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 recommendations for a 3 year old

I was recently asked for suggestions by a parent of a 3 year old.

There are a lot of different resources I could suggest, but they really depend on the child and the parents. The main question for customization is about the parents: what are their starting assumptions about math/math learning and how much do they want to engage on selecting/planning activities?

For example, if a parent doesn't really get the growth mindset, I would advise a heavy dose of Jo Boaler. If the parent wants open explorations and can build their own specific tasks, maybe the Vi Hart videos are good inspiration.

That aside, there are a few resources/products good enough that I’m willing to give blanket recommendations:

1. Lots of tools for measuring. Playing with measuring has so many benefits, I can’t list them all, but some of the highlights are (a) seeing math and numbers all around us, (b) tactile engagement, (c) inherent process of comparison, and (d) natural connection with language as the kids and parents talk about what they are measuring/why. The links I've provided just show examples, I am not necessarily recommending them over other versions.
1. Set of plastic measuring cups (imperial units and fractions)
2. Tape measure (we just used standard adult tape measures, but as a recommendation, you need to be careful about tape measures that have fast return springs for cutting or catching small fingers)
3. Balance scale and set of standard weights (this math balance is a good option and one we bought)
4. Timer (we liked this one)
5. For older kids, a step counter, GPS wrist-watch showing speed, thermometer, pH meter, electricity meter are all interesting additional measuring devices.
2. Talking Math with your Kids:
1. E-book
2. Blog. I recommend reading all the posts, I think they are a superset of the material in the e-book, so this is a better resource unless you want the “curated” highlights. This link goes directly to posts tagged 3 years old.
3. Tiling toys and shapes book in the TMWYK store. I particularly like Which on doesn’t belong? A better shapes book.
3. Denise Gaskin’s Playful Math books: these talk about general habits and methods in an intro section, then specific activities (mostly games) in the rest of the book.
4. I got a lot out of these storybooks (free to print) with my kids: CSMP Math Storybooks.
5. Standard gambling tools: playing cards and dice (I like pound-o-dice for the assorted colors, sizes, shapes)

There are some computer games/systems, a lot of board games, and mechanical puzzles, but the stuff above is where I think parents should start for young children.

What do you think of my recommendations? Any additions you think are worth adding to make a top 10?

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

Puzzling puzzlers

Preparing for classes today, we went through our lists and links to gather a list of puzzles for the kids. Hopefully, this will help short circuit the work we have to do next time.

• Eric Friedman Puzzle Collection (highly recommended by Mathpickle)
• Cryptarithm: shape substitution version, basic ones, 
• Funny dice 
• SolveMe Mobiles
• Make your own coin system
• Multiplication: Miller's Puzzle from Dudeney
• Find the Factors
• Multiples of 7 and asymmetric information: What's my number from Tanya Khovanova
• Nonagrams and battleship puzzles
• Logic Grid puzzles: Two dimensional starters, Logic-Puzzles, 
• Distortions logic puzzle (nice twist on truth tellers or liars)
• Questions from an adult, general public, math contest: Math Mama Writes
• Math is Fun collection
• Of course, there are a lot of great puzzles on Mathpickle
• Sudoku
• Number arrangements: magic squares, "magic" triangles and stars
• Zukei geometry puzzles (find the "hidden" shape)
• Area maze, K-maze, B-cross, Shikaku, Dokoeq (all from Naoki Inaba)
• Verruca Values

Our math curriculum

Sasha Fradkin (who writes one of our favorite blogs) asked a question about the curriculum we use. My reply was getting long, so I decided to make it into a separate post.

Do we use an existing curriculum or are we making our own?

We are doing a mix. My wife prefers to have a linear curriculum as a guide and fall-back, in case there wasn't time to plan anything more customized. She currently uses:

• RightStart/Abacus Math: ok, but not exceptional curriculum, highlight is the extensive use of physical manipulatives.  
• Beast Academy workbooks: wrote more extensively about this in a review before. I think these create good jumping-off points for really fun conversations. 
• DreamBox: for consolidation of standard skills, our enthusiasm for this is waning, rather than waxing right now (noted in same review as BA). 

