The 146th Playful Math Education Blog Carnival

Welcome to the146th Playful Math Education Blog Carnival!

What kind of math does the number 146 make?

146 is the 6th octahedral number because 6(2·6² + 1)/2 = 146.
That means that 1² + 2² + 3² + 4² + 5² + 6² + 5² + 4² + 3² + 2² + 1² = 146.

Base ten number 146 looks interesting when it is written in some other bases:
146₁₀ = 123₁₁ because 1(11²) + 2(11¹) + 3(11º) = 146, and
146₁₀ = 222₈ because 2(8² + 8¹ + 8º) = 146.

The factors of 146 are 1, 2, 73, and 146. Coincidentally, 1 + 2 + 73 + 146 = 222.

You can read other ways 146 and the numbers from 121 to 150 make math at Pat Ballew’s Math Day of the Year Facts.

Here are the attractions at this month’s carnival. Click on one to be transported right there!

Notice Patterns, Wonder, Create Math!

Graphics that let us notice patterns and wonder about them are fun, but students don’t have to wait for some teacher somewhere to make them. Denise Gaskins, the original playful math carnival creator, reminds us that students can be Math-Makers, and she invites them to have their creations published! Check out some student creations that have already been published.

Carrot Ranch noticed that Maths Is Everywhere: Clocks, Numbers and place value, patterns and algebra, measurement and geometry, probability and statistics, and much MORE.

Anna noticed something cool about the multiplication table. Can you notice it, too?

Mathematical Art

Nisha-designs decorated mugs with some lovely Abstract Geometric Circle Triangle Art.

K’s Dreamscape has a tutorial for you to make Simple Geometric Art! using cardboard, paint, paintbrushes, and painter’s tape.

Kreativekavya of Fremont forum uses circles, lines, and rectangles in Geometric Art!

Dianna Kolawole shares bright geometry art by Maranda Russel in Wordless Wednesday Geometric Art.

RobertLovesPi made a beautiful Pentadecagon and Its Diagonals.

FracTad’s Ractopia describes how to Create a Geometric Eye Using Desmos.

Sarah Carter of MathEqualsLove has created a gorgeous 3-colored origami Harlequin Cube and shows pictures of the steps taken in her post.

Karmen of Gallery K has made math digitally in some stunning Geometric Art.

Tessellation Art

Tessellation Art is the subject of Bumbastories’ What Four?

RobertLovesPi regularly publishes tessellations like Two Versions of a Tessellation Featuring Regular Hexagons, Regular Pentagons, and Tetraconcave, Equilateral Octagons.

I especially liked how his A Tessellation of Regular Hexagons, Golden Triangles, and Rhombi turned out. It seems to change depending on where you focus your eyes.

Mathematical Photography

You can make math using a camera! Marlene Frankel, A Photo’s Worth searched for and captured lots of geometry in Lens-Artists Photo Challenge #141 Geometry.

Tina Schell of Travels and Trifles photographed some geometric examples of the Fibonacci sequence.

Oh, the Places We See found geometry everywhere but carefully selected some geometric photographs from around the world.

Jazzersten photographed More Greek Geometric Art at the Museum of Fine Arts.

Estimation Booth

Steve Wyborney has engaging Esti-Mystery puzzles ready for every day for the rest of the school year!

Chasing Unicorns humorously blogs about Organizing Jelly Beans. How many jelly beans can you eat each day to keep yourself below the estimate of refined sugar consumed per American per day?

Within 1%, how long is the hypotenuse of this right triangle? If certain criteria are met, John D. Cook’s blog post, Hypotenuse Approximation, can help you be the first to find the correct answer and win the prize.

Fractions, Ratios, and Decimals

Henri Picciotto of Henri’s Math Education Blog updates us on how to use fraction rectangles to help students make sense of adding, subtracting, or comparing fractions with different denominators.

1001 Math Problems shares an engaging and delicious Chocolate Problem involving fractions.

Third-grade teacher, Ms Victor, couldn’t help but see fractions while eating lunch in When Your Teacher Brain Is on Overdrive.

Jillian Starr shares how to transition from unit fractions to more complex fractions in Teaching Fractions Through One Whole.

Do you bake using ratios instead of measuring cups? Kat from the Lily Cafe does and will show you how to use ratios and a scale to Make Flatbread. What a tasty way to make math!

Duane Habecker of The Other Math, More Than What’s in the Textbook invites you to solve ratio problems using Tape Diagrams.

Read how much laughter can be had learning long division involving decimals in FiveHundredaDay’s post It’s not them, it’s me.

Carnival Games

Ajitadeshmukh shares the game, The Number Detective [Spying the number]. This is a game that uses an ordinary deck of playing cards and reinforces the concepts of adding, subtracting, multiplying, and/or dividing. It can be played by children in early elementary grades and up.

Primary Ideas shared how well a game of Noggle (Number Boggle) went when it was played in a Google session remotely.

Anna of one+epsilon designed a logic game called Dot, Dot Poof! Here’s a bonus: Kids 6 and up might inadvertently learn a little linear algebra playing it, too!

A Game of Linear Equations by Bethany of MathGeekMama will help students find solutions to their problems!

MakeMathNotSuck blogged about Theresa Wills’s Playing Interactive Math Games for middle grades. It is really exciting that these games can be played in real-time with a partner.

Wendi Bernau made an Easter Egg Hunt Escape Room for her 15- and 17-year-old kids. The escape room included puzzles based on their current schoolwork. The 17-year-old had to solve a puzzle that required calculations, graphing, and trigonometry. The kids liked the escape room so much that they are already talking about doing it again next year.

Harsh Sharma writes about How Math Games and Puzzles Improves Brain Activity. It turns out that Failing/Losing is as important in brain development as Succeeding/Winning is!

Hands-on Math

The Pi Project lets you listen in on the delightful conversation about knitting and fairies and the place value police in The Beauty of Base Ten Blocks.

Melissa Packwood of The Florida Reading Coach blogged about some Affordable Math Manipulatives that can assist students in learning mathematics. shares some ideas to Make Your Own Maths Manipulatives.

House of Mirrors

Reflections are important topics in geometry and coordinate math. Our House of Mirrors is full of fascinating reflections.

Ted Jennings, shared a beautiful picture of an alligator and a turtle in Reflections.

Hannah Michaela of CoC-GetFit gives a geometric definition of a mirror image, shares a few examples in pictures and a thoughtful poem about mirrors and reflections in Mirror Image.

Beth of I didn’t have my glasses on made math by photographing a reflection that is happening at the front and the back of a pond in Argo Park.

Ritva Sillanmäki wrote a poem and made math by photographing a reflection that happens on the left and the right side of a river.

Bushboys World has several amazing pictures of birds in See My Reflection.

I shared a couple of puzzles where the squares of two numbers look like they are looking in a mirror.

Museum of Mathematics

All over the world math is being made on this day, April 28. Pat’s Blog shares some famous ways math has been made in the past On This Day in Math, April 28th.

David Campbell of Culturico writes about the beloved Louis Carroll in Portrait of a mathematician in love with the art of writing.

Indrajit RoyChoudhury tells us about Bhaskaracharya, a 12th century Indian mathematician and astronomer in Arjuna’s Arrows and Algebra. Bhaskaracharya discovered differential calculus 500 years before the births of either Newton or Leibniz.

