Welcome to the 176th Playful Math Carnival—a number rich with intrigue and hidden patterns! Last month, the carnival was hosted by my friend Della, over at The Beauty of Play. She knows how to both play and instill beauty in education. Go over there. You won’t be sorry.
What’s So Special About 176?
We will not break with tradition. Before we begin the Playful Math Carnival proper, let’s dive into the fascinating numerical landscape of 176.
176, an even and composite number, invites exploration with its diverse factors: 1, 2, 4, 8, 11, 16, 22, 44, 88, and 176. Factor it using a Prime Climb representation, and you’ll reveal its “colors” as four orange spaces for 2 and one red space for 11, adding a visual layer to its factorization.
But does it fit the mold of perfection? Almost, but not quite! Its factors sum to 196, giving it “abundant” status, as the total exceeds the number itself.
12 is an example of an abundant number.
And what of happiness? Yes, 176 is a happy number! With repeated summing of the squares of its digits, we eventually land on 1, sealing its fate as a joyful integer in a journey as winding as the blog carnival itself.
Represented in Roman numerals, 176 is CLXXVI, and in binary, 10110000. Curiously, this number and its square, 30,976, and cube, 5,451,776, end with the same digits, 76—an elegant consistency across powers.176 is not in the Fibonacci sequence nor is it a triangle number. If you were wondering.
Counting to 176 might take around 28 seconds, but in this carnival, we’re just getting started. Let the celebration of the 176th Playful Math Carnival begin.
The Blog Carnival
In preparation for the 176th Playful Math Carnival, I began, as always, with a visit to a familiar math community – Twitter’s #MTBOS. This sparked a journey down a Fibonacci rabbit hole—an irresistible path for any math lover, especially with Fibonacci Day approaching on November 23. So, in the spirit of that iconic sequence, I chose Fibonacci as our theme for this carnival.It started with this post from Mathigon:
The number patter 1, 1, 2, 3, 5, 8, 13, … doesn’t just describe the growth of a rabbit population (under certain idealized conditions). The pattern also appears in sunflowers, pinecones, and many other places in nature! Explore more at https://t.co/LKFj4enWdOpic.twitter.com/QHOnljd9wr
Don’t just play the video. There is an excellent lesson on rabbit breading and the Fibonacci Sequence. It’s part of a larger group of lessons on sequencing and patterns. You can click the link in the link above or you can find it here.
Tom from Tom Rocks Maths and master of the Numberphile YouTube channel posted the Frog Problem. What might this have to do with Fibonacci? Or Tribonacci?
www.freepik.com
A frog wants to cross 100 lily pads, and is able to jump 1 or 2 spaces at a time. How many different ways are there to cross (where the order matters)?
Extension problem – what if the frog can jump 1, 2 or 3 spaces at a time?
Who is Fibonacci Anyway?
Your kids don’t have to be in middle school or high school to begin exploring Fibonacci. Sarah McClelland from Little Bins for Little Hands posted a series of playful activities for exploring the Fibonacci Sequence including: Zentangles and the Fibonacci Spiral.
The folks over at Creative Star Learning celebrated Fibonacci Day last year by exploring Fibonacci in Nature. In that post you’ll find a discussion of Romanesco Broccoli that is both a fractal and displays the Fibonacci pattern and introduced me to The Fib.
Jennifer Quellette at Artstechnica does a deeper give into Romanesco Broccoli and it’s patterns. Linda Li at Garden Betty does too, but she will also tell you where to find it and how to cook it.
I had never heard of A Fib before. So, off I went. Denise Gaskins posted about Fibs on her site. So I found myself, along with Alice, deep down the rabbit hole. I started at the Farm School with Fib Foolery. What is a Fib? Greg Pincus, the inventor, says, “A Fib is a six line, 20 syllable poem in which each line gets its syllable count from following the Fibonacci sequence. This means the six lines have a syllable count of 1, 1, 2, 3, 5, and 8 respectively. Some would say the first number of the Fibonacci sequence is actually a zero… so imagine every Fib starting with a beat of silence.”
The FIB
by Greg Pincus
OneSmall,Precise,Poetic,Spiraling mixture:Math plus poetry yields the Fib.
