Math Teachers at Play #71 (with 71 links)

Back in 2009, the first time I hosted a Math Teachers at Play Blog Carnival post, we were at #11. Seems like those smaller numbers almost always had something interesting going on. For 71, it's a stretch...

What's Special About This Number has:

• 71 divides the sum of all the primes before it (i.e., 2 + 3 + 5 + 7 + 11 + ... + 67 is divisible by 71)
• 71 = (4! + 4.4)/.4 (representation of numbers using only four 4's)
• 71 = 36² - 35² = 36 + 35
• 71 - 1 = 1 x 2 x 5 x 7 and 71 + 1 = 3 x 4 x 6 products of partitions of consecutive numbers)
• 71² = 7! + 1
• 71² = 2⁷ + 17³ (sum of prime powers of two prime numbers)
• 71³ = 357911 (consecutive odd numbers) 

 Number Gossip has:

• 71 is the only two-digit number n such that (nⁿ-n!)/n is prime.
• 71 is the 2^nd Google number. The n^th Google number is the first n-digit prime found in the decimal expansion of e. They are named Google numbers because of the unusual hiring ad that Google put up.

2, 71, 271, 4523, 74713, ...

71, Mark Gonyea

71, Brent Yorgey

And here are a few images of the number 71 itself. Mark Gonyea is a designer. Brent Yorgey and Richard Schwartz are mathematicians. Posters are available for the numbers 1 to 100 from each of these artists.

Richard Schwartz

Math teachers at play know that math is best learned when the student is thoroughly engaged, through their body, their imagination (story-telling), or the world of games. I've started out this month's post with those three categories. (Most of the submissions this month described hands-on, or feet-on, activities. It's as if there had been a theme agreed upon without anyone mentioning it.) Some of the following posts are from submissions, and others are posts that I wanted to share from my internet wanderings. This post has 71 links. (You might need to digest it in smaller bites.) Enjoy!

Learning with Our Bodies

Julie, at Highhill Education, shares her family's mandala art, and the geometry they learned while doing it. (Julie is based in Germany. In the U.S., I've found some of the best inexpensive books come from Dover - here are some of their mandala coloring books.)

Jennifer Bardsley, at Teaching My Baby to Read, works (plays) with her son, exploring rotational symmetry using cookie cutters and flour.

Ticia, at Adventures in Mommydom, shares her hands-on fraction lessons.

Margo Gentile, at Margo's Math and More, was inspired by all those snow days, and created some wonderful mazes in the snow for her dog and kids to navigate. (Bummer! I don't have any snow here in California to try this in.)

Lilac, at Learners in Bloom, wrote Combinatorics in Kindergarten, her story of making clothes for the bears, so her daughters could count how many outfits Little Bear could wear. She also made a Feed the Clown game to help her daughters have fun practicing basic addition facts.

Maria Droujkova, at Moebius Noodles, describes how a student combined ideas from the Moebius Noodles book, making mirror books to create fractal stars.

Nicora, at Bridging the Gap, suggests that we should let students break things to help them learn fractions. "Give them lots and lots of experiences where they have to break up things evenly and share things fairly." Discuss, put back together, discuss some more.

Steven Strogatz describes the math found in our cowlicks and fingerprints.

Storytelling

Denise, at Let's Play Math!, shares the power of stories: "The mere hint of fantasy adventure can change graphing equations from boring to cool." She used  Dan Wekselgreene's inequalities lesson based on the adventures of Ohio Jones. (Denise has written extensively about Fibonacci and Alexandria Jones. I think they must all be related somehow.)

Advertisers tell stories to convince us to buy what they're selling. Often their stories are deceptive. Mr. Honner describes how Prudential's 'oldest person you know' ad subtly points in the wrong direction.

There is a story mathematicians like to tell, of how the young Carl Gauss was asked, along with his classmates, to add up the numbers from 1 to 100, perhaps to give the schoolmaster a bit of time to relax. As the story goes, Carl saw a nice trick, and wrote just the answer down, turning it in almost immediately. Brian Hayes was curious about the historical accuracy of this story, and researched it. His article (in American Scientist) is quite intriguing.

Alexandre Borovik has written a delightful story about Anthony the Ant, and his discoveries of his world (a piece of paper, which gets folded into a cube).

Games & Puzzles

John Golden, at Math Hombre, has a games page that looks marvelous!

Did you know that some coins cost more to make than their face value?! Dan, at Math for Love, used this to make a math lesson, and one of his students came up with a great question. The puzzle she posed is whether or not you can come up with coins that are worth one dollar and cost one dollar to make. And another coins puzzle, from Nathan Kraft, at Out Rockin' Constantly: Which is worth more, a pound of quarters, or a pound of dimes?

I don't know if this counts as a puzzle - more like a problem-solving challenge. Nat Banting, at Musing Mathematically, asks what pattern the wet part of a tire makes as the tire rolls along.

How fast can you decide how many dots you saw? (The creators of this simple game have not left their names. They write: "Subitizing is the ability to immediately recognize the quantity of a small number of objects without counting. Research has shown subitizing to be foundational to basic arithmetic and other math skills. Many children who struggle with basic math also have trouble subitizing.")

Mike, at Spiked Math, has created a very visual puzzle.

Greg Ross, at Futility Closet, has given us the lovely puzzle you see below. You might word it differently, depending on your student's stamina and mathematical sophistication. (Instead of asking for proof, maybe just ask them to find all the midpoints first. Then ask if they can pick points so none of the midpoints will occur at an intersection. Finally, see if they can figure out why five points will always produce at least one line whose midpoint is on an intersection.)