Papannasons  has written an essential biography of 20th Century Indian mathematician Srinivasa Ramanujan. Knowcusp reviews the movie about Ramanujan in The Man Who Knew Infinity: A tale of one of the Greatest Mathematicians of all times. While (Roughly) Daily mentions him and several other great mathematics in “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

LA of Waking up on the Wrong Side of 50 is featured in the controversial current events area of the Math Museum in Anything Can Happen Friday: Math. LA includes the actual newsletter in which Oregon instructs its math teachers to allow for more than one correct answer. LA is upset thinking that now Oregon math teachers must accept incorrect math like 2 + 2 = 79. Perhaps Oregon is just welcoming some of Denise Gaskins’ math rebels who might say that 2 + 2 = 79 – 75, or some other of the infinite number of possible non-simplified yet still very much correct answers.

Likewise, the College Fix reported that Oregon math teachers have been instructed to let their students show their work by making TikToks, silent videos, or cartoons about the math they are learning, in other words, let students make their own math. I think about Ramanujan who taught himself math from an old textbook and then created his own mathematical symbols and terminology when he dreamed up more advanced mathematics. Later when he was told he needed to prove his ingenious mathematical formulas with rigorous proofs, did it help him or restrict him?

Esther Brunat has “curated a collection of Math TikToks” that now belong in a modern Museum of Math.

Adding, Subtracting, Multiplying, Dividing, Etc.

Have you ever experience joy when skiing? Bill McCallum of Illustrative Math compares that feeling to being fluent adding and subtracting numbers.

Laura of Riddle From the Middle describes why third-grade students often struggle with determining which operation to use in SoCs – the right teacher makes a world of difference.

Tess M Perko of River to Humility has written a sweet short story: The Imagination Grandpa Story 3: The Multiplication Staircase.

With doses of frustration and humor, Joseph Nebus of NebusResearch explains why No, You Can’t Say What 6/2(1+2) Equals.

Bethany of MathGeekMama shares her game that makes learning order of operations fun and not impossible!

Math Story Time and Other Books

1 + 1 + 1 = 3. Any number greater than one can be partitioned in a similar fashion. Patricia Nozell reviews a perfect picture book, I Am One: A Book of Action by Susan Verde. A little math can be learned while one person works with another and another to make the world a better place.

Writing this post has introduced me to Perfect Picture Book Fridays. Susanna Leonard Hill reviewed Little Ewe: The Story of One Lost Sheep, by Laura Sassi. Your 3- to 5-year-old will love counting logs, frogs, and other rhyming nouns as you read this book together.

Sue Heavenrich of Sally’s Bookshelf blogged about Bracelets for Bina’s Brothers, a picture book about estimation for 3-6-year-olds, and concluded that Math + Art > Numbers. Activities to make the math in the book more meaningful are also included in the blog post.

Wrenbeth22 of Miss Beth has a Book reviewed The Boy Who Loved Math by Deborah Heligman and LeUyen Pham. This is the story of Paul Erdös, a famous twentieth-century mathematician who made friends all over the world by sharing the math he loved.

Darlene Beck-Jacobson reviewed three biographical storybooks: Queen of Physics by Teresa Robeson, Code Breaker, Spy Hunter by Laurie Wallmark, and Counting on Katherine by Helaine Becker in Celebrate Girls and Women in STEM Day with Some Great Books.

Patricia Tilton of Children’s Books Heal reviewed Wonder Women of Science by Tiera Fletcher and Ginger Rue as part of Women’s History Month. The book is perfect for 9 to 12-year-olds. She also made me aware that Nerdy Book Club reviewed the same book. From that review, I learned the delightful true story of a human calculator named Tiera Fletcher that I am anxious for you to read as well!

MikesMathPage tells us that James Tanton’s Solve This book is full of incredible math projects to do with kids. In this post, he and his son explore a little topology in Going back to James Tantons’ amazing Möbius Strip cutting project.

In Monday’s Math Madness, Willow Croft thoroughly enjoys a 15th-century maritime manuscript called The Book of Michael of Rhodes. There is a lot of math in the book, but even if the reader doesn’t like math much, it won’t take away from the thrilling adventure. It is suitable for high school students and older.

Kelly Darke of MathBookMagic and FairyMathMother would like you to know about Math Book Wisdom: An Early Math Resource Book. It isn’t a book to read to kids, but it is filled with math wisdom for the parents and teachers who teach children.

Crow Intelligence reviewed a book that interests me a lot:  Playing with Infinity – Mathematical Explorations and Excursions by Rózsa Péter.  I only need to decide if I will read it in English or try to get through it with the little bit of  Hungarian I know!

The Enchanted Tweeting Room

Jo Morgan blogs about some wonderful ideas for teaching Place Value Tool, Powers, Simple Linear Graphs and more that she’s found on Twitter and elsewhere in 5 Maths Gems #143.

On Mondays, MathEqualsLove blogs about many must-read tweets she finds on Twitter. You will want to check out Volume 80, and Volume 81.

The Whispering Spot

Imagine someone whispering at a spot inside a building and someone else clear across the room being able to listen to them clearly! Such a whispering spot exists at this carnival! See what happens when three math teachers teach by listening to their students:

When a student didn’t understand a mathematical concept, he broke a rule by leaving the classroom. Kaneka Turner of BlackWomenRockMath details how she listened to the student with her ears, her eyes, and her heart in The Art of Listening. By so doing, she successfully helped him make the connections needed to understand the lesson while simultaneously letting him know he was truly understood. What trajectory would his life be on now, if she had not listened as she did?

The Heinemann Blog features an interview between Marilyn Burns and Lucy Calkins on Listening to Learn. By listening to the interview or reading its transcript, you can learn how Marilyn Burns interviews individual students and listens to them to advance their understanding of mathematics.

In the second half of Bill Davidson’s podcast interview with Robin Ramos, she describes how she scripts questions and listens to not just individual students but to a classroom of students at the same time!

Listening is key anytime we talk with a math maker. You can read Life Through a Mathematician’s Eyes’ interview of an up-and-coming mathematician: Akshay Thakur for the Inspirational Corner.

Of course, teachers need to be listened to as well. See Research Minutes’ Teacher Stress and Burnout in the Wake of Covid 19.

Poetry Corner and Some Trigonometry

In Math Makers Write a Poem, Denise Gaskins gives us some ideas and examples of student-written mathematical poetry.

I also have found some examples of people making math by writing poetry. Even if a poem speaks negatively about math, it gives us all an opportunity to LISTEN to students and meet them where they are.

Trigonometry for Dogs is a short, sweet poem by Lyna Galliara.

My heart broke when I read Looking at Love Lost, by murisopsis of A Different Perspective. It is a poem about falling out of love with mathematics in high school beginning with trigonometry. Simply saying Trig is Easy doesn’t help and only makes a person not feel heard. Perhaps Wyrd Smythe’s Explanation of Trig Basics might have been helpful?

Puzzles shares an easy Easter Egg Sudoku Puzzle that even preschoolers can do.

Puzzle a Day challenges us to solve A Mathematical Multiplication Puzzle with a six-digit product without using a calculator. I can attest that it can be done!

De Graw Publishing’s blog gives us Number Problems and Easy Sudoku Puzzles for Kids: Math and Logic Games Problems for Children.

