Playful Math Education Carnival #119

Welcome to Playful Math Education Carnival #119! Just to be clear, that exclamation is to express excitement, not factorial. Fortunately, you will have a bit of time before there's any danger of confusing this post with the edition (119 factorial).

Anyway, only the very coolest folks get to handle a MTaP edition that can be written with a factorial. And I just realized how close (and yet how far) I was to such glory.

119 Fun facts

• 119 is the number to call for emergency services... in parts of Asia (wiki reference).
• Of course, 119 backwards is 911 which is the US emergency services phone number
• 119 is aspiring, the sequence formed by summing proper factors ends with a perfect number.
• 119 isn't prime, but it almost feels like it
• 119 = 7 x 17. I don't think products of consecutive primes ending in 7 has a name, but maybe it should?

Do you have other fun facts about 119? Please? Please?

Dedication

I'm saddened to note the passing of  Alexander Bogomolny this month, and I dedicate the edition of the  carnival to him. The material he developed and made available on his site

https://www.cut-the-knot.org/ is truly amazing and remains with us for our benefit.

Miscellaneous

Aperiodical has an article from Benjamin Leis on the Big Internet Math-off.

Something James Propp wrote as part of the Big Internet Math-off: A pair of shorts. 

Elementary

I was reminded of the game of Chomp! in Shecky Riemann's linkfest (most of which isn't elementary level, but worth investigating).

Cathy O'Neil tells her mathematician origin story. I hope all our kids can have an empowering math experience like this.

Discussion of a "square dancing" puzzle from Mike Lawler: part 1 and part 2. I think there is a lot more to explore here and hope some of you will write parts 3 and beyond... 

I always love game discussions. Set is a game you probably all know, but in case you don't here's an intro and a deeper analysis in the Aperiodical.

Pat Ballew writes about divisibility rules. Pat also discusses a fun XKCD in prime time fun.

I'm delighted at how this starts with something many take for granted (12 hour vs 24 hour time of day conventions) and then builds a fun exploration.

Middle school

Have you been waiting for someone to write the perfect post giving you an introduction to tons of Desmos activities? Well, Mary Bourrasa has done it for you.

Michael Pershan tweeted a pointer to a nice collection of logic puzzles on puzzling stackexchange.

Denise Gaskins pointed out a past note about factor trees and some cute wordplay from Danica McKellar's book: prime numbers are like monkeys.

This segues directly into a review of two number theory books by Ben Leis (also the author of the Big Internet Math off post above) in which he discusses some other visualizations beyond factor trees: 

High school

Mr Honner: math around us and parallel lines cut congruent arcs on a circle in a nice picture. 

Ben Orlin invents and illustrates a new adage that there are no puddles in mathematics, only oceans in disguise. 

More advanced

Mathematical theorems you had no idea existed because they are false: https://www.facebook.com/BestTheorems/

Have fun finding counterexamples. Also, link disproves the conjecture that there is nothing worthwhile on facebook.

The Scientific American Blog has been running these columns on "my favorite theorem." Go back and take a look (I think this was their first one): Amie Wilkinson's favorite theorem.

A fascinating discussion of the Fields' Medal and some ideas about what it should be supporting. 

What was the score? Maybe the sum of scores was 119?

Math Teachers at Play Blog Carnival #113

Welcome everyone to the 113th Math Teachers at Play blog carnival! As usual, I'm lucky to have the best month to curate. 113 is the prime MTaP because:

• 113 is prime
• all permutations of the digits are prime
• all 2 digit subsets of the digits are prime
• the product of the digits is prime
• the sum of the digits is prime
• 113₍₄₎ (113 in base 4, the smallest base that is sensible) is also prime!
• 2¹¹³ - 2 is divisible by 113. Wow! (see Jordan Ellenberg's Favorite Theorem)

October is for Play

October is a perfect month to talk about mathy play because of:

Later this week is the largest international games convention in Essen, Germany. I'm jealous of any of you who get to go. For the rest of us, I've collected a bunch of great math games and explorations to keep us happy.

Before we move off Spiel, though, take a look at the logo above again.

What do you notice?

What do you wonder?

Click this button to compare with another version of the logo:

Elementary

In the spirit of Malke Rosenfeld's Math in Your Feet, Mrs. Miracle's post about beat passing games can inspire a whole-body exploration of patterns. I like expanding beyond visual patterns and the fact even very small children can create their own beat pattern.

