Welcome to the 142nd 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.

Seriously, plan on coming back to this post several times. There’s so much playful math to enjoy!

By tradition, we start the carnival with a puzzle/activity in honor of our 142nd edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

## Activity: Planar Graphs

According to the OEIS Wiki, 142 is “the number of planar graphs with six vertices.”

What does that mean?

And how can our students play with it?

A *planar graph* is a set of vertices connected (or not) by edges. Each edge links two vertices, and the edges cannot intersect each other. The graph doesn’t have to be fully connected, and individual vertices may float free.

Children can model planar graphs with three-dimensional constructions using small balls of playdough (vertices) connected by toothpicks (edges).

Let’s start with something smaller than 142. If you roll four balls of playdough, how many different ways can you connect them? The picture shows five possibilities. How many more can you find?

Sort your planar graphs into categories. How are they similar? How are they different?

A wise mathematician once said, “Learning is having new questions to ask.” How many different questions can you think of to ask about planar graphs?

Play the Planarity game to untangle connected planar graphs (or check your phone store for a similar app).

Or play Sprouts, a pencil-and-paper planar-graph game.

For deeper study, elementary and middle-school students will enjoy Joel David Hamkins’s Graph coloring & chromatic numbers and Graph theory for kids. Older students can dive into Oscar Levin’s *Discrete Mathematics: An Open Introduction*. Here’s the section on planar graphs.

*[“Geöffneter Berg” by Paul Klee, 1914.]*

## Contents

And now, on to the Carnival’s main attraction: the blog posts.

If you’d like to skip directly to your area of interest, click one of these links.

- Talking Math with Kids
- Exploring Elementary Arithmetic
- Adventuring into Algebra and Geometry
- Scaling the Slopes of High School Math
- Enjoying Recreational Puzzles and Math Art
- Teaching with Wisdom and Grace
- Credits, and a Cry for Help

Art images below are from the 2020 Bridges Conference Gallery.

*[“Four Dodecahedra” by Ulrich Mikloweit.]*

## Talking Math with Kids

“Math is not just adding, subtracting, multiplying and dividing. It is the mystery of numbers within numbers and the discovery of how numbers keep changing the world.”—Savannah Sanders, Never Give Up

- Christopher Danielson (@Trianglemancsd) celebrates the math of Eggs in cartons and relates a story to demonstrate the value of follow-up questions.

- Kelly Darke (@KellyDarkeMath) creates some beautiful Math Book Magic Dreams.

- The Early Math Collaborative (@eriksonmath) gathers ideas for preschool geometry in Chores for Kids Can Help Them Find Math Around the House.

- Ann Carlyle offers 15 playful math ideas for preschool and kindergarten in The Value of Virtual Learning: Focusing on Meaningful Play.

- The anonymous Urban Mama relates her son’s experience with different aspects of counting in Math Books We Love.

- Herbert Ginsburg shares 3 Interactive Storybook Apps for Playful Early Math Learning.

- Amber Domoradzki and son invent a new Math Game: Get to 6.

- For homeschoolers, Hana (@PepperandPine) begins a series of video lessons on resources for studying The Quality of Numbers 1-12, Waldorf-style. If you do these lessons, be sure to add Christopher Danielson’s One is one … or is it? video.

- Savannah Sanders (age 9yo) describes her experience with math in Never Give Up (@BlackWomenMath).

*[“Los tres Amigos” by Zdenka Guadarrama.]*

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## Exploring Elementary Arithmetic

“Almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication … the way kids think it should.)”—Pat Ballew, Division of Fractions by the Alien Method

- Michael Minas (@mminas8) and son Nash practice addition and strategic thinking with their new game MathemaTic Tac Toe. Check out their entire game collection at LoveMaths.me/Games.

- Tracy Proffitt (@tracyjoproffitt) and Kent Haines (@GamesYoungMinds) share the printable card game Round ‘Em Up for rounding numbers to the nearest hundred.

