Domino: A Shape With Merit

May 26, 2016

Paper Cut Dominoes

Paper Cut Dominoes

The domino shape just doesn’t get enough respect.

You are likely familiar with the fact that the game of dominoes is played with tiles that are like two squares on top of each other. Much to my delight and surprise, about a year ago, mathematician Justin Lanier enlightened me to the fact that this rectangle, which is twice as long as it is wide, has a specific name and that name is DOMINO.

(pause. paste in Matt Vaudrey’s reaction…

…ok, continue with post)

This may not seems like such a grand reason to celebrate, but, over the past year, by being able to pull this definition out of my back pocket, both my teaching and my structural problem solving have improved and deepened.  There is something about calling something by its proper name that has value.


students 2002

students 2002

For decades, with thousands of children who have done bookmaking with me, I’ve been trying to figure out how to get students to decorate their books using geometric shapes in a meaningful way. A meaningful way means, to me, getting them to really understand and exploit the characteristics of the shapes. Thousands of children for decades means I can try out one idea after the other, attend to nuance, and maybe even figure out a thing or two.

Cover Made by First graders 2010

Cover Made by First graders 2010

I’ve known for a long time that starting with squares can help reign in the chaos of making decorations for books. I actually use to cut out and distribute piles of squares to students so that they could work with squares.

Lots of cut squares

Lots of cut squares

Yeah, I would spend hours cutting little squares out brightly colored card stock. Even now it’s painful looking at these squares, remembering those late nights of cutting cutting cutting… and then the students would want to  smear glue all over their books and sprinkle the squares like candy on them and, when the glue dried, half of the squares would fall off because the gluing was sub-prime. Ok, I’m sort of exaggerating:  Many turned out beautifully, but I still didn’t feel like the kids were getting the most out of the activity: they weren’t making the connection that I was hoping for with the square.

Dominoes, cut into squares, then half squares, which turn out to be a scaled down domino, then halved again to make small squares

Dominoes, cut into squares, then half squares, which turn out to be a scaled down domino, then halved again to make small squares

Eventually I realized that I could give students the double-square shape, though at this point I didn’t know that it was called at domino. I show students how to line up two of these double-square perpendicular to each other,  from which they cut “on the line” that separates the colors, and snip, snip, they have four squares. If they cut the square again, this time by eye, they have scaled down rectangles, and they can then cut these in half to make baby squares. Yes, baby squares. These are young children and young children like baby squares. I also talk to the about rotating the square, cutting it from point to point to make two triangles, and then cutting the triangles in half to make baby triangles.

Dominoes cut and arranged

Dominoes, cut

Now just this year, during this teaching season, I have a made a point to introduce the double-square shape by name. The students I work with are shape savvy: they know what a rectangle is and they know that the square is a special kind of rectangle, but their eyes light up when I tell them a that a double-square shape is a special shape too, a domino. Does it make a difference for them to call it by name? I answer that with a resounding YES.

Emperor Penguin Book

Lily’s Emperor Penguin Book

It’s been like night and day. Students seem to honor the shape and special qualities of the domino. I know this because, with the hundreds of students that I’ve worked with this year, I saw, for the first time, the overwhelming majority of students being at ease with the idea of working with just the shapes that I talked with them about. There was so much less arbitrary cutting of paper, so much less just slapping down whatever shape that emerged from a hastily cut paper strip.

Thinking hands

Thinking hands

Across the board, with the introduction to the domino, students approached this part of the project with a sophistication that I had never seen before.

pages in progress with cut paper decoration

pages in progress with cut paper decoration

Instead of having to wade through photographs to show the type of work that I think is most valuable to show, I am finding it hard to choose between all the interesting work that these students are making.

Mostly Green

Mostly Green

I look at these and I can sense that the student is connected to the underlying structure. If you don’t know what I mean by that, just look again at these photos.

It seems to me that I even have metric which tells me that this is not just something I’m imagining. One of the other techniques that I show students is how to make paper spring.  This is a challenge for most students, but a worthwhile one, as they love  how it gets used in their books. Though not a domino shape, it is made by having an awareness that the two strips of paper they start with are of equal width, and I show them how to weave and rotate the strips down into a square.

Working hands

Working hands, paper spring made with paper strips that are 8 times as wide they are  long, which, is  I am told by mathematician Paul Salomon,  an octomino

After first showing them the techniques of decoration with the domino it was astounding how quickly the students understood how to do the spring.

Paper Spring, completed

Paper Spring, completed  Interesting fact: my page on How to Make a Paper Spring has been visited more than 16K times

In fact, in each class, these parts of my presentation went so quickly and smoothly that I kept checking my notes, thinking I had missed some part of the lesson because there was so much extra time.

Needless to say, I have much respect for the domino.

For the last few months I have been casually, now more seriously, studying a folded paper structure in Chinese Folk Art, called the Zhen Xian Boa. My plan is to write a number of posts on this structure. It’s not hard to figure out that part of my fascination with the structure is that, from beginning to end, there’s the domino.

Zhen Xian Boa , Chinese Folk Art

Zhen Xian Boa , Chinese Folk Art

Here’s some links to look at about the Domino

Two from Wikipedia:

And here’s a wonderful paper written by Paul Salomon and Justin Lanier, about the ways to think about dominoes and other polyominoes

Justin Lanier just shared this link to a video after reading this post. Perfect.






One Response to “Domino: A Shape With Merit”

  1. Hannie Says:

    I love reading your posts but would prefer to be in the classroom!

    Liked by 1 person

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