For some reason, this old Christopher Danielson post resurfaced on my RSS reader. While it is an old one, I hadn't seen it before, so maybe you missed it too or will appreciate reading it again: Armholes. Maybe it is easier to be patient while waiting for the kids to get dressed if we are also exploring math at the same time?

How many holes?

An online math competition for elementary kids: BRICS Math. I like using math competition questions as a jumping off point for further conversations and explorations. Sometimes the questions have natural extensions (what if we changed this number?) and other times we just talk about what the kids found interesting about the question or what it made them think about.

Iva Sallay (who has hosted the last edition of MTaP) makes a Halloween 10 Frame (just in time!)

Here are some wonderful images and gifs from Gábor Damásdi. They could be a good prompt for Notice & wonder for young kids and older ones:

• Fountain
• A tesselation pattern
• An islamic pattern

AO Fradkin talks about a tricky game that helps develop mathematical language: A figure with pointy things...
This question: why do we bother with defined terms and mathematical language fits nicely with Chasing Number Sense's exploration of the definition of a polygon: Polygon is a shape that is really big.

Denise Gaskins, the wonderful unifying force behind this blog carnival, reminds us of 30+ things to do with a 100 chart. I have one more to add: our family first learned to play Go on a 100 chart with some small blue cubes and bananagram tiles, before we had a chance to buy our first dedicated board. Here's an old snap from the beginning of the year:

Middle School

If you like chained fraction puzzles (we do!) and you like thinking about concrete manipulatives (we do!!) then you'll enjoy this post from Bridget Dunbar: Thinking in th Concrete.

Another post from Iva Sallay uses candy to teach equation solving: Solve for X with candy. With Iva's help, we're certainly ready for a mathy Halloween.

Presh Talwalkar at Mind Your Decisions occasionally posts viral puzzles with some nice explanations. I enjoyed this one (octagon in a paralellogram) because it fit with a problem solving strategy we've been practicing recently: test a special case. Here, we tried starting with a square, then discussed whether that was really a "special case" or fit the general situation.

Mike Lawler, as usual, has some fun posts, this time I picked out his videos talking about Tim Gowers's intransitive dice.

Different from Tim Gowers's dice?

Huge jars of coins are wonderful, for so many reasons. Kristen (Mind of an April Fool) shares a fun 3-act lesson: Sassy Cents. Our family has gotten a lot of mileage out of doing notice and wonder at home with similar 3-act lessons.

Jim Propp contributed to the excitement of Global Math Week with a sort of History of Exploding Dots. I have an especially warm feeling for this story because he includes mention of the "minicomputer" idea created by Frederique Papy. These minicomputers figured prominently in my own elementary math education.

High School/More advanced

Continuing with the theme above around language, definitions, precision and math, Mr Orr gives us 3 Desmos Activities for Talkers & Drawers.

Curiousa Mathematica shares a Putnam exam question that is actually very accessible: Spots on a ball. Try to think about it before reading the solution.

Patrick Honner talks about some of the math related to gerrymandering in Wasted Votes. In his discussion, he describes a game that sounds very much like Mathpickle's A Little Bit of Aggression (pdf here.)

SolveMyMaths has done a series this month on trig identities. These build step-by-step pictures to help understand what is going on, for example, in the angle addition formulas. Take a look (they are easier to understand than this final step picture, but it is one of my favorites):

Teaching Resources

For all of us who sometimes have to find a math curriculum for our kids, David Wees has created his checklist of necessary characteristics: Questions about Curriculum.

What makes a good school? Jane Mouse (in russian) explains that there is no "best" school. For those of us who teach our own kids, one interpretation is that we should try to expose the kids to a variety of modes and styles. Also, what is working now might change over time.

Sam Shah offers a number of hacks for making your own material. My personal recommendation is for you to try this out with your kids: make problems together and discuss the process. What makes a good question? What makes a hard vs an easy question? Can you create problems with only one, more than one, or no answers?

Resourceaholic (Jo) has, you guessed it, a presentation on resources: Power of Six presentation. Be sure to take a look at Jo's Resource Library for ideas when you need secondary school material.

I'm sure there are a lot of other great posts with families and teachers sharing their math games and explorations. Please add comments to let me know about your favorites from the month (or older ones)!

Math Teachers At Play Carnival #110 Summer Vacation Edition

Hello again math folks! I've been in the middle of a major transition, moving between Asia and North America, so haven't really had time to post recently. Putting together this month's carnival was a nice opportunity to see what everyone else has been writing about and get some new ideas!

