Art and Math · geometry and paper · Paper Toy



gyrobifastigium (plural gyrobifastigia or gyrobifastigiums)

  1. A polyhedral solid formed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is noted for being one of the few regular polyhedra that packs in three dimensions.
Art and Math · Arts in Education

Paper, Books and Math Workshop

There’s this overlap of paperfolding, bookmaking and math that’s been in my sights for sometime now. Next month The Center For Book Arts has me on the schedule to share my interest with educators.

It’s a natural fit: Fold a piece of paper in half a couple of times and you’ve got a book. There you have it, all this things I’ve been thinking about in one sentence.

Bookmaking by Paula Beardell Krieg


Why math and book arts, you might ask? CBA asked me to propose a course for educators. Over the years I’ve taught classroom bookmaking dozens of times, though my focus during those years was literacy. In recent years it has occured to me, as I visit many schools and work with hundreds of different students each year, that teachers have loads of support for teaching literacy.

Teaching math, on the other hand, can be more challenging. In my desire to stay relevant, the hands-on projects I’ve been designing for classrooms had evolved towards supporting math curriculum and math thinking. And, oh yeah, I love this work.

Everyone folds paper, many people teach math, less people make books. Not too many people have a strong relationship with all three, Basically, I want to be teaching this workshop because otherwise I doubt it will exist.

Here’s what I know about offering a workshop for teachers:

  • They want content that they can use on Monday.
  • They do not have time to do special prep which requires more than the school copy machine.
  • They don’t have easy access to special materials.
  • They like having a handy resource folder.
  • They want their students to be learn and be happy.
Equivalent Fractions
Equivalent Fractions

Here’s how I know about math:

  • I play with math kind of obsessively….never got the memo about math being scary
  • I went through K -12 math three times, once as a student, twice as a parent.
  • I’ve been working in schools, discussing math projects with teachers and math coaches. We discuss standards and curriculum goals, and I talk to students about the math they are learning.
  • I’ve been rather passionately working on deepening my math knowledge in workshops, conferences, and connecting with math educators through their writing and through the #MTbos and #iteachmath communities on twitter as @PaulaKreig

In this one-day PD workshop I plan on focusing on deepening connections, doing hands-on, classroom friendly projects that address areas of math that will help students create strong foundation for future learning.

For instance, we’ll be working with number lines in a way that is both interactive and which illuminates patterns. I’ve been working out ways of presenting number lines in ways which delight kids. The sounds of discovery that come from students when they start seeing what I show them has been one of the most beautiful sounds of my career with kids. I will also have hands-on ways of showing the number line that moves from natural numbers to negative numbers and beyond.

Since the number line stays with children, evolving from finger counting through the coordinate plane (and beyond) my focus here in not only to use the number line, but also to elevate it as an important tool that they have reason to embrace. One of my favorite responses, which informs my work with number lines, was from a first graders who told me that they looked at number lines earlier in the year and now they were done with that.

How to make an origami Pocket by Paula Krieg
Click to enlarge for reading or printing

We’ll also be doing some work with perimeter and area. What frustrates me about students’ learning here is that they often mix up perimeter and area, not remembering which is which. One of the projects we’ll be making is a perimeter-pocket. We’ll make an origami pocket, which in itself is a wonderful lesson as is goes from being a square to a triangle to a trapezoid to a pentagon. But this perimeter pocket will have a ruler embedded in the structure and a string in the pocket to use to measure around things. Then there will be an area-rug book, with lift the flap peek-a-boo images hiding under the area rugs. Fun, easy, memorable!

I love that shapes are part of school curriculum at an early age. There are numerous projects I have in my toolbox that compose and decompose shapes.

I have some nice peek-a-boo projects that address the same composing and decomposing concept with numbers.

Symmetry will also get the attention that it’s due. Sure, we’ll talk about lines of symmetry by way of making pop-ups, but we’ll also look at the idea of symmetry as a it relates to equations, which can be seen as numerical symmetry. One of my exciting discoveries has been how naturally young kids grasp the idea of symmetry, and how well they they can connect it to equations.


A one hundred cents flower
A one hundred cents flower

We’ll even do some hands-on play with play money. I do these one-hundred cents designs with kids which gives then lots of practice with money, as well as practice counting by fives and tens and twenty-fives.

I could go on and on here, but I think you get the idea. And, actually, that’s part of my aim for the day, not just to present projects that can be immediately used by teachers in the classroom, but also that the idea of making the curriculum more hands on will inspire teachers to create their own simply made projects.

