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.

Make a Difference - Math Game

 I once again am getting to teach the math game design seminar (at some point they'll realize it's too fun to count for my workload) and I wanted to try and capture my design thinking on a promising new game.

Phil Shapiro shared on Bluesky his math pairs game. A randomized list of 1 to 100 where you find pairs that add up to 100. 

For whatever reason, that made me wonder about a game finding differences. In elementary there's often a default to subtraction=take-away (Separate Result Unknown si parlez vous CGI) (I don't speak French) So a game that focused on the difference would be a good thing. I thought, what if you roll a die and need to find a pair that is that far apart?

For kids I like a number board that has a structure, so kids can use patterns to find what they want. (Nothing against Phil's game, where the Where's Waldo feeling is a lot of the fun.) My first try was a double spiral.

It was a lovely pattern, and I liked how it put small and large numbers together. As the game play evolved, it became clear that I needed a normal grid. 

The other thing you can see here is pretty typical for me when I have a mechanic idea. Try out the mechanic and worry about the win condition as you go. The above image was my first try. I asked on Bluesky who won, and Phil responded probably yellow, since it seemed to have more territory. Biggest block of squares? Longest path? I stuck with that for a while. Eventually I realized the game is about differences, the win condition should be, too. What if the path with the biggest difference won?

Mechanically I really liked that. Then there's an advantage for the first player. And it raises questions: what's a path? I thought it should be only edge to edge, but it became too easy to cut someone off. Having squares connect corner to corner gave some of that Blokus energy. I did wonder about the sum of two paths, but that's unnecessarily complicated.

I'm still trying different play rules. Should one of your new squares have to be adjacent to one of your old squares? Currently I'm saying no, because that makes more interaction possible as well as opening up more strategy with more choice.

The board was 9x9 originally because I wanted that double spiral, so it had to be odd x odd. I can see this being on a hundred board. Great representation, and I love to have kids spend time with it. I like that +/-9 are above each other, because 10s are often comfortable already, and it feels like 9s still makes for lots of interesting patterns. It does make a game around 20 turns - which is long for 2nd & 3rd grade. Although kids play Joe Schwartz's Hundred Board Game (definitely a Best of Math Games awardee; video explaining it) is more turns, but the turns are quicker. 

Why I think this is worth developing is because as I play, I have to think! Looking for pairs, any in good strategic placement, what is possible... a lot to consider. Too much for middle elementary? I hate to underestimate the players. Towards the end of the game, there are surprisingly frequent times that you can't take the number you would first take. The number rolled makes a difference in play, as well as providing some variance that helps with surprise.

One thing that came up is what if there's a tie? Then the winner is the person with the biggest difference on their second path that doesn't overlap their first path. Maybe a second path that doesn't cross their first?

What should keep you playing after you have a maximal chain? I thought about a bonus for being the last player to play. But that feels fussy. How else can I make people care about finishing? Maybe they don't have to?

Current rules text: 

Two teams. Roll a 10 sided die (0=10) or flip a Tiny Polka Dot card. High number goes first and gets 10. Team 2 rolls and takes 90 and then 90 minus their number. 

On your turn, find two numbers whose difference is the number you rolled and color them in your color. After 10 and 90, teams can choose any pair of numbers with the difference they rolled.

Game ends when both teams have to pass because there is no pair with that difference. Both teams draw a path connecting the biggest difference that can find. Squares connect edge to edge or corner to corner. Winner is the team to have the biggest difference in their path.  For example if team 1 makes a path from 10 to 71 (71-10=61) and team 2 makes a path from 24 to 90 (90-24=66) team 2 wins! If tied, the winner is the team with the longest 2nd path that doesn't overlap their first.

The 10 and 90 start is trying to remove that first turn advantage. It's also is a step towards understanding strategy, which can be nice to bake into the rules.

Definitely want to try with a d20 as well. Maybe as a 4th and 5th grade variation? On a 0 with Tiny Polka Dot cards, you could be allowed to pick a single number - which definitely could be useful. Another variation for high school + players could be the sum of two paths victory rule.

Current game board. If you try, I would love to hear what you think. I'll definitely play with my games seminar, and maybe with my elementary preservice teachers &/or 2nd and 3rd graders.

Two games with the most recent rules. 10/90 start is working well. 

PS. I make some references here to the criteria I use for thinking about games. Definitely a part of my design thinking.

PPS. I also like the name, which is unusual for me, but am open to suggestions.