## Domino: A Shape With Merit

### May 26, 2016

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

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.

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

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

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

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.

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

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

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

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, 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  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

Two from Wikipedia:

https://en.m.wikipedia.org/wiki/Domino_(mathematics)

https://en.m.wikipedia.org/wiki/Domino_tiling

And here’s a wonderful paper written by Paul Salomon and Justin Lanier, about the ways to think about dominoes and other polyominoes  https://mathmunch.files.wordpress.com/2015/08/moves-proposal.pdf

## November 20: Weblog Birthday, Six Years

### November 20, 2015

Dolls by Angela, age 6-ish

This blog became public six years ago today. I had been writing in secret for a while, trying to get the hang of WordPress, before first hitting the Publish Public setting. To celebrate, I’m posting photos of work done by each member of my little family, and pointing out my most visited pages.

Birthday Bird, by my son John

The images on my blog that have gotten the most clicks are:

How to Make an Origami Pamphlet : 3750 clicks.  My number one favorite book to makes has the most clicks. This elegant little structure is known also know as a  zine, hot dog booklet, eight page pamphlet as well as by many other names.

Origami Pocket Tutorial:  3647 clicks.  This is slightly different from an origami cup because it is sized to hold a little book.

Modified Pamphlet Stitch Booklet:  3006 clicks.  A good structure to make when needle and thread seems too cumbersome.

How to Make a Six-Sided Snowflake from a Paper Napkin: 2859 clicks. I made this image quickly one evening after trying to explain how to do this on the phone to my good friend Cynthia in Minnesota. There are so many tutorials on how to make a snowflake that I never thought mine would rise to the towards the top of google search results, but it has!

Box decorated by my husband

My most visited posts are:

‘Tis the Season to Make Paper Snowflakes    34,689 visitors. These views are mostly seasonal.

Elementary Nature Printing   17,198 visitors. I think it’s Pinterest that drives views to this page.

How to Make a Paper Spring  14,164 visitors. There’s not much competition on search engines for this structure. Even so, I think that the tutorial page on this post is one of my best and I doubt anyone will make one that’s better. Blush.

How to Make an Origami Pamphlet  13,674 visitors. This post, by the way, was published 10 days shy of six years ago. Nearly everyday this post is visited. Yesterday it got 16 views.

The remains on the inside of a Conch Shell, egg tempera, by me, today.

So, that’s it, my six-year wrap up. Next year maybe I will write about my own favorite posts. Thanks for visiting. It means the world to me to know you’ve stopped by.

## How to Make a Paper Spring

### March 14, 2010

How to Make a Paper Spring

Here is the How-to-Make-a-Paper-Spring hand-out that goes along with the photos in the previous post.  Not only is this little structure delightful and satisfying for children to make, but I have also been told that it is a great OT (Occupational Therapy) activity which helps develop fine motor skills, and supports eye/hand development.  In any case it is fun!

Years ago, before there was a glut of ad-filled “how-to”s on the internet, I came across another version of the paper spring;  the page was translated from German.  Instead of using two strips of paper, the author described using one long narrow strip of paper, which he folded in the middle  to create a right angle.  He proceeded in a way that is consistent with my directions.   This way of making the spring is less colorful, but, otherwise, it is an elegant structure.  The author of the article called the paper spring a Mouse Ladder.   Mouse Ladder.  I like that name.

I also made a B&W Paper Spring  hand-out.

## Paper Springs

### March 7, 2010

Paper Springs

There is nothing that I teach that attracts the focus of young classroom students more than the promise of a paper spring. Nothing that I teach  elicits more ‘oohs’ and  ‘aahs’ than when I show-off a completed spring.  Although it is a small element (about 1″ x 1″ x 2″)  the colors and the movement of it delights children.

Paper Spring Line-Up

My next post will be a how-to handout on how to make a paper spring.  But, simply put, attach, at a right angle, the ends of two equal strips of paper. Alternately fold one strip over the other until you run out of paper.  That’s all.  If you like, take a look  on page 31 of a book I co-authored for Scholastic; here you will find more complete directions.  Or wait unti my next post, which will be an instruction sheet for the paper spring.

The way that I usually use this structure with children is to place it on a page under something that can be enlivened by a bit of movement.  Often, for example, at the back of the book, where there is a blurb about the student/author, I will ask students to cut out a tracing of their hand which they then  mount on a spring, so that the hand waves.

There are many other ways to use paper springs, including gluing them together, end to end.  A word of caution:  if you leave strips of paper behind for children, after having mentioned the bit about the paper chain, when you visit the classroom next you are likely find a paper chain that has aspirations of reaching China.