We also use the CCSS math standards as a reference. I periodically check against the standards to see whether we are missing anything. If so, I will go to the Georgia Standards of Excellence, read through their activities for the related unit, and pick a couple that seem fun.

If I were forced to use only a single source, GSE would be my recommendation.

About Georgia Standards of Excellence
As far as K-5 math, this is a really awesome resource with a ton of great activities. For some reason, we find the webpage organization a bit confusing, so here is our quick recipe.  

To get to the great activities, I click the expansion menu in the right-hand box for the grade level of interest, then look at the curriculum map for that grade. I find the topic of interest, then click the link for the unit that covers that topic. The unit doc includes a lot of teacher background, which I mostly skip and focus on the activity descriptions.

Games and explorations
My personal preference is much less structured. I really like games and explorations and spend a lot of time exploring math activities on the MTBoS. Most of what I do with the kids is inspired by something I saw while doing my own play.

That said, there are some sources that are so good, we are essentially going through all of their activities:

1. Mathpickle. Cannot say enough positive about this.
2. Peter Liljedah's numeracy activities. Now that I notice them, I bet his good problems, card tricks, and resources pages will all have gems as well. 
3. NRICH and Wild Maths.

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.

Logic Puzzle collection and Dropping Phones

Recently, Mathbabe put out a request for riddles. There are some good links in the comments:

• The Riddler on FiveThirtyEight
• Logic Puzzles on Math is Fun
• Alex Bellos at The Guardian
• Riddles.com (naturally)
• Puzzling.stackexchange.com (is there a stack exchange for everything now?!)
• Authors cited: Dudeney, Gardner, Polya, Smullyan, Vaderlind. 

We've played with puzzles from almost all of these sources in the past, but this was a good opportunity to put together a nice list.

The one that stood out to me was on FiveThirtyEight. That's a site I read frequently, especially during this US election season ... and I was totally unaware of The Riddler feature. We kicked off with the oldest puzzle from their archive: Best way to drop a smartphone.

Breaking stuff

Right away, J1 and J2 loved the theme and were into questions about whether they could somehow keep the phones, if they weren't broken, or utterly destroy them, if they were broken. We talked about setting an upper bound with a very simple strategy of starting on the 1st floor and working up each floor. That's not a great answer, but it got them into modifications and improving strategies.

Through the conversation, it was interesting to see them start with the idea that the drops for one of the phones would be de minimis and could be ignored, but then start to pay attention to that aspect. Also, they had to grapple with the idea of balancing the number of drops that would be required in different cases.

While we didn't get to the optimal strategy, but the kids managed to get a version that, at worst, would take 19 drops for the 100 story building. Their intuition was based around taking the square root of 100. They could see that this probably wasn't the best answer, since there were still cases that, at worst, would take 10 drops and others that, at worst, would take 19.

Can you do better?

Smaller before bigger

The puzzle page poses the same challenge for a 1000 story building. However, we found something interesting when working on the version for a 10 story building: there is  strategy where all worst cases take the same number of drops, but it is not the optimal strategy!

It is a little hard to write about this without disclosing the strategy, but here's a hint:

• 4 + 3 + 2 + 1 = 10 and 5 + 4 = 9
• We are always allowed to assume that the phones will break if dropped from the top floor of the building

This led to another extension: what size buildings will have the same issues as for a 10 story building?

Probability comes in

Another extension is to think about the expected number of drops required and strategies that minimize this. Crucially, this extension introduces the idea about our prior beliefs about the sturdiness of the phones: where do we think the phones are likely to break, what is our confidence?

We didn't pursue this extension very far, but it did lead to some interesting conversations about terminal velocities. For those who want to follow that thread, this (other) stack exchange thread might suit you: How to figure out height to achieve terminal velocity.