Sarah Carter of MathEqualsLove shares a new puzzle in Number Ball Puzzles by Naoki Inaba. She translates the rules from Japanese to English so that you can have some idea where to put the missing numbers in the puzzle. Be warned, for the bigger puzzles, you might need to use your eraser a lot.

Sara also shared a sequence puzzle. Her students have enjoyed predicting the next letter in the sequence.

Maggie Heffernam suggested to Brian Marks of Yummy Math that he write a math activity when a real-life man was paid in greasy pennies.

Bedtime Math has a musical mathematical puzzle for you in Mile-Long Xylophone.

Math Teaching Strategies

Some teachers have half of their students in class and half remote over zoom. Keeping the at-home kids engaged can be difficult. Libo Valencia of Fresh Ideas for Teaching has six proven strategies to engage students in these hybrid classes.

You or your students can easily make Original Which One Doesn’t Belong puzzles!

Dan Draper of Opinions Nobody Asked For explores Area Models and Grid Method.

Probability and Statistics

Joseph Nebus of Another Blog, Meanwhile posts humorous statistics every Saturday like this cumulative bar graph showing Star Wars Movies versus Star Trek Movies. His vertical axis is a hoot.

Mr. Rowlandson of Pondering Planning in Mathematics has been Thinking About Probability Trees. Do you add or multiply the fractional probabilities? His blog post spells out what to do.

Athletes are constantly making math. Greg Pattridge of Always in the Middle writes about the statistics produced with every play in It’s a Numbers Game! Baseball.

Lunatic Laboratories uses alliteration to tell a tale of tails in One-tailed vs. two-tailed tests in statistics.

Did you know that if you get 11,000 steps a day, you will walk a million steps every quarter and just over 4 million steps a year? LisaFeatherstone had a daily goal of walking 10,000 steps and still made the 4-million steps goal. She used a spreadsheet to track the data her fitbit gave her and wrote a formula to predict when she would meet her goal.

Lvonlanken of The Shy Genealogist analyses the data she’s collected to determine which John Smith is her ancestor in Sorting the Land Records. Some genealogical programs will provide you with all kinds of statistics from your family tree. See the stats the Chiddicks Family found in My Family Tree in Numbers. I was pleased that they didn’t simply accept every statistic. They made predictions of the results and compared their predictions with the statistics the program produced.

MSCNM uses probability and statistics to answer the question Should You Buy a Lottery Ticket?

Blue Ribbons

Jo Morgan of Resourceaholic recently celebrated seven years of blogging by reviewing the very best teaching ideas and resources from the previous year and naming the winners of her (Maths) Gem Awards. Check it out!

The pandemic hasn’t stopped some people from doing good. Leila Zerai writes for LondonNewsOnline about a Student Winning the Prestigious Lewisham Mayor’s Award for Offering Free Online Maths Tuition.

A short story, Advanced Word Problems in Portal Math, is a finalist in the Nebula Best Short Story Contest. The reviewer didn’t care for the story because the math references were hard to understand. Let me tell you a little secret: I think that’s the way it was meant to be because I didn’t get the math references either! The story was just a fun way to make math. Another example of purposeful over-our-heads math was in a Barnaby comic. I know how to find the determinant of a two by two matrix and how to multiply binomials, but I look forward to Joseph Nebus explaining that comic sometime soon. It is still a funny comic even if I don’t fully understand it yet.

Math Memes and Comics

Joseph Nebus of Nebusresearch explains the mathematics of a comic in Where Else Is a Tetrahedron’s Centroid.

Design a Carnival

I hope you had a wonderful time at this month’s carnival! This month the Carnival of Mathematics #192 was hosted at Eddie’s Math & Calculator Blog. Perhaps you would like to design your own carnival.

Simran M Karkera of MSCNM tells the story of a girl who loved math that used trigonometry and calculus to design a roller coaster that thrilled her previously-mocking friends in A Mathematical Ride!

Last month the 145th Playful Math Carnival was hosted by Mathhombre. Perhaps you would like to host the next carnival or one later in the year. You don’t have to go overboard like I probably did. I was having so much fun, I couldn’t stop myself! To volunteer to host a carnival go to Denise Gaskins’ Carnival Volunteer Page.

1622 A Blue Egg for Your Easter Basket

Today’s Puzzle:

These somewhat tricky level-5 puzzles are probably better suited for middle school and up than younger kids. Use logic on every step and you should be able to find its unique solution.

Math Eggs from Twitter:

Here are some Easter egg puzzles I saw on Twitter. Some are perfect for the littles and others are for older kids. Easter egg hunts can be fun for anyone of any age.

Factors of 1622:

  • 1622 is a composite number.
  • Prime factorization: 1622 = 2 × 811.
  • 1622 has no exponents greater than 1 in its prime factorization, so √1622 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1622 has exactly 4 factors.
  • The factors of 1622 are outlined with their factor pair partners in the graphic below.

More about the Number 1622:

1622 is the sum of four consecutive numbers:
409 + 410 + 411 + 412 = 1622.

Welcome to the 140th Playful Math Education Blog Carnival!

What’s Special about 140?

Ladies and Gentlemen, welcome to the 140th Playful Math Education Blog Carnival! Feast your eyes on the number, 140, the 7th member of the famous Square Pyramidal Number family. Watch as 140 performs these amazing feats:

140 is the 7th square pyramidal number because
(7³/3) +(7²/2) + (7/6) = 140.

140 has twelve factors and will now use them to make a lovely factor rainbow:

For140’s next trick, see what happens when it is divided by six of its non-factors:

Finally, 140 is the fourth harmonic divisor number, and Wolfram Math World even uses 140 to explain what a harmonic divisor number is.

Now let’s move on to the blog entries for this month’s carnival:

Children’s Literature and Math

Kelly Darke of Math Book Magic wrote a post about a brand new entry in children’s literature, The Boy Who Dreamed of Infinity.  This is the story of the great Indian mathematician, Ramanujan. The book is available for us to read now, but Kelly was able to read it for the first time last year. I felt so much joy inside of me as I read first her reaction to the description of the book and later to the book itself. I am grateful that the mere idea of that book prompted Kelly to create a blog to share the magic of good mathematical children’s literature.

Rhapsody in Books Weblog tells us about, Raye Montague, an African-American girl born in 1935. She was told repeatedly that her race and her gender would prevent her from becoming the mathematician that she dreamed of becoming. She didn’t heed her naysayers. The Girl with a Mind for Math: The Story of Raye Montague tells her inspiring story.

On world Tessellation Day, TheKittyCats blog introduced us to Tessellations!  Children will enjoy looking at the illustrations in the book and won’t even realize they are learning some math in the process unless someone shares that secret.

Games and Math

Alan Parr of Established 1962 explains how to play Dotty Six, a game played with a tic tac toe grid and a die. Because he asked himself, “What If Not?”, he was able to suggest some mathematically interesting variations of the game as well.

Mrs. O’Brien started her blog, Math Epiphany, very recently-in May 2020. She has written two posts about using games to learn mathematics. She writes about card games in Summer Math Fun: Math Games, and about board games in Summer Math Fun: Good Old Fashioned Games.

Denise Gaskins shares a math game that lets children complexify expressions or equations rather than simplify them. Making the expression or equation a little more complex than it was before can be great fun and wonderfully educational. Check out The Best Math Game Ever!

