Welcome to the 147th edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.

Bookmark this post, so you can take your time browsing. There’s so much playful math to enjoy!

By tradition, we start the carnival with a puzzle in honor of our 147th edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

## Puzzle #147: Iva Sallay’s Great Hexahex

Iva Sallay (@findthefactors) has hosted many delightful editions of the Playful Math Carnival, including the recent #146, and she wrote a *fantastic* post about how to bring the carnival to your own blog:

In that post, she tells the story of a mathematical contortionist, the Great Hexahex:

Hexahex knows how to bend himself into 82 different “free” positions. He wants to stretch himself a little bit and add 65 more “one-sided” positions for a total of 147 contortions in his repertoire.

Reflections (mirror images) count as distinct “one-sided” positions if they are not mere rotations of any other position.

Below is a graphic showing those first 82 positions (and a link to download a printable version), followed by their reflections across a horizontal line. Put an X above the 17 positions in the bottom three rows that *do not* qualify as distinct. Then count up the remaining shapes.

You will see that Hexahex can indeed contort himself 147 ways!

## Continuing Our Dodeca-versary Party

As I mentioned back in Carnival #144, this year is the 12th anniversary of the Playful Math Education Blog Carnival (originally titled Math Teachers at Play). We’ve shared so many wonderful, playful, creative math ideas through the years that I hate to lose track of them all.

Since Iva did such a wonderful job of collecting and linking to current mathy treasures in her “You Can Host” article, I decided to make this edition another blast from the past.

The posts in this 147th edition are drawn from our second year of publication. We had 12 monthly carnivals from February 2010 to January 2011, with 11 different hosts:

- Dan Mackinnon (@mathrecreation) hosted our first-birthday carnival with #23.
- I hosted #24.
- Riley Lark (@rileylark) hosted #25 and the sing-along carnival #32.
- John Golden (@mathhombre) hosted #26.
- The psuedonymous Math Mom hosted #27.
- Alexander Bogomolny (@CutTheKnotMath) hosted #28.
- Jason Dyer (@jdyer) hosted #29.
- Jonathan Halabi (@Jd2718x) hosted #30.
- Homeschooler Misty hosted #31.
- Lisa Henry (@lmhenry9) closed out the calendar year with #33.
- Finally, Guillermo Bautista (@jr_bautista) wrapped up the carnival’s second year with #34.

## Contents

And now, on to the main attraction: the blog posts. If you’d like to skip directly to your area of interest, click one of these links.

- Playing with Preschool and Early Elementary Math
- Exploring Elementary Arithmetic
- Adventuring into Algebra and Geometry
- Scaling the Slopes of High School Math
- Enjoying Recreational Puzzles and Problems
- Teaching with Wisdom and Grace
- Giving Credit Where It’s Due

Would you like to the Playful Math Carnival on your blog? We’d love to have you join us! Click for details:

## Playing with Preschool and Early Elementary Math

- In #30, Clemencia Rosado used pennies and paperclips to practice counting in M for Math Games.

- In #24, homeschooler Andrea introduced a fun card game for early addition facts.

- In #25, homeschooler Kendra created a new Addition Math Game.

- In #23, Kendra played Place Value War with her sons.

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## Exploring Elementary Arithmetic

- In #30, Michelle Martin helped her 4th- and 5th-grade students visualize large numbers in Every Vote Counts.

- In #31, Rebecca Zook (@zooktutoring) shared a simple but effective mnemonic illustration: Gallon man to the rescue!

- In #26, Cynthia Hockman-Chupp (@love2learn2day) gave students a reason to discuss angles, polygons, and more in Table TOP Math: Integrating Math, Art, and Vocabulary. (That link goes to a later, reposted version of the article. And here are instructions for making one type of origami top.)

- In #25, I posted my Cookie Factory Guide to Long Division.

- In #27, I wrote Hobbit Math: Elementary Problem Solving 5th Grade, which later became part of my book
*Word Problems from Literature*.

- In #32, John Cook (@JohnDCook) explained the trick to determining Divisibility by 7.

- In #34, John Golden (@mathhombre) regaled us with a medieval tale of integer adventures: The Quest for the Holy Snail.

#### Learning Math Through Games

- In #24, my daughter Teresa shared her favorite online Math Games for Learning Times.

- In #32, John Golden (@mathhombre) introduced his Unit Rummy game.

- In #23, Brent Yorgey explained the game of Divisor Nim.

- In #27, John field-tested one of my favorite games: Decimal Point Pickle.

