This is a joint blog post by Kara Shane Colley, the author of My Hundred Friends that is soon to be published at Natural Math, and Dr. Maria Droujkova, the founder of Natural Math. We’ll be posting periodic updates about the book on the blog.

I like how it’s funny and learnful.
Elizabeth, 4th grade reader

Maria: In My Hundred Friends by Kara Shane Colley, the math of factors and multiples is friendship. It’s a cozy little world, inviting readers to play and explore. From April 27 to June 27, head to My Hundred Friends crowdfunding campaign to preorder hardcover copies for your family or group, reserve your complimentary PDF, and support mathematical storytelling!

Kara: While editing the book, I spent several afternoons reading early drafts of My Hundred Friends with a Montessori class of 4th, 5th, and 6th graders. The teacher gave me a corner of the room and set me up with pairs of students.

I greeted the children and explained, “I’m writing a book and I’d like your feedback on it. I’d like you to read it and tell me what you like, what you don’t like, what’s confusing. I’ve illustrated the book so far, but I am not going to be the actual illustrator.” I handed them my sketchbook, and they turned to page 1.

In the first few pages, the children noticed right away how excited the numbers are to see themselves in the other numbers. On page 4 in the bottom right corner was 4’s prime factorization, 2 ✕ 2, with Two exclaiming, “Look! It’s me and me again.” On page 8 in the bottom right corner was 8’s prime factorization, 2 ✕ 2 ✕ 2, with Two doing excited cartwheels and exclaiming, “Wow, it’s me! And me! And me!”

Quoting Two’s line, Beck, a 4th grade reader, said, “‘It’s me!’ I’ve never seen a book like that.”

The numbers being numbers have inherent, repetitive structure just bursting to come out. The children turned the page from page 8 to page 9. In the bottom right corner, they saw 9’s prime factorization, 3 ✕ 3. The children chuckled to themselves when they saw Three exclaim, “Look! It’s me and me again.”

And it wasn’t very long before the children reached page 16, where they saw 16’s prime factorization, 2 ✕ 2 ✕ 2 ✕ 2. They smiled and laughed again. Helen, a 6th grade reader, told me, “I like 2. He’s always there!” Theo, another 6th grade reader, remarked, “I like the patterns in the book. I feel rewarded.”

Maria: Humans are social creatures, and big areas of our brains are devoted to people and relationships. Our imaginations spark playful bonds with characters and toys. What if we could use all that social reasoning to boost our mathematical power? Yet merely adding a face and limbs to a number won’t give it enough life for a good story or game. My favorite authors create their number-characters and storylines with deep care for each number’s mathematical behavior. In beloved books like The Cat in Numberland and shows like Numberblocks, numbers are persons, and everyday adventures spring from their mathematical personalities.

Kara: The children also liked the connections to daily life. When they reached page 13 with its connections to bagels, 4th grade readers Beck and Elizabeth became animated, telling me that the bagel should be “tastier-looking” and that I should “toast it and color it.” Based on this feedback, the bagel did become tastier as I made edits to the book. (I can’t wait to see illustrator Coley Nielsen’s bagel! Yum!)

The children interacted with the drawings in ways that I didn’t expect, reaching out to grab the bagel or tracing the path of a jumping number-character. They touched the pages, brushing their hands over the number-characters, almost like they were petting them. Theo said, “Your drawings are so cute, so sweet!”

Maria: Why do we care so deeply about children touching their mathematics? Because that physical play develops excellent habits of mind. Children learn to order numbers in space to hunt for complex patterns. They use visual structures like graphs and tables to solve problems, or read data from clever diagrams such as factor trees in My Hundred Friends. Our society is full of complex number patterns and data. Early, happy familiarity with organizing numbers in space helps every child.

Kara: I usually spent about 15 minutes with one pair of children, and then the next pair of children came to my corner. Many pairs just read the book straight through, starting at page 1 and going forward. One pair of children, Theo and Pascale, reached page 24 and noticed that the number-character Twenty-Four had one strip of brown at the bottom. Pascale, a 4th grade reader, asked, “What’s brown?” She paused and then wondered, “ Is it 12?” as they flipped back to page 12. Pascale guessed, “6?” as they flipped back to page 6. “No, it’s 3!” she exclaimed as they landed on page 3.

They were on to something! They had picked up some sort of pattern with the colors. From that point forward, Theo and Pascale started reading with an eye towards what was going on with the colors. I didn’t explain the role of colors in the book; they were having fun trying to figure it out. I held myself back, asking a few open-ended questions, like “What do you notice?” or “What do you see?” 

This was such an exciting moment for me. I had created a world where kids were exploring and asking themselves math questions. They were playing with the number-characters, playing with math. At the end of our session together, Elizabeth remarked, “I like how it’s funny and learnful.” That felt like high praise! 

If you have a child, a classroom, or a fondness for cozy math, I invite you to come explore My Hundred Friends. What number connections might your child make? How might your student pretend-play with their favorite number-characters? 

This is a joint blog post by Kara Shane Colley, the author of My Hundred Friends that is soon to be published at Natural Math, and Dr. Maria Droujkova, the founder of Natural Math. We’ll be posting periodic updates about the book on the blog.

“The chalkboard appeared when Two wanted to draw on it.”

Maria: In My Hundred Friends by Kara Shane Colley, the math of factors and multiples is friendship. It’s a cozy little world, inviting readers to play and explore. From April 27 to June 27, head to My Hundred Friends crowdfunding campaign to preorder hardcover copies for your family or group, reserve your complimentary PDF, and support math joy for all!

