I’ve been interested in creating fractions projects for kids exactly as long as I’ve been working with children in schools (decades). This year, after enjoying,messing around with a hexagon/golden ratio project I wondered if I could modify the idea of using scaled hexagons to help fourth graders make better sense of fractions. My first attempt at this didn’t work out so well.
Making 1, or 100% out of two halves
I gave students hexagons that were scaled to 1, one-half, a third, a fourth, a fifth, a sixth, an eighth, a tenth, and twelfths. The task was to pair and arrange them so they would span the length of a whole, aka 100% across.
1/3 + 1/6 + 1/6 + 1/3 = 1
The project went okay, but it just didn’t snap for me.
I’ve been thinking about how to improve this project. Today I had a chance to work with a small group of kids. I tried a new approach that worked so much better. What was especially great was that it included making a simple book. Yay!
Fractions for a Book
I started kids off with a hexagon that was labeled 1/2. I explained about how the lengths we would be looking at would be the horizontal or vertical length of the hexagon (I didn’t use these words, rather gestured what I meant). Then we layered the hexagon with equivalencies. Here you can see two 1/12ths equals 1/6, three 1/6ths equals 1/2, and 1/6th and two 1/12ths equals 1/3.
Equivalent Fractions
Nice, right? Snap!
The books we made were just two sheets of paper folded in half, bound with yarn using a modified pamphlet stitch.
Equivalent Fractions Book
What’s great about using hexagons for this project is that you can still see the labels of the lower layers as the equivalencies are built up. The adults in the room had a bit of trouble with accepting that the hexagons were scaled (similar) versions of each other, but the kids had no problem with it. This reinforces my notion that children have a better intuitive understanding of scale than do adults.
This is the way I explain the scaling to adults: We all know what half a candy bar looks like. That’s one way of thinking of one-half. But when we say a child is half the size of the parent, we don’t envision the child to be half a parent, like they were half a candy bar. Instead, we envision them smaller than the parent in their height as well as width. This explanation seems to work.
The bullseye view of fractions
After doing a bunch of equivalencies, this child decided to nest her fractions.
Okay then. Here, what’s obvious is the hierarchy of the hexagons that are scaled by fractions. Nice!
This project can use a bit more refinement, but this is as far as I’m going with it right now.
I’m including PDF of the hexagons. The labeling includes the colors of the paper I use for printing. Yeah, it’s lots of files. Welcome to my life.
I still have some holiday cards hanging around, giving me pleasure each time I see them. For instance, there’s this one from Judith. Looks like she got it at the Museum for Modern Art. It’s shiny and sparkly, many layered and changes with the light.
From Ed
This one is from Ed, It’s like a little box, though it collapses flat. There’s two layers inside, This one also changes quite a bit with the light, which I love. I particularly like the little dog in the corner, and the way she’s bonding with the snowman. Or the bird? This card was designed, then cut and assembled by Ed’s own hands.
From Joan
Another homemade card, from my friends Joan and Hank. Joan delights me every year with her drawings. She printed this on her home copy machine, then added embellishments by hand. Notice how my little bird statue it hoping to bond with the trumpet.
From Helen
There’s nothing handmade about this card with the deer and snowman, but I like the expression and the gesture of the snowman so much that this card from my sister-in-law has stayed on display.
In December Susan Joy Share and I took a ride together from NYC to New Haven and visited the studio of the pop-up card company Up With Paper. We were mesmerized by just about everything about our time with them. They gifted us some cards, including this multilayered unicorn card which becomes even more magical when the spot that says PRESS is pressed: lights flicker on behind the unicorn. This is quite lovely. It sits on sort of dark shelf in my house, so the full effect of the lights is always a possibility. (Thank you, Monica!)
Also from Up with Paper
This cluster of gingerbread houses is also from Up With Paper. The pop-up in the middle is surprising to me, as I don’t generally think about making a house from center-aligned criss-cross. This criss-cross structure has tabs on it that extend to become the flanking gingerbread houses. Very beautiful, very elegant, very clever. I actually got a box of eight of these. I gave away a few. The first one I gave away was to Fabio, who ended up being our taxi driver for each of our stops during Susan’s and my day in New Haven. If you go to New Haven you must contact Ada’s Taxi service and ask for Fabio. You must.
From the Cooper Hewitt shop
The day after Susan and I were in New Haven, we spent the day in NYC, mostly at the Cooper Hewitt Museum. I bought this snowflake card in the museum shop. I love how the slits in the snowflake allow the snowflakes to rotate.
From Joe and Ilisha
This strange-looking rectangle arrived in the mail. It’s all cut paper and tabs and slits. I don’t think there’s a drop of glue in this. It moves! When it moves, those odd little shapes line up!
From Joe and Ilisha
Snap!
I don’t really think it’s time to put this one away. Maybe I will put it away when we achieve world peace.
Then there’s this lovely image to maybe put away.
By Katelyn, Angela, and Paula
Us girls colored this Christmas Eve and Christmas Day. It sits in a corner near our couch.
This gives me such pleasure every time I see it that I don’t think I will be putting it away yet.
Scaling Shapes, designed by Professor Chaim Goodman- Strauss, Visions of the Universe by Alex Bellos & Edmund Harriss
I’ve always loved coloring books. Is this true for every kid? I really don’t know. Are coloring books for kids even around much anymore?
When the coloring books for adults became all the rage a few years ago, it made perfect sense to me, although most of the titles on the market didn’t interest me much. I like sense an underlying rigorous structure, but one that is not simply symmetrical. Naturally, then, I fell in love with the books by Alex Bellos and Edmund Harris.
