This is a joint blog post by Kara Shane Colley, the author of My Hundred Friends 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.

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 their 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.