State Fair of Texas’s Grade 6 STEM games. I don’t know whether there’s a more recent edition. But also imagine that tasks like counting the traffic flow or thinking about what energies are shown at different times in a ride do not age.
Dorney Park, in northeastern Pennsylvania, was never my home park, but it was close. And I’ve had the chance to visit several times. People with Kutztown University, regional high schools, and Dorney Park prepared Coaster Quest – Geometry. These include a lot of observations and measurements all tied to specific rides at the park. (And a side fact, fun for me: Dorney Park’s carousel used to be at Lake Lansing Amusement Park, a few miles from me. Lake Lansing’s park closed in 1972, and the carousel spent several decades at Cedar Point in Ohio before moving to Pennsylvania. The old carousel building at Lake Lansing still stands, though, and I happened to be there a few weeks ago.)
A 2018 posting on Social Mathematics asks: Do height restrictions matter to safety on Roller Coasters? Of course they do, or else we’d have more roller coasters that allowed mice to ride. The question is how much the size restriction matters, and how sensitive that dependence is. So the leading question is a classic example of applying mathematics to the real world. This includes practical subtleties like if a person 39.5 inches tall could ride safely, is it fair to round that off to 40 inches? It also includes the struggle to work out how dangerous an amusement park is.
Speaking from my experience as a rider and lover of amusement parks: don’t try to plead someone’s “close enough”. You’re putting an unfair burden on the ride operator. Accept the rules as posted. Everybody who loves amusement parks has their disappointment stories; accept yours in good grace.
This leads me into planning amusement park fun. School Specialty’s blog particularly offers PLAY & PLAN: Amusement Park. This is a guide to building an amusement park activity packet for any primary school level. It includes, by the way, some mention of the historical and cultural aspects. That falls outside my focus on mathematics with a side of science here. But there is a wealth of culture in amusement parks, in their rides, their attractions, and their policies.
Let me resume the fun, by looking to imaginary amusement parks. TeachEngineering’s Amusement Park Ride: Ups and Downs in Design designs and builds model “roller coasters”. This from foam tubes, toothpicks, masking tape, and marbles. It’s easier to build a ride in Roller Coaster Tycoon but that will always lack some of the thrill of having a real thing that doesn’t quite do what you want. The builders of Son Of Beast had the same frustration.
The Brunswick (Ohio) City Schools published a nice Amusement Park Map Project. It also introduces students to coordinate systems. This by having them lay out and design their own amusement park. It includes introductions to basic shapes. I am surprised reading the requirements that merry-go-rounds aren’t included, as circles. I am delighted that the plan calls for eight to ten roller coasters and a petting zoo, though. That plan works for me.
I mentioned yesterday Iva Sallay’s hosting of the 140th Playful Math Education Blog Carnival. This is a collection of pieces of educational, recreational, or otherwise just delightful mathematics posts. I’d said I hoped I might have the energy to host one again this year and, you know? Denise Gaskins, who organizes this monthly event, took me up on the offer.
So, if you write, or read, or are just aware of a good mathematics or mathematics-related blog, please, leave me a comment! I’ll need all the help I can get finding things worth sharing. Anything that you’ve learned from, or that’s delighted you, is worth it. It’ll teach and delight other people too.
Iva Sallay’s Find the Factors recreational mathematics puzzle blog is once again host to the Playful Math Education Blog Carnival. It’s the 140th installment of this blog series, this month, and you might find several pieces of mathematics instruction or trivia or recreation or just play worth reading. It’s been a while since I had the energy to host one but I hope to be able to again this year. Enjoy, I hope.
Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.
In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.
That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …
This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.
Fermat’s Two-Squares Theorem, discussed here, is not the famous one about . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.
What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.
I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?
Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.
These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.
I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.
Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.