- Dan Draper (@MrDraperMaths) takes rounding to a deeper and much richer level in Rounding Defamiliarisation.

- MathCurious (@curious_math) creates a Google slides version of their Block It! multiplication game.

- Iva Sallay’s (@findthefactors) blog is full of multiplication puzzles that are fun for students and adults. Can you solve her Mystery Message? Here are the first three clues.

- Miss Konstantine (@giftedHKO) tweaks the Countdown game to help her students practice Laws of Arithmetic.

- Brian Marks (@Yummymath) and Leslie Lewis investigate some creepy measurements in Vampire Bats.

- The Math Solutions (@Math_Solutions) team celebrates National Chocolate Day with a variety of activities featuriong Chocolate and Math.

- Pat Ballew (@ballew_pat) explains Division of Fractions by the Alien Method.

- Sam Blatherwick (@BlatherwickSam) expands on Don Steward’s proportion puzzles in Ways with Boxes.

*[“Concise Lesson in Uniform Partitions” by Conan Chadbourne.]*

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## Adventuring into Algebra and Geometry

“Working with a student who has not yet been taught simultaneous equations or ‘rules’ of solving equations with two unknowns is fun — for the student as well as for the teacher. Both of us are experiencing Aha moments in almost every session!”—Rupesh Gesota, Math Coach blog

- Paula Beardell Krieg (@PaulaKrieg) shows how students can use the concept of a radius to create beautiful math art on a Circular Grid made with Straight Lines.

- For my entry in the carnival, I’ve shared a simple but fun exploration of Parallel and Perpendicular Art.

- Erick Lee (@TheErickLee) asks students to apply their geometry skills with The Birdhouse Challenge.

- Ivars Peterson (@mathtourist) celebrates beautiful symmetry in Geometry out of Latvia and Africa.

- I enjoyed Rupesh Gesota’s gentle way of dealing with a student’s misconceptions in “Sir, the equation is correct. But I am wondering as to how we got it by the Method which is incorrect?” I hope he posts the follow-up lesson so we can see how the story ends.

- Sue VanHattum (@suevanhattum) takes an algebraic dive into the meaning of Division by 0.

- Jonathan Halabi (@Jd2718x) poses Two Geometry Puzzles about polygons.

- Mike Lawler (@mikeandallie) and son work through two proofs in What a kid learning geometry can look like – studying the power of a point.

*[“Tangent to a flower” by Jana Kopfová.]*

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## Scaling the Slopes of High School Math

“We must have empathy, and we must have humility, and we must understand what we have done badly in the past. We must catch ourselves when we repeat the patterns that brought us to past evils. We must do more than only calculate. “—Joseph Nebus, My All 2020 Mathematics A to Z: Statistics

- Greg Ross (@_FutilityCloset) discusses the math of vampires in Resource Management.

- Colleen Young (@ColleenYoung) collects resources for teaching The Modulus Function and Transformations of Functions.

- Colin Beveridge (@icecolbeveridge) calls on the magic of trigonometric identities to banish a devilish problem about An Unexpected Golden Ratio.

- I love visual proofs! Burkard Polster (@Mathologer) asks What does this prove? Some of the most gorgeous visual “shrink” proofs ever invented.

- Joseph Nebus (@nebusj) continues his fascinating A to Z posts with a look at the dangerous history of Statistics.

- Mike Lawler (@mikeandallie) and son delve into A fun – and surprising – statistics project inspired by Freya Holmér’s geometry twitter thread.

- The Mathematics News Snapshots for High School (MNS) project offers insight into modern mathematical puzzles like Catalan’s Conjecture.

- And don’t miss the 186th Carnival of Mathematics.

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## Enjoying Recreational Puzzles and Math Art

“This is the wonderful thing about just thinking and playing with half-formed thoughts: often exciting ideas will flash into your brain when you least expect them.”—James Tanton, Math and Cats

- Nrich Maths (@nrichmaths) offers a wide variety of puzzles for students to submit solutions. Check out the Primary Live Problems for ages 5–11 and Secondary Live Problems for ages 11–18.