Evolution of the cover

Kara: In making drafts of the cover, you can see that some aspects of the cover stayed the same. On every draft, there are number-characters hanging from the title, sitting on the title, running up a path, and cheeky number Seven trying to get our attention!

It was my publisher Maria’s idea that the number-characters should be doing math on the cover. In the illustrator Coley Nielsen’s cover, you can see Two building a cubic model with blocks, Fifty-Five playing tag with their factors, and Seventeen proudly displaying their factor rainbow.

Maria: It is said that sci-fi isn’t about the future and fantasy isn’t about magic: it’s all about our life, here and now. Where does math come from? We make mathematics as we live our lives, create, and discover together. Kara and the number-characters built such a friendly village with their mathematics! Imagine that. Let’s go!

Evolution of page 9

Kara: Collaborating with Coley has been exhilarating. The process of working with her helped me develop some ideas that weren’t totally fleshed out in my head. As you can see in the early draft, I originally thought of the pages of the story as loose leaf paper with a horizontal line at the top of the page and holes punched down the left side. As you can see on the second draft, the horizontal line at the top remained but the hole punches were gone. As we prepared to send over the second draft for Coley to draw her version, I realized that the whole book was like a math journal; it was a scrapbook that I made of my hundred friends. Because it was a scrapbook, I asked Coley to draw graph paper and make it look like it was taped onto the page.

Maria: Our numbers, our math – is it out there, hiding in plain sight all around us, to be discovered? Or does mathematics come alive by our creativity, the same way pretend-play toys or fiction characters do? The readers can see Kara’s and Coley’s art either way.

Evolution of page 12

Kara: Another evolution of the world occurred as Coley began to draw drafts of page 12. In my mind, I was the one writing “12 is one dozen” and I was the one who drew the egg carton. Coley drew a couple of versions of the egg carton for us and that made me realize that I didn’t want a totally realistic egg carton. I wanted a rough “sketch-like” version of the egg carton.

On the other hand, the chalkboard at the bottom was part of the imaginary number-character world. I didn’t draw the chalkboard. The number-characters came to play in the math scrapbook, and the chalkboard appeared when Two wanted to draw on it.

Come check out My Hundred Friends. The book is a kind, welcoming math gathering. We invite you to come, hang out, and meet a few new friends.

By Sophia Zhanissov (grade 6) and Maria Droujkova

Some math activities are so rich in mathematical connections, inspirations, and energy! If mathematical routines were people, Growing Patterns would be a pillar of the community. Growing Patterns routine connects several key ideas in computer science, algebra, geometry, and beyond. Moreover, all that is doable at the pre-algebra level; you’ll be essentially learning algebraic reasoning from scratch. Here’s what we did over the hours of the project, in a nutshell:

1. Admired Fawn Nguyen’s (last name is pronounced “win”) whimsical collection of hundreds of growing patterns at VisualPatterns.org and decided to add to it!
2. Made our own visual growing patterns using graph paper, colored pens, and wooden cubes.
3. Using spreadsheets, scripted recursive formulas for a few starter number sequences: counting numbers, even numbers, odd numbers, and alternating numbers (1, -1, 1, -1…).
4. Scripted recursive formulas for the three patterns we’d like to add to the collection.
5. Wrote stories explaining the three patterns.
6. Scripted closed formulas for n-th entries in the three patterns.
7. Looked at the corresponding functions using a grapher (Desmos).

The drawings, stories, functions, formulas, and scripts are all different representations of the same infinite, growing sequences. They help us understand the mathematical nature of the sequence and express its growth in a single closed formula. That formula, for us, evokes many geometric, algebraic, and computer-science connections, kind of like e=mc^2 packs a vast amount of physics for those who know how to derive it.

A Very Long Pool

A very long pool is my pattern, where the water is some blue squares(or opposite), the Rim is the green squares making an equal shell around the water(or opposite once again), and the stairs is the singular green square on the right side of the pool. The “very long pool” is essentially a pool that grows by the second, making it (technically) infinite or just very long.

The number of squares and recursive formulas for the water, rim, stairs, and the total space of the pool:

Closed formula for the Nth total space: N*6+4.

Heart

Growing emotions is one of my patterns, it is a heart that continually grows forever or infinitly. The yellow squares inside is the inner heart, where it beats and grows in size, and the purple squares outside is the outer heart, that protects the inner heart from dangers. Given that is grows infinitly, it isn’t strong enough to be on it’s own, and is very sensitive(emotional) and prone to damage. Together they grow and grow to the point where there is no stopping point.

The number of squares and recursive formulas for the inner heart, the outer heart, and the total number of squares:

Closed formula for the Nth total space, outer heart: N*4+6; inner heart 2*N^2+4*N-3.

Road to… Nothing

The road to nothing is my pattern, where there is a road, a very long road, or you could just say, infinite. Most patterns go infinite if there is no stopping point, so does this pattern. This road leads to, you guessed it, nothing… If there is no stopping it, there is no end which means it leads to nothing. It may come across one or two things, but it doesn’t stop going, it leads no one, once again. It has no purpose, it is useless you may call it, it has no emotions, unlike the growing emotions pattern, so it has no feelings, so say whatever you want to it, as it won’t even respond. But anyways, the yellow part is some holes in the road, and the purple is some stepping pads that make up the road.

The number of squares and recursive formulas for the green parts, pink parts, and total squares in the road:

Note that in this example, the closed formula for the total number of squares in the road to nothing is quite easy. But alternating recursive patterns by color took some doing, the way we did them!

Closed formula for the Nth total bricks in the road: N*3.