Do you know how to play Dara, Five Field Kono, Mu Torere, Pong Hau K’I, Shisima, or Triangle Peg Solitaire? I’ve never even heard of these games before. Mark Chubb of Thinking Mathematically introduces all of these games in Math Games-building a foundation for mathematical reasoning. One of the games is illustrated in the tweet below.

Math and Paper-Folding

How is making a pouch out of a newspaper doing math? Paula Krieg and John Golden explain that it can be more than simply taking measurements and using rulers in Pouch: Something from Almost Nothing #3.

Did you know that paper folding can help kids understand systems of linear equations?

Or paper folding can even help kids understand Knot Theory or Topology?

Math and Art

What creatures can be seen in these numbers? Click on the video to find out what one artist saw!

We can always count on Robert Loves Pi to create dazzling and beautiful geometric art. Here he tells us how relaxing it is to produce it:

Photographer Ming Thein shares 13 discussion-evoking photos in Photo essay: The texture of geometry. I’d love to hear some of those discussions!

Quilts can be stunning examples of mathematical art. Aby Dolinger of Abyquilts has created a quilt pattern she calls “Math Whiz,” and this mathy quilt was featured on the July/August 2020 COVER of Quiltmaker magazine! Congratulations Aby!

Geometry and Trigonometry

Laura of Mathsux² has written an explanation and created a video to take the mystery out of trigonometric ratios in How to Use SOHCAHTOA.

Jo Morgan’s website is filled with resources to make teaching and learning math more effective, and yet she always finds room for more ideas! She recently created her 133rd Maths Gems Post that included some playful ways to look at angles in a circle and areas of rectangles inside of a larger rectangle. . . Now to segway into word problems. . . If Jo writes two Maths Gems a month and her July 30th post was number 133, and the Playful Math Blog Carnival comes out ten times a year and this is the 140th post, when will the Maths Gems number and the Playful Math Blog number be the same number?

Word Problems

I love the giant Sequoia Trees. This blog post has story problem suggestions about Sequoia Trees for every age group:

A-Hundred-Years-Ago Blog explores some Hundred-year-old Food-related Math Problems when large oranges were only 60 cents a dozen. Go back in time and enjoy solving these with your students! Let them compare them with word problems from the 21st century.


Alan Paar of established1962 shares his last experience helping kids play with math before the lockdown. It was a series of puzzles that made A Lesson That Will Stay With Me. He has used these adventures for 30 years and kids enjoy them so much. He was especially glad that these puzzle adventures will be their last memories of attending primary school. They was so much better than Key Stage 2 SATs.

The Find the Factors puzzles I create are a playful way to get to know the multiplication table better. This level 3 puzzle can be solved by considering the factors of 30 and 90 where only factor pairs with numbers from 1 to 10 are used. After those factors are found, write them in the appropriate cells then work your way down the puzzle row by row using logic until all of the factors are found.

Singapore Maths Tuition shares an “average” math puzzle with a twist that might baffle all but those kids who enjoy math but find little challenge in traditional math work. No worries for the rest of us; a good explanation is also included.

Mathematical Humor

Emily’s Post tells a timely math joke about three ducks that will teach while it delights children in Modern Math.

Joseph Nebus has a humor blog in which he wrote a humorous post he titled What your Favorite Polygon Says about You. I’m not sure what my favorite polygon is, but I will carefully consider all the possibilities.

The Bored Side of the Phone shares a couple of stand-up-comedy-worthy jokes about Mathematics in The Truth About Maths.


Natural Numbers:

The counting numbers/whole number set has been further categorized! Can you imagine how? Read all about it in Publications de BOULAY’s New Whole Numbers classification. As you learned about the set of ultimate numbers, how well did your imagination serve you?

Rational and Irrational Numbers: Mike of Mike’s Math Page gives us a front-row seat observing how he teaches his sons about mathematics in Sharing John Urschel’s great video on rational and irrational numbers with my son.

Imaginary Numbers: Every year Joseph Nebus lets his readers chose mathematical topics for each letter of the alphabet, and then he writes a post about each of those A to Z topics. For 2020 he wrote a serious essay on imaginary numbers that playfully included some comics about some numbers that you can imagine.

Life skills and Math

Ladybugs or ladybirds want to teach you some math concepts. Come out in the garden with the lesson plans provided by DogwoodDays in Garden Schooling: Ladybird Maths and see what you learn!

Cooking is an important life skill and a fun way to learn about fractions and other math concepts. The For-Health blog featured a post kids and adults can enjoy together: How to Cut Down Recipes: What’s Half of ½ cup, ¾ cup, 2/3 cup and More. Verifying the given measurement equivalents can be great fun for kids so do let that happen!

How can we make our lives be as well-balanced as an equation? That’s a good question for high school students to consider. A life coach’s advice on how to find success in life is given using mathematical symbols and vocabulary in Mathematics of Life, Learn from Math symbols.

Corona Virus Math

In Wheel of Theodorus – Distance Learning Edition, MrJoyce180’s shares his students’ work creating their own, and I do mean their own, Wheels of Theodorus. All of this creating occurred virtually during the lockdown. He shares both the successes and the failures. This was one of my favorite discoveries while I created this carnival.

I didn’t have my glasses on questions Cosco’s mathematical reasoning of cake buying and serving in Let (a few of) them eat cake! Can you formulate a word problem from this post?


When Disney produces a direct to video sequel, will a Roman numeral, an Arabic numeral, or neither most likely appear in the title? Even young children will be able to explore that topic with Joseph Nebus in this Statistics Saturday Post.

World Affairs uses cleverly represented graphs to help us understand The Math of How China Surpasses USA in 5 Years. Understanding the math behind the graphs could help us improve our situation.

Poetry and Math

Beginning with irrational numbers, Prerna’s Blog uses mathematical and poetic language to describe the Mathematics of My Mind.

Math+Life connects math with life by writing poetry. After you read Set in Stone the mathematics of sets is explained followed by how they relate to life. Do we place limits on children or adults when we categorize them into sets of different types of people?

Making Math More Inclusive

Please read Sunil Singh’s powerful and thought-provoking post, How to Begin Bringing Rich and Inclusive Math History Resources Inside K to 12 Classrooms.

I am also pleased to introduce you to the brand-new BlackWomenRockMath Blog. Their first post is The Brilliance Hiding in Plain Sight in which three women share their sobering math stories. Thankfully, they each were able to overcome negative early experiences in learning mathematics to make worthwhile contributions to mathematics education today.

Mathematics Carnivals and Amusements:

Every Monday Denise Gaskins invites you over for a Morning Coffee. There she will direct you to other mathematics blogs for your edification and amusement.

There is also a Carnival of Mathematics that may interest you. The current (184th) Carnival is hosted at Tom Rocks Maths.

I really liked putting this month’s carnival together, and I hope you have enjoyed reading it as well. Feel free to stop by and hang out whenever you’d like.

The previous Playful Math Education Blog Carnival #139 was hosted by Math Mama Writes. Be sure to check it out if you haven’t already.

I am already looking forward to the next Playful Math Education Blog Carnival which will be hosted by Joseph Nebus of Nebus Research.

