December 2


Playful Math Education Blog Carnival #143

This is the 143rd Playful Math Education Blog Carnival and the first one on my new website. This carnival was started by Denise Gaskins many years ago, and it's still going.  Given that this is the November/December Carnival, the theme this month is winter math play.

Last month's carnival was hosted by Joseph Nebus. While you are there you might want to check out his 2020 Mathematics A to Z series

Playful Math Blog Carnival 143 - By the Numbers

It's been tradition that each blog carnival starts by sharing some information on the blog carnival number. This month's number is 143, which is a composite number with the prime factors of 11 and 13. It's also the sum of 7 consecutive primes 11 + 13 + 17 + 19 + 23 + 29 + 31.

It's also a self-number. What in the world is a self-number you ask? A self-number is a number that cannot be written as the sum of any other natural number and it's digits. For example, let's start with 9. Because we are starting with 9, we are going to call it the generator.  If we sum the natural number in the generator and it's digits we get:

9   +  9 = 18

The first 9 is the natural number, the second 9 is the digit. We add those together and we get 18. Eighteen is NOT a self-number. If we use 18 as the generator we get:

18 + 1 + 8 = 27

The first number we have above is 18, that's the natural number. And then we add the digits 1 and 8 to it to get 27. This means that 27 is NOT a self-number.  The natural number 143 has no generators, therefore, it is a self-number. There are 5 self-numbers less than 10. Are they odd or even? Why?

Winter - By the Numbers

It doesn't matter where you look, there are always things to count and lots of things to wonder about. How many snowflakes in a snowman? How many points are on a star? How many turkeys do we eat every year? Does anyone actually eat all the fruitcake that is sold each year? Here is winter and her holidays by the numbers:

$38,993.59: PNC calculates the cost of the 12 Days of Christmas every year. Except, I couldn't find it this year.  The Institute of Mathematics and Applications put out a fantastic post about the math inside the 12 Days of Christmas - it turns out some surprising patterns. 

 1.6 billion:  The number of  Christmas Cards we purchase each year.

15,000:  The number of Christmas Tree farms in the US. 

15-20 million: Christmas Trees  sold each year in the US.

8: Days and nights of Hanukkah, and a number of days that a one-day supply of oil miraculously burned during the time of the rededication of the temple by the Maccabees.

19: Number of celebrities mentioned in Adam Sandler’s Hanukkah Song.

84 Inches: 24 hour snowfall record for the US. Recorded at the Crestview maintenance station of the California Highway Department on Jan. 14-15, 1952.

194 inches: Largest single storm snowfall  recorded at the Norden train depot near Donner Summit, in the Sierra Nevada, April 20-23, 1880.

6: Number of points on a snowflake.

180 billion to 10 quintillion: Molecules of water in a snowflake.

3.1 mph: Rate at which snowflakes fall.

Games and Puzzles

As I've gotten older and see how my kids learn, I spend way more time playing games and doing puzzles which force my kids to think and not just blurt out correct answers. Correct answers do not mean that students understand. I've collected a few winter and holiday related Games and Puzzles to help you get your students thinking this month.

It is the season of Advent and Mathigon's Advent  countdown puzzles are a lot of fun.  Denise Gaskins has collected an entire list of different math related advent calendars. All of them are worthy of exploring.

Holiday Scavenger Hunt from Math Curious for practicing multiples, primes and square numbers. 

David over at What's New made this 3D present packing puzzle which should keep youngsters busy for several hours. 

Toni from Our Family Code created a whole lesson on Pascals Triangle and Christmas Trees which leads directly to Lacy's Cuisenaire Rod partitioning activity.  If you want to take this even further for older students or if you are generally curious, you'll find this excellent post on The 12 Days Christmas and Pascal's Triangle at The Conversation by Michael Rose - I had not discovered the petals on my own or read it anywhere else. This is the kind of math that gets kids excited about math. It is ripe for wondering all kinds of things, even if there are no immediate answers. And it's shocking, delightfully shocking. No child should get through school without having explored this kind of thing. 