After reading about Fibs, I jumped back on Twiiter’s #MTBOS and searched for Fibonacci. They did not disappoint. Nearly every blog post had a mention of Vi Hart’s videos on spirals. Including this one, by Math Hombre. He did an entire week on spirals with his class. Vi Hart’s first video is below. There is also a part 2 and part 3.
Massimo shared a post on orbital patterns, pointing out—as so many have—that the Fibonacci sequence seems to crop up everywhere. This sparked a fascinating reply about bubble patterns and whales. I was skeptical but did a quick search, and sure enough, the footage is real.
Begin at 2:08, drone footage of humpback whales, shows a remarkable example of Fibonacci spirals in nature. In a phenomenon known as bubble-net feeding, groups of four or five humpbacks dive down, releasing streams of bubbles in a spiral formation at precise moments and in a perfect sequence to trap fish. This coordinated spiral pattern follows the Fibonacci sequence, drawing these whales into a natural rhythm echoed in seashells, flower petals, and now, bubble nets.
The deeper I go down this Fibonacci rabbit hole, the more it feels like math is woven into the very fabric of life. If we let math become boring, we’re missing a profound opportunity—math is full of beauty and wonder, enough to keep us captivated for more than a lifetime. Surely, we can make it interesting for just 13 short years!
What if you want some lesson plans for Fibonacci? We all have that documenting to do. It is hard to measure exploring, noticing and wondering. The Fractal Society has an excellent set of Fibonacci lesson plans designed for students from 3rd to 12th grade, though I’d say even 2nd graders would enjoy and benefit from them.Susan Ardia over at Math Wisdom collected a slew of Fibonacci resources for elementary and high school that include nature, music, the stock market and more. Transum.org has an online tool called the Fibonacci Quest. It resembles Tom’s Numberphile’s frog lily pad problem. And it was interesting to play around with. Tina from Tina’s Dynamic Homeschool posted several living books on Fibonacci and invites you to make Fibonacci Lemonade.
Arthur Benjamin expresses my feelings about why we should study math. This is his famous video on the Fibonacci series. No nature, no stock market. Just patterns inside the series itself. It is a side hole in the rabbit hole.
Daniel Mentrard noted that the diagonals of Pascal’s Triangle happen to be the Fibonacci sequence. Isn’t that interesting?
I am a huge fan of Cuisenaire Rods. It just so happens that if you partition Cuisenaire Rods, you end up atPascal’s Triangle. If you have some base ten blocks like Math-U-See or Cuisenaire, this is a great activity. In any event, isn’t that curious? But that is a rabbit hole for a different time. Thank you for joining me for the 176th Playful Math Carnival. If you’d like to host a carnival, you can sign up with Denise Gaskins, creator of the carnival, math game extraordinaire, and my fairy math mother here.
Welcome to the 167th Playful Math Blog Carnival, except it’s not really a blog carnival because blogs are a dying breed. It’s more of a internet/social media carnival. That’s not the point really, it’s about playful math and there is still plenty of that on the web.
Last month, the carnival was hosted by Sue at Math Mama. If you missed it, you should check it out.
The Number 167
It’s traditional to start the carnival with a discussion on the carnival number: 167. Number 167 is a Happy Number. I know, I know – what in the world is a happy number? Does the existence of happy numbers mean there are sad numbers? Yes, sad numbers do exist. According to Wikipedia, “In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit.” Let’s try this, I’m a bit skeptical but off we go:
Just being honest here. This isn’t looking like we are going to get to one, but let’s try it again.
Nearly done. I can see the end! One more time.
That was a delightful surprise. Sad or unhappy numbers don’t offer this little surprise. 167 is full of surprises.
For instance, we can insert a couple of symbols and get a correct equation: 1 + 6 = 7. Isn’t that fun? This would make a great exploration for learners: Using only complements inside of and including 10, how many of these types of numbers could you create? How would you know when you had them all?
167 is a strictly non-palindromic number, which means it can’t be written the same forward and backwards in any base between 2 & 165. The reason we stop at 165 is because every number n (in this case 167) is the palindrome 11 in the base n-1 (so 167 – 1 = 166).