Perhaps you would like to volunteer to host one of the carnivals? Contact Denise Gaskins to get on the carnival calendar! I can’t wait to see what you put together!

131 Playful Math Carnival

Welcome to the Playful Math Education Blog Carnival featuring the amazing prime number 131, whose digits can mutate into other prime numbers right before your eyes!

131, a permutable prime number

make science GIFs like this at MakeaGif
Yessiree, 131 is prime, and so is 113 and 331. Do I need to mention that 3, 11, 13, and 31 are also prime numbers?
131’s next trick happens when you add up all the 2-digit PRIMES that begin with a 4:
41 + 43 + 47 = 131.
Because 131is a palindrome, it reads the same forwards and backward. Here’s another trick: 131 is 65 in BASE 21 and 56 in BASE 25.


We have many different attractions this month. You can go to any category quickly here:

Carnival Attractions:


You’ve heard of the three R’s, reading, and writing and ‘rithmetic, but what is arithmetic? Joseph Nebus shares a few comics about basic arithmetic and explains what they mean:

Arithmetic is also television’s Lisa Simpson’s favorite subject in school and she will miss it greatly as she recovers from the mumps. In this blog post, Safi explains Dr. Hibbert’s comforting words to her about polygons, hypotenuses, and Euclidean algorithms.


You can always count on Robert Loves Pi to produce a beautiful and complex geometric design. This one he calls Two Rhombic Polyhedra with Tessellated Faces. Here’s another one:

Paula Beardell Krieg helped students create big, beautiful geometric artwork and origami in Summer Projects with Teens.

Also, check out Paula’s Paper, Books, and Math Workshop for many more ways to learn math through art.

Big Prize, Little Chance of Winning

Several years ago Mental Floss wrote about carnival games that offer big prizes but have very little chance of being won. This carnival has a couple of those as well. They are called unsolved math problems. Even if winning probably isn’t going to happen, that doesn’t mean the games and activities aren’t fun. Explaining Science updates us on a very famous unsolved problem, The Goldbach’s Conjecture. Supercomputers have worked on it, but we are no closer to a solution.

In A Neat Unsolved Problem in Number Theory That Kids Can Explore, Mike’s Math Page explores the new-to-me Collatz conjecture that for every positive n, the sum 3 + 8n will equal a perfect square plus an even number. It’s a simple enough conjecture for kids to understand and it is fascinating, yet mathematicians have not been able to prove or disprove it yet!

Creative Writing

Subha laxmi Moharana (Angel Subu) writes creatively about some tough topics in high school mathematics in Math Poem. I think her words could be turned into a rap.

Poetrywithmathematics shares Doug Norton’s lovely mathematical poem Take a Chance on Me.

What if graphs were self-conscious about their looks? High School aged students can consider that thought as they read the imaginative blog post, To Infinity and Beyond.


There’s a cozy classroom place that promotes mathematics in Our New Math Space. It was designed for older students by Continuous Everywhere But Differentiable Nowhere and includes many pictures.

Have you considered displaying a weekly math joke? MathEqualsLove shares a fun joke and a puzzle for kids to gather around and enjoy.

Factoring Quadratics

Super Safi uses another episode from the Simpsons to teach about the quadratic formula.

Food for Thought

Anybody can cook or do math. Really? What does that even mean? Math4Love explains both in What We Mean When We Say, “Anyone Can Do Math.”
Math with Bad Drawings makes a similar point in The Adventures of Captain Math.


Joyful Parenting made a simple kindergarten-age counting game and called it Snack Math, but even older kids might enjoy figuring out exactly how many crackers are required to play the game.

How many are in the jar. What is a good estimate? Add Steve Wyborney’s clues one by one to get an even better estimate. He has 51 New Esti-Mysteries that also happen to teach several different math concepts.

For older students, Kent Haines a free game he calls Last Factor Loses. I played it a few times with a student. Making prime factorization a game really did make it more fun.


Bn11nb enjoys the geometry of architecture. The pictures in this post are worth a look and could be an inspiration to your students.

House of Mirrors (Reflecting on Mathematics Teaching)

We often reflect on the effectiveness of our teaching methods. Sometimes we are advised to require students to use more strategies. We might ask them to notice or wonder about a concept. These two thoughtful posts will certainly give you cause for reflection:

“The More Strategies, the Better?

Noticing and Wondering: A powerful tool for assessment


Robert Kaplinsky shares ten things he’s embarrassed to tell you. Has he been reading your mind and mine?


What is your favorite part of a cupcake? What if you could buy just that part? What if you wanted to put a whole cupcake together? How much would that cost? Your child can learn about money and decimals exploring those answers with Mathgeekmama’s  Money Math Problems.

Museum of Mathematics

Beads can be a fun manipulative when learning mathematics. Joseph Nebus has begun his 2019 Mathematics A-Z series by writing about the Japanese abacus. He compares it to a slide rule and the Chinese abacus. He also describes how to use it to add, subtract, and multiply numbers. Students could have some fun using it to understand place value, too.

Life Through a Mathematician’s Eyes is giving museum tours in A History of Mathematics-August. K-12 students could be fascinated by the mathematical relics from the Smithsonian founded in August 1846 as well as the Seven Bridges of Königsberg solved by Euler in August 1735.

Pumpkin Patch

Erin of Sixth Bloom’s Pumpkin Math-Preschool Activity will engage your little ones as they learn to count and sort pumpkin-shaped macaroni or candies.

They will also love decomposing numbers using pumpkin seeds and  Mathgeekmama’s cute Pumpkin cards.


Digital Educators Alliance offers free posters of admirable women in math and related fields:

While Sara Van Derwerf set of 112 New Math Fail Posters will delight students as they notice and wonder about and LEARN from grown-ups’ computing mistakes.


7Puzzle gives some clues about a 3-digit number. Can you figure out what it is?



Alan Parr writes about a newspaper puzzle called Evens Puzzles. He suggests that students can make their own and hints that he has thought up several variations of it. I look forward to reading about those!



American Calendars for September had more than a week’s worth of palindromes. Would palindromes make a good puzzle? Yes! Print off a 100 chart and try Denise Gaskins’s A Puzzle for Palindromes. Also, check out her new Morning Coffee feature each week for more math teaching tips.

Next Month’s Carnival

That’s it for this month’s Math Education Blog Carnival. The 132nd Carnival will be next month at Arithmophobia No More. Would you like to share a post or host the carnival? Go to Let’s Play Math for details!

127th Playful Math Education Blog Carnival

Ladies and gentlemen welcome to the Playful Math Education Blog Carnival featuring the incredible number 127 of the famous Mersenne Prime family! Let’s give a big hand to. . . . .

2²-1 = 3, the smallest single Mersenne Prime;
2³-1 = 7, the smallest double Mersenne Prime;
2⁷-1 = 127, the smallest triple Mersenne Prime; and finally
39-digit 2¹²⁷-1, the smallest quadruple Mersenne Prime!

It took NINETEEN YEARS (1857 to 1876) for Édouard Lucas to test Mersenne Prime Number 2¹²⁷-1 BY HAND to successfully verify that it is indeed a prime number. It is the largest Mersenne Prime that has ever been verified by hand calculations!