As you scan through the links I've highlighted, please don't get too fixated on the grade level splits. These are really approximate and I expect you will find worthwhile activities for all ages in every section.

In Memoriam: Maryam Mirzakhani

On the 14th of July, Maryam Mirzakhani passed away. She was the first woman to win the Fields Medal. It would be wonderful if you could do some exploration in her honor this month. One of her areas of research was on pool tables. Here are some places to get an idea of the way mathematicians have been inspired by this game:

• Elliptical pool (from Numberphile)
• The Illumination Problem (also Numberphile)
• A snooker (problem 3 from the 2013 International Tournament of Young Mathematicians)
• Wolfram MathWorld Discussion of Alhazen's Billiard Problem that indicates the link with optics

If you find other kid-friendly projects related to Mirzakhani's work, please tell me in the comments!

Some 110 facts

This was the best number carnival to be able to host, because:

• 110 = 10 * 11. That means it is pronic, the product of two consecutive integers.
• 110 looks suspiciously like a binary number. Binary 110 = decimal 6. Decimal 110 = Binary 1101110, which I like to read as 110 1 110
• Because it has an odd number of 1s in its binary expansion, 110 is odious
• 110 is a Harshad number because it is divisible by the sum of its digits
• The element with atomic number 110 is Darmstadtium (Ds).
• 110 is the number of millions of dollars spent in March for a Basquiat painting, the highest amount paid at auction for a work by an American artist.

A number talks picture that caught my eye

I'm not sure there is anything especially 110 about this picture, but there are a lot of mathematical questions to ask and things to observe here:

In a related vein, if you and your kids need some mesmerizing math gifs, take a look at Symmetry.

Elementary skills

Denise Gaskins has written a lot to help parents engage playfully and mathematically with their kids. In this blog post, she has collected highlights that are great with young students and worth remembering for older ones, too: How to Talk Math with Your Kids.

I love board games and think there is still tremendous value in the physical games that electronic versions miss. Here's an example from Sasha Fradkin, where cleaning up after playing gives us a chance to think about whether skip counting is just a chant or if the words mean something: Skip counting or word skipping

While she's at it, Sasha Fradkin also has a nice puzzle activity with Numicons. I would think of this as a progression step toward tangram and other dissection puzzles.

Which one doesn't belong is a math meme you should know already. If you don't, ask in the comments and I'll point you in the right direction. Christopher Danielson has recently introduced Which Poster Doesn't Belong? While you are visiting his blog, enjoy his story about The Three Year Old Who is Not a Monster.

Exploding Shapes is a catalyst for notice and wonder from The Math Forum. I really like this because here are many different directions to go and no single "right" answer. Also, let's give a cheer because it looks like this recent set of posts shows the math forum folks have returned to posting nice conversation starters.

Swine on a Line by Jim Propp is a nice game/puzzle that seems a great companion to James Tanton's Exploding Dots. Hmm, maybe July 4th inspired me to look for lots of explosions...?

Middle school(ish)

Rupesh Gesota starts with a nice puzzle and shows us how it was analyzed by several different students: One Puzzle, Many Students, Many Approaches. I particularly like how the introduction to the puzzle encourages us to think of different methods.

Mike Lawler has done a huge number of really great explorations with his kids. Here are some recent projects with books from the Park City Mathematics Institute: Playing Around. If you haven't been following Mike and his kids, I really encourage you to go through his past posts.This blog is fantastic for great projects and connections with other resources.

Curious Cheetah shows us several ways to calculate square roots. I would say, like long division, the value isn't in memorizing the algorithms, but understanding how they work and using them to play with numbers.

Manan Shah has a couple of nice summer explorations. The first is an excursion into the digits of prime numbers: Prime Numbers. The second is a coin flipping and gambling game to ponder during these warm vacation months: Summer Excursion Coin Flipping.

There are other, problem-based, posts on Benjamin Leis's blog, but this one made me jealous of his recent purchase of the A Decade of the Berkeley Math Circle.

High school/more advanced

Thinking Inside the Box, Simon Gregg takes a new look at a familiar shape, the cube. His comment about the exploration really nicely captures something that is beautiful about mathematical exploration: "I came back to a familiar place from an unfamiliar starting place."

A cute absolute value game now appears as a nicely animated game: Absolute Value. I think this is a nice simplification and implementation of the original game.

There's an improv game where the players have to switch between movie genres. Film noire or "hard boiled detective" comes up every time. This TedEd video could introduce fractals and this film genre at the same time.