Not only is 167 prime, but it is a safe prime. That means it is the sum of a doubled prime plus 1 or 2(133) + 1. Not sure what this has to do with qualities of safeness, but if you do, let me know in the comments section. It is also a full reptend prime. Not sure I’ve wrapped my head around full reptend primes and Fermat’s Quotient. But this I get: 7 is also full reptend prime. That means there is a 6-digit number such that when you multiply it by any other integer, the digits are exactly the same and in the same order but start in a different place. The number for 7 is 142,857. Now go ahead and multiply it by integers 1 – 9. Notice what happens. This is the stuff that makes math cool. The same thing is true for 167 – but the number is 166 digits long and I don’t know what it is, nor can I get my head wrapped around that.
On To Other Mathy Goodness
While #MTBOS (MathTwitterBlogOSphere) isn’t what it used to be, there is plenty of great stuff to be found on social media. Though the write ups are a bit shorter these days.
This is just some of the fun, playful math stuff I found from the August 2023 chit chat across the web. There was a lot of buzz on what’s left of the #MTBOS on Twitter about this game Chris Hogbin created called Number Hive. This is a seriously addictive game. It’s free and no ads. Yeah. I dropped Sudoku on my phone for this little gem. It’s math facts meets the game ‘GO’ or ‘4 in a Row’ but requires a lot more strategy. There’s an app, however you can also buy a full set of game boards (super cheap) to do in-person, 2-player games. And there’s even Number Hive Tournament Rules. The Number Hive boards mean the game can be played at math clubs, classrooms, homeschool co-ops, and family game nights. Just warning you now, you’ll be fighting with your kids over the app.
This is a different version of “4 in a Row” created by Joe Swartz at Exit10A. I learned about it because Robert Kaplinski is playing this with his elementary students.
It seems like it’s the season for new math books on math play. There are so many inexpensive resources to add to the curriculum to make math playful. This is just what I found posted online in August.This wasn’t exactly in August, but it’s close. Ben Orlin, from Math with Bad Drawings, has published a free companion to his book Math Games with Bad Drawings, called Solo Flights. It’s all about how to turn the games in his book into games you can play alone. If you aren’t following Ben Orlin, you should be. This post isn’t from August either, but the sale on the Super Tangrams ends in August (so I think that counts) on Henri Picciotto’s Mathed site. Since there is no other place to get Super Tangrams, you might want to check it out and all of the other tangram material that he’s published – much of it free. Tangrams are fabulous math fun and depending on the puzzle, can be both simple and quite challenging. Excellent spatial problem-solving activity and, of course, play.Super Tangrams:
There’s a lot of buzz surrounding the book Math Play by Libo Valenciaand I’ll probably be picking it up – because you just can’t have enough books on playing math and this looks interesting.
The folks over at Abacus created the very best recipe for a bubble mixture. There are still enough humid days to make it worth putting the recipe together plus winter is coming. Plus, according to them, this recipe consistenly makes large, long lasting bubbles (one lasting 24 hours). How is that for the intersection of math, science and play?
Over on Twitter, Abacus also dropped this mesmerizing video. Geometry and music. You can’t avert your eyes.
Speaking of geometry, Karen Campe wrote an excellent article on using the area model for working out hard to grasp math ideas. We are big fans of the area model. The post does a great job of using both a drawn model and one that uses manipulatives. For those of you who’ve seen those YouTube videos making fun of the area model as if it’s this weird thing common core dreamed up – you’ll find it in Ray’s Arithmetic published 1885. Did you ever wonder where we got the long division symbol?
If you don’t have enough to explore and need some more, James Myklebust-Hampshire tweeted about the sembl app. It’s a free app that contains over 500 tasks that are designed to provoke thinking.
What big ideas could you uncover with your class while playing ‘Behind the Yellow Door?’ Before the last blue door disappears, ask ‘what might be behind each door?”Available for free, alongside 500 other tasks, at https://t.co/AE5E7hWgZr#iteachmath#mtbospic.twitter.com/8gZi4I9tue
We’re nearly done with our August Playful Math Blog Carnival. Before you go, check out Denise Gaskins’ Math Monday GameStrike Out. Denise is the one who started this carnival and keeps it going. You can subscribe and get all kinds of playful math activities in your inbox or get one of her highly recommended playful math books. Also, let her know if you’d be interested in hosting a Playful Math Blog Carnival.
Oh, before we go, The International Day of Mathematics is March 14th, 2024. What does that have to do with August this year? Well, they just announced next year’s theme. It’s Playing with Math. Thought I’d let you know since this is the Playful Math Blog Carnival and all.
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.
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 stormsnowfall 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Stars
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.
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.
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.