Lure of the Labyrinth and Mathpickle

I'm puzzled why it has taken so long for me to hear about these two resources. Seriously, folks, have I not been immersed in elementary-age math activities for the last several years, trawling through other people's favorites lists and doing my own searches? How could both of these have been (until now) undiscovered gems?

Mathpickle

Going beyond the silly name, this is a site with fantastic games, puzzles, and comments about learning and teaching math. The site is well structured and almost every link is a gem, so it will be easy to find worthwhile material. If you need any extra help, just try Games organized by Subject.

Note: at mathpickle, don't get caught worrying too much about "grade level" labels, especially don't skip things you think might be "too young" for your student. Once they have the (minimal) knowledge to get into an activity, there is usually no ceiling blocking further or deeper exploration or understanding.

Little Bit of Aggression: Part of J1's curriculum while at home sick last week

Some quotes/philosophy
Mathpickle is the work of Gordon Hamilton who aims to "get curricular unsolved problems into classrooms worldwide – one for each grade K-12."  This might remind you of the interesting post by Lior Practher from last september: Unsolved Problems with the Common Core.

Gordon's comments about board games strongly resonate with me:

A small fraction of games work well in the classroom. They must be resilient to damage and loss. They must be easy to teach and quick to play and put away. In the classroom budget, they must compete against cheap electronic games. 

A large fraction of games work well at home – that’s why parents must take the lead on establishing a culture of board gaming.  

Somewhere on the site, Gordon mentions that he is now developing some educational apps; I'm keen to see what he creates!

Lure of the Labyrinth

Gordon also has great taste in other resources, some of which are listed in his collection of Inspired People.  Within this sub-site, he has very high praise for Scot Osterweil's Lure of the Labyrinth.
The two older Js and I have been playing and are hooked.

Inside is a game world with a set of challenges related to a running story and a collection of (at least) nine different puzzle/game types that develop a range of mathematical ideas.

Some of the cool things about this game:

1. Fun challenges that get harder in interesting ways
2. Cartoon monsters and an interesting story context
3. The minimum of exposition. Challenges are not really explained, so you get to experiment to figure out what is going on. The designers found a great middle ground between "I know exactly what I need to do" and "I have no clue." Something on the order of "I know I can try these things, but am not totally sure what will happen."
4. Navigating around the game world is interesting, in and of itself. I was particularly delighted by the lay-out of the "triangle" wing (the last wing of the factory you unlock).
5. The game basics are high quality: good graphics, background music, interface
6. Free!

You have no idea what this is, but will be able to figure it out!

A classic puzzle with some gross-out twists

A minor, minor, nit:

In case the game designers ever read this.....

First, thanks so much! We love the game and really appreciate all the time, effort, and attention that must have gone into creating it.

Second, go back and read those previous two sentences again!

Third, well, since you persisted in reading this far . . . the number of actions allowed in the Managers' Cafeteria is a bit mean. As illustrated in the screen cap above, you have 12 actions to correctly specify and place 12 orders. That means no mistakes are tolerated, not if you mis-enter a serving size or accidentally forget to change the type of food. We understand that mistakes require a redo in other challenges (mineshaft, gardens, advanced testing labs) but:

1. none of those require the same shear number of clicks as the Managers' Cafeteria
2. It feels like a large number of clicks in MC are committed (they lock you into a path without possible change) while far fewer in the other challenges are so decisive
3. Many of the MC clicks are just mechanical, they don't bear on the level of understanding of the challenge.

Our suggestion is simple: add 3 more "lights" (allowed attempts) to MC.

15 piece tangram (more math from trash)

who: J0 and J2
what did I use: a postcard advertising some property for sale

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Note: when I originally wrote this, J2 and J1 were busy doing something else. By the time I had half composed this post, J2 had noticed what I was doing and started playing with the tangrams again. More evidence that the easiest way to get kids into an activity is just to have it out and available or doing it yourself.

