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 2nd Google number. The nth 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 (4.3x10
19) 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.
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.
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
Recursion (links to other collections of math links)
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.
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!
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