Holiday and Winter Themed Math Art

We love to make salt dough ornaments at our house. If you follow the directions in the linked post, including giving them a coat of white spray paint, they will make a fabulous base for these  Zentangle ornaments from Craftwhack.  I've been mezmerized by the repeating patterns and simplicity of Zentangle drawings for some time. If you want to explore some of the math behind Zentangles Isabel at Math in Zentangles has your back.

Photo Courtesy of  Craft Whack.

String Art and Mathematics

Long before I discovered string art, I learned about modular multiplication on a circle.  The gist is that you start with 10 points on a circle and you plot the times table on the circle using the last digit of the product. If you were doing the 4's times table, you would draw a line connecting 4 -> 8 -> 2-> 6 -> 0 -> 4  and the pattern starts all over again with 4. The patterns change depending on how many points you have on your circle and the multiplier.

If you have a 35 point circle, then you just skip count around the circle instead of plotting the last digit. It's interesting to see different patterns emerge.

 Wolfram has a nifty applet that allows you to play around with both the multiplier and the number of points on the circle. It's pretty mesmerizing to watch the patterns emerge. 

Photo Credit: Wolfram

Someone got the idea to turn this whole thing into holiday themed string art. Which also makes for a great math activity.

The folks over at Geometry Expressions put out this PDF on the math behind string art and it will help you create lesson plans or math talks based on your student's string art creations. 

I find it helpful to see someone else do what I'm attempting to do. I found this video tutorial on making string art greeting cards. 

 Schooltime Snippets made these Christmas Tree Cards for Little Folks.  I like them because they use yard instead of thread and even very young children can participate in this math activity.

For the more daring , Christel at GoosArt Embroidery has some lovely patterns for all occasions including the holidays. And Vicki Neale talks about the math behind her cards here. 

I really love this version of a tree from Hubart.

Malke over at Math in Unexpected Places created a bunch of modular multiplication stars that don't require a needle but use cardboard with slits to hold the string rather than holes. These will make nice ornaments instead of greeting cards. 

Snowflakes and Other Cold Things

In case you didn't know, December 27th is the National Make A Cut Out Snowflake Day. It seems a bit strange to limit the snowflakes to just cut out snowflakes given that we are taking the entire day to celebrate. 

Photo Credit: Simon Beck

If you've never seen Simon Beck's geometric snow and sand art, you should check out his Facebook page. I noticed he's using some Zentangle patterns in his sand art now. 

Before we make Snowflakes, let's first find out why it snows.

Karen Cox from Pre-Kinders made some very nice pattern block snowflake mats.  These are great for pre-k and kindergarten crowd. At Frugal Fun For Boys, Sarah made some templates for pattern block snowflakes. Using the cutout templates you can  create your own designs, save them, and display them.

Della at The Beauty of Play  introduced paper quilling into their homeschool. She made this beautiful quilled paper snowflake. You should check out her blog just for the photography.

 If you've never done paper quilling before, you can learn the basics at the Papery Craftery, and they also have some wonderful patterns for paper snowflakes.

Photo Credit: The Beauty of Play

We can't have a conversation about snowflakes without a discussion of the Koch Snowflake. Evelyn Lamb wrote a lively piece over at Scientific America on the Koch Snowflake and it's applications - including giant pecan pies.  

I get pretty excited about fractals and tessalations. And low and behold, the Koch Snowflake tessalates. The best visuals of Koch Tessellations was put out by the University of Groningen.

If you know of a good lesson plan for studying Koch Snowflakes for the elementary set, please leave a link in the comments. 

Several iterations in the construction of the Koch snowflake. Credit: António Miguel de Campos Wikimedia

I found the motherlode of Starwars, Harry Potter, and Frozen snowflake patters  over on Anthony Herra Designs. These are very well done and a lot of fun. He's been putting out new designs since 2012.  

Be forewarned - you are going to need an Exacto or similar knife, but these were magical for my 10 year old son.

First  Palatte has some easy to follow and beautiful Snowflake Patterns.  Just fold along the lines and cut. 