Ladies and gentlemen, Prime number 127 has one more amazing feat up its sleeve that you will have to see to believe:

In this month’s blog carnival posts, the amazing Desmos will delight and astound young and old alike. There will even be an Easter egg hunt! The blog post links (in turquoise) are joined by several links from Twitter (in blue-violet) and a few from other places such as Youtube (in red). Stay as long as you like and ENJOY what the many carnival participates have to offer in 20 different amusement areas!

Art and Mathematics

  1. Stephanie showed off her colorful and impressive Tessellation Math Art Wall on twitter.
  2. David Petro used all 84 pieces of a 21st-century pattern block set to create a lovely symmetrical design.
  3. After this year’s very long winter, I’m especially glad Colleen Young collected some lovely and amazing springtime Desmos drawings and gifs in It’s Springtime. . . .
  4. Do you see mathematics everywhere? Continuous-Everywhere-but-Differentiable-Nowhere can and does, even on a student’s shirt. Read the story and see the t-shirt design replication in Desmos in Going Off the Beaten Path.
  5. Wanting to have your students create an art project in Desmos? 1ntegration-by-Parts has given that assignment many times and has links for student directions to help them meet your expectations in Desmos Art Project (Update).
  6. You must click on the turtle face link in Desmos Art! With just ONE equation, a magnificently detailed turtle face was produced in the Desmos calculator. I was so impressed. I tweaked that single equation by changing the number 16 to 7.29, added some color restrictions as well as equations to make a mouth and some eyes to produce my own Desmos Art piece I call Blue-eyed Beauty.

Classwork/Homework that is enjoyable

  1. Elementary-school-age students will enjoy Desmos’s Polygraph activity given to second graders that Matt Vaudrey shared.
  2. I-Speak-Math has a mathematics homework solution students LOVE. Read about it in Meaningful Homework and CPM.
  3. Jennifer Michaelailis has a pro tip on how to keep students who need a little extra help in class from feeling defeated.
  4. If you want a free math education gathering in your area, here’s how to get one started. Also, check out Denise Gaskins’s resources to keep the group going.

Creative Writing

  1. Many people have a personal story to tell that explains why they love mathematics. Through a Mathematician’s Eyes opens up and shares her experiences in My Story. What obstacles did she face? How did she feel about enjoying a subject so many others hated?
  2. Philip Jose Pacis played with some math vocabulary and wrote a poem he titled Fractions about a fractured relationship.
  3. Calendars and time are mathematical topics. How many other mathematical terms do you see in Maggie C.’s poem On Time? Do any of those words have more than one meaning?


  1. Math Geek Mama has come up with Decimals on a Number Line Game and includes everything you need to teach this concept with complete confidence.
  2. Would you like to play Decimal Pickle? On Twitter, Mrs. Unger explained how to play it as well as a few of its variations.

Desmos First Aid Station?

  1. There’s no need to call the paramedics when it’s time to learn about parametric functions. Suzanne Van Oy has come to the rescue! She sees a lot of value in parametric functions and their graphs. Why is she so excited about them? What’s all the fuss? Suzanne answers those questions and more in Why Parametrics?.
  2. Suzanne Van Oy also recently put an incredible spinning Desmos Birthday Cake on twitter. How does she make Desmos gifs that don’t look like they need to be put on life support? Six months ago she blogged about how to do it in Making a Great Desmos Gif. She certainly knows what she’s doing!
  3. Sometimes Desmos doesn’t do what you expect. Your work might need some first aid. DesmosGraph (Unofficial)’s post: Desmos Traps: Why Is It Not Working may have the diagnosis and cure you seek.
  4. Although this post from Gold & Ratios is two years old, I still wanted it included on this list. It teaches how to add color and texture to Desmos.

Eggstraordinary Mathematical Easter Egg Hunt from Twitter

  1. Cliff Pickover shared Don M. Jacobs equation for an egg laying on its side.
  2. Tamás Görbe shared an equation for an egg that is standing up on its wide end.
  3. Get out your markers and start coloring Paula Beardell Krieg’s graphic of an egg and Three eggs! Paula also shows how she colored one of her eggs.
  4. Mathigon shared a gif of a colorful, nine-piece, tangram-like egg puzzle that can also transform into a rabbit and a goose.
  5. Robert Bosch shared a TSP art depiction of an egg and a chick that have a few mathematical traits in common.
  6. Ha! Doesn’t this always happen? I found more fabulous eggs the day after the hunt: Daniel Mentrard’s eggs made in Geogebra,
  7. And these Eggsponenential eggs created by Traci Jackson!

Exponents and Exponential Functions

  1. Christopher Danielson posted a question about exponents on Twitter that generated a lot of thinking from adults. I’m sure it would do the same for kids who understand a little bit about exponents.
  2. Jongarland6 was able to get ELL students conversing with each other in English about exponential functions. How it was done is described in Desmos Sparking Academic Conversations.


  1. Math Play Day gives ideas from 20SomethingKids and 1KookyTeacher about playful ways first-grade kids learned about fractions.
  2. Mathgeek Mama published some adorable free Equivalent Fraction Robot Puzzles.
  3. A recently released YouTube video has a little girl teaching about fractions in Maths 4 Kids’ Fractions of Shapes and Fraction Vocabulary.
  4. Here’s another one featuring the same little girl: Fractions of Amounts Using the Bar Model
  5. Every carnival has food concessions. You can have a lesson at breakfast on fractions based on CTSPEDMATHDUDE’s post Sausage Fractions: Real Life Example.
  6. When teaching fraction division, should you start with rules or diagrams? Filling the Pail speaks from experience in Fraction Division and explains the advantages and disadvantages of both approaches.

Games and Educational Toys

  1. On the spur of the moment, I came up with a very quick Yahtzee variation that I played with one of my students. We counted the number of rolls it took us to get a Yahtzee. Lowest score wins. He beat me badly every round as the graphic above this category attests, but on a different day, Lady Luck was with me more than him.
  2. Denise Gaskin also has a tried and true Yahtzee game variation that she calls Six Hundred. You only need to print her directions and scoring sheets, provide six 6-sided die and pen or pencils, and you’ll be ready to make memories in more ways than one.
  3. Autism Awareness Week was earlier in April. In this post, Special Educational Resources Blog reviewed three games made by Orchard Toys: Money Match Cafe (teaches about money), Look and Find Jigsaws (teaches number and letter recognition), and Bus Stop (uses processes like 3-2+4-1 to figure out how many kids are on the bus when the bus arrives at the bus station.)
  4. The Mathematical Tourist shared how to play a game called Clobber. The game has been around since 2001, but the best strategy to win the game is still a mystery. I’m sure getting clobbered will be just as much fun as winning.


  1. What did an insightful five-year-old tell Paula Beardell Krieg about triangles?
  2. Simon Gregg showed some pictures demonstrating how students had fun exploring squares in several different ways.
  3. Similar geometric shapes line up and beg you to compare their dilations in Paula Beardell Krieg’s post About Halfway There.
  4. Robert Loves Pi creates wonderful 3-D rotating digital geometric models. This one he calls the Twelfth Stellation of the Triakis Tetrahedron.
  5. Desmos is a great tool, but sometimes I need to actually see how people use it to teach. does exactly that by sharing some pictures of Desmos Geometry being presented to a class of 10th graders.