Math Education

Crystal Wagner, at Triumphant Learning, knows that problem solving is at the heart of mathematics, and gives some guidelines and resources for keeping problem-solving at the heart of your math lessons. (I'd like to add two more resources to her list: The Art of Problem Posing, by Stephen Brown, who advocates for students to pose problems, like this. And for advanced math students, The Art and Craft of Problem Solving, by Paul Zeitz.)

Jenny, at Elementary, My Dear, Or Far From It, describes the benefits of confusion.  She also linked to this piece by Jeffrey McClurken, and this piece by Annie Murphy Paul. (It's hard for students in the U.S. to understand how useful confusion is - as a stage in learning anything new. I want to share all their ideas with my college students. Maybe I'll write a post consolidating it all...)

Dan Finkel, at Math for Love, gives an inspiring description (using his goats for comparison) between the fearful learner, who leaves others in charge, and the adventurous learner who takes charge both of their own learning and of the math problem at hand.

Mama Squirrel, at Dewey's Treehouse, wrote How I became a Math Teacher? to describe her journey into math teaching and her thoughts about the matter.

Cathy, at Math Babe, interviews the lead writer of the common core math standards.

Megan Hayes-Golding describes why ranking tasks are especially valuable as learning tools.

Alexandre Borovik, at the De Morgan Forum points to a paper [pdf] by Herbert Wilf, who argues that there is no useful math education research out there. The abstract states:

We examine a number of papers and a book, all of which have been cited, by people who are knowledgeable in the field, as being good examples of “research in mathematics education.” We find specific serious flaws, indeed fatal flaws, in all of them, so that no conclusions of any interest follow as a result of any of the “research” that is reported in these works. We have found no evidence that the research paradigm, involving test and control groups, randomized trials, etc., which is invaluable in the life sciences, is of any use whatever in studying mathematics education and we urge that it be abandoned, in favor of human-to-human discourse about how we can improve curricula and teaching.

Also at the De Morgan Forum are the results of a study that found that practice at "guesstimating" can speed up math ability.  (Hmm, isn't this research on math education? Maybe Wilf has different sorts of research in mind. I'd enjoy discussing his paper with anyone interested.) And another: Can you imagine a whole post on 3 - 1 = 2? Alexandre Borovik has translated a paper originally written in Russian by Igor Arnold, which gives 20 different problems that all boil down to finding 3 - 1.

Bruno Reddy, at Mr. Reddy's Math Blog, posted some interesting videos from a workshop he attended. If you teach students with limited English language proficiency, you may find this valuable.

Geoff Krall, at Emergent Math, is thinking about how to make his classroom a safe place for taking risks. That's a common theme at Cheesemonkey's blog.

Visual Math

Dan Walsh, at Dan's Geometrical Curiosities, saw this, and just had to figure it out. The mathematical description of what's happening is called curves of pursuit.

Here's your chance to make a bit of mathematical art: at weavesilk.com. Ahh...

If you were going to try to figure out how far it is to the horizon, what sort of picture would you draw? Bryan Meyer, at Doing Mathematics, thinks we can learn a lot about students' thinking by discussing the pictures they draw to solve problems with.

The two mathematicians described at the beginning of this post, Brent Yorgey and Richard Schwartz, have made their images to help people visualize factors, prime numbers, and composite numbers. Jeffrey Ventrella has the same goal with his composite number tree and his book, Divisor Drips and Square Root Waves (link is to a fascinating online version). There is also this intriguing Prime Number Patterns applet, by Jason Davies.

News

A simulation of glacier movement can be run backwards to predict where things were in the past. When the remains of some hikers who were lost almost 90 years ago were recently found, the simulation was used to figure out where they most likely were when they died.

Rachel Thomas, at +plus magazine, writes about the math of bubbles, which inspired the architecture of the National Aquatic Centre in Beijing, built for the Olympics.

The Rubik's cube has over 43 quintillion (4.3x10¹⁹) positions. It has recently been shown that there is a way to move it to each different position in sequence, without ever repeating a position. (This is called a Hamiltonian circuit.) Thanks to Robert Talbert for pointing this out on his Casting out Nines blog.

Caroline Chen has written a very readable account of the strange proof of the ABC Conjecture.

The Journal of Humanistic Mathematics has some interesting articles in its current issue, including one on Gallileo and Aristotles' Wheel [pdf] describing a paradox and how mathematicians think about it.

Joselle Kehoe, at Mathematics Rising, writes about brain research showing that the part of the brain that deals with counting deals also (and earlier in our evolution) with representing the fingers.

(from Pat Bellew)

Valentine's Day Math

Mr. Honner wishes us all a Happy Permutation Day.

Laura, at Math for Grownups, shared this quickie video (she calls if a gif, my son calls these vines) valentine.

A Few More Tidbits

Math Fun Facts  (list and home)

The 142857 Times Table

Video Helpers for Algorithms and Problem-Solving (from Prairie Creek Community School)

Recursion (links to other collections of math links)

Brie Finegold, at Blog on Math Blogs, has ideas about how to get your friend to like math.

Math Munch comes out weekly. Anna Weltman, Justin Lanier, and Paul Salomon say: "We write Math Munch to help more kids find something mathematical that they love." Here's a post I liked on art and math, one of their favorite topics, I think. And this post is full of puzzles.

Don’t miss the 107^th Carnival of Mathematics (our sister blog carnival).

That rounds up this edition of the Math Teachers at Play carnival. I hope you enjoyed the ride.
The next installment of our carnival will open sometime during the second week of March. If you would like to contribute, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Past posts and future hosts can be found on our blog carnival information page. We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the Math Teachers at Play blog carnival, please speak up!