Patterns of the Universe and Visions of the Universe by Alex Bellos and Edmund Harriss, published by The Experiment
In the interest of full disclosure, I have not been asked to write this post by anyone connected to these books. In fact, the reason I am writing this now is because I am working on something else that won’t be ready to write about for awhile, so I thought that, with the holiday season upon us, the colorfulness of writing about these books seems timely.
Cascading Waves, Read the Sines, Patterns of the Universe by Alex Bellos and Edmund Harriss
I love these books for all the same reasons that I’ve always loved coloring books. They let me mess around with color without having to think too much about form. Just experimenting with color combinations without thinking about much else is a great thing for me to do when I’m wanting to make something but not feeling particularly ambitious. I think this cascading waves is my favorite of all the pages. If the authors happen to be reading this, please make more versions of this one for me, please.
Polyhedra, Cascading Waves, Read the Sines, Patterns of the Universe by Alex Bellos and Edmund Harriss
I do not recommend using crayons in these books. The image above in started as crayon only, but it didn’t have a look that I liked until I added those crackly lines made with markers. Crayons are amazing when used on paper that is a bit rough, just like the newsprint paper in traditional coloring books, but seems to me that they aren’t well suited for the smooth papers of adult coloring books.
Modular Forms, Visions of the Universe by Alexs Bellos & Edmund Harriss
Colored pencils work well on these pages, but be advised that not all colored pencils are created equal. The best value for colored pencils are the Crayola brand, but Prismacolor Colored Pencils are just about the only upgrade that makes sense. If I’m using colored pencils on these pages, it’s Prismacolor that I’m using. However, If I am making drawings of my own, using my own designs, I generally use my absolute favorite colored pencils, made by Caran D’Ache.
Petit Fourier 1, Patterns of the Universe by Alex Bellos and Edmund Harriss
Sometimes mixing up the Prismacolors with markers works out best. This page above, which references color bands used by Georgia O’Keefe in one of her paintings, was done with both Prismacolors and brush markers (an expensive Japanese brand, SAI).
Read the Sines, Patterns of the Universe by Alex Bellos and Edmund Harriss
I love the way Sharpies look on these pages. Yes, they bleed through like crazy, so I always put a protective piece under the page that I am working on. Otherwise, I don’t care that they bleed through the page. The publishers of these books were kind enough to keep the verso side of the spread mostly blank, so I don’t lose anything that I’m not willing to lose in the interest of using exactly the materials that I want to use.
The Collatz Conjecture, Visions of the Universe by Alexs Bellos & Edmund Harriss
My favorite pages, so far, are one that are not full of tiny, tiny details, like the one above, These tiny details require more attention than is comfortable for me.
The Kolakoski Sequence, Visions of the Universe by Alexs Bellos & Edmund Harriss
This Kolakoski Sequence has just about the right balance for me of detail and broad areas of color. I’m generally more interested in the description of the form after I’ve colored it, but in this case it was already obscured by the bleed through from the previous image. Oh, well, I still have the internet.
Mechanical Curves, Patterns of the Universe by Alex Bellos and Edmund Harriss
Here’s another one done with both markers and pencils. I didn’t find this uncolored page very compelling, so I challenged myself to use color to make it more interesting to me.
detail Mechanical Curves, Patterns of the Universe by Alex Bellos and Edmund Harriss
I was pretty happy wit the way this one turned out.
Dodecaplex, Patterns of the Universe by Alex Bellos and Edmund Harriss
The thing I like the most about these books is that I can pick them up and put them down with impunity. I don’t have to remember what I was thinking about or where to start. During the years of raising children and never having any extended amounts of time to work on anything, it still makes me feel happy to have something to do that doesn’t suffer from interruptions.
Spiral Tile,Patterns of the Universe by Alex Bellos and Edmund Harriss
This spiral is the first one I did in these books. Took quite a few sitting before I considered to be finished, but that’s fine.
Co-Prime Numbers, Patterns of the Universe by Alex Bellos and Edmund Harriss
Here I got to play with trying to create a illusion that these circles were not in a precisely straight line. Sadly, in this photo that illusion seems to be mostly lost, but on the page they do look wiggly.
Graph Theory, Visions of the Universe by Alexs Bellos & Edmund Harriss
Not all drawings that I do come out great. Some come out well but don’t photograph well, like this one, which I love, but doesn’t seem to like the camera much. Sharpies love the camera.
Those funny shapes to the right of the page are called Sphericons. They are peculiar, funny, and I will making some to give away and some to hang on our tree. They are pesky to try to make out of paper, hence it will be many more days before they get a post of their own. But am happy to have finally made a post about my favorite coloring books.
OH, and if the editors or authors are taking requests, how about a coloring book along the same themes as these, made for kids and printed like old-fashioned coloring books, on newsprint, so that crayons would be the preferred medium? A mathy coloring book for kids, with large, not-particularly detailed patterns sounds like a great idea to me.
Making things out of paper seems to be something we do during the holiday season.
Here’s some projects that I’ve written about with links to the full posts that explain them, that seem appropriate for the season.
This first one, the Spiraling Ornaments was surprisingly well-liked. All year, when I visited different places, I’d see ones that people I know had made.
Next, directions for a six-sided snowflake. My big tip is to use paper napkins, as they already are the right shape: no extra prep needed! Also, paper napkins cut quite easily. They are perfect for snowflakes.
Festive Jumping Jacks are quite fun. I’ve made these with kids just a few times, as all the knot tying makes this an intense project for anything more than a small group, but so worth the effort!