Michael Pershan puzzles over two measures of steepness in his trigonometry class: When Measures of Steepness Disagree. I really like the questions he raises about how to use two different scales that measure the same concept, but are not linearly related.

Also, Michael links to the New Zealand Avalanche Advisory, with a nice graphic showing a case where the greatest danger of avalanche is in the middle of a slope range:

Dave Richeson breaks down an impressive rainbow photo:

Fair sharing is a really interesting theme to motivate a lot of great math. Tanya Khovanova looks at a couple of fair sharing problems and strategies in Fair share sequences.

Also, check out the sister carnival to this one: The Carnival of Mathematics over at The Aperiodical.

Techniques for teaching

This post is an old classic, but I've been reminded of it because it is used in a workshop that I frequently attend: using student reflections.

Have you visited NRICH recently? No?!?! Go over now (here's the link) and find a really cool activity to do with your kids. Seriously!

Some tips on giving feedback: Effective feedback for deeper learning.

Using Desmos to check your work: Desmos is the new back of the book.

A thought piece on the modern role of teachers: Teachers Sow Thirst for Learning. If you can read Indonesian (which I can't) you may find some other interesting pieces here on math education.

A special announcement

James Tanton is leading a project for a world-wide week of math this fall. Please take a look at the project page Global Math Week

Math Teachers at Play #100 - (Blog Carnival)

Wow, the 100th Math Teachers at Play! Such an honor to put together this milestone edition. Thanks to Denise Gaskins for creating and managing this great resource. Let her know if you are interested in hosting in the future.

I asked the 3J's for observations and facts about 100:

• It is written with a 1 and two 0s
• Square number (10 x 10)
• It is a sum of two squares 64 + 36
• It is a sum of two primes in several ways: 97 + 3, 89 + 11, 83 + 17, 71 + 29, 53 + 47
• 1100100 in binary
• 100 has 9 factors, which sounds like a lot, but is not an anti-prime
• 100 is the start of a 26 term Collatz Sequence: 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 (little program for Collatz play)

Number Gossip revealed that it is a practical, powerful, happy, odious number. Hmm, a jolly dictator?

Here's a very familiar 100 in our daily life:

Unfortunately, this 100 makes far fewer appearances:

A 100 puzzle

To get started, here's a tricky puzzle that, semi-famously, has been a Google interview question. This phrasing comes from Good Riddles Now:

There are 100 prisoners lining up to go to jail. Each prisoner is wearing a hat that is either black or white. The prisoners don't know their own hat color, just the hat color of those in front of them in line (the first prisoner in line can't see anyone's hat and the last prisoner can see everyone's hat except their own). Starting from the back, one of the guards asks each prisoner what color their hat is. If they are correct they get to go free but if they are wrong they go to jail. 

If the prisoners get to discuss a plan, how can at least 99 of them be saved?

Talking about math
I blended ages here because I've noticed a lot of great cross-age pollination. The little ones are often very engaged and surprisingly insightful on topics that seem much more mature, while elementary conversations often touch some deep concepts.

AO Fradkin and her daughter discussed "how long is 3 minutes?" Hard to get deeper than the nature of time and how our perception depends on context (or does it?)

A view of Fermat's role in Fermat's Last Theorem from Mathematical Enchantments that nicely touching the fact of changing tastes in mathematical research. What mathematical preferences do you and your kids have?

Some tidbits for the Math is Everywhere meme:

1. Using big data to analyze story arcs.
2. Life through a Mathematician's Eyes has two posts on What a Mathematician should see in Amsterdam and Visiting Amsterdam like a Mathematician

I can never resist an icosahedron picture and there's a nice one in the first Amsterdam post:

Enjoy a joke anecdote about mathematical precision from Curiousa Mathematica. I like to tell elementary kids these kinds of jokes and see how they respond. It is a delight when they get it, but I also like to laugh at their deadpan expressions when they don't understand. 

Finally, a great way to start talking about math is to do math where you'll be seen doing it. This might require an emergency math kit (from Solve My Maths.)

Crafts and Constructions

Generally, these are accessible to young children, but also have some deep mathematics that offer something for any of us to explore.

Something our kids recently made: Flextangles craft activity (3d flexagons)

We love optical illusions and Sugihara's Illusion is a fantastic one. This post gives an explanation, then the next provides a printable. Agamographs are an accessible craft with a similar idea (instructions on Babble Dabble Do).

Benjamin Leis writes about an eye-catching decomposition and recomposition puzzle to start some exploration: hinged polygons.