If you don't want to cut out snowflakes you can calculate the sum of a snowflake with David Nacin at Quadrata and the Sum of  A Snowflake Puzzle.

Clarissa Grandi over at Artful Maths made these wonderful Origami Snowflakes, she includes links to patterns and video tutorials. 

Credit: Artful Maths

And of course, it would be a terrible thing to forget Vi Hart and her hexaflexaflakes. These are a jump from hexaflexagons but only a little bit.  We have made hexflexaritos (the burrito version) - I don't recommend the peanut butter and jelly variety. The jelly doesn't stay put and gets pretty oozy. I cannot say much about hexaflexa bologna flakes. We don't eat bologna - but if you do, let us know how it went. 

Paper Crafts and Origami 

When I think of the holidays and math art, I always go straight to polyhedra. It doesn't matter which shape you choose, they will all make lovely ornaments. But if you aren't sure what to create, I've gathered some resources for you.

Vi Hart made a lovely pattern for a dodecahedron 12 Days of Christmas Ornament. On this page you'll also find her collected Christmas posts, which include both music and math. 

At Woo!Jr, they've made some cute pyramid shaped ornaments which are a bit easier to handle for younger crowd than the more complex polyhedra shapes.

Borromean Rings

Paula Beardell Krieg from Bookzoompa - Made these wonderful ornaments from Borromean Rings. There is an updated twitter file which is more mathematically accurate.

These look like polyhedra but they're not. Paula made a nice video to show you how to fold them.  While you are there, check out her other work. 


Many moons ago, I resided in Pennsylvania. There was nothing better than heading to Bethlehem, the Christmas City, to tour stores, see the lights, and visit Christkindlmarkt. One of my favorite stores was the Moravian Book Store. It had a wonderful collection of unique  home items, furniture, Christmas decorations and a fantastic kids selection. You'll notice, in the windows, the famous Moravian Star. 

Moravian Stars are more than just Christmas Decorations. They started out as a math lesson. If you'd like to know the very mathy history behind the Moravian Star, you can jump on over to the Morning Call

Moravian Book Shop Bethlehem, Pennsylvania. 

Jennifer, over at Jennifer Maker, made a beautiful pattern for a 20 pointed Moravian Star tree topper. It's lovely, but looks complicated. For those of you who aren't quite as adventureous, you can try your hand at the Froebel star here.

The five pointed star, which is a type of polygon, has a special number hidden inside, a study of the 5-pointed star would make a great starting point for other math explorations about that special number. If you just want to make some 5 pointed stars the best tutorial I found was on Homemade Gifts Made Easy.

If you drop over at Muslin and Merlot you'll find these great 4 pointed star treat boxes.

Christmas Trees

Gathering Beauty founder Emma created a tutorial for making easy origami Christmas Trees.  I'm pretty fond of easy. But if you are more daring...

These trees are made of six sheets of folded paper and are an intermediate project. If you've never completed an origami project before, you might want to do the above tree before you attempt this project. I have to admit that Jo's trees are enchanting to look at. 

Last Words - Almost

This was probably my favorite Playful Math Education Blog Carnival to write. There is so much out there to explore that I'm sure I've barely scratched the surface.

We'll be Celebrating National Cut Out a Snowflake Day, and try our hand at Koch Tile tessellations. We'll be making a lot of Polyhedra Ornaments using Stacy Speyer's book Cubes and Things. 

Credit: Edmund Harriss Patterns of the Universe, a mathematical coloring book by Edmund Harriss and Alex Bellos

Come back because I'm going to keep adding to this post. So if you have a blog post that should be included, drop me a line in the comments.

If you want to host a Playful Math Education Blog Carnival, or have a playful math post that you think belongs in a blog carnival, contact Denise at Let's Play Math.


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Notice, Wonder, Discover

Notice, Wonder, Discover
  • My goodness! this is a fantastic post Sonya. You made a wonderful thread to combine a lot of good activities and thinking.

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    Caleb Gattegno

    Educator, Mathematician 

    Only awareness is educable. . ."