#### Fun with Fractions and Ratios

- In #28, Alexander Bogomolny (@CutTheKnotMath) observed that there is a way to naturally introduce addition of fractions without a formal definition.

- In #32, I explored The Secret of Egyptian Fractions and James Tanton’s explanation. On my other blog, I shared Don Cohen’s Infinite Series: How to Add Up a Gazillion Fractions.

- In #30, Pat Ballew (@ballew_pat) taught Division of Fractions by the Alien Method.

- In #24, Jason Dyer (@jdyer) challenged us all with his Conceptual fraction test.

- In #26, Ian Byrd (@IanAByrd) found an interesting way to review ratios in An Apple Stock Math Project.

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## Adventuring into Algebra and Geometry

#### Algebra Activities

- In #24, Nick Yates (@nyates314) created two (free) printable algebra puzzles in Puzzling Out Some Quadratics.

- In #25, Dan Wekselgreene (@dwekselgreene) shared an Ohio Jones systems of inequalities lesson, though the Internet Archive couldn’t recover the actual files. But good news! Dan’s turned the story into a Desmos activity, and he says you can still download the original Word-doc files here: Lesson 14 and Lesson 15.

- In #29, Guillermo Bautista (@jr_bautista) drew on the cheesboard puzzle to help students Make Sense of Exponential Growth.

#### Getting into Geometry

- In #24, Erlina Ronda (@math4teaching) offered a simple investigation activity about polygons to help students Make connections through math investigation.

- In #25, Dan Mackinnon (@mathrecreation) played “What do you notice?” with Three Circles and came up with some fun connections.

- In #28, John Golden (@mathhombre) celebrated Full Circle Day with an investigation of similarity.

- In #24, Steven Strogatz (@stevenstrogatz) explained the Pythagorean Theorem in Square Dancing, part of his delicious Elements of Math series.

- In #32, Kate Nowak (@k8nowak) channelled her inner Dan Brown to motivate Special Right Triangles.

- In #34, Mimi Yang (@untilnextstop) tempted us with food and geometry in Salvadoran Customs and Math.

- In #26, my daughter Teresa described her exploration of taxicab geometry.

#### Algebra and Geometry Games

- In #24, John Golden (@mathhombre) shared a variety of fun geometry games for K-8 teachers.

- In #31, John was Playing Math with the Prism Power Game.

- In #25, John described the graphing strategy game Coordinate Connect.

- In #30, Sue VanHattum (@suevanhattum) introduced graphing with a function game in A Good Day to Do Without the Textbook.

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## Scaling the Slopes of High School Math

#### Advanced Algebra and Geometry

- In #24, John Cook (@JohnDCook) tried writing a Twitterable proof: Euclid’s infinitely many primes. Be sure to enjoy the comments!

- In #30, Rebecca Zook (@zooktutoring) created a graphic-mnemonic device: An Easy Way to Remember How Logarithmic Notation Works. See also Kate Nowak’s (@k8nowak) classic post on Introducing Logs.

- In #25, Sue VanHattum (@suevanhattum) shared her Murder Mystery: A Project for Logarithms.

- In #24, Jonathan Halabi (@Jd2718x) was Having fun teaching Σ of geometric series. (Follow-up post here: Proof by taxation, and you might have fun with this puzzle.)

#### Cruising on to Calculus

- In #27, Pat Ballew (@ballew_pat) stumped his pre-calc students by asking the simple question, “Given Two Points, write the equation of a line containing the two points…” but in
*three*dimensions.

- In #27, Sue VanHattum (@suevanhattum) tried Sneaking Up On the Fundamental Theorem of Calculus. It’s a lovely story, though the pictures have gone internet-AWOL.

- In #24, Riley Lark (@rileylark) discussed One way to convince kids of finite infinites.

- In #23, several bloggers took time to celebrate e Day. My favorite was Pat Ballew’s (@ballew_pat) e-day and Andy Jackson.

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## Enjoying Recreational Puzzles and Problems

#### Sharing Math with Friends

- In #29, Sol Lederman described how to impress your friends with magical math: 12 Cent Math Trick — even kids can do it!

- In #29, Sue VanHattum (@suevanhattum) shared a short video about her Richmond Math Salon. What a delightful way to play math with other families!

- In #30, I was Planning a New Math Club of games, activities, and calculus for young people.

- In #31, I wrote a fictional story about how we organized A Homeschool Math Carnival.

#### Book Reviews

- In #31, Becky Johnston discussed
*The Number Devil*and the challenge of raising a gifted child in On Prima Donnas and Rutabagas, and Fear.

