Playful Math Carnival 183

Welcome to the 183rd edition of the Playful Math Carnival. Begun by Denise Gaskins, previously known as Math Teachers at Play. If you'd be interested in hosting, just drop her a line. Previous editions are listed here! The most recent was Sue Van Hattum's Storytelling Carnival.

183 is a semiprime - though not too mysteriously, as 3 is a factor.

183 squares in this quarter circle. (Cf. A001182) 

New to me was Legendre's Three Square Theorem. Every number can be written as the sum of three squares, unless it is of the form 4^k(8*m+7). (Number theorists are wizards.) 183 = 8*22+7 so it cannot be. Also then, 175 and 191 cannot. Are there any closer integers to 183 which cannot? 

The totient of a counting number is how many natural numbers less than it are relatively prime. The totient of 9 (8, 7, 5, 4, 2, 1) is 6. The totient of 6 (5, 1) is 2. The totient of 2 is 1. Since 9 = 6+2+1, it's the sum of its iterated totients, which makes it a perfect totient number. So is 183! What is its totient sequence? I'll get you started, ϕ(183)=120. 

Projective planes are a fascinating structure that grew out of the study of perspective. One way to generate them is from a finite field. A finite field of order p^n (p a prime) has a projective plane of (p^n)^2+p^n+1 points. The Fano Plane on Z_2 is the most famous example. In our case, 183=13^2+13+1! What's the one before and the one after in this pattern?

183 is also in the sequence n^2-n+1, Hogben's oddly called central polygonal numbers. That means it's on the diagonal of the Ulam spiral. Which side will it be on?

Do you have a favorite 183 fact I've missed?

This edition is dedicated to those regularly producing and sharing new content. 

Regulars

Karen Campe is the definition of regular! Ever month a calendar of problems, and more besides. For example, December 2025 and her Pythagorean Squares activity.

Jim Propp does a monthly deep dive into a different topic each month for his mathematical enchantments column. So carefully crafted and well written. Consider this on Matrix Multiplication. Is it ugly?

Monthly? Pshaw! Pat Bellew has a daily post: on On This Day in Math. But also many other tidbits, like what about monkeys typing Shakespeare? (Of course, some wit pointed out that we already produced a primate who wrote Shakespeare. Shakespeare.)

Denise Gaskins gets close to daily, with her weekly Monday Math Games and Thinking Thursday and other posts besides.

Jenna Laib is a treasure trove. In addition to regular, amazing Slow Reveal Graphs, like this one on butterfly wingspans, she writes deep think pieces like this on culture and identity in math class.

David Petro does a regular feature with lists of resources as well.

Colleen Young does a feature of her own gathering links from all over, but also other posts, like these nifty RISP problems.

Kristin Gray covers a variety of topics, but I particularly enjoyed her list of math stories.

Maths for Humans has a variety of content, but I enjoyed this reflection on learners' problem solving.

Cambridge Mathematics does a monthly Espresso summarizing math ed research, and other features like their new podcast.

The Futility Closet isn't just math, but has amazing tidbits on a frequent basis. Like Lee Sallow's geomagic squares. 

I'm new to Dr. Austin's Maths, but she adds good problems on a regular basis.

Substack

Dylan Kane weekly shares his deep thoughts on math, teaching and teaching math. Here he pitches gradual increase of difficulty, vs release of responsibility.

Fawn Nguyen is back writing again. Some refreshed old hits, some all new content. Always amazing.  Here she's writing about when students write the questions.

Dan Meyer writes a lot about edtech especially AI, but still does some straight teaching talk, too. Like teaching is harder when the math is easier.

Grading for Growth is an alternative grading blog, featuring two of my colleagues among others. Here's Robert Talbert writing about the Four Pillars of Alternative Grading.

Videos

Christina Tondevold's got a regular Buildring Math Minds video series. I thought this one was a big one on the purpose of math learning, Doing and Thinking.

Meet a Mathematician introduces us to mathematicians and math teachers, like the cool Greg Lakey.

Kyle Pearce & Jon Orr have a Make Math Moments video more than once a week, tackling issues like moving towards efficiency.

Watch everything Howie Hua makes. Here he's arguing that 3x8 and 8x3 are different.

Here and There

Grant Snider is a brilliant cartoonist, getting a bit mathy on occasion. Especially with this bit on infinity.