Kira Zelbo offers up a free booklet that introduces isometric dot paper for drawing and visualizing three dimensional forms: Spatial Learning. Education Realist recently documented his experience using isometric grid paper with his class in Great Moments in Teaching.

Elementary Explorations and Middle School Mastery

Denise Gaskins (have you heard of her?) stimulates a discussion of favorite puzzle books in her review of Lilac Mohr's Math and Magic In Wonderland.

AO Fradkin again, helping some early elementary kids in discovering the triangle inequality.

John Golden serves another ace with his Wimbledon Game.

An old post from Sue Downing caught my attention: these place value cups are a great idea and make me think (fondly) of combination bike locks.

Addition Boomerang is a Mathpickle activity we played recently with our 1st-4th graders. When can you make 100? With that target, of course we had to include it in this MTaP!

Math Minds has ants-on-the-brain in 100 hungry ants. Trust me, it is better than ants in the pants.

Second graders explore proofs based on a geometric investigation: Squarable Numbers

Another post from Sue Downing that came in handy recently: "Is that all there are?" Multiplication facts. Now that you know the multiplication facts aren't scary, get some practice by playing with 

anti-primes (Numberphile). We got a lot of value out of seeking the anti-prime (er, highly composite number) that follows 24.

Number bracelet investigation. Our school and family have looked at this several times. Another great extension is to work in bases other than 10. Another way to understand it is working modulo 10 (or whatever integer you choose to set to 0).

Baseball, with its large collection of player and team stats, offers a natural entry point for mathematical conversations. Mashup Math walks through some examples in Mathematics of Major League Baseball. For a related take on this idea, see Mathpickle's Introducing Stats to Younger Children.

High School Adventures

Mrs E shares a lesson plan focused on getting kids to analyze and critique advertisements at Mrs E Teaches Math. An important life skill on its own and Mrs E sees it as a helpful bridge into writing proofs.

What happens when you are living inside a word problem? Math in real life.

I don't think this is a recent addition from Dan Meyer, but his Finals Week three-act seemed more fun to do during summer, away from the normal stresses of the school year.

There are some elementary and sophisticated ways to think about divisibility rules. In this post Curious Cheetah hits a bunch at one time.

"Stoichiometry is the math of chemistry," according to Amy Roediger. Here is her explanation and a discussion of teaching methods, parts 1, 2, and 3

Manan Shah contributes his approach to dealing with Annoying Function Notation. Can you make sense of these contrasting pairs:

Also, make sure to check out Manan's curation of the latest Carnival of Mathematics.

Po Shen Lo and Mike Lawler: a great combination... and Mike gives us two posts with PSL (first and second)!

Step into some difficult probability, statistics, and forecasting with Big Thompson Flood.

Singapore Maths Tuition walks us through a vector calculation to find the foot of a perpendicular from a point and to a line.

Ben Vitalis has a constant stream of interesting challenges, most accessible to algebra students: Odds equal Evens.

Puzzling Recreations

Math Arguments makes a surprising re-appearance to post a nice probability dice from Ben Orlin.

Lisa Winer (star of the 99th MTaP) talks about her plexer puzzles. Aside from being fun, these are a good place to practice Notice & Wonder.

Our family has recently become fans of the Futility Closet podcast, especially their lateral thinking puzzles.

Reminder: one great thing to do with any puzzle is CREATE YOUR OWN!!!

Teaching Tips

Which one doesn't belong (WODB) is a great format for a rich discussion. WODB.ca has a really nice collection of mathematical WODBs. I was reminded of this resource by this delightful WODB of WODB from John Golden.

Joe Schwartz continues a MTBoS theme of getting fixes for worksheets.

Amy Roediger put together a good collection of resources as she prepared to lead a

coding camp. A Recursive Process has a further discussion of the links between coding and math, including some more resources.

The folks at the Mind Research Institute have put together a summer reading list of 9 Enlightening Summer Reads for Math Teachers. The list mixes sci-fi and books about teaching. If you don't know the Mind Research Institute, they are behind/linked with ST Math, an on-line elementary grades math system that I really like. I had the opportunity to trial their system years ago and recently went through their free demo.  Now, if only someone there would be willing to get in touch and tell me how I could subscribe for my kids to use .....!

TMC16
Twitter Math Camp 2016 was just held and there are a lot of math teachers blogging about their experiences. If the posts I've gathered above aren't enough for you, I suggest hitting J Fairbanks's blog 8 is My Lucky Number for 10 (wow!) posts about the convention and further links.