- In #24, Cap Khoury (@tweedcap) reviewed Mathematical Fiction: Riot at the Calc Exam and Reality Conditions. “I think there’s a lot more intrinsic humor in mathematics than many people expect…”

#### Counting, Probability, and Discrete Math

- In #29, I shared some of my favorite links for an Introduction to Probability.

- In #31, Guillermo Bautista (@jr_bautista) explained how to Use Mathematics to Win the Lottery.

- In #34, Brian Hayes (@bit_player) revealed the difficulty of calculating real-world probabilities in The prime twins conjecture.

- In #28, Sam Alexander (@XamuelA) explored some puzzling application of König’s Lemma.

#### And Plenty of Puzzles

- In #27, Math Mom’s students tackled Dr. Mike’s Big Numbers Contest.

- I love challenging my math club students (and myself!) to create their own story problems. In #34, I posted Babymath: Story Problem Challenge III.

- In #33, Patrick Vennebush (@pvennebush) posted his playful Math Jokes 4 Mathy Folks Final Exam.

- How many different types of triangles can you identify? In #33, John Golden (@mathhombre) shared a Triangle Puzzle.

- In #24, Alison Kiddle explored A cutting-cubes problem, and some responses.

- In #33, I posed a mathematical Football Puzzle.

- In #26, Jonathan Halabi (@Jd2718x) challenged his algebra 2 students with a tricky Puzzle: continued root. (And don’t miss Part 2!)

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## Teaching with Wisdom and Grace

#### Understanding Mathematics

- In #24, Fëanor translated a delightful story in which a hapless student struggles with a challenging word problem and “Nikolai Nosov demonstrates why he is possibly the finest expositor of mathematical pedagogy in children’s literature anywhere.” Check it out: Vitya’s Maths.

- In #24, Michael Paul Goldenberg (@MichaelPaulGol1) reviewed a fascinating book in which Terezinha Nunes and Peter Bryant Dole Out The Multiplicative Harshness.

- In #33, Sue VanHattum (@suevanhattum) pondered Eigenstuff and the problem of fear in learning math.

#### Understanding Education

- In #29, Whit Ford (@MathMaine) offered advice for students to Learn the Game of Learning.

- In #30, John Golden (@mathhombre) posed the question, Is Math Creative?

- In #30, Beatrice Ekoko (@BeatriceEkoko) wrote about Mindful Mathematics.

- In #30, Jonathan Halabi (@Jd2718x) shared his Outlook on teaching math.

- In #34, David Wees (@davidwees) exposed A Fundamental Flaw in Mathematics Education.

#### Homeschooling Math

- In #29, John Golden (@mathhombre) offered guidelines for homeschoolers on choosing a math program. This post contains one of my favorite quotes on teaching: “The toughest thing for a homeschooler is the same as for a school teacher — shifting from a weak tea vision of math being grinding calculations to a rich frothy mug of math as an active way of thinking.”

- In #24, Kendra posted tips for homeschoolers who dream of teaching elementary math without a textbook (perhaps using some of the activities and games above): Organizing a Math Lesson.

- In #34, Cynthia Hockman-Chupp (@love2learn2day) linked to a year’s worth of playful math ideas for homeschoolers.

#### Playing in the Classroom

- In #24, Dan Wekselgreene (@dwekselgreene) noticed his students’ struggle to remember what they’ve learned, and he gave review a creative twist in Language and Retention of Math Concepts.

- In #26, Dan Meyer (@ddmeyer) suggested that Math Class Needs a Makeover and went viral in the teacher community.

#### Teacher Tricks and Tips

- In #24, Joanne Jacobs (@JoanneLeeJacobs) encouraged teachers to ask for informal proofs (even in elementary school) in Why is this answer right?, a post prompted by David Ginsburg’s short article Don’t Tell Students to Show Their Work—Make Them!

- In #27, David Ginsburg (@CoachGinsburg) tackled the lack of “sanity checking” in Estimation Before Computation. (And reminded host Math Mom of her own Calculator Rant post.)

- In #31, Mimi Yang (@untilnextstop) helped her students understand the rules of logical deduction in No more p’s and q’s!

- In #26, Sue VanHattum (@suevanhattum) explained to her students How To Read a Math Book.

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## Giving Credit Where It’s Due

Carnival graphics are from DepositPhotos.com. The Hexahex puzzle was created by Iva Sallay.

And that rounds up this edition of the Playful Math Education Blog Carnival. I hope you enjoyed the ride!

The next installment of our carnival will cover July and August at Kelly Darke’s Math Book Magic blog. Visit our blog carnival information page for more details.

**We need 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 Playful Math Education Blog Carnival, please speak up!