15 piece tangram

In Bangkok, we get a lot of junk mail touting property viewings. One postcard was just the right size and durability to use for a 15 piece tangram set. I would encourage you to make and play with your own 15 piece set (or a 7 piece, if you don't already have one).

In fact, you should do what I didn't do this time: work with your kids to cut apart the original square or have them make their own sets by themselves. Even just following the diagram is a geometry experience as well as an exercise in scaling.  Our card was 15cm on a side, the diagram I used gave dimensions based on a 3 inch side.

I was a little concerned about the dissection of the central square because of the circular curves. I traced out the dissection with a ballpoint pen that made a slight indentation in the card, then traced over that outline with the tip of a very sharp knife. It isn't perfect, but I'm very pleased with the result:

The image adds some hints when we want to reform the basic square. Good or bad?

I was, perhaps, remiss in not providing full credit to the tangram book we are using. here it is, in all its Dover Publications glory, Tangrams 330 Puzzles by Ronald C Read.  Looking over my shoulder, J2 said, "the book actually had 334 puzzles." So, boom, 4 free puzzles!

Platonic Solids Defying Gravity I

Unrelated, J2 made a temporary installation of mathematical art today in the kitchen: Gravity Defying Platonic Solids I. Except for the obvious one, he also took the pictures included here:

The Miller's Puzzle (a multiplication investigation)

who: J1 and J2
what did we use: good old pencil and paper
when: intermittently through the weekend as they recovered from being ill

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Denise Gaskins (of Let's Play Math, you know, one of my top recommendations for parents) included a game from Dudeney's puzzle book: The Canterbury Puzzles. The game is a good one which we will use in an upcoming class, but I won't steal her thunder on that one, so go subscribe to her newsletter and find out for yourself.

I'm always excited to see a new source and had never heard of Dudeney or the book, so I went to look. My suggestion is to skip the introduction and go right to the puzzles. One of the first for our play is the Miller's puzzle. Nine numbered sacks are arranged as below:

You notice that 7x28 = 196, but 34 x 5 is not 196. Can you swap sacks, keeping the 1-2-3-2-1 arrangement, so that the products on both sides are the same number in the middle?

Bonus: those sacks are heavy, so how can you get a solution swapping as few sacks as possible?

Our initial attempts

First, the kids were intrigued and suspicious: is 7x28 really 196 and is 34x5 really not 196?

196 is, of course, an old friend: the square of 14. J2 was really excited to see it appear in this puzzle and the factorization 7 x 28 was a nice complement to his identification of 14 x 14.

Second, their inherent sense of efficiency kicked in and they wanted to see if they could fix the problem by simply changing the three bags on the right side. In itself, that was a good discussion about:

• What are all of the possibilities?
• How do we know, without multiplying, that 43x5 and 45x3 won't be 196?
• What are 43x5, 35 x 4, 53x4, 45x3, 54x3?
• What parity do we get when multiplying an even and an odd? When checking our answers, we can at least look to see if we got the right parity.

Their third step was to try a couple of other simple products with the numbers 1 to 9. Overall, this part of the play had some good conversation and a lot of multiplication practice.

A way forward

I didn't give solutions, but I did ask some questions to help guide the investigation further:

1. Where could the 1 go?
2. Where could the 5 go?
3. Where could the odd digits go?

Don't just take those questions in turn, think about them, then see if your answers to one question give new insight into the others.

Another Idea

I like to look for calculation short-cuts and encourage you to do the same. That gives rise to another mini-hint: what are the relationships between the following number pairs and how can you use that to simplify your search for a solution:

1&2, 2&4, 3&6, 4&8
1&3, 2&6, 3&9

Also, isn't it sad that this cute sequence isn't a solution: 4 x 54 = 216 = 8 x 27? So close . . .

Finally, once you have found solutions, you have opened a new can of worms: how many sack moves does it take to go from the Miller's original arrangement to your solution? How can you tell that you've found the solution with the least moves?

An unrelated picture

A slightly more related picture