Giving Back

  1. When Women Inspire gives Three Worthy Reasons to Teach Charity to Your Kids. One of those reasons is that they will naturally learn the mathematics of money management.
  2. Read the impressive CBS News account of how a STEM Robotics team made a toddler wheelchair for a  two-year-old whose family couldn’t afford one.
  3. LMS Life Skills was practically speechless! Her class designed quilts blocks by using linear equations. Then the class made two quilts and donated them!

Linear Equations

  1. Wheeler’s Thoughts on Teaching used a bank balance problem to teach about solving a system of linear equations. The students were able to think about the problem and work on it with much fewer hints from their teacher. That makes teaching math much more fun for the students as well as the teacher.
  2. Jeff Lay created an Easter egg hunt activity to review linear equations, and he is happy to share the google docs he made with you.
  3. Ms. Wheeler exclaimed that sometimes crayons and paper do the trick while her glass created stain class art.
  4. Coincidentally, Ian Maclellan also had his class produce some stain glass art with linear equations.
  5. Alicia Phillips shared one of her student’s projects that used only linear equations and was made on Desmos.

Literature and Mathematics

  1. Imagine this carnival ride: a catapult that will send you flying through the air! Lana Pavlova and Meredith Wilkes have assisted Math Book Magic in creating the perfect design of an unforgettable carnival ride in Play with Your Math with Little Pea. How far will this catapult take you?
  2. Erikson Institute writes how Anno’s Flea Market by Mitsumasa Anno, Which Would You Rather Be? By Willaim Steig, and Whose Shoes? By Stephen R. Swinburne are Three Books That Encourage Simple Graph Explorations with Young Ones.
  3. Life Through a Mathematicians Eyes loves to curl up with a good book that features mathematics. Find out which books she has gathered and plans to read in her MathReadathon.
  4. Kelly Anne Garner received several must-have mathematics in literature book suggestions from Twitter to build a fabulous math library. Check out the whole thread.

Museum of Mathematics

  1. Chirag Mittal took charge of April’s birthday celebration of Leonard Euler. Did you know that Euler is credited with being the first to use letters from our alphabet and the Greek’s alphabet to represent some very important functions and numbers: Σ, f(x), e, i,  and π?
  2. Alan Paar of Established 1962 has put together a tour of Wendover School and the way teachers taught and students there learned several mathematical topics and other subjects from 1868 to 1930.
  3. Jo Morgan retweeted a tweet that caught my eye and was, therefore, the catalyst for bringing  MathigonOrg’s expansive and interactiveTimeline of Mathematics to this month’s carnival.

Number Theory

  1. The number 127 is a centered hexagonal number as demonstrated by the graphic above.
  2. Ramblings of a Writer recently marveled about how many things come in fours in her post Exploring the Number Four.
  3. Dr. Helen J. Williams has pictures from a very playful session on “Fiveness”.
  4. Science Switch had a few things to write about Belphegor’s Prime number, 1000000000000066600000000000001, in The Most Evil Number.

Optical Illusions

  1. When there was a day off from school in the middle of the week, BMore Energy found plenty of kid’s activities in Manhattan’s Museum of Illusions.
  2. Love Travelling takes us on a trip to see the fun-filled Vilnius Museum of Illusions. There is so much to see there!
  3. While Matematickcom shows how to make a paper optical illusion that you can make yourself in very little time.


  1. Math with P. Nik gives instructions and several examples of his Three Elastic Bands puzzles. He said they were easy to make, so I made the one at the top of this category. Follow P. Nik’s instructions and you probably won’t need to click on the tiny answer key under the puzzle.
  2. When Simona Prilogan of Fiat Lux writes a number puzzle, it is much more than it appears.  You have to study patterns inside the puzzles to figure out what the relationship really is. Give her Wednesday Math Puzzle a try!
  3. Simona included a bonus, information about Bolsover Castle, in her Monday Math Puzzle. You will find two different Math Puzzles in the middle of reading about the Castle!
  4. This clever tie matching exercise from Math with P. Nik feels more like a puzzle than a worksheet. Can you match the graph families with the correct equation families?

Statistics and Probability

  1. Yes, you can do statistics in Desmos! You can make Normal, Poisson and Binomial distributions and even graph box and whisker plots in Desmos! Colleen Young shows you what that looks like in her post Statistics with Desmos.
  2. Does El Niño play much of a part in rising global temperatures? In New Kid in Town, Open Mind answers that question and includes line graphs to help us visualize global temperature data collected since 1979.
  3. This year Easter occurred on April 21st. That seemed rather late to me, but it isn’t the lastest it could be. In Joseph Nebus’s post, What Dates Are Most Likely for Easter?, he’ll direct you to a post he wrote two years ago where all the data is lined up to figure out the probability.

Telling Time

  1. What time is it? There is more than one valid way to give the correct time, and one way should not be labeled as a smarter way to give the time than the others. That’s the message given in Dan Meyer’s Don’t Teach Math “the Smart Way”. He even suggests a lovely game from Desmos to get kids talking about telling time.
  2. After a long winter with snow causing several school days to be replaced with “e-learning days,” Educational Technology in Action wrote about using that same Desmos talking time activity in Desmos for meaningful e-learning days.


  1. On Twitter, Jo Morgan shared a photo that truly enhanced the 1679 definition of a Rhombus.
  2. Joseph Nebus of NebusResearch regularly writes about mathematics-themed comics. Here is a comic about the difference in definitions of vertex and apex. It also has a graph theory puzzle and three other comics about story problems involving addition and subtraction, slope intercept form, and paradoxes.
  3. What does the word Asymptotic mean? Hazel Clementine shared a catchy musical definition.

Thanks for coming to this month’s carnival! I hope you enjoyed it. I had a wonderful time hunting for goodies to put in the carnival and organizing it. I felt like I was on an Easter egg hunt looking for the best eggs!

Math Misery? will host May’s Playful Math Education Blog Carnival. Perhaps YOU will consider contacting Denise Gaskins and volunteering to host a future carnival! There are two open dates in the summer still available this year.

You can also visit The 126th Playful Math Education Blog carnival hosted by Math Mama Writes. . . or the 157th Carnival of Mathematics hosted by Lines Curves Spirals for more mathematical adventures!

1371 Today is a Good Day to Review Proof by Induction

0² = 0
1²  = 1

Does that pattern hold for all natural numbers? Could we claim that n²  = n?

Yes, we can, and I’ve written a proof to prove it! The proof uses a valuable concept in mathematics called induction. I remember being introduced to proofs by induction when I was in Junior High. Nowadays, if it is not part of Common Core, it wouldn’t be taught much anymore. Nevertheless, I will use it here to prove that n² = n.

Using a similar proof, we can also prove that n³ = n, n⁴ = n, n⁵ = n, n⁶ = n, and so forth!

Today is the perfect day to review how to use proof by induction so try your hand at proving at least one of those mathematical statements on your own. Use the same steps in my example: prove true for n=1, assume true for n = k, prove true for k + 1, write your conclusion. then have a very Happy April Fools’ Day, Everyone!

Today is also a very good day to review that (x + y)² = x² +2xy + y²  and NOT x²  + y², a very common error students make. Confession: I remember making that exact error in high school when I definitely should have known better. Using induction to prove something in mathematics is a valid technique, but if you use invalid equations like
(x + y)³ = x³ + y³, you will make invalid conclusions. Thus, today might also be a good day to review the binomial theorem and Pascal’s triangle. (Pascal’s triangle has numbers in its interior, not just 1’s going down the sides, after all.)