Dan Scher is the leading person I know on Web Sketchpad. Here he shares a new idea Algebra Mazes.

You never know what Tanya Kovanhova is going to write about, but here's a math magic trick.

Sue Van Hattum is writing a terrific math series, getting at big ideas from an accessible angle. She'd love some beta readers... And she has put together a collection of teaching gems from the books.

I've been enjoying the art from the JMM/Bridges 2026 show.

Crazy cool Möbius band explorer from Ben Sparks.

Those fool Teaching Like Ted Lasso guys are back with a new episode on Joy and Softball. 

Sophia (FractalKitty) leads #mathober every year. Lots of interesting bits of math, art and coding were shared. 

Puzzles/Games:

• Karen Campe suggested two. One Up, a kind of number maze.
• Jigsy is the other. That's had an open tab since I first saw it! A geometric puzzle where you can scale the pieces. So clever. I appreciate the levels of difficulty.
• Dive also gets a permanent tab. It's like 2048 for factors.
• A Mathblr user came up with these neat irojirai puzzles on coloring a square.
• I gathered 13 fraction games from a teacher ed course. 
• I also tried to make a tangram like puzzle with 30-60-90 triangles. Here's some examples.

A list of 183 cm tall celebrities? I don't know all these people.

And on that note, I'm out! Hope you found something to get you playing with math.

13 Fraction Games

 As a part of Michigan's restructured elementary certification, we got the opportunity to restructure our two math ed courses for elementary into an introduction with geometry and measurement (Statistics has a companion course on data and statistics.), Number and Operations, Fractions and Decimals, and Early Childhood Mathematics. Most students will take all four, depending on grade band certification, and those with a math emphasis have 3 more. (I'm teaching the statistics one next semester for the first time in Winter 2026.) I taught the fractions course for the first time this fall, and thought I'd share the fraction games we used. It's a community-based learning course, so we met at an elementary school and taught kids twice a week, one 4th and one 5th grade lesson. So these are kid tested! Mostly they are shared elsewhere or modifications of games I hope you know already. Games without a creator name are mine or my variation on another game, though most of these are pretty simple so I'm sure they're out there from others!

The course begins thinking about fair share problems in context. 3 brownies to share between 2 friends. How much does each get? And trickier and trickier from there. The pictures learners make from these problems are our first models.

Then we start exploring a more formal linear model of fraction bars for 1/2, 1/4, 1/8 and 1/16. Sometimes I'll have them make their own from construction paper, sometimes cut and decorate from a print out. On card stock if that's a possibility. With these two classic Marilyn Burns games, there's so much to notice. Composition and decomposition, equivalence and more. But they start where I want to start. Fractions are quantities in relation to a whole. Fraction numbers are inherently confusing. A 1 and a 4 and it means something different... division is involved somehow? 

Fraction Cover Up
By Marilyn Burns

Fraction Kit for each player/team.
Die with ½, ¼, ⅛, ⅛, 1/16, 1/16 or a spinner set to the same.

Put the whole down. The goal is to cover up the whole EXACTLY with the pieces you roll.

On your turn: roll or spin. Add that size piece to the whole. If the piece is too large to fit in the space remaining, pass the turn.

First team to cover exactly wins.

Example: You have ½, ⅛, ¼… and you roll ¼. Too big, pass the turn.

Fraction Uncover
By Marilyn Burns

Fraction Kit for each player/team.

Die with ½, ¼, ⅛, ⅛, 1/16, 1/16 or a spinner set to the same.

Cover the whole with two 1/2s. The goal is to uncover the entire whole exactly.

On your turn roll the die. If you have that size piece, remove it. If you don’t have that size, you have an option. You can break up a piece into any pieces which equal it. You can also choose to keep the pieces you’ve got.

Example: You have two 1/2s, and roll a ¼. Bummer! You choose to replace a ½ with ¼, ⅛, and two 1/16. 

Example: You’re down to just a ¼ and 1/16. You roll a ⅛. Bummer! Do you break up the ¼ or keep it?

Extras:

• video explaining Fraction Cover Up
• handout with these two games

Eighths vs Sixteenths

One team gets the eighths and the other the sixteenths from the fraction kit. The eighths team has a spinner with 1, 2, 3, 1, 2, 3, and the sixteenths has a regular 1-6 die. On their turn, they spin or roll and get that many pieces. Trying to make exactly one. If you need 2/8 and spin ⅜, you pass your turn.