This is a rich, wide open inquiry into what we can do together to make math real.

The Center for Book Arts, NYC, Saturday, October 26



Art and Math · Math and Book Arts

Kindergarten Folder for Making Math

Making Symmetry
Making Symmetry

There are some kindergarten teachers I’ve been working with for years. This year I’ve worked with them to create a math-centered book project for their young students. I launched this with a small class earlier this season then repeated it with about 4 groups, total of about 70 kindergarteners, this past week. It went well.

Actually I’m so delighted with how it went that it’s almost embarrassing.

Making the folder
Making the folder

We made a folder out of a long strip of paper, 35″ x  7.5.”

Folder with four compartments
Folder with four compartments


I put some score lines in to help these 6 years olds get started but they made most of the folds themselves. I make a big deal about how to fold paper.

The folder is basically a four page accordion, with pockets for a different math activity in each of the pockets.

The first pocket has a paper with peek-a-boo flaps to help kids visualize the composition of groups of numbers. This was an  unusual folded structure, but they caught on really quickly, as you can see in the video clip below.

After the folding comes the cutting


Cutting the peek-a-boo flaps
Cutting the peek-a-boo flaps

Then the coloring…

Compositions of the number 4
Compositions of the number 4

…finally they used these images to become more familiar with number compositions. We made these cards for the numbers 2, 3, 4 and 5.

Here’s how it looked watch kids use these to learn their number facts:

Okay, so that was for one pocket.

In another pocket there were squares that the students cut out. I used these to talk about symmetry.

Symmetry with cards
Symmetry with cards

Where one student placed a card on their side of the midline (pencil)  another student mirrored the placement. Seeing symmetry is important in math as students as it is a non-numerical way for them to experience the balance that an equation like 2 +3 = 5 expresses.


I extended this symmetry activity beyond the cards in their pockets. We used items around the classroom to create symmetrical designs, something my twitter community liked and retweeted generously.

We also did a project using beads, reminiscent of an abacus, to make groups of 10.

Separating10 beads into groups
Separating 10 beads into groups

The idea here is to give kids another way to interact with ways to make groupings of 10, contributing to their fluency and grasp of combinations of numbers.

Bead counting book
Bead counting cards


Finally, we did a fortune-teller, aka chatterbox, which many of us made when we were children.

Fortune Teller, Chatterbox
Fortune Teller, Chatterbox


Of course the insides were math themed, using their sight words, too.

Fortune Teller template
Fortune Teller template

Here’s a little clip of the kids playing with these. They absolutely loved this toy.

At first I had a hard time trying to teach this structure to kindergarteners. Once I realized that if I taught it after I worked with them on the symmetry part of this project, the folding would then make more sense to them. It turned out to not be nearly as hard to show them as it originally seemed to be.

The final touch was putting hands on the covers. Literally.

Front and back covers
Front and back covers

Since the kindergarten math curriculum emphasizes using fingers for counting, it seemed highly appropriate to decorate the covers this way.

Whew! What a week!

I was able to meet with each class for a little over an hour three times each.

Looking forward to repeating this project with other groups.

Also, now I want to create something like this for first graders! That’s what I will be working on this week.



Art and Math · design · Geometric Drawings

Copy, Rotate, Reflect and add eggs

In the middle of my arts-in-ed season I’ve kept trying to find time to mess around, trying to make beautiful images.

Today I started a wonderful, week-long math activity folder project with four classes of kindergarten students, am barely able to stay awake right now, but I’ve been wanting to at least throw these  images into my blog here.

I started doing this some time before Easter. Just wanted to make something. Started with a graph that I was able to reduce to just these few lines:

Then I copied, rotated and reflected these lines and came up with a nice tiled surface.

I honestly just loved this image. Parts of it I expected, other parts came as a surprise.

Spent lots of time coloring it in. Mostly used watercolor brushes, SAI Japanese Traditional Colors, but also used some colored pencils.

When it was done, I didn’t much care for the finished result.

It was okay, but didn’t make me as happy as I would have liked.

But then I started playing with it. Put it into Photoshop, isolated squares….


…then did some copying, rotating and reflecting…

I kept coming up with all sorts of stuff that surprised me.


I kept trying out different combinations…

… and then because Easter was on my mind I started wondering if I could map these on to eggs in Illustrator.

Turns out the answer was yes.


These were so fun to do.

I liked how the watercolor translated so well in to the digital environment.

Was very surprised that I ended up with these eggs. But very happy.

OK, that’s it for now. Gotta get ready for tomorrow with kindergarten!