My post today was inspired by a post written by Sara Van Der Werf titled Why I’ve Started Teaching the FOIL Method Again. In her post, she not only plays a great April Fools’ joke on her readers, but she explains a tried and true way to multiply binomials and other polynomials.

I read her post exactly one year ago today, and since then, I have been waiting for April Fools’ Day to roll around again so that I could share this post with you. It is my hope that you will enjoy my little prank and learn a little mathematics from it as well.

Now I’ll write a little bit about the number 1371:

  • 1371 is a composite number.
  • Prime factorization: 1371 = 3 × 457
  • 1371 has no exponents greater than 1 in its prime factorization, so √1371 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1371 has exactly 4 factors.
  • The factors of 1371 are outlined with their factor pair partners in the graphic below.

1371 is the hypotenuse of a Pythagorean triple:
504-1275-1371 which is 3 times (168-425-457) informs us that 1² + 37² + 1² = 1371, and there’s no April Fooling about that!

1366 Fractions Acting Improperly

In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form.

On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.

This first tweet has a link explaining why it is improper to use the term improper fraction:

I hope that you will consider not labeling any fraction as improper, as well!

Now I’ll write a little bit about the number 1366:

  • 1366 is a composite number.
  • Prime factorization: 1366 = 2 × 683
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1366 has exactly 4 factors.
  • Factors of 1366: 1, 2, 683, 1366
  • Factor pairs: 1366 = 1 × 1366 or 2 × 683
  • 1366 has no square factors that allow its square root to be simplified. √1366 ≈ 36.95944

1366 is also the sum of the twenty-six prime number from 5 to 107. Do you know what all those prime numbers are?

1350 Logic is at the Heart of This Puzzle

Today’s Puzzle:

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

If you need some tips on how to get started on this puzzle, check out this video:

Factors of 1350:

  • 1350 is a composite number.
  • Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
  • The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
  • Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
  • Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
  • Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

1350 has twelve factor pairs. One of the factor pairs adds up to ­75, and a different one subtracts to 75. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 1350:

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

Facts about and Factors of 2019

Here’s a countdown you can use to ring in the New Year:

Countdown to 2019

make science GIFs like this at MakeaGif

2019 is the sum of consecutive numbers three different ways:
1009 + 1010 = 2019
672 + 673 + 674 = 2019
334 + 335 + 336 + 337 + 338 + 339 = 2019

There is one way that 2019 is the sum of consecutive odd numbers:
671 + 673 + 675 = 2019

2019 is the difference of two squares two different ways:
338² – 335² = 2019
1010² – 1009² = 2019

2019 is the sum of three squares nine different ways:
43² + 13² + 1² = 2019
43² + 11² + 7² = 2019
41² + 17² + 7² = 2019
41² + 13² + 13² = 2019
37² + 25² + 5² = 2019
37² + 23² + 11² = 2019
37² + 19² + 17² = 2019
35² + 25² + 13² = 2019
31² + 23² + 23² = 2019

2019 is the hypotenuse of a Pythagorean triple:
1155-1656-2019 so 1155² + 1656² = 2019²

2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2¹ + 2⁰ = 2019

2019 is a palindrome in a couple of bases:
It’s 5B5 in BASE 19 (B is 11 base 10) because 5(19²) + 11(19) + 5(1) = 2019,
and 3C3 in BASE 24 (C is 12 base 10) because 3(24²) + 12(24) + 3(1) = 2019

Every year has factors that often catch people by surprise. Today I would like to give you my predictions for the factors of 2019:
2019 will have four positive factors: 1, 3, 673, and 2019
However, 2019 will also have four negative factors: -1, -3, -673, and -2019

Which factors, positive or negative, will be your focus in the coming year?

Finally, I’ll share some mathematics-related 2019 and New Year tweets that I’ve seen on twitter. Some of these tweets have links that contain even more facts about the number 2019.

That tweet inspired me to make my own 3 × 3 Magic Square where every number is different but every row, column and diagonal totals the same number:



And finally, here is my contribution to 2019 twitter:


825 Quarters Make Dividing by 25 Easy

Numbers ending in 00, 25, 50, and 75 can be divided evenly by 25. How much is 825 divided by 25? That quotient is the same as the answer to “how many quarters are in $8.25?” (A quarter is ¼ of a dollar and is written .25 or 25¢.)

You probably could visualize the answer in your head even if I hadn’t included a picture! That’s why I often ask kids the how-many-quarters question when they are stumped dividing something by 25 . It seems that kids are always able to give the quotient after that question. Notice that “8.25 ÷ .25 =” and  “8 ¼ ÷ ¼ =” have the same answer, too. You can also ask that how-many-quarters question to find the answer when something is divided by .25 or ¼.

It would almost be as easy to divide $8.26 or $8.39 by 25. The quotient would be the same as the problem above but with some loose change becoming the remainder. Using money for division problems could even help kids better understand dividends, divisors, quotients, and remainders.

Here’s an example of a how-many-quarters type question that will help you divide by 75, .75 or ¾.

We can count and see that there are 11 sets of 3 quarters in $8.25. That means that $8.25 ÷ .75 is 11. It also means that 8¼ ÷ ¾ = 11.

Dividing by fractions can be a very abstract concept for students, and they may ask questions like, “What does 8¼ ÷ 1¼ even mean?” Again quarters come to the rescue! 5 quarters can be so much more friendly than 1¼ is. Shorthand for 5 quarters is the fraction, 5/4. Since they have the same denominators, dividing 8¼ by 1¼ is the same as dividing 33 by 5:

Kids think money is more fun than math, but money is just a subset of mathematics which is full of lots of other fun topics, too. Here are a couple of ways other educators have used money to teach a math topic.

Jen of Beyond Tradit’l Math shared her way to teach subtracting decimals using money. Her method will surely captivate any child who tries it and even make regrouping fun:


Robert Kaplinsky uses several very short videos to keep students engaged without them actually touching any money. Check out the replies, too. Paula Beardell Krieg’s excellent $1.00 art project is there, and that would be fun for anyone 2nd grade or older to do:


Now back to the number 825.

Clearly 825 has to be divisible by both 5 AND 3 in order for (821, 823, 827, 829) to be the fourth prime decade, which it is.

  • The last digit of 825 is 5, so it is divisible by 5.
  • 8 + 2 + 5 = 15, a multiple of 3, so 825 is divisible by 3.
  • 8 – 2 + 5 = 11, so 825 is divisible by 11.

All numbers ending in 00, 25, 50, or 75 can have their square roots simplified. If you were trying to simplify √825, you could visualize quarters in your mind to easily divide 825 by 25. Then √825 = (√25)(√33) = 5√33

Here is 825’s factoring information:

  • 825 is a composite number.
  • Prime factorization: 825 = 3 × 5 × 5 × 11, which can be written 825 = 3 × 5² × 11
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 825 has exactly 12 factors.
  • Factors of 825: 1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825
  • Factor pairs: 825 = 1 × 825, 3 × 275, 5 × 165, 11 × 75, 15 × 55, or 25 × 33
  • Taking the factor pair with the largest square number factor, we get √825 = (√25)(√33) = 5√33 ≈ 28.72281