- James Tanton (@jamestanton) is writing a new weekly National Math Festival (@NatMathFestival) newsletter with delightful puzzles inspired by student questions. Check out the back issues here.

- If you’d like your students to think about how math affects politics, there’s still time to play with The Gerrymander Math Project before next week’s Election Day.

- Art is one of my favorite ways to play with math ideas. Sophia Wood (@fractalkitty) hosts the Mathober Doodles challenge. There’s still time to catch the last few days, or you can just ignore the dates and start at the beginning.

- Don’t forget Annie Perkins’s (@anniek_p) Math Art Challenge — an evergreen resource of mathematical inspiration for your students.

- Paula Beardell Krieg (@PaulaKrieg) is Making Art by Making Rules. And experimenting with variations of a rule in Knot Obsessed.

- Numberphile (@numberphile) plays around with Colouring Knots.

- James Propp (@JimPropp) explains fields of characteristic two in When 1+1 Equals 0.

- Pat Ballew (@ballew_pat) investigates the
*arbelos*in The Shoemaker’s Knife Cuts Beautiful Math Across the Centuries.

*[“Five 7 Crossing Links” by Annie Perkins.]*

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## Teaching with Wisdom and Grace

“Thinking iseverythingin mathematics. Thinking is where joy is to be found … A math class should be an environment where students feel free to share their thinking, and feel no shame about brainstorming.”—Francis Su, Teach math like you’d teach writing

- Tisha Jones (@tishateaches) discovers The Beauty of Mathematics in her teaching.

- Simon Gregg (@Simon_Gregg) is intrigued by questions about preschool math in Seeing the mathematical: Filling.

- Sarah Eason (@SarahHEason) and Susan Levine (@LevineSusan) investigate ways to encourage young children’s learning about math at home in Family Math in Research and Practice: Where Do We Go from Here?

- Michael Minas (@mminas8) discusses the value of Using Maths Games During Remote Learning (or anytime).

- Katherin Cartwright (@kath_cartwright) offers two delightful posts about teaching with games: What’s your favourite maths game? and the follow-up Favourite maths games: Big ideas in small packages.

- Tracy Proffitt (@tracyjoproffitt) and Ruth Erquiaga (@math6teacher) explore best practices for helping students solve problems on the Math Before Breakfast podcast: Part 1 and Part 2.

- Fawn Nguyen (@fawnpnguyen) builds student engagement and encourages creative math thinking by Changing Up Popular Warm-Up Routines.

- Jenna Laib (@jennalaib) helps her sixth-grade students learn the value of failure and frustration with Johnny Upgrade.

- Standardized testing can drive a parent to despair. But there’s hope in my latest Homeschool Math FAQ post: Help, I’ve Ruined My Daughter.

- The Mathematics Teachers Association of the Nova Scotia Teachers Union (@MTA_NS) publishes all of their Mathematics Matters newsletters online for free. The Saskatchewan Mathematics Teachers’ Society (@SMTSca) posts each issue of The Variable. And the British Columbia Association of Math Teachers (@BCAMT) shares their eVector newsletter, too. Great resources full of creative math!

- JoAnne Growney (@MathyPoems) links to several Poetry-math resources — for classrooms and for fun.

- Ben Blum-Smith (@benblumsmith) reflects on A K-pop dance routine and the false dilemma of concept vs. procedure.

- David Bressoud (@dbressoud) examines The Strange Role of Calculus in the United States.

- Francis Su (@mathyawp) writes about how to Teach math like you’d teach writing.

*[“Red Cylinder with self intersections” by Gabriele Meyer.]*

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## Credits, and a Cry for Help

Art images are from the 2020 Bridges Conference Gallery. Quotes are from the linked blog posts.

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 is planned for late November or early December at Arithmophobia No More 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!

Wow! Packed to the rafters.

The online math community is a rich resource!