On the replay, switch sides.

Be sure to ask teams how much they have and how much they need. This works on iteration, adding with like denominators, and sums to 1.

Another early fraction game I love that can be repeated later is I Spy a Fraction.

I Spy a Fraction

Stand in a circle so you can see one another.

One person in the group says I see something true about ____ of us, filling in the blank with a fraction. Others, starting to the right of the spy, try to guess what characteristic you’re describing. Once someone has guessed (or there are no more guesses) invite another person to spy a fraction. (I often start with glasses.) To choose a next spy, you can go around the circle in order, or have the person who guesses correctly be the next spy. Play several rounds.

Depending on student experience, you can try giving a reduced fraction. You may have to specify whether you are included or excluded to get a total that can have simplified fractions. Students new to the game or to fractions, keep fractions unreduced.

Fraction More

One game board.
Each team needs 12-16 squares of one color. (Or any markers you can tell apart!)

Before playing the game the first time, it is a great idea to see what some of the fractions look like. Roll the die, and practice making 1/16, 1/8, 1/4 and 1/2. Look for how repeating those amounts divides the board into that many pieces.

Start: both teams roll the fraction die (or spinner marked ½, ¼, ⅛, ⅛, 1/16, 1/16). The SMALLER roll goes first and fills that fraction of the squares.

Then teams take turns rolling the die and adding that fraction of their color to the grid. If there is not enough room, you lose your turn.

When the grid is full, the team with more than half wins.

Look for opportunities to describe the state of the board with fractions. What fraction of the whole board is blue? Is empty? Is red? Is filled? We’re looking for being able to use fraction names for quantities sensibly.

16ths Nim

Draw a 4x4 grid. 

On each team’s turn, they can fill in one, two or three sixteenths. Count as they fill in the total part of the board filled.

The team that fills the board LOSES! Losing team chooses to go first or second in the next game.

Good for naming amounts, and getting to iterate. Kids typically find this pretty engaging. A bit easier than the fraction bar Nim. 

Kids almost immediately started drawing other size grids, which is awesome! Work on naming those fractions.

The Deck

Over the course of the semester, the teachers work on building a deck of fraction cards. At first just number cards and bar model, then adding area and discrete models and more fractions over the course of the semester. (Here's a GeoGebra applet I made to help.)

Concentration (Memory)

A deck of about 20 fraction cards from two halves of an index card. On one half of the card, the symbol for the fraction, like 12, and on the other a fraction bar representation. 

I might make: ½, ⅓, ⅔, ¼, 2/4, ¾, ⅙. 2/6, 3/6, 4/6, ⅚, ⅛, 2/8, ⅜, 4/8, ⅝, 6/8, ⅞ or a subset of those.

Lay out the cards in a grid. On a player’s turn, they flip over 2 cards. If they match, they score them. In memory/concentration at home, you probably take another turn, but I recommend NOT doing that in school games, as it leads to less turns for the kids you want practicing more!

There is some strategy to concentration, in terms of flipping over a new card or known card first. You might talk about why you’re doing what you’re doing as you play. 

It sometimes helps to have the kids make all the pairs before the first game, so they can see the matches.

This game was surprisingly popular with the kids, with them often requesting it up until the end of the semester.

Fraction Go Fish

Use your full deck of cards. At least 8 cards for each player.

Deal each player 4 cards. On your turn, ask ONE player for one fraction. If they have it, they give it to you. If they don’t they tell you to GO FISH, and you draw another card. At the end of your turn, if you have a match, you can play one match. No extra turns!

Play continues until someone goes out, or you run out of cards. If you run out of cards, everyone gets one more chance to ask for a card.

Be clear that you can match two pictures if they show the same fraction.

Equally popular with memory. Some groups focused on a set of numbers and one model for each number so players new if they were asking for a number or a picture.

1s Go Fish

Instead of looking for different representations, players look for fractions that add to one. Eg. ¾ and ¼. They don’t have to be the same representation. As usual, it’s good to practice making pairs before trying in the game. Make sure that all the cards in the deck have a match! Remove any that don’t.

Fraction Path

Make a path with 6 spaces on it, and a clearly marked start and finish. The goal is to fill in your path from smallest to biggest fraction. Once you place a fraction, you cannot move it.

Players take turns drawing a card. If you can, you have to write it in a spot. If you don’t have a spot that works, you lose your turn.

First player to fill their path from smallest to largest wins.

Great after activity: make a number line 0 to 1 with the numbers in your path, placed as accurately as possible.

I am a huge fan of path games. Once learners are comfortable with the quantities and representations, it's great to move on to comparison, and even relative magnitude like the number line post activity. You can adjust the number of spaces, but 6 made for a game quick enough for our instructional time frame.

More or Less

Players each have a hand of 3 cards. On your turn, you call whether more or less wins. Players choose a card and hold it out face down. Everybody shows their card at the same time. If there’s a tie for least or most, just those players play another card from their hand with the same rule. Draw back up to 3 cards. After once through the deck, players with the most cards win.

Nothing wrong with War for number comparison, but this simple modification adds a lot of choice and strategy.

Fraction Dice War

Using the fraction dice from earlier in the semester. Teams roll two dice (or one twice). The team with a higher total gets a point. First team to five points wins. 

Very simple. Encourage pictures to compare, or use the 16 grid and counters to show the fractions. A good early adding with different denominators game because you only have to change one one of the addends.

A Little Bigger

Materials: a deck of fraction cards.

Deal 5 cards to each player or team.

Player to the left of the dealer plays their smallest card. 

Each subsequent player has to play a card greater or equal to that card. For example, if the card is ¼, you could play a picture of ¼ or a ⅓ or ½ or…

If you can’t play a bigger card, draw a card or take the top card off the pile.

When no one can play a bigger card, discard the stack. The last player who played starts the next stack with their smallest fraction.

First person out is the winner! (Or play until there is only one player left.)

Here We Go

Next semester I have a new class working with the same kids, so we'll be needing some new games. I'll still have them make a deck of fraction cards. But we'll need new games for fraction equivalence and more operations practice. 

What are the fraction games you like? Share some back.

Jaxon's Part Way

Illustration by Ysabela Golden

0/12

The sun is up, the door is open, the path beckons.

Jaxon sighed deeply. 

They had a long way to go, and not starting the journey felt more free. They didn't have to go. Yet. But even one step, started the journey. They thought of the start as 0/12 of the way, for some reason. But it could be 0/1, 0/how many steps, 0/hours it will take. But as long as they didn't start, they could just not go. Once the journey was begun, it remained begun. 

Jaxon stepped out of the door. It's begun! Some small measure past 0/12. Or maybe it was a big measure, as hard as it was to begin.

And then actually they did step back, because they had forgotten their water bottle. As usual. 3/7 times this week. 

1/4

Smooth sailing, if sailing was walking. Hmm, if sailing was walking, their staff was the rudder? Their hat was the sail? "I guess there's a reason metaphors aren't allegories." 

"I never met a 4 I didn't like."

"Hello!" said Jaxon, "have you met many fours?"

The ... cat? they wanted to say ... was on a branch overhead, and Jaxon never would have noticed them if they hadn't spoken, which was a good argument against catness as they had never met a talking cat. Or, if cats were generally capable of speech, a cat Jaxon had met had never spoken to them before. Maybe they had met many talking cats!

"Petit 4s, what 4s, where 4s... speaking of which, where 4 art thou?" asked the Cat?, "and who 4 art thou, as well?"

"Jaxon, I am, and I think I'm about a quarter of the way."

"Quarter master! Pleased to meet you."

"And you are? And are you a cat?"

"I am Checkershire. I'm not so much a cat as an obvious reference." 

"A reference to -"

"A reference 4," Checkershire interupted. "I'm here 4 a question. Without an answer, I'm afraid you can not pass."

"Is that a threat?"

"It's more of a service. With an answer I'll open a sweet shortcut."

Hmm. "I'm not sure there are shortcuts on a journey."

"If you get where you're going, then the journey's done," Checkershire advised. "Are you game?"

"OK, I'll try."

"Trying is the start, young one. If you take my path, my compatriot awaits you a nickel short of half way. How far have you to go?" 

"A nickel? 5? 5 less than half, so maybe 45?" 

"Be that your answer?" the not cat asked.

"Just thinking out loud," Jaxon said. "But 45 what? Halfway sounds like a portion, so maybe I need an answer less than a half. A nickel is 5 cents, 5 hundredths of a dollar. So that's a twentieth."

"Is that your answer?" 

"Just a waypoint! A twentieth less than a half, which is 10 twentieths, so 9 twentieths. Not my answer! Be 4 you ask." The creature which had started to ask, just smiled. "Give a person time to think. So 9/20 is possible, but I'm not starting from zero, well I did start from zero... wait. What was the question?"

"If you take my path, my compatriot awaits you a nickel short of half way. How far have you to go?" 

"Ah hah! Not where is your compatriot, but how far do I have to go. I told you I was a quarter of the way, which is 5 twentieths, so I am 4 twentieths away! That is my... wait, let me check. I'm 1 fourths, which is 5 twentieths. 4 twentieths more would make 9, then one nickel, 5 hundredths or 1 twentieth more for 10 twentieths, which is halfway!" Whew! Jaxon felt that buzz of things clicking together.

9/20

"Hey! I think I can see halfway from here," Jaxon exclaimed, as they topped the hill. They'd be glad for some down time as it had been a lot of climbing.

"In a rush?"

"Who's that?" Jaxon was caught out, looking around.

"Weren't you talking to me? I certainly didn't mean to startle you."

Finally Jaxon located it. A wee red ... imp?  "Well, hello!"

"Hello yourself. Some call me Tenlodectl, a diminutive yet distinguished daemon of numerical persuasion."

"A demon! Keep away!" Jaxon cried in alarm.

The imp held up calming red hands. "Easy, there! I am no threat. Not a demon, a daemon. Related, but well intentioned. We tempt people -"

"Don't tempt me!"

"- with intellectual pursuits, I was going to say. Puzzles, connundrums, a nice poem... rather fun at a party, if I say so myself. Are you headed to a party?"

"Well, that sounds, not dangerous at least."

"To be honest, there are puzzles that will lead to a lifelong obsession. That's a dream. Speaking of which, is this a dream?"

"I think I'm awake." Jaxon pinched themselves. "Definitely awake. If that works! I am on a journey."

"A journey! All well and good. I have stepping stones for you." With a complicated tiny gesture, Tenlodectl, seemed to summon stones from the ground. Each was labeled with a fraction. Even one below Jaxon's feet, labeled 0/1. The others were 1/5, 2/5, 3/5, 4/5, 1/4, 2/4, 3/4, 1/3. 2/3, 1/2. 

"Thank you, I guess. But I was getting along fine." And raised their foot, about to step onto 1/2. 

"HOLD!" shouted the imp. "A mistep here will cost you a nap, during which I will give you delightful dreams of Farey numbers." 

"A forced nap? Faerie numbers?"

"Farey."

Jaxon couldn't hear a difference. "But my journey! Maybe I'll just walk around." 

"Well..." Tenlodectl looked sheepish. "I'm afraid that's not an option, either. You have to solve my puzzle, or suffer the sleepquences. The stones must be traversed in order."

"What's the order?" Jaxon reasonably asked. 

"I can't tell you that!" Tenlodectl said. "It's a puzzle!"

"OK." Jaxon thought there was no use in protesting what seems to be their fate. I"n order, you said." Tenlodectl nodded encouragingly. "That could be a lot of things. Alphabetical. But numerical order seems most likely." And with that, stepped strongly onto 1/5th. Still awake. Jaxon knew a fifth was smaller than a fourth or a third, because a whole split into more parts means the parts are smaller. But what about a fourth, a third and 2 fifths. All less than a half... and a third is more than a fourth. But where does 2 fifths fall? "I used twentieths before, talking to Checkershire."

"Ooh I like that cat!" 

"I knew it! I asked if they were a cat."

"I just meant it like, cool cat, hipster, ..."

"Oh. Anyhow, a fourth, split into 5 pieces would be twentieths, so 5 twentieths. But the fifths into four pieces, so 4+4, 8 twentieths." Jaxon stepped onto the 1/4. No nap! "But what about the third? Twentieths are no help. Hey! I split fourths into 5 pieces and 5ths into 4 pieces. Fifths into 3 pieces would be fifteenths, and thirds into 5 fifteenths. So 5 fifteenths and 3+3, 6 fifteenths. So close." Jaxon toed onto the 1/3, glancing at the daemon. Ten was not giving any hints! So Jaxon strode from there to 2/5. Awake! "2/4 and 1/2 - they're the same. Better take 1/2 to be safe." Awake!

Jaxon looked back. "I think I can reverse it! If I'm half way, then it's like a mirror." And Jaxon stepped confidently onto 3/5, 2/3 and 4/5, then hopped both feet onto 1/1. "Tah dah, Tenlodectl." 

"Well done, Jaxon. No necessary nap, but may I bring you a number dream some night?"

"Please do!" And with that, the imp was gone and the path continued.

2/3

The post-daemon path was rather uneventful by comparison. Lovely sights, though. Perfect little binary bushes, each branch branching twice. A hill where they could see Funville. Lots of playgrounds! And then they heard it.

"174,000."

"139,500."

"227,000!"

Then Jaxon saw a distinguished looking bearded person in great turquoise robes with a hand resting on the trunk of a small tree, gazing up into the branches. "78,000, little tree. Keep growing!"

"Hello!" Jaxon said. "What is the meaning of these fabulous numbers?"

"I am counting, or rather, estimating the leaves." Bowing deeply, he said, "I am he who counts."

Jaxon bowed back. "I am Jaxon, currently journeying. Pleased to meet you." 

"And I you, my friend. Would you care to count leaves with me?" 

"Unfortunately, I think I should continue on." But they noticed the path diverged in the wood. And it might make a big difference. "By any chance, do you know which path I should take?"

"Indeed I do, for I have calculated their length. If I am correct, both lead to your destination."

"Excellent!" And Jaxon prepared to flip a coin to pick.

"However," the counter countered, "one is approximately 7 thirds the length of the other." 

"Which is the shorter?" Jaxon noted the slightly sad countenance of the counter and guessed, "or do I have to solve a problem to get the information?"

"You mean you get to consider a question! As you may have surmised, I am a enthusiast and appreciater of computation. As such, I just wish to know, were you to take the longer path, how much longer would your journey be than if you took the shorter path?"

Jaxon almost blurted out 7/3, thinking that the answer had already been provided. "It will help me to think out loud. May I?"

"Please do! Mathematical thinking is my very favorite thing."

"I thought maybe you had given me the answer, 7/3. But that was comparing the path remaining, and you asked for the whole journey. The right path would be a 1/3 of the whole journey, so does that mean I want 7/3 of a third? A third in three pieces makes ninths, and then seven of those would be 7/9."

"Is this, then, your answer? Truly you understand the partitioning of parts."

"I don't think it is my answer." Jaxon picked up a stick and broke it into thirds. They were tempted to break it into the significant parts, 1/4, 9/20 and 2/3 but that wouldn't help this problem. They got another stick and broke it into the size of the third. Broke that into three pieces to make ninths. And then laid out 7 ninths. "So the whole path is 2 thirds and 7 ninths. So 6 ninths and 7 ninths, 13 ninths."

"Is this then the answer?" asked a smiling counter. 

"Wait. How does 13 ninths compare to 3 thirds? Wait! 3 thirds is 9 ninths, so it's comparing 13 ninths to 9 ninths, so it is just 13 to 9. So yes it is my answer! 4 ninths longer, which is almost half again as long!"

"Well done, Sahib! Please continue to your right."

12/12

As Jaxon rounded the last curve, they were surprised to see they were back home. A whole journey just to get where they started! Although, they thought, isn't that usually the case. Regardless, it was quite a day of amazing sights. They prepared a bowl of minutes and seconds for their watch dog, and some crackers split into fourths, just the size to match their cheese cut into sixteenths. How many crackers did they need?

"I'm just going to eat until I find out."

World Tessellation Day 2025

Emily Grosvenor, when she wrote her sweet children's book Tess Elation, had the idea to start World Tessellation Day, selecting M. C. Escher's birthday, June 17 (1898), as the logical day. That was 2016, making this the 10th!

From the man himself: I found two both labeled Development II, woodcut 1939.

The best place for regularly finding mathematically interesting tessellations that I have is the Mathematical Tiling and Tessellation Group on Facebook. (Right?)

Miki Imura had a recent series there on her modulo wrinkle tiling

and has a website where you can play with it. So non-intuitive - would make a great physical tile with which to play!

Michael Cheshire blew me away with this WOOD INLAY

And moderator Ghee Bom Kim regularly amazes with fractal and radial tessellations.

Also Robert Fathauer, Dominic Pons, Michael Sterling Helso, Alan LeBudde, and more. Well worth your time.

Alex Romero, of Tile Farmer, nicely sent along a few of his recent made with TF: 

More BlueSky math folks:

Ally Kraus, a fiber artist

Joy Sprinkles does a lot of AI art but also hand drawn things like:

Alejandro Gallardo makes so many beautiful GeoGebra tilings, often inspired by historical tessellations.

Richard Connors McConochie. I'm a sucker for a cat tessellation.

MisterCorzi share the classic Pringles tessellation from Theo Rooden.

From other places:

Gábor Damásdi had a beautiful post of "unusual tilings."

RobertLovesPi has many tessellations, often colored in MS Paint! (Find here on Bluesky)

Tudi regularly posts wearable Penrose tiles on Redbubble...

Xavier Golden is a HS art teacher, and his students had some sweet tessellations.

Most of the tessellations I make or find I post on Tumblr. Here's my top 3 of my own this year. Mostly I'm interested in making interactive ones where you can play, so the first link is to the GeoGebra and the second to what I did with it.

Cairo tessellation, pictures.

Two turn pentagonal tessellation, pictures, inspired by a Michael Helso post.

Pentagon hexagons tessellation, pictures, from a Gábor Damásdi dissection of a hexagon.

Did I say 3? One more.

Got any tessellations you've been making? Send them my way!

We'll leave with this amazing photo by Dan Kelley from the FB group, a sweet radial tessellation in metal!

Star Patterns - a Mathart Lesson

One of my favorite courses to teach is our embedded elementary whole number and operation teacher prep course. Typically they teach every course day, twice a week. This semester, one section taught 1st and 3rd, and the other 2nd and 3rd. Pairs of teachers work with small groups of elementary learners for 30-50 minutes. Mostly our lessons are a number talk (or other instructional routine; WODB, same but different, Slow Reveal Graph, ...), a story problem and a game. But we try to do a couple of other formats, a three act lesson, a data collection lesson, etc.  At the end of the semester, I like to have a math and art lesson as a kind of celebration, and to get to experience making with the kids.

In the past, we've done Hundred Face (handout), but time-wise that was not going to work this year. I do recommend that lesson highly - always fun, so many ways to see 100 and think about exchanging different combinations to make the same number, great opportunity to do real art with significant restraints.

When I was thinking of a new lesson for the 1st and 2nd grades, I wanted something that involved counting, and could be done pretty quickly for the class with the time limit. 

Eventually I landed on string art. I like this in general, I love all the things there are to notice. The mechanism is simple enough to communicate to young learners: choose a skip number and a place to start, connect those vertices, repeat. 

I made a GeoGebra applet for the teachers to play with, and, if time allowed, they could share with kids. We got the teachers to make some conjectures about the string art patterns from their noticing and playing. When does the pattern hit every node? When are there gaps? What difference does a smaller vs a larger skip number make?

For a couple years, I've been lucky to be able to do these lessons with the teachers during school spring break and have my son Xavier join us. He's a high school art teacher and the author/artist of our graphic novel. He talked to the teachers about line, as an element of art, and developed this slide show to share about Emma Kunz and Bridget Riley.

We talked about different ways to use the pattern to make art. Xavier suggested coloring so that not two adjacent areas shared a color.

Teacher art was interesting:

One of the things that I love about combining math and art is being able to address mistakes as happy little accidents. We adjust, incorporate it, use fix up strategies.

The lessons went very well with the 1st and 2nd grades, most kids were highly engaged. When we finished, we set out all the art work and did a gallery walk. Then the artists shared something they liked about someone else's work. Lots of great noticing, and they were very pleased when another kid found something in their work. There were a lot of great predictions about where the count was going to take them next, and some kids were confident enough to just draw the pattern. The physical skills of using the ruler and coloring were relevant. Xavier had given the teachers some tips about careful coloring and precision.

Altogether, this is one I'll use again. It would be interesting as two part lesson in middle school, with the first an investigation into the patterns, with connections to factors and multiples. The math was great, and the preservice teachers were impressed with the kids' interest, focus and creativity. As usual, the art connection engaged kids who are less interested in traditional math lessons.

Lesson Plan Crib Notes

Part 1

Share your examples of rectangle string art. What do they notice and wonder?

Part 2

Have them pick a starting grid and a skip number. (Handout) Help them get the first few lines. As they work, see what they notice about the pattern. Help them use the grid to count - especially on the 10x10 square. The bigger a factor the skip number shares with the total perimeter (40 for 10x10, 42 for 9x12) the fewer lines until it meets up with itself. So skip 13 will hit every point, skip 12 will have fewer lines. Pick any starting point - some people like starting in a corner - and continue connecting lines to the point your skip number of points away. You can do colored lines, black ink or pencil. Talk to them about the counting math, patterns they notice, what they see. DON’T WORRY ABOUT PERFECTION.

Part 3

Decorate!

Summary

Find out what they like about their art. Get a picture. Tell them something positive that you saw about them or their work. Be authentic!

Playful Math Carnival 180

 May I March from April?

I was supposed to post the March/April Playful Math Carnival, but it's May! May the 4th even, happy Star Wars Day to them that celebrate it.

180 is a pretty amazing math number. 18 divisors, more than any smaller number. 18 divisors also makes it refactorable, divisible by the Very abundant, as you might guess. Harshad (or Niven) also, divisible by the sum of its digits. The sum of two squares (both squares of divisors!) For Euler's totient function, 

Of course, 180 is probably most famous in math for being the sum of the angles in a triangle, or have the degrees of a full turn. How would you prove the triangle relation? (I tell my math history students that is one of the few theorems every math major should be able to prove.)

Speaking of triangles and math history, Pat Bellew discusses Heron and his formulas.

Chris Luzniak read a book that made him realize he needs Math Therapy. Chris is hosts Debate Math podcast with Rob Baier. I loved their episode on comparing teaching reading and math with married couple Courtney and Ryan Flessner.

Denise Gaskins, the home and creator of this here blog carnival, had a math journaling post with three elementary math games. Also don't miss her Math Game Mondays.

Ann Elise Record shared a great padlet of math games. It includes a link to her podcast, discussing meaningful math games with Dr. Nicki Newton.

Rachel Lambert shared the start of some research into mathematical games and their use with teachers. Really exciting and I can't wait to see where it goes!

Howie Hua, modern master of math memes, had the fun Tom and Jerry meme above show up in a reddit Explain the Joke thread. Speaking of Howie, this Star Wars math made me think of another of his memes.

Chalkdust, one of my favorite math periodicals, had an article looking at the discrete math underlying Sudoku. (While you're there, be sure to check out Dear Dirichlet, the funniest mathiest advice column ever.)

One of the great math events this spring has been showings of Counted Out, a documentary examining the importance of math and math education in modern life, centering the work of Robert Moses. Here you can read more about the movie and the people featured. I have never had a stronger endorsement for an education documentary.

The delight of March for me was Ayliean MacDonald's Math Art March. 

One idea I tried out for Math Art March was a pattern themed Exquisite Corpse game. This is an art game where you fold a paper and each subsequent artist can only see the very end of the previous artist's work, and draws off of that.

Jenna Laib writes about Anderson's Endless Zeroes, an elementary math investigation into a unit conversion problem.

Daniel Scher created a sweet dynamic applet to use sliding rulers to think about integer addition and subtraction.

I was pretty happy with this Escherized version of a hexagon dissection I saw. Play yourself in GeoGebra. 

I've just started on these, but Arula Ratnakar writes mathematical fiction at ClarkesWorld.

The two most recent math books I'm most excited about were The Five Sides of Marjorie Rice, about one of my favorite mathematicians, and How Did You Count?, another great Christopher Danielson book that makes the reader the mathematician.

We'll close with the math blog-o-sphere's most reliable writer, Dylan Kane, who took a break from deep thinking about learning and teaching to share a fun folding problem from Play With Your Math.

Sorry again this was so delayed! If you're interested in hosting the Playful Math Carnival, give it a go! Share what you've loved. The previous was at Denise's Let's Play Math, and the next might also be Denise. 

Coming up on my blog this month will be two elementary math and art activities, and some great new math games from my senior seminar.

To close, I think I have to share one of the Star Wars Standards of Mathematical Practice memes that Dave